Preface

Low-frequency acoustic energy released within the Earth's interior propagates through several types of seismic waves categorized by body waves or surface waves. The importance of seismic wave research lies not only in our ability to understand and predict earthquakes and tsunamis, but it also reveals information on the Earth's composition and features in much the same way as it led to the discovery of Mohorovicic's discontinuity. As our theoretical understanding of the physics behind seismic waves has grown, physical and numerical modeling have greatly advanced and now augment applied seismology for better prediction and engineering practices. This has led to some novel applications such as using artificially-induced shocks for exploration of the Earth's subsurface and seismic stimulation for increasing the productivity of oil wells. This book demonstrates the latest techniques and advances in seismic wave analysis from the theoretical approach, data acquisition and interpretation, to analyses and numerical simulations, as well as research applications. The major topics in this book cover the aspects on seismic wave propagation, characteristics of their velocities and attenuation, deformation process of the Earth's medium, seismic source process and tectonic dynamics with relating observations, as well as propagation modeling of seismic waves.

#### **Dr. Masaki Kanao**

Associate Professor, National Institute of Polar Research, Midori-cho, Tachikawa-shi, Tokyo, Japan

#### **Dr. Genti Toyokuni**

Department of Geophysics, Tohoku University, Sendai, Miyagi, Japan

Chapter 1

Abstract

1. Introduction

the key consideration.

1

Wensheng Zhang

A High-Order Finite Volume

on Unstructured Meshes

effectiveness and good adaptability to complex topography.

Keywords: numerical solutions, computational seismology, 3D elastic wave, wave propagation, high-order finite volume method, unstructured meshes

Wave propagation based on wave equations has important applications in geophysics. It is usually used as a powerful tool to detect the structures of reservoir. Thus solving wave equations efficiently and accurately is always an important research topic. There are several types of numerical methods to solve wave equations, for example, the finite difference (FD) method [1, 2], the pseudo-spectral (PS) method [3, 4], the finite element (FE) method [5–9], the spectral element (SE) method [10–14], the discontinuous Galerkin (DG) method [15–18], and the finite volume (FV) method [19–22]. Each numerical method has its own inherent advantages and disadvantages. For example, the FD method is efficient and relatively easy to implement, but the inherent restriction of using regular meshes limits its application to complex topography. The FE method has good adaptability to complex topography, but it has huge computational cost. In this chapter, the FV method is

In order to simulate wave propagation on unstructured meshes efficiently, the FV method is a good choice due to its high computational efficiency and good

Method for 3D Elastic Modelling

In this chapter, a new efficient high-order finite volume method for 3D elastic modelling on unstructured meshes is developed. The stencil for the high-order polynomial reconstruction is generated by subdividing the relative coarse

tetrahedrons. The reconstruction on the stencil is performed by using cell-averaged quantities represented by the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, the new method has a very local property like the discontinuous Galerkin method. Furthermore, it can be written as an inner-split computational scheme which is beneficial to reducing computational amount. The reconstruction matrix is invertible and remains unchanged for all tetrahedrons, and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. The high-order accuracy in time is obtained by the Runge-Kutta method. Numerical computations including a 3D real model with complex topography demonstrate the

#### Chapter 1

## A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

Wensheng Zhang

#### Abstract

In this chapter, a new efficient high-order finite volume method for 3D elastic modelling on unstructured meshes is developed. The stencil for the high-order polynomial reconstruction is generated by subdividing the relative coarse tetrahedrons. The reconstruction on the stencil is performed by using cell-averaged quantities represented by the hierarchical orthonormal basis functions. Unlike the traditional high-order finite volume method, the new method has a very local property like the discontinuous Galerkin method. Furthermore, it can be written as an inner-split computational scheme which is beneficial to reducing computational amount. The reconstruction matrix is invertible and remains unchanged for all tetrahedrons, and thus it can be pre-computed and stored before time evolution. These special advantages facilitate the parallelization and high-order computations. The high-order accuracy in time is obtained by the Runge-Kutta method. Numerical computations including a 3D real model with complex topography demonstrate the effectiveness and good adaptability to complex topography.

Keywords: numerical solutions, computational seismology, 3D elastic wave, wave propagation, high-order finite volume method, unstructured meshes

#### 1. Introduction

Wave propagation based on wave equations has important applications in geophysics. It is usually used as a powerful tool to detect the structures of reservoir. Thus solving wave equations efficiently and accurately is always an important research topic. There are several types of numerical methods to solve wave equations, for example, the finite difference (FD) method [1, 2], the pseudo-spectral (PS) method [3, 4], the finite element (FE) method [5–9], the spectral element (SE) method [10–14], the discontinuous Galerkin (DG) method [15–18], and the finite volume (FV) method [19–22]. Each numerical method has its own inherent advantages and disadvantages. For example, the FD method is efficient and relatively easy to implement, but the inherent restriction of using regular meshes limits its application to complex topography. The FE method has good adaptability to complex topography, but it has huge computational cost. In this chapter, the FV method is the key consideration.

In order to simulate wave propagation on unstructured meshes efficiently, the FV method is a good choice due to its high computational efficiency and good

adaptability to complex geometry. In this chapter an efficient FV method for 3D elastic wave simulation on unstructured meshes is developed. It incorporates some nice features from the DG and FV methods [15–17, 19, 20, 23] and the spectral FV (SFV) method [24–26]. In our method, the computational domain is first meshed with relative coarse tetrahedral elements in 3D or triangle elements in 2D. Then, each element is further divided as a collection of finer subelements to form a stencil. The high-order polynomial reconstruction is performed on this stencil by using local cell-averaged values on the finer elements. The resulting reconstruction matrix on all coarse elements remains unchanged, and it can be pre-computed before time evolution. Moreover, the method can be written as an inner-split computational scheme. These two advantages of our method are very beneficial to enhancing the parallelization and reducing computational cost.

∂u ∂t

eigenvalues si of matrices A, B, and C and are given by

vp ¼

where g ¼ g1; ⋯; g<sup>9</sup>

where

respectively.

xi; yi ; zi

Figure 1.

3

2.2 The generation of a stencil

tetrahedral elements Tð Þ <sup>m</sup> :

� �<sup>T</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86400

<sup>þ</sup> <sup>A</sup> <sup>∂</sup><sup>u</sup> ∂x

A, B, and C are all 9 � 9 matrices and can be obtained obviously [27]. The propagation velocities of the elastic waves are determined by the

<sup>þ</sup> <sup>B</sup> <sup>∂</sup><sup>u</sup> ∂y

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

s<sup>1</sup> ¼ �vp, s<sup>2</sup> ¼ �vs, s<sup>3</sup> ¼ �vs, s<sup>4</sup> ¼ s<sup>5</sup> ¼ s<sup>6</sup> ¼ 0, s<sup>7</sup> ¼ vs, s<sup>8</sup> ¼ vs, s<sup>9</sup> ¼ vp,

ffiffiffiffiffiffiffiffiffiffiffiffiffi λ þ 2μ ρ

, vs ¼

are the velocities of the compression (P) wave and the shear (S) wave velocities,

Suppose that the 3D computational domain Ω is meshed by NE conforming

In practical computations, the integrals in the FV scheme on physical tetrahedral element Tð Þ <sup>m</sup> are usually changed to be computed on its reference element. Figure 1

Ω ¼ ⋃ NE m¼1

shows a physical tetrahedron <sup>T</sup>ð Þ <sup>m</sup> in the physical system, and <sup>x</sup> � <sup>y</sup> � <sup>z</sup> is transformed into a reference element TE in the reference system ξ � η � ζ. Let

� � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; 4 be the coordinates of physical element <sup>T</sup>ð Þ <sup>m</sup> . The transformations between x � y � z system and ξ � η � ζ system will be given in the final

The physical element <sup>T</sup>ð Þ <sup>m</sup> (left) in the physical coordinate system <sup>x</sup> � <sup>y</sup> � <sup>z</sup> is transformed into a reference

element TE (right) in the reference coordinate system ξ � η � ζ.

ffiffiffi μ ρ r

Tð Þ <sup>m</sup> : (5)

s

<sup>þ</sup> <sup>C</sup>∂<sup>u</sup>

,<sup>u</sup> <sup>¼</sup> <sup>σ</sup>xx; <sup>σ</sup>yy; <sup>σ</sup>zz; <sup>σ</sup>xy; <sup>σ</sup>yz; <sup>σ</sup>xz; <sup>u</sup>; <sup>v</sup>; <sup>w</sup> � �<sup>T</sup>

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>g</sup>, (2)

, and the matrices

(3)

(4)

The rest of this chapter is organized as follows. In Section 2, the theory is described in detail. In Section 3, numerical results are given to illustrate the effectiveness of our method. Finally, the conclusion is given in Section 4.

#### 2. Theory

#### 2.1 The governing equation

The three-dimensional (3D) elastic wave equation with external sources in velocity-stress formulation can be written as the following system [1, 15]:

$$\begin{cases} \frac{\partial \sigma\_{\infty}}{\partial t} - (\lambda + 2\mu) \frac{\partial u}{\partial x} - \lambda \frac{\partial v}{\partial y} - \lambda \frac{\partial w}{\partial x} = \mathbf{g}\_{1}, \\ \frac{\partial \sigma\_{\infty}}{\partial t} - \lambda \frac{\partial u}{\partial x} - (\lambda + 2\mu) \frac{\partial v}{\partial y} - \lambda \frac{\partial w}{\partial x} = \mathbf{g}\_{2}, \\ \frac{\partial \sigma\_{\infty}}{\partial t} - \lambda \frac{\partial u}{\partial x} - \lambda \frac{\partial v}{\partial y} - (\lambda + 2\mu) \frac{\partial w}{\partial x} = \mathbf{g}\_{3}, \\ \frac{\partial \sigma\_{\infty}}{\partial t} - \mu \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right) = \mathbf{g}\_{4}, \\ \frac{\partial \sigma\_{yx}}{\partial t} - \mu \left(\frac{\partial v}{\partial x} + \frac{\partial w}{\partial y}\right) = \mathbf{g}\_{5}, \\ \frac{\partial \sigma\_{x}}{\partial t} - \mu \left(\frac{\partial u}{\partial x} + \frac{\partial w}{\partial x}\right) = \mathbf{g}\_{6}, \\ \rho \frac{\partial u}{\partial t} - \frac{\partial \sigma\_{\infty}}{\partial x} - \frac{\partial \sigma\_{xy}}{\partial y} - \frac{\partial \sigma\_{xx}}{\partial x} = \rho \mathbf{g}\_{7}, \\ \rho \frac{\partial v}{\partial t} - \frac{\partial \sigma\_{xy}}{\partial x} - \frac{\partial \sigma\_{yy}}{\partial y} - \frac{\partial \sigma\_{xz}}{\partial x} = \rho \mathbf{g}\_{8}, \\ \rho \frac{\partial w}{\partial t} - \frac{\partial w}{\partial x} - \frac{\partial w}{\partial y$$

where u, v, and w are the wavefield of particle velocities in x, y, and z directions, respectively; λ and μ are the Lamé coefficients and ρ is the density; gi ð Þ x; y; z; t are the known sources; σxx, σyy, and σzz are the normal stress components while σxy, σxz, and σyz are the shear stresses. For the convenient of discussion, we rewrite Eq. (1) as the following compact form:

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

$$\frac{\partial \mathbf{u}}{\partial t} + A \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + B \frac{\partial \mathbf{u}}{\partial \mathbf{y}} + C \frac{\partial \mathbf{u}}{\partial \mathbf{z}} = \mathbf{g}. \tag{2}$$

where g ¼ g1; ⋯; g<sup>9</sup> � �<sup>T</sup> ,<sup>u</sup> <sup>¼</sup> <sup>σ</sup>xx; <sup>σ</sup>yy; <sup>σ</sup>zz; <sup>σ</sup>xy; <sup>σ</sup>yz; <sup>σ</sup>xz; <sup>u</sup>; <sup>v</sup>; <sup>w</sup> � �<sup>T</sup> , and the matrices A, B, and C are all 9 � 9 matrices and can be obtained obviously [27].

The propagation velocities of the elastic waves are determined by the eigenvalues si of matrices A, B, and C and are given by

$$\begin{aligned} \mathfrak{s}\_1 = -v\_p, \quad \mathfrak{s}\_2 = -v\_o, \quad \mathfrak{s}\_3 = -v\_o, \quad \mathfrak{s}\_4 = \mathfrak{s}\_5 = \mathfrak{s}\_6 = \mathfrak{d}, \quad \mathfrak{s}\_7 = v\_o, \quad \mathfrak{s}\_8 = v\_o, \quad \mathfrak{s}\_9 = v\_p, \end{aligned} \tag{3}$$

where

adaptability to complex geometry. In this chapter an efficient FV method for 3D elastic wave simulation on unstructured meshes is developed. It incorporates some nice features from the DG and FV methods [15–17, 19, 20, 23] and the spectral FV (SFV) method [24–26]. In our method, the computational domain is first meshed with relative coarse tetrahedral elements in 3D or triangle elements in 2D. Then, each element is further divided as a collection of finer subelements to form a stencil. The high-order polynomial reconstruction is performed on this stencil by using local cell-averaged values on the finer elements. The resulting reconstruction matrix on all coarse elements remains unchanged, and it can be pre-computed before time evolution. Moreover, the method can be written as an inner-split computational scheme. These two advantages of our method are very beneficial to enhancing the

The rest of this chapter is organized as follows. In Section 2, the theory is described in detail. In Section 3, numerical results are given to illustrate the effectiveness of our method. Finally, the conclusion is given in Section 4.

The three-dimensional (3D) elastic wave equation with external sources in

∂u <sup>∂</sup><sup>x</sup> � <sup>λ</sup>

<sup>∂</sup><sup>x</sup> � ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup>

∂v ∂y ∂v ∂y � λ ∂w <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>g</sup>1,

∂v ∂y � λ ∂w <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>g</sup>2,

� ð Þ λ þ 2μ

¼ g4,

¼ g5,

¼ g6,

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ρ</sup>g7,

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ρ</sup>g8,

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>ρ</sup>g9,

<sup>∂</sup><sup>y</sup> � <sup>∂</sup>σxz

<sup>∂</sup><sup>y</sup> � <sup>∂</sup>σyz

<sup>∂</sup><sup>y</sup> � <sup>∂</sup>σzz

where u, v, and w are the wavefield of particle velocities in x, y, and z directions,

the known sources; σxx, σyy, and σzz are the normal stress components while σxy, σxz, and σyz are the shear stresses. For the convenient of discussion, we rewrite Eq. (1) as

∂w <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>g</sup>3,

(1)

ð Þ x; y; z; t are

velocity-stress formulation can be written as the following system [1, 15]:

<sup>∂</sup><sup>t</sup> � ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup>

∂u

∂u <sup>∂</sup><sup>x</sup> � <sup>λ</sup>

∂v ∂x þ ∂u ∂y � �

∂v ∂z þ ∂w ∂y � �

∂u ∂z þ ∂w ∂x � �

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>σxy

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>σyy

<sup>∂</sup><sup>x</sup> � <sup>∂</sup>σyz

respectively; λ and μ are the Lamé coefficients and ρ is the density; gi

� <sup>∂</sup>σxx

� <sup>∂</sup>σxy

parallelization and reducing computational cost.

∂σxx

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

∂σyy <sup>∂</sup><sup>t</sup> � <sup>λ</sup>

∂σzz <sup>∂</sup><sup>t</sup> � <sup>λ</sup>

∂σxy <sup>∂</sup><sup>t</sup> � <sup>μ</sup>

∂σyz <sup>∂</sup><sup>t</sup> � <sup>μ</sup>

∂σxz <sup>∂</sup><sup>t</sup> � <sup>μ</sup>

ρ ∂u ∂t

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ρ ∂v ∂t

ρ ∂w <sup>∂</sup><sup>t</sup> � <sup>∂</sup>σxz

the following compact form:

2

2. Theory

2.1 The governing equation

Seismic Waves - Probing Earth System

$$
v\_p = \sqrt{\frac{\lambda + 2\mu}{\rho}}, \quad v\_s = \sqrt{\frac{\mu}{\rho}} \tag{4}$$

are the velocities of the compression (P) wave and the shear (S) wave velocities, respectively.

#### 2.2 The generation of a stencil

Suppose that the 3D computational domain Ω is meshed by NE conforming tetrahedral elements Tð Þ <sup>m</sup> :

$$\Omega = \bigcup\_{m=1}^{N\_E} T^{(m)}.\tag{5}$$

In practical computations, the integrals in the FV scheme on physical tetrahedral element Tð Þ <sup>m</sup> are usually changed to be computed on its reference element. Figure 1 shows a physical tetrahedron <sup>T</sup>ð Þ <sup>m</sup> in the physical system, and <sup>x</sup> � <sup>y</sup> � <sup>z</sup> is transformed into a reference element TE in the reference system ξ � η � ζ. Let xi; yi ; zi � � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; 4 be the coordinates of physical element <sup>T</sup>ð Þ <sup>m</sup> . The transformations between x � y � z system and ξ � η � ζ system will be given in the final

#### Figure 1.

The physical element <sup>T</sup>ð Þ <sup>m</sup> (left) in the physical coordinate system <sup>x</sup> � <sup>y</sup> � <sup>z</sup> is transformed into a reference element TE (right) in the reference coordinate system ξ � η � ζ.

subsection of Section 2. For convenience, let x ¼ ð Þ x; y; z and ξ ¼ ð Þ ξ; η; ζ . And denote the transformation from ξ � η � ζ system to x � y � z system by

$$\mathbf{x} = \mathbf{x}\left(T^{(m)}, \boldsymbol{\xi}\right),\tag{6}$$

and its corresponding inverse transformation by

$$\mathfrak{k} = \mathfrak{k}\left(T^{(m)}, \mathbf{x}\right). \tag{7}$$

The detailed expressions of the transformations (6) and (7) will be given in Section 2.5.

Inside each TE the solutions of Eq. (2) are approximated numerically by using a linear combination of polynomial basis functions ϕlð Þ ξ; η; ζ and the time-dependent coefficients w^ ð Þ <sup>m</sup> <sup>l</sup> ð Þt :

$$\mathbf{u}^{(m)}(\xi,\eta,\zeta,t) = \sum\_{l=1}^{N\_p} \hat{\mathbf{w}}\_l^{(m)}(t)\phi\_l(\xi,\eta,\zeta),\tag{8}$$

where Np is the degree of freedom of a complete polynomial.

In order to construct a high-order polynomial, we need to choose a stencil. Traditionally, the elements being adjacent to the element Tð Þ <sup>m</sup> are selected to form a stencil. In [20] three types of stencils, i.e., the central stencil, the primary sector stencil, and the reverse stencil, are investigated. These stencils usually choose 2N neighbors for the 3D reconstruction. Here N is the degree of a complete polynomial. Due to geometrical issues, the reconstruction matrix resulting from these stencils may be not invertible. This may happen when all elements are aligned in a straight line [20]. In the following, we propose to partition Tð Þ <sup>m</sup> or in fact its corresponding reference element TE into finer subelements to form a stencil. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. Moreover, this algorithm is easy to implement especially in 3D and for all elements whether they are internal or boundary elements.

Let Ne be the number of subelements in Tð Þ <sup>m</sup> after subdividing. For a complete polynomial of degree N in 3D, a reconstruction requires at least Np subelements, where

$$N\_p = (N+\mathbf{1})(N+\mathbf{2})(N+\mathbf{3})/\mathbf{6}.\tag{9}$$

2.3 The high-order polynomial reconstruction

<sup>u</sup>ð Þ m kð Þ <sup>¼</sup> <sup>1</sup>

∣Tð Þ m kð Þ ∣

ð

Tð Þ m kð Þ

nience. The reconstruction requires integral conservation for uð Þ <sup>m</sup> in each

stencil designed above, we have

Figure 2.

Table 1.

six tetrahedrons.

the cell-averaged quantities, i.e.,

subelement Tð Þ m kð Þ , i.e.,

5

The high-order polynomial is reconstructed in each element Tð Þ <sup>m</sup> or TE. For the

The degree of a complete polynomial N and its corresponding degrees of freedom Np are listed. Correspondingly, the number of uniform segments M on each edge and the number of subelements Ne are also listed.

The stencil obtained by subdividing the reference element TE into M3 <sup>¼</sup> <sup>3</sup><sup>3</sup> tetrahedral subelements, where M ¼ 3 is the number of uniform segments on each edge of TE. Note that a small subcubic (red) in TE consists of

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

DOI: http://dx.doi.org/10.5772/intechopen.86400

N 12 3 4 Np 4 10 20 35 M 23 3 4 Ne 8 27 27 64

where <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne is the index for subelements in <sup>T</sup>ð Þ <sup>m</sup> . The FV method will use

to reconstruct a high-order polynomial, where ∣Tð Þ m kð Þ ∣ represents the volume of the subelement Tð Þ m kð Þ . The time variable t in uð Þ <sup>m</sup> is omitted for discussion conve-

Tð Þ m kð Þ , (10)

<sup>u</sup>ð Þ <sup>m</sup> ð Þ <sup>x</sup> dV, k <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne, (11)

<sup>T</sup>ð Þ <sup>m</sup> <sup>¼</sup> <sup>⋃</sup> Ne k¼1

In our algorithm, we guarantee Ne is always greater than Np. As shown in Figure 2, we divide each edge of the reference element TE into M uniform segments. Thus we have Ne ≔ M<sup>3</sup> tetrahedral subelements in TE. Note that a small subcubic in TE consists of six tetrahedrons. With the transformations of Eqs. (6) and (7), we denote all subelements in <sup>T</sup>ð Þ <sup>m</sup> for a fixed <sup>m</sup> by <sup>T</sup>ð Þ m kð Þ for <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne. In Table 1, the degree of a complete polynomial N and its corresponding degrees of freedom Np are listed. Correspondingly, the number of M and Ne are also listed in Table 1. This algorithm for generating the stencil is easily implemented for all coarse tetrahedrons. Moreover, the reconstruction matrix resulting from this stencil is always invertible and remains unchanged for all elements <sup>T</sup>ð Þ <sup>m</sup> for <sup>m</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, NE. Note that the reconstruction matrix may be not invertible if all elements are aligned on a straight line [15]. However, this will not happen here for our algorithm.

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

#### Figure 2.

subsection of Section 2. For convenience, let x ¼ ð Þ x; y; z and ξ ¼ ð Þ ξ; η; ζ . And denote the transformation from ξ � η � ζ system to x � y � z system by

> <sup>x</sup> <sup>¼</sup> <sup>x</sup> <sup>T</sup>ð Þ <sup>m</sup> ; <sup>ξ</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup> <sup>T</sup>ð Þ <sup>m</sup> ; <sup>x</sup> 

The detailed expressions of the transformations (6) and (7) will be given in

Np

l¼1

In order to construct a high-order polynomial, we need to choose a stencil. Traditionally, the elements being adjacent to the element Tð Þ <sup>m</sup> are selected to form a stencil. In [20] three types of stencils, i.e., the central stencil, the primary sector stencil, and the reverse stencil, are investigated. These stencils usually choose 2N neighbors for the 3D reconstruction. Here N is the degree of a complete polynomial. Due to geometrical issues, the reconstruction matrix resulting from these stencils may be not invertible. This may happen when all elements are aligned in a straight line [20]. In the following, we propose to partition Tð Þ <sup>m</sup> or in fact its corresponding reference element TE into finer subelements to form a stencil. The subdivision algorithm guarantees the number of subelements is greater than the degrees of freedom of a complete polynomial. Moreover, this algorithm is easy to implement especially in 3D and for all elements whether they are internal or boundary

Let Ne be the number of subelements in Tð Þ <sup>m</sup> after subdividing. For a complete polynomial of degree N in 3D, a reconstruction requires at least Np subelements,

In our algorithm, we guarantee Ne is always greater than Np. As shown in Figure 2, we divide each edge of the reference element TE into M uniform segments. Thus we have Ne ≔ M<sup>3</sup> tetrahedral subelements in TE. Note that a small subcubic in TE consists of six tetrahedrons. With the transformations of Eqs. (6) and (7), we denote all subelements in <sup>T</sup>ð Þ <sup>m</sup> for a fixed <sup>m</sup> by <sup>T</sup>ð Þ m kð Þ for <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne. In Table 1, the degree of a complete polynomial N and its corresponding degrees of freedom Np are listed. Correspondingly, the number of M and Ne are also listed in Table 1. This algorithm for generating the stencil is easily implemented for all coarse tetrahedrons. Moreover, the reconstruction matrix resulting from this stencil is always invertible and remains unchanged for all elements <sup>T</sup>ð Þ <sup>m</sup> for <sup>m</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, NE. Note that the reconstruction matrix may be not invertible if all elements are aligned on a straight line [15]. However, this will not happen here for our algorithm.

Np ¼ ð Þ N þ 1 ð Þ N þ 2 ð Þ N þ 3 =6: (9)

Inside each TE the solutions of Eq. (2) are approximated numerically by using a linear combination of polynomial basis functions ϕlð Þ ξ; η; ζ and the time-dependent

w^ ð Þ <sup>m</sup>

and its corresponding inverse transformation by

<sup>u</sup>ð Þ <sup>m</sup> ð Þ¼ <sup>ξ</sup>; <sup>η</sup>; <sup>ζ</sup>; <sup>t</sup> <sup>∑</sup>

where Np is the degree of freedom of a complete polynomial.

Section 2.5.

elements.

where

4

coefficients w^ ð Þ <sup>m</sup>

<sup>l</sup> ð Þt :

Seismic Waves - Probing Earth System

, (6)

: (7)

<sup>l</sup> ð Þt ϕlð Þ ξ; η; ζ , (8)

The stencil obtained by subdividing the reference element TE into M3 <sup>¼</sup> <sup>3</sup><sup>3</sup> tetrahedral subelements, where M ¼ 3 is the number of uniform segments on each edge of TE. Note that a small subcubic (red) in TE consists of six tetrahedrons.


Table 1.

The degree of a complete polynomial N and its corresponding degrees of freedom Np are listed. Correspondingly, the number of uniform segments M on each edge and the number of subelements Ne are also listed.

#### 2.3 The high-order polynomial reconstruction

The high-order polynomial is reconstructed in each element Tð Þ <sup>m</sup> or TE. For the stencil designed above, we have

$$T^{(m)} = \bigcup\_{k=1}^{N\_\epsilon} T^{(m(k))},\tag{10}$$

where <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne is the index for subelements in <sup>T</sup>ð Þ <sup>m</sup> . The FV method will use the cell-averaged quantities, i.e.,

$$\overline{\mathbf{u}}^{(m(k))} = \frac{1}{|T^{(m(k))}|} \int\_{T^{(m(k))}} \mathbf{u}^{(m)}(\mathbf{x})dV, \quad k = \mathbf{1}, \cdots, N\_{\epsilon} \tag{11}$$

to reconstruct a high-order polynomial, where ∣Tð Þ m kð Þ ∣ represents the volume of the subelement Tð Þ m kð Þ . The time variable t in uð Þ <sup>m</sup> is omitted for discussion convenience. The reconstruction requires integral conservation for uð Þ <sup>m</sup> in each subelement Tð Þ m kð Þ , i.e.,

$$\begin{aligned} \int\_{T^{(m(k))}} \mathbf{u}^{(m)} \left( \mathbf{x} \left( T^{(m)}, \boldsymbol{\xi} \right) \right) dV &= |T^{(m(k))}| \overline{\mathbf{u}}^{(m(k))}, \\ \forall T^{(m(k))} \subset T^{(m)}, \quad k &= 1, \cdots, N\_{\epsilon}. \end{aligned} \tag{12}$$

ffiffiffi 6 <sup>p</sup> <sup>w</sup>^ ð Þ <sup>m</sup>

<sup>2</sup>GTG �R<sup>T</sup> R 0

! w^

[19, 20, 27]. And the system can be written as

DOI: http://dx.doi.org/10.5772/intechopen.86400

<sup>R</sup> <sup>¼</sup> ffiffiffi 6 <sup>p</sup> ; <sup>0</sup>; <sup>⋯</sup>; <sup>0</sup>

struction matrix [19, 20].

ð

Tð Þ m kð Þ

∂u ∂t dV þ ð

2.4 The spatial discrete formulation

Tð Þ m kð Þ

<sup>F</sup><sup>h</sup> <sup>¼</sup> <sup>1</sup> 2

> þ 1 2

ator of the eigenvalues given in Eq. (3), i.e.,

with eigenvalues in Eq. (3), i.e.,

7

<sup>∣</sup>Að Þ m kð Þ <sup>∣</sup> <sup>¼</sup> <sup>R</sup>∣Λ∣R�<sup>1</sup>

Using Eq. (8) and integration by parts yield

ð

Tð Þ m kð Þ

<sup>A</sup> <sup>∂</sup><sup>u</sup> ∂x <sup>þ</sup> <sup>B</sup> <sup>∂</sup><sup>u</sup> ∂y

∂u ∂t dV þ ð

T Að Þ m kð Þ þ jAð Þ m kð Þ <sup>j</sup> � �T�<sup>1</sup><sup>∑</sup>

T Að Þ m kð Þ � jAð Þ m kð Þ <sup>j</sup> � �T�<sup>1</sup><sup>∑</sup>

where mj is the index number of coarse tetrahedral element neighboring subelement Tð Þ m kð Þ . The notation ∣Að Þ m kð Þ ∣ denotes applying the absolute value oper-

where R is the matrix and its columns are made up of the eigenvectors associated

<sup>1</sup> ¼ ∑ Ne k¼1

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

With the constraint, Eq. (15) is solved by the Lagrange multiplier method

λp

where <sup>λ</sup><sup>p</sup> is the Lagrangian multiplier and both <sup>R</sup> and <sup>R</sup><sup>~</sup> are 1 � Ne matrices:

The coefficient matrix on the left-hand side of Eq. (19) is the so-called recon-

We now derive the semi-discrete finite volume scheme based on Eqs. (2) and (8). Integrating over each subelement Tð Þ m kð Þ on both sides of Eq. (2), we have

> <sup>þ</sup> <sup>C</sup>∂<sup>u</sup> ∂z

> > ∂Tð Þ m kð Þ

where dS denotes the infinitesimal element in the face integral and F<sup>h</sup> is the numerical flux, and we adopt the widely used Godunov flux [15, 19, 20, 23]

� �, <sup>R</sup><sup>~</sup> <sup>¼</sup> <sup>1</sup>

!

uð Þ m kð Þ Ne

> <sup>¼</sup> <sup>2</sup>G<sup>T</sup><sup>u</sup> R~u

> > Ne

� �dV <sup>¼</sup> <sup>0</sup>, k <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne: (21)

Np

l¼1 w^ ð Þ <sup>m</sup> <sup>l</sup> <sup>ϕ</sup>ð Þ <sup>m</sup> l

Np

<sup>w</sup>^ ð Þ mj <sup>l</sup> <sup>ϕ</sup>ð Þ mj <sup>l</sup> ,

, ∣Λ∣ ¼ diagð Þ js1j; ⋯; js9j , (24)

l¼1

; <sup>⋯</sup>; <sup>1</sup> Ne

!

: (18)

, (19)

� �: (20)

<sup>F</sup>hdS <sup>¼</sup> <sup>0</sup>, (22)

(23)

To solve the reconstruction problem, inspired by the DG method [15–17, 23, 28, 29], we use hierarchical orthogonal basis functions. The basis functions ϕlð Þ ξ; η; ζ of a complete polynomial of degree N (N ¼ 1; 2; 3; 4) in the reference coordinate system can be found in [27]. We remark that the basis functions are orthonormal and satisfy the following property:

$$\int\_{T\_E} \phi\_l(\xi, \eta, \zeta) d\xi d\eta d\zeta = \begin{cases} \frac{\sqrt{6}}{6}, & l = \mathbf{1}, \\ 0, & l \neq \mathbf{1}. \end{cases} \tag{13}$$

Transforming equation (12) in the physical coordinate system x � y � z into the reference coordinate system ξ � η � ζ and noticing Eq. (8), we obtain

$$\sum\_{l=1}^{N\_p} \left( \int\_{\mathcal{T}^{(m(k))}} \phi\_l(\xi, \eta, \zeta) d\xi d\eta d\zeta \right) \hat{\mathbf{w}}\_l^{(m)} = |\tilde{T}^{(m(k))}| \overline{\mathbf{u}}^{(m(k))}, \tag{14}$$
 
$$\forall \tilde{T}(m(k)) \subset \tilde{T}(m) = T\_E, \quad k = \mathbf{1}, \cdots, N\_e$$

where T m <sup>~</sup> ð Þ is in fact the reference element TE and Tmk <sup>~</sup> ð Þ ð Þ is the transformed element corresponding to the subelement Tð Þ m kð Þ .

The integration in Eq. (14) over Tmk <sup>~</sup> ð Þ ð Þ in <sup>ξ</sup> system can be computed efficiently if it is performed over its reference element in a second reference system ~ξ. Denote the transformation from <sup>~</sup><sup>ξ</sup> to <sup>ξ</sup> and its inverse by <sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup> Tmk <sup>~</sup> ð Þ ð Þ ; <sup>~</sup>ξ<sup>Þ</sup> � and <sup>~</sup><sup>ξ</sup> <sup>¼</sup> <sup>~</sup><sup>ξ</sup> Tmk <sup>~</sup> ð Þ ð Þ ; <sup>ξ</sup><sup>Þ</sup> � , respectively. Transforming Eq. (14) into <sup>~</sup><sup>ξ</sup> system and rewriting the result as a compact form, we have

$$G\hat{\mathbf{w}} = \overline{\mathbf{u}},\tag{15}$$

where G is the Ne � Np matrix with entries Gkl given by

$$\mathbf{G}\_{kl} = \frac{1}{|T\_E|} \left( \int\_{T\_E} \phi\_l \left( \xi \left( \bar{T}^{(m(k))}, \bar{\xi} \right) \right) d\bar{\xi} d\bar{\eta} d\tilde{\zeta} \right), \qquad k = 1, \cdots, N\_e; \ l = 1, \cdots, N\_p,\tag{16}$$

and

$$\overline{\mathbf{u}} := \left( \overline{\mathbf{u}}^{(m(1))}, \,\overline{\mathbf{u}}^{(m(2))}, \dots, \overline{\mathbf{u}}^{(m(N\_\epsilon))} \right)^T, \quad \hat{\mathbf{w}} := \left( \hat{\mathbf{w}}\_1^{(m)}, \hat{\mathbf{w}}\_2^{(m)}, \dots, \hat{\mathbf{w}}\_{N\_p}^{(m)} \right)^T. \tag{17}$$

We need at least Np subelements in the stencil since the reconstructed number of degrees of freedom is Np. As listed in Table 1, Ne subelements are used to form the stencil. Note that Ne is definitely larger than Np, which is helpful to improve the reconstruction robustness [20, 21]. Thus Eq. (15) is an overdetermined problem. We use the constrained least squared technique to solve it.

From the orthogonality of basis functions and the property of Eq. (13), we remark that Eq. (15) is subject to the following constraint condition [27]:

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

$$\sqrt{\mathbf{\hat{6}}} \hat{\mathbf{w}}\_1^{(m)} = \sum\_{k=1}^{N\_\epsilon} \frac{\overline{\mathbf{u}}^{(m(k))}}{N\_\epsilon}. \tag{18}$$

With the constraint, Eq. (15) is solved by the Lagrange multiplier method [19, 20, 27]. And the system can be written as

$$
\begin{pmatrix} 2G^T G & -R^T \\ & \mathbf{0} \end{pmatrix} \begin{pmatrix} \hat{\mathbf{w}} \\ \boldsymbol{\lambda}\_p \end{pmatrix} = \begin{pmatrix} 2G^T \overline{\mathbf{u}} \\ \boldsymbol{\tilde{R}} \overline{\mathbf{u}} \end{pmatrix}, \tag{19}
$$

where <sup>λ</sup><sup>p</sup> is the Lagrangian multiplier and both <sup>R</sup> and <sup>R</sup><sup>~</sup> are 1 � Ne matrices:

$$R = \left(\sqrt{\mathbf{\tilde{6}}}, \mathbf{0}, \dots, \mathbf{0}\right), \qquad \tilde{R} = \left(\frac{1}{N\_{\epsilon}}, \dots, \frac{1}{N\_{\epsilon}}\right). \tag{20}$$

The coefficient matrix on the left-hand side of Eq. (19) is the so-called reconstruction matrix [19, 20].

#### 2.4 The spatial discrete formulation

ð

Seismic Waves - Probing Earth System

Tð Þ m kð Þ

orthonormal and satisfy the following property:

ð TE

∑ Np ð

element corresponding to the subelement Tð Þ m kð Þ .

<sup>ϕ</sup><sup>l</sup> <sup>ξ</sup> <sup>T</sup><sup>~</sup> ð Þ m kð Þ ; <sup>~</sup><sup>ξ</sup> � � � �

u ≔ uð Þ <sup>m</sup>ð Þ<sup>1</sup> ; ; uð Þ <sup>m</sup>ð Þ<sup>2</sup> ; ⋯; uð Þ m Nð Þ<sup>e</sup> � �<sup>T</sup>

� �

We use the constrained least squared technique to solve it.

l¼1

the result as a compact form, we have

ð TE

Gkl <sup>¼</sup> <sup>1</sup> ∣TE∣

and

6

uð Þ <sup>m</sup> x Tð Þ <sup>m</sup> ; ξ � � � �

dV <sup>¼</sup> <sup>∣</sup>Tð Þ m kð Þ <sup>∣</sup>uð Þ m kð Þ ,

(12)

(13)

(14)

<sup>∀</sup>Tð Þ m kð Þ <sup>⊂</sup>Tð Þ <sup>m</sup> , k <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne:

ffiffiffi 6 p

8 ><

>:

w^ ð Þ <sup>m</sup>

<sup>∀</sup>Tmk <sup>~</sup> ð Þ ð Þ <sup>⊂</sup>T m <sup>~</sup> ð Þ¼ TE, k <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne,

Transforming equation (12) in the physical coordinate system x � y � z into the

where T m <sup>~</sup> ð Þ is in fact the reference element TE and Tmk <sup>~</sup> ð Þ ð Þ is the transformed

The integration in Eq. (14) over Tmk <sup>~</sup> ð Þ ð Þ in <sup>ξ</sup> system can be computed efficiently

if it is performed over its reference element in a second reference system ~ξ. Denote the transformation from <sup>~</sup><sup>ξ</sup> to <sup>ξ</sup> and its inverse by <sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup> Tmk <sup>~</sup> ð Þ ð Þ ; <sup>~</sup>ξ<sup>Þ</sup> � and <sup>~</sup><sup>ξ</sup> <sup>¼</sup> <sup>~</sup><sup>ξ</sup> Tmk <sup>~</sup> ð Þ ð Þ ; <sup>ξ</sup><sup>Þ</sup> � , respectively. Transforming Eq. (14) into <sup>~</sup><sup>ξ</sup> system and rewriting

d~ξd~ηd~ζ

, w^ ≔ w^ ð Þ <sup>m</sup>

We need at least Np subelements in the stencil since the reconstructed number of degrees of freedom is Np. As listed in Table 1, Ne subelements are used to form the stencil. Note that Ne is definitely larger than Np, which is helpful to improve the reconstruction robustness [20, 21]. Thus Eq. (15) is an overdetermined problem.

From the orthogonality of basis functions and the property of Eq. (13), we

remark that Eq. (15) is subject to the following constraint condition [27]:

<sup>1</sup> ; <sup>w</sup>^ ð Þ <sup>m</sup>

<sup>6</sup> , l <sup>¼</sup> <sup>1</sup>,

0, l 6¼ 1:

<sup>l</sup> <sup>¼</sup> <sup>∣</sup>T<sup>~</sup> ð Þ m kð Þ <sup>∣</sup>uð Þ m kð Þ ,

Gw^ ¼ u, (15)

, k ¼ 1, ⋯, Ne; l ¼ 1, ⋯, Np, (16)

<sup>2</sup> ; <sup>⋯</sup>; <sup>w</sup>^ ð Þ <sup>m</sup> Np

: (17)

� �<sup>T</sup>

To solve the reconstruction problem, inspired by the DG method [15–17, 23,

28, 29], we use hierarchical orthogonal basis functions. The basis functions ϕlð Þ ξ; η; ζ of a complete polynomial of degree N (N ¼ 1; 2; 3; 4) in the reference coordinate system can be found in [27]. We remark that the basis functions are

ϕlð Þ ξ; η; ζ dξdηdζ ¼

reference coordinate system ξ � η � ζ and noticing Eq. (8), we obtain

<sup>T</sup><sup>~</sup> ð Þ m kð Þ <sup>ϕ</sup>lð Þ <sup>ξ</sup>; <sup>η</sup>; <sup>ζ</sup> <sup>d</sup>ξdηd<sup>ζ</sup> � �

where G is the Ne � Np matrix with entries Gkl given by

We now derive the semi-discrete finite volume scheme based on Eqs. (2) and (8). Integrating over each subelement Tð Þ m kð Þ on both sides of Eq. (2), we have

$$\int\_{T^{(\mathfrak{m}(k))}} \frac{\partial \mathbf{u}}{\partial t} dV + \int\_{T^{(\mathfrak{m}(k))}} \left( A \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + B \frac{\partial \mathbf{u}}{\partial \mathbf{y}} + C \frac{\partial \mathbf{u}}{\partial \mathbf{z}} \right) dV = \mathbf{0}, \qquad k = \mathbf{1}, \cdots, N\_{\epsilon}. \tag{21}$$

Using Eq. (8) and integration by parts yield

$$\int\_{T^{(m(k))}} \frac{\partial \mathbf{u}}{\partial t} dV + \int\_{\partial T^{(m(k))}} \mathbf{F}^b dS = \mathbf{0},\tag{22}$$

where dS denotes the infinitesimal element in the face integral and F<sup>h</sup> is the numerical flux, and we adopt the widely used Godunov flux [15, 19, 20, 23]

$$\begin{split} \mathbf{F}^{h} &= \frac{1}{2} T \Big( A^{(m(k))} + |A^{(m(k))}| \Big) T^{-1} \sum\_{l=1}^{N\_p} \hat{\mathbf{w}}\_l^{(m)} \phi\_l^{(m)} \\ &+ \frac{1}{2} T \Big( A^{(m(k))} - |A^{(m(k))}| \Big) T^{-1} \sum\_{l=1}^{N\_p} \hat{\mathbf{w}}\_l^{(m\_j)} \phi\_l^{(m\_j)}, \end{split} \tag{23}$$

where mj is the index number of coarse tetrahedral element neighboring subelement Tð Þ m kð Þ . The notation ∣Að Þ m kð Þ ∣ denotes applying the absolute value operator of the eigenvalues given in Eq. (3), i.e.,

$$|A^{(m(k))}| = R|\Lambda|R^{-1}, \qquad |\Lambda| = \text{diag}(|\mathfrak{s}\_1|, \dots, |\mathfrak{s}\_\emptyset|), \tag{24}$$

where R is the matrix and its columns are made up of the eigenvectors associated with eigenvalues in Eq. (3), i.e.,

$$R = \begin{pmatrix} \lambda + 2\mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda + 2\mu \\ \lambda & 0 & 0 & 0 & 1 & 0 & 0 & 0 & \lambda \\ \lambda & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \lambda \\ 0 & \mu & 0 & 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \mu & 0 & 0 & 0 & \mu & 0 & 0 \\ v\_p & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -v\_p \\ 0 & v\_s & 0 & 0 & 0 & 0 & 0 & -v\_s & 0 \\ 0 & 0 & v\_t & 0 & 0 & 0 & -v\_t & 0 & 0 \end{pmatrix}. \tag{25}$$

Fþ,i,p l ¼ ð

<sup>∂</sup>ð Þ TE <sup>j</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86400

2.5 The time discretization

condition of the same number of elements [27].

<sup>k</sup>ð Þ<sup>1</sup> <sup>¼</sup> <sup>L</sup> <sup>u</sup><sup>n</sup>; ; <sup>t</sup>

<sup>k</sup>ð Þ<sup>2</sup> <sup>¼</sup> <sup>L</sup> <sup>u</sup><sup>n</sup> <sup>þ</sup>

<sup>k</sup>ð Þ<sup>3</sup> <sup>¼</sup> <sup>L</sup> <sup>u</sup><sup>n</sup> <sup>þ</sup>

<sup>u</sup><sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>u</sup><sup>n</sup> <sup>þ</sup>

<sup>u</sup>ð Þ <sup>0</sup> <sup>¼</sup> <sup>u</sup>n,

<sup>u</sup>ð Þ <sup>n</sup>þ<sup>1</sup> <sup>¼</sup> <sup>p</sup>ð Þ<sup>5</sup> :

8 < :

Table 2.

9

<sup>ϕ</sup><sup>l</sup> <sup>ξ</sup> Tmi <sup>~</sup> ð Þ ð Þ ; <sup>~</sup>ξð Þ<sup>i</sup> <sup>~</sup>χð Þ <sup>p</sup> ; <sup>~</sup>τð Þ <sup>p</sup> � � � � � � <sup>d</sup>χdτ, i <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>; <sup>p</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>:

where χ and τ are the face parameters. The transformation of the face parameters χ and τ to the face parameters ~χ and ~τ in the neighbor tetrahedron depends on the orientation of the neighbor face with respect to the local face of the considered tetrahedron. And the mapping is given in Table 2. For a given tetrahedral mesh with the known indices i and p, there are only 4 of 12 possible matrices Fþ,i,p per element [15, 20]. Comparing with the traditional FV method, the method with the splitting form described above has much less computations of face integrations. Note that only our proposed FV method can be written as a splitting form. Theoretical analysis shows our method can save about half computational time under the

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

Equation (27) is in fact a semi-discrete ordinary differential equation (ODE) system. In order to solve it formally, we denote the spatial semi-discrete part in Eq. (27) by a linear operator L. Then Eq. (27) can be written as a concise ODE form:

dt <sup>¼</sup> <sup>L</sup>ð Þ <sup>u</sup>; <sup>t</sup> : (32)

du

Traditionally, the classic fourth-order explicit RK (ERK) method

1 2

1 2 Δtkð Þ<sup>1</sup> ; t

Δtkð Þ<sup>2</sup> ; t

� �,

<sup>k</sup>ð Þ<sup>i</sup> <sup>¼</sup> aikð Þ <sup>i</sup>�<sup>1</sup> <sup>þ</sup> <sup>Δ</sup>tL <sup>p</sup>ð Þ <sup>i</sup>�<sup>1</sup> ; <sup>t</sup>

p 12 3 ~χ τ 1 � χ � τ χ ~τ χτ 1 � χ � τ

Transformation of the face parameters χ and τ to the face parameters ~χ and ~τ.

� �,

� �,

n þ 1 2 Δt

n þ 1 2 Δt

<sup>n</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup>

can be applied to advance u from u<sup>n</sup> to u<sup>n</sup>þ1. Here Δt is the time step. Now we

<sup>p</sup>ð Þ<sup>i</sup> <sup>¼</sup> <sup>p</sup>ð Þ <sup>i</sup>�<sup>1</sup> <sup>þ</sup> bikð Þ<sup>i</sup> , i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, <sup>5</sup>,

<sup>6</sup> <sup>Δ</sup><sup>t</sup> <sup>k</sup>ð Þ<sup>1</sup> <sup>þ</sup> <sup>2</sup>kð Þ<sup>2</sup> <sup>þ</sup> <sup>2</sup>kð Þ<sup>3</sup> <sup>þ</sup> <sup>k</sup>ð Þ <sup>4</sup> � �

<sup>n</sup> <sup>þ</sup> ciΔ<sup>t</sup> � �,

<sup>n</sup> ð Þ,

<sup>k</sup>ð Þ <sup>4</sup> <sup>¼</sup> <sup>L</sup> <sup>u</sup><sup>n</sup> <sup>þ</sup> <sup>Δ</sup>tkð Þ<sup>3</sup> ; <sup>t</sup>

use the low-storage version of ERK (LSERK) to solve Eq. (32):

1

(31)

(33)

(34)

And T is the rotation matrix given by

$$T = \begin{pmatrix} n\_x^2 & s\_x^2 & t\_x^2 & 2n\_x s\_x & 2s\_x t\_x & 2n\_x t\_x \\\\ n\_y^2 & s\_y^2 & t\_y^2 & 2n\_y s\_y & 2s\_y t\_y & 2n\_y t\_y \\\\ n\_z^2 & s\_z^2 & t\_x^2 & 2n\_z s\_x & 2s\_z t\_z & 2n\_x t\_x \\\\ n\_y n\_x & s\_y s\_x & t\_y t\_x & n\_y s\_x + n\_x s\_y & s\_y t\_x + s\_x t\_y & n\_y t\_x + n\_x t\_y \\\\ n\_z n\_y & s\_x s\_y & t\_z t\_y & n\_x s\_y + n\_y s\_z & s\_x t\_y + s\_y t\_z & n\_x t\_y + n\_y t\_x \\\\ n\_z n\_x & s\_x s\_x & t\_z t\_x & n\_x s\_x + n\_x s\_z & s\_x t\_x + s\_x t\_z & n\_x t\_x + n\_x t\_z \end{pmatrix},\tag{26}$$

where nx; ny; nz � � is the normal vector of the face and sx; sy; sz � � and tx; ty; tz � � are the two tangential vectors. T�<sup>1</sup> denotes the inverse of T.

Inserting Eqs. (23) into (22) and rewriting the result into a splitting form of easy computation in the reference system ξ, we have

$$\frac{\partial}{\partial t}\overline{\mathbf{u}}^{(m(k))}|T^{(m(k))}| + \sum\_{j=1}^{4} \mathbf{F}\_j^h = \mathbf{0} \tag{27}$$

with

$$\mathbf{F}\_j^h = T^j A^{(m(k))} \left( T^j \right)^{-1} |\mathbf{S}\_j| \sum\_{l=1}^{N\_p} F\_l^{-,j} \hat{\mathbf{w}}\_l^{(m)}, \qquad m = m\_j,\tag{28}$$

and

$$\begin{split} \mathbf{F}\_{j}^{h} &= \frac{\mathbf{1}}{2} T^{j} \Big( \mathbf{A}^{(m(k))} + |\mathbf{A}^{(m(k))}| \Big) \big( T^{j} \big)^{-1} |\mathbf{S}\_{j}| \sum\_{l=1}^{N\_{p}} F\_{l}^{-,j} \hat{\mathbf{w}}^{(m)} \\ &+ \frac{\mathbf{1}}{2} T^{j} \Big( \mathbf{A}^{(m(k))} - |\mathbf{A}^{(m(k))}| \Big) \big( T^{j} \big)^{-1} |\mathbf{S}\_{j}| \sum\_{l=1}^{N\_{p}} F\_{l}^{+,i,p} \hat{\mathbf{w}}\_{l}^{(m\_{l})}, \quad m \neq m\_{j}, \end{split} \tag{29}$$

where Sj is the area of the <sup>j</sup>-th ð Þ <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup> face of subelement <sup>T</sup>ð Þ m kð Þ . <sup>F</sup>�,j <sup>l</sup> and F<sup>þ</sup>,i,p <sup>l</sup> are the left flux matrix and the right state flux matrix, respectively, which are given by

$$F\_{l}^{-,j} = \int\_{\vartheta(T\_{E})\_{j}} \phi\_{l}(\mathfrak{F}(\tilde{T}(m(j)), \tilde{\mathfrak{F}}(j)(\chi, \mathfrak{r}))) d\chi d\mathfrak{r}, \quad j = 1, 2, 3, 4,\tag{30}$$

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

$$F\_{l}^{+,i,p} = \int\_{\delta(T\_{E})\_{j}} \phi\_{l}\Big(\mathsf{f}\Big(\bar{T}(m(i)), \bar{\mathsf{f}}(i)\Big(\bar{\chi}^{(p)}, \bar{\mathsf{f}}^{(p)}\Big)\Big)\Big)d\chi d\mathsf{r}, \quad i = 1, 2, 3, 4; \qquad p = 1, 2, 3. \tag{31}$$

where χ and τ are the face parameters. The transformation of the face parameters χ and τ to the face parameters ~χ and ~τ in the neighbor tetrahedron depends on the orientation of the neighbor face with respect to the local face of the considered tetrahedron. And the mapping is given in Table 2. For a given tetrahedral mesh with the known indices i and p, there are only 4 of 12 possible matrices Fþ,i,p per element [15, 20]. Comparing with the traditional FV method, the method with the splitting form described above has much less computations of face integrations. Note that only our proposed FV method can be written as a splitting form. Theoretical analysis shows our method can save about half computational time under the condition of the same number of elements [27].

#### 2.5 The time discretization

R ¼

n2 <sup>x</sup> s 2 <sup>x</sup> t 2

0

BBBBBBBBBBB@

where nx; ny; nz

with

and

F<sup>þ</sup>,i,p

8

given by

Fh <sup>j</sup> <sup>¼</sup> <sup>1</sup> 2

> þ 1 2

F�,j <sup>l</sup> ¼ ð

<sup>∂</sup>ð Þ TE <sup>j</sup>

n2 <sup>y</sup> s 2 <sup>y</sup> t 2

n2 <sup>z</sup> s 2 <sup>z</sup> t 2

T ¼

0

Seismic Waves - Probing Earth System

BBBBBBBBBBBBBBBBB@

And T is the rotation matrix given by

λ þ 2μ 0 0000 0 0 λ þ 2μ λ 00010 0 0 λ λ 0 0001 0 0 λ 0 μ 0000 0 μ 0 0 0 0 100 0 0 0 0 0 μ 000 μ 0 0 vp 0 0000 0 0 �vp 0 vs 0000 0 �vs 0 0 0 vs 000 �vs 0 0

<sup>x</sup> 2nxsx 2sxtx 2nxtx

<sup>y</sup> 2nysy 2syty 2nyty

<sup>z</sup> 2nzsz 2sztz 2nztz

Inserting Eqs. (23) into (22) and rewriting the result into a splitting form of easy

4 j¼1 Fh

nynx sysx tytx nysx þ nxsy sytx þ sxty nytx þ nxty nzny szsy tzty nzsy þ nysz szty þ sytz nzty þ nytz nznx szsx tztx nzsx þ nxsz sztx þ sxtz nztx þ nxtz

� � is the normal vector of the face and sx; sy; sz

<sup>u</sup>ð Þ m kð Þ <sup>∣</sup>Tð Þ m kð Þ <sup>∣</sup> <sup>þ</sup> <sup>∑</sup>

∣Sj∣ ∑ Np

T<sup>j</sup> � ��<sup>1</sup>

T<sup>j</sup> � ��<sup>1</sup>

where Sj is the area of the <sup>j</sup>-th ð Þ <sup>j</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup> face of subelement <sup>T</sup>ð Þ m kð Þ . <sup>F</sup>�,j

<sup>l</sup> are the left flux matrix and the right state flux matrix, respectively, which are

l¼1 F�,j <sup>l</sup> <sup>w</sup>^ ð Þ <sup>m</sup>

∣Sj∣∑ Np

l¼1 F�,j <sup>l</sup> <sup>w</sup>^ ð Þ <sup>m</sup>

∣Sj∣∑ Np

l¼1

F<sup>þ</sup>,i,p <sup>l</sup> <sup>w</sup>^ ð Þ mj

<sup>~</sup> ð Þ ð Þ ; <sup>~</sup>ξð Þ<sup>j</sup> ð ÞÞ <sup>χ</sup>; <sup>τ</sup> � �dχdτ, j <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>, � (30)

the two tangential vectors. T�<sup>1</sup> denotes the inverse of T.

∂ ∂t

Að Þ m kð Þ T<sup>j</sup> � ��<sup>1</sup>

computation in the reference system ξ, we have

Fh <sup>j</sup> <sup>¼</sup> <sup>T</sup><sup>j</sup>

<sup>T</sup><sup>j</sup> <sup>A</sup>ð Þ m kð Þ þ jAð Þ m kð Þ <sup>j</sup> � �

<sup>T</sup><sup>j</sup> <sup>A</sup>ð Þ m kð Þ � jAð Þ m kð Þ <sup>j</sup> � �

ϕ<sup>l</sup> ξ Tmj

1

CCCCCCCCCCCCCCCCCA

1

CCCCCCCCCCCA

� � and tx; ty; tz

<sup>j</sup> ¼ 0 (27)

<sup>l</sup> , m ¼ mj, (28)

<sup>l</sup> , m 6¼ mj,

, (26)

� � are

(29)

<sup>l</sup> and

: (25)

Equation (27) is in fact a semi-discrete ordinary differential equation (ODE) system. In order to solve it formally, we denote the spatial semi-discrete part in Eq. (27) by a linear operator L. Then Eq. (27) can be written as a concise ODE form:

$$\frac{d\mathbf{u}}{dt} = L(\mathbf{u}, t). \tag{32}$$

Traditionally, the classic fourth-order explicit RK (ERK) method

$$\begin{aligned} \mathbf{k}^{(1)} &= L(\mathbf{u}^n, t^n), \\ \mathbf{k}^{(2)} &= L\left(\mathbf{u}^n + \frac{1}{2}\Delta t \mathbf{k}^{(1)}, t^n + \frac{1}{2}\Delta t\right), \\ \mathbf{k}^{(3)} &= L\left(\mathbf{u}^n + \frac{1}{2}\Delta t \mathbf{k}^{(2)}, t^n + \frac{1}{2}\Delta t\right), \\ \mathbf{k}^{(4)} &= L\left(\mathbf{u}^n + \Delta t \mathbf{k}^{(3)}, t^n + \Delta t\right), \\ \mathbf{u}^{n+1} &= \mathbf{u}^n + \frac{1}{6}\Delta t \left(\mathbf{k}^{(1)} + 2\mathbf{k}^{(2)} + 2\mathbf{k}^{(3)} + \mathbf{k}^{(4)}\right) \end{aligned} \tag{33}$$

can be applied to advance u from u<sup>n</sup> to u<sup>n</sup>þ1. Here Δt is the time step. Now we use the low-storage version of ERK (LSERK) to solve Eq. (32):

$$\begin{aligned} \mathbf{u}^{(0)} &= \mathbf{u}^{n}, \\ \begin{cases} \mathbf{k}^{(i)} = a\_i \mathbf{k}^{(i-1)} + \Delta t L\left(\mathbf{p}^{(i-1)}, t^n + c\_i \Delta t\right), \\ \mathbf{p}^{(i)} = \mathbf{p}^{(i-1)} + b\_i \mathbf{k}^{(i)}, & i = 1, \cdots, 5, \end{cases} \\ \mathbf{u}^{(n+1)} &= \mathbf{p}^{(5)}. \end{aligned} \tag{34}$$


Table 2. Transformation of the face parameters χ and τ to the face parameters ~χ and ~τ.

As we can see the LSERK only requires one additional storage level, while ERK has four. The coefficients required in Eq. (34) are listed in Table 3 [30].

As to the stability condition, it is controlled by the Courant-Friedrichs-Lewy (CFL) condition [15, 19];

$$
\Delta t \le \frac{1}{2N + 1} \frac{h\_{\text{min}}}{v\_p},
\tag{35}
$$

<sup>ξ</sup> <sup>¼</sup> <sup>∣</sup> <sup>J</sup>1<sup>∣</sup> ∣ J∣

x � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86400

x<sup>2</sup> � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z � z<sup>1</sup>

to ξ � η � ζ system is defined by

8 ><

>:

Tmk <sup>~</sup> ð Þ ð Þ in <sup>ξ</sup> � <sup>η</sup> � <sup>ζ</sup> system.

3. Numerical computations

3D models.

11

where

� � � � � � �

> � � � � � � �

∣ J1∣ ¼

∣ J3∣ ¼

, <sup>η</sup> <sup>¼</sup> <sup>∣</sup> <sup>J</sup>2<sup>∣</sup>

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

� � � � � � �

� � � � � � �

equal to six times the volume of the tetrahedron element Tð Þ <sup>m</sup> .

<sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup> from Eq. (41) by Cramer ruler similarly. Denote

� � � � � � �

<sup>∣</sup>~J<sup>∣</sup> <sup>¼</sup>

∣ J∣

� � � � � � �

� � � � � � �

, ∣ J2∣ ¼

, ∣ J∣ ¼

Note that J is the determinant of the Jacobian matrix of the transformation being

The coordinate transformation from the second reference coordinate <sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup>

<sup>ξ</sup> <sup>¼</sup> <sup>ξ</sup><sup>1</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> � <sup>ξ</sup><sup>1</sup> ð Þ~<sup>ξ</sup> <sup>þ</sup> <sup>ξ</sup><sup>3</sup> � <sup>ξ</sup><sup>1</sup> ð Þ~<sup>η</sup> <sup>þ</sup> <sup>ξ</sup><sup>4</sup> � <sup>ξ</sup><sup>1</sup> ð Þ~ζ, <sup>η</sup> <sup>¼</sup> <sup>η</sup><sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> � <sup>η</sup><sup>1</sup> ð Þ~<sup>ξ</sup> <sup>þ</sup> <sup>η</sup><sup>3</sup> � <sup>η</sup><sup>1</sup> ð Þ~<sup>η</sup> <sup>þ</sup> <sup>η</sup><sup>4</sup> � <sup>η</sup><sup>1</sup> ð Þ~ζ, <sup>ζ</sup> <sup>¼</sup> <sup>ζ</sup><sup>1</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> � <sup>ζ</sup><sup>1</sup> ð Þ~<sup>ξ</sup> <sup>þ</sup> <sup>ζ</sup><sup>3</sup> � <sup>ζ</sup><sup>1</sup> ð Þ~<sup>η</sup> <sup>þ</sup> <sup>ζ</sup><sup>4</sup> � <sup>ζ</sup><sup>1</sup> ð Þ~ζ, :

then the transform from <sup>ξ</sup> � <sup>η</sup> � <sup>ζ</sup> system to <sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup> system can be solved for

ξ<sup>2</sup> � ξ<sup>1</sup> ξ<sup>3</sup> � ξ<sup>1</sup> ξ<sup>4</sup> � ξ<sup>1</sup> η<sup>2</sup> � η<sup>1</sup> η<sup>3</sup> � η<sup>1</sup> η<sup>4</sup> � η<sup>1</sup> ζ<sup>2</sup> � ζ<sup>1</sup> ζ<sup>3</sup> � ζ<sup>1</sup> ζ<sup>4</sup> � ζ<sup>1</sup>

which is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the subelement Tmk <sup>~</sup> ð Þ ð Þ for <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne. In Eqs. (41) and (42), ξi; ηi; ζ<sup>i</sup> ð Þ for i ¼ 1; 2; 3; 4, denote the vertex coordinates of

In this section we give three numerical examples to illustrate the performance of the developed method above. The convergence test of the proposed method can be found in [27]. Though the method is developed for the 3D case, it can be simplified to 2D without essential difficulty. The principle is the same. The first example is a test for a 2D model with uneven topography. The other two examples are for two

Example 1. The first example is a two-layered model with the inclined interface

z∈½ � �1:6km; 1:8km . The surface of the model is uneven to imitate the real topography. The vp and vs velocities are 3000 m=s and 2000 m=s in the upper layer and 2400 m=s and 1600 m=s in the lower layer, respectively. The densities ρ are 2200 kg=m<sup>3</sup> and 1800 kg=m<sup>3</sup> in the upper and lower layer, respectively. Figure 3b is the coarser triangular meshes for this model. A coarser version of the mesh is

shown in Figure 3a. The range of the model is x∈½ � �1:6km; 1:6km and

, <sup>ζ</sup> <sup>¼</sup> <sup>∣</sup> <sup>J</sup>3<sup>∣</sup>

∣ J∣

x<sup>2</sup> � x<sup>1</sup> x � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup>

x<sup>2</sup> � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup>

> � � � � � � �

, (42)

, (38)

� � � � � � �

� � � � � � �

, (39)

: (40)

(41)

where vp is the P wave velocity and hmin is the minimum diameter of the circumcircles of tetrahedral elements. This condition is a necessary condition for discrete stability, and a bit more restrictive form is actually used in numerical computations.

The absorbing boundary conditions (ABCs) in computations are required as the computational domain is finite. There are two typical ABCs to be adopted here. One is flux type ABCs [16, 19]. That is to say, the following numerical flux in Eq. (23) at all tetrahedral faces that coincide with domain boundary

$$\mathbf{F}^{h} = \frac{1}{2}T\left(A^{(m(k))} + |A^{(m(k))}|\right)T^{-1}\sum\_{l=1}^{N\_p} \hat{\mathbf{w}}\_l^{(m)} \phi\_l^{(m)},\tag{36}$$

which allows only for outgoing waves and is equivalent to the first order ABCs. Though the absorbing effects of this method vary the angles of incidence, it is still effective in many cases [19]. The advantage of this type ABCs is that it merged into the FVM framework naturally and there is almost no additional computational cost. Another type is the perfectly matched layer (PML) technique originally developed by [31], which is very popular in recent more 10 years.

#### 2.6 Coordinate transformation

The transformation between different coordinate systems is frequently used. For ease of reading, we present the formulations here. Let xi; yi ; zi � � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; 4 be the coordinates of a physical element. The transformation from ξ � η � ζ system to x � y � z system is defined by

$$\begin{cases} \mathbf{x} = \mathbf{x}\_1 + (\mathbf{x}\_2 - \mathbf{x}\_1)\boldsymbol{\xi} + (\mathbf{x}\_3 - \mathbf{x}\_1)\boldsymbol{\eta} + (\mathbf{x}\_4 - \mathbf{x}\_1)\boldsymbol{\zeta}, \\ \mathbf{y} = \mathbf{y}\_1 + (\mathbf{y}\_2 - \mathbf{y}\_1)\boldsymbol{\xi} + (\mathbf{y}\_3 - \mathbf{y}\_1)\boldsymbol{\eta} + (\mathbf{y}\_4 - \mathbf{y}\_1)\boldsymbol{\zeta}, \\ \mathbf{z} = \mathbf{z}\_1 + (\mathbf{z}\_2 - \mathbf{z}\_1)\boldsymbol{\xi} + (\mathbf{z}\_3 - \mathbf{z}\_1)\boldsymbol{\eta} + (\mathbf{z}\_4 - \mathbf{z}\_1)\boldsymbol{\zeta}, \end{cases} \tag{37}$$

then the transformation from x � y � z system to ξ � η � ζ system can be solved for ξ, η and ζ from Eq. (37) by the Cramer ruler, i.e.,


Table 3. Coefficients for the low-storage five-stage fourth-order ERK method. A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

$$\xi = \frac{|J\_1|}{|J|}, \qquad \eta = \frac{|J\_2|}{|J|}, \qquad \zeta = \frac{|J\_3|}{|J|}, \tag{38}$$

where

As we can see the LSERK only requires one additional storage level, while ERK

As to the stability condition, it is controlled by the Courant-Friedrichs-Lewy

1 2N þ 1

where vp is the P wave velocity and hmin is the minimum diameter of the circumcircles of tetrahedral elements. This condition is a necessary condition for discrete stability, and a bit more restrictive form is actually used in numerical

The absorbing boundary conditions (ABCs) in computations are required as the computational domain is finite. There are two typical ABCs to be adopted here. One is flux type ABCs [16, 19]. That is to say, the following numerical flux in Eq. (23) at

which allows only for outgoing waves and is equivalent to the first order ABCs. Though the absorbing effects of this method vary the angles of incidence, it is still effective in many cases [19]. The advantage of this type ABCs is that it merged into the FVM framework naturally and there is almost no additional computational cost. Another type is the perfectly matched layer (PML) technique originally developed

The transformation between different coordinate systems is frequently used. For

the coordinates of a physical element. The transformation from ξ � η � ζ system to

x ¼ x<sup>1</sup> þ ð Þ x<sup>2</sup> � x<sup>1</sup> ξ þ ð Þ x<sup>3</sup> � x<sup>1</sup> η þ ð Þ x<sup>4</sup> � x<sup>1</sup> ζ,

z ¼ z<sup>1</sup> þ ð Þ z<sup>2</sup> � z<sup>1</sup> ξ þ ð Þ z<sup>3</sup> � z<sup>1</sup> η þ ð Þ z<sup>4</sup> � z<sup>1</sup> ζ,

 �0.4178904744998519 0.3792103129996273 0.1496590219992291 �1.1921516946426769 0.8229550293869817 0.3704009573642048 �1.6977846924715279 0.6994504559491221 0.6222557631344432 �1.5141834442571558 0.1530572479681520 0.9582821306746903

then the transformation from x � y � z system to ξ � η � ζ system can be solved

� �<sup>η</sup> <sup>þ</sup> <sup>y</sup><sup>4</sup> � <sup>y</sup><sup>1</sup>

� �<sup>ξ</sup> <sup>þ</sup> <sup>y</sup><sup>3</sup> � <sup>y</sup><sup>1</sup>

i ai bi ci 1 0 0.1496590219992291 0

T�<sup>1</sup> ∑ Np

l¼1

w^ ð Þ <sup>m</sup> <sup>l</sup> <sup>ϕ</sup>ð Þ <sup>m</sup>

; zi

� �ζ,

� � for <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; 4 be

(37)

hmin vp

, (35)

<sup>l</sup> , (36)

has four. The coefficients required in Eq. (34) are listed in Table 3 [30].

Δt ≤

T Að Þ m kð Þ þ jAð Þ m kð Þ <sup>j</sup> � �

all tetrahedral faces that coincide with domain boundary

by [31], which is very popular in recent more 10 years.

ease of reading, we present the formulations here. Let xi; yi

y ¼ y<sup>1</sup> þ y<sup>2</sup> � y<sup>1</sup>

for ξ, η and ζ from Eq. (37) by the Cramer ruler, i.e.,

Coefficients for the low-storage five-stage fourth-order ERK method.

<sup>F</sup><sup>h</sup> <sup>¼</sup> <sup>1</sup> 2

2.6 Coordinate transformation

x � y � z system is defined by

Table 3.

10

8 ><

>:

(CFL) condition [15, 19];

Seismic Waves - Probing Earth System

computations.

∣ J1∣ ¼ x � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup> � � � � � � � � � � � � � � , ∣ J2∣ ¼ x<sup>2</sup> � x<sup>1</sup> x � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup> � � � � � � � � � � � � � � , (39) ∣ J3∣ ¼ x<sup>2</sup> � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z � z<sup>1</sup> � � � � � � � � � � � � � � , ∣ J∣ ¼ x<sup>2</sup> � x<sup>1</sup> x<sup>3</sup> � x<sup>1</sup> x<sup>4</sup> � x<sup>1</sup> y<sup>2</sup> � y<sup>1</sup> y<sup>3</sup> � y<sup>1</sup> y<sup>4</sup> � y<sup>1</sup> z<sup>2</sup> � z<sup>1</sup> z<sup>3</sup> � z<sup>1</sup> z<sup>4</sup> � z<sup>1</sup> � � � � � � � � � � � � � � : (40)

Note that J is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the tetrahedron element Tð Þ <sup>m</sup> .

The coordinate transformation from the second reference coordinate <sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup> to ξ � η � ζ system is defined by

$$\begin{cases} \xi = \xi\_1 + (\xi\_2 - \xi\_1)\tilde{\xi} + (\xi\_3 - \xi\_1)\bar{\eta} + (\xi\_4 - \xi\_1)\tilde{\xi}, \\ \eta = \eta\_1 + (\eta\_2 - \eta\_1)\tilde{\xi} + (\eta\_3 - \eta\_1)\bar{\eta} + (\eta\_4 - \eta\_1)\tilde{\xi}, \\ \zeta = \zeta\_1 + (\zeta\_2 - \zeta\_1)\tilde{\xi} + (\zeta\_3 - \zeta\_1)\bar{\eta} + (\zeta\_4 - \zeta\_1)\tilde{\zeta}, \end{cases} \tag{41}$$

then the transform from <sup>ξ</sup> � <sup>η</sup> � <sup>ζ</sup> system to <sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup> system can be solved for <sup>~</sup><sup>ξ</sup> � <sup>~</sup><sup>η</sup> � <sup>~</sup><sup>ζ</sup> from Eq. (41) by Cramer ruler similarly. Denote

$$|\tilde{f}| = \begin{vmatrix} \xi\_2 - \xi\_1 & \xi\_3 - \xi\_1 & \xi\_4 - \xi\_1 \\ \eta\_2 - \eta\_1 & \eta\_3 - \eta\_1 & \eta\_4 - \eta\_1 \\ \zeta\_2 - \zeta\_1 & \zeta\_3 - \zeta\_1 & \zeta\_4 - \zeta\_1 \end{vmatrix},\tag{42}$$

which is the determinant of the Jacobian matrix of the transformation being equal to six times the volume of the subelement Tmk <sup>~</sup> ð Þ ð Þ for <sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, Ne. In Eqs. (41) and (42), ξi; ηi; ζ<sup>i</sup> ð Þ for i ¼ 1; 2; 3; 4, denote the vertex coordinates of Tmk <sup>~</sup> ð Þ ð Þ in <sup>ξ</sup> � <sup>η</sup> � <sup>ζ</sup> system.

#### 3. Numerical computations

In this section we give three numerical examples to illustrate the performance of the developed method above. The convergence test of the proposed method can be found in [27]. Though the method is developed for the 3D case, it can be simplified to 2D without essential difficulty. The principle is the same. The first example is a test for a 2D model with uneven topography. The other two examples are for two 3D models.

Example 1. The first example is a two-layered model with the inclined interface shown in Figure 3a. The range of the model is x∈½ � �1:6km; 1:6km and z∈½ � �1:6km; 1:8km . The surface of the model is uneven to imitate the real topography. The vp and vs velocities are 3000 m=s and 2000 m=s in the upper layer and 2400 m=s and 1600 m=s in the lower layer, respectively. The densities ρ are 2200 kg=m<sup>3</sup> and 1800 kg=m<sup>3</sup> in the upper and lower layer, respectively. Figure 3b is the coarser triangular meshes for this model. A coarser version of the mesh is

Figure 3. A two-layered model with curved surface topography (a) and the triangular meshes (b).

shown here as the finest mesh in computations cannot be seen clearly. The triangular meshes can fit the curve topography very well. Note that none triangular element crosses the interface. In computations the P<sup>4</sup> polynomial reconstruction is applied. The computational domain is meshed by 113472 coarse elements. Each coarse element is subdivided into 25 subelements further. So there are 2,836,800 fine elements totally. The time step is <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>5</sup> s. The source is located at ð Þ¼ x; z ð Þ 0; 0:2km with time history

$$f(t) = -2a(t - t\_0)e^{-a(t - t\_0)^2}, \quad t\_0 = 0.08, \quad a = \left(\pi f\_0\right)^2,\tag{43}$$

Example 2. The second example is a cuboid model. The physical size of the model is ð Þ x; y; z ∈½ �� 0; 2km ½ �� 0; 2km ½ � 0; 1km . The model and its unstructured tetrahedral meshes are shown in Figure 6. There are totally 836,612 coarse tetrahedrons to mesh the model. A coarser mesh is shown as the actual mesh in computations is too fine to see clearly. Each coarse tetrahedron is subdivided into Ne ¼ 27 subelements as we adopt P<sup>3</sup> polynomial reconstruction. The parameters for λ, μ, and ρ are 109 Pa, 10<sup>9</sup> Pa, and 1000 kg=m3. The time step in computations is 10�<sup>4</sup> s. The source is located in the center of the model with time history given by

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

DOI: http://dx.doi.org/10.5772/intechopen.86400

Snapshots of u component (a) and v component (b) at propagation time 0:30s.

f tðÞ¼ sin 40 ð Þ πt e

GHz main frequency.

Figure 6.

13

A cubic model and its unstructured tetrahedral meshes.

Figure 5.

It is applied to the u component. The 3D snapshots of u, v, and w components at propagation time 0:42 s are shown in Figure 7. From these figures, we can clearly see two types of waves, i.e., the compressive wave and the shear wave. The splitting PML in nonconvolutional form is adopted here [32], and the boundary reflections are absorbed obviously and effectively. The message passing interface (MPI) parallelization based on spatial domain decomposition is applied. The CPU time for extrapolation 1000 time steps is about 33, 310 s with 128 processors each with 2.6

�100t 2

: (45)

where f <sup>0</sup> ¼ 20 Hz is the main frequency. In order to simulate point source excitation, a spatial local distribution function defined by

$$G(\mathbf{x}) = \begin{cases} \exp\left(-7||\mathbf{x} - \mathbf{x}\_0||\_2^2 / r\_0^2\right), & ||\mathbf{x} - \mathbf{x}\_0||\_2^2 \lessapprox r\_0^2, \\ 0, & ||\mathbf{x} - \mathbf{x}\_0||\_2^2 > r\_0^2, \end{cases} \tag{44}$$

is applied, where x<sup>0</sup> ¼ x0; y0; z<sup>0</sup> � � are positions of the source center. The source is added to the u component; that is to say, all source terms except g<sup>7</sup> in Eq. (1) are all zero. Figure 4 is the snapshots of u and v components at propagation time 0:25 s. Figure 5 is the snapshots of u and v components at propagation time 0:30 s. We can see the P wave and S wave propagate toward out of the model. The reflected and transmitted waves due to the tilted physical interface are also very clear. These are the expected physical phenomena of wave propagation in elastic media.

Figure 4. Snapshots of u component (a) and v component (b) at propagation time 0:25s.

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

Figure 5. Snapshots of u component (a) and v component (b) at propagation time 0:30s.

Example 2. The second example is a cuboid model. The physical size of the model is ð Þ x; y; z ∈½ �� 0; 2km ½ �� 0; 2km ½ � 0; 1km . The model and its unstructured tetrahedral meshes are shown in Figure 6. There are totally 836,612 coarse tetrahedrons to mesh the model. A coarser mesh is shown as the actual mesh in computations is too fine to see clearly. Each coarse tetrahedron is subdivided into Ne ¼ 27 subelements as we adopt P<sup>3</sup> polynomial reconstruction. The parameters for λ, μ, and ρ are 109 Pa, 10<sup>9</sup> Pa, and 1000 kg=m3. The time step in computations is 10�<sup>4</sup> s. The source is located in the center of the model with time history given by

$$f(t) = \sin\left(40\pi t\right)e^{-100t^2}.\tag{45}$$

It is applied to the u component. The 3D snapshots of u, v, and w components at propagation time 0:42 s are shown in Figure 7. From these figures, we can clearly see two types of waves, i.e., the compressive wave and the shear wave. The splitting PML in nonconvolutional form is adopted here [32], and the boundary reflections are absorbed obviously and effectively. The message passing interface (MPI) parallelization based on spatial domain decomposition is applied. The CPU time for extrapolation 1000 time steps is about 33, 310 s with 128 processors each with 2.6 GHz main frequency.

Figure 6. A cubic model and its unstructured tetrahedral meshes.

shown here as the finest mesh in computations cannot be seen clearly. The

A two-layered model with curved surface topography (a) and the triangular meshes (b).

�αð Þ <sup>t</sup>�t<sup>0</sup> <sup>2</sup>

exp �7k k x � x<sup>0</sup>

the expected physical phenomena of wave propagation in elastic media.

Snapshots of u component (a) and v component (b) at propagation time 0:25s.

where f <sup>0</sup> ¼ 20 Hz is the main frequency. In order to simulate point source

� �

2 2=r<sup>2</sup> 0

0, k k x � x<sup>0</sup>

is added to the u component; that is to say, all source terms except g<sup>7</sup> in Eq. (1) are all zero. Figure 4 is the snapshots of u and v components at propagation time 0:25 s. Figure 5 is the snapshots of u and v components at propagation time 0:30 s. We can see the P wave and S wave propagate toward out of the model. The reflected and transmitted waves due to the tilted physical interface are also very clear. These are

ð Þ¼ x; z ð Þ 0; 0:2km with time history

Seismic Waves - Probing Earth System

Figure 3.

Figure 4.

12

Gð Þ¼ x

is applied, where x<sup>0</sup> ¼ x0; y0; z<sup>0</sup>

f tðÞ¼�2αð Þ t � t<sup>0</sup> e

8 < :

excitation, a spatial local distribution function defined by

triangular meshes can fit the curve topography very well. Note that none triangular element crosses the interface. In computations the P<sup>4</sup> polynomial reconstruction is applied. The computational domain is meshed by 113472 coarse elements. Each coarse element is subdivided into 25 subelements further. So there are 2,836,800 fine elements totally. The time step is <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>5</sup> � <sup>10</sup>�<sup>5</sup> s. The source is located at

, t<sup>0</sup> ¼ 0:08, α ¼ πf <sup>0</sup>

, k k x � x<sup>0</sup>

� � are positions of the source center. The source

� �<sup>2</sup>

2 <sup>2</sup> ⩽r<sup>2</sup> 0,

2 <sup>2</sup> >r<sup>2</sup> 0,

, (43)

(44)

Figure 7.

The 3D snapshots of u component (a), v component (b), and w component (c) at propagation time 0:42 s in a cuboid model. The source is located in the center of the model.

Example 3. The third example is a real geological model in China. As shown in Figure 8a, it has a very complex topography. The physical scope of the model is x∈½ � 0; 2:0km , y∈½ � 0; 3:5km , and z∈ ½ � 0; 1:1km . The corresponding 3D mesh is shown in Figure 8b. A coarser version of the mesh is given as the actual mesh in computations is too fine to see clearly in the figure. The model is meshed with 210,701 relative coarse tetrahedral elements. Each coarse tetrahedron is subdivided into Ne ¼ 64 subelements as we adopt P<sup>4</sup> polynomial reconstruction, and thus there are 13,484,864 fine elements totally. The time step Δt is 10�<sup>4</sup> s. The source is situated at x0; y0; z<sup>0</sup> <sup>¼</sup> ð Þ <sup>750</sup>m; <sup>1300</sup>m; <sup>300</sup><sup>m</sup> with the same time history in Eq. (45). The media velocities of vp and vs are vp ¼ 3000 m=s and vs ¼ 2000 m=s. The MPI parallelization based on spatial domain decomposition is applied. The

nonconvolutional splitting PML [32] is adopted. The 3D snapshots of u, v, and w components at propagation time 0:80 s are shown in Figure 9. The CPU time for extrapolation 10,000 time steps is 100, 449 s with 256 processors each with 2.6 GHz main frequency. From Figure 9, we can see clearly the propagation of P wave

3D snapshots of u, v, and w components at propagation time 0:80 s in a real 3D model. The results are obtained by the method in this chapter with P<sup>4</sup> reconstruction. (a) u component, (b) v component,

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

DOI: http://dx.doi.org/10.5772/intechopen.86400

A new efficient high-order finite volume method for the 3D elastic wave simulation on unstructured meshes has been developed. It combines the advantages of the DG method and the traditional FV method. It adapts irregular topography very well. The reconstruction stencil is generated by refining each coarse tetrahedron which can be implemented effectively for all tetrahedrons whether they are internal or boundary elements. The hierarchical orthogonal basis functions are exploited to perform the high-order polynomial reconstruction on the stencil. The resulting reconstruction matrix remains unchanged for all tetrahedrons and can be precomputed and stored before time evolution. The method preserves a very local property like the DG method, while it has high computational efficiency like the FV method. These advantages facilitate 3D large-scale parallel computations. Numerical computations including a 3D real physical model show its good performance. The method also can be expected to solve other linear hyperbolic equations without

and S wave.

Figure 9.

(c) w component.

4. Conclusions

essential difficulty.

15

Figure 8. A real 3D model with complex topography. (a) model and (b) unstructured tetrahedral meshes.

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

Figure 9.

Example 3. The third example is a real geological model in China. As shown in Figure 8a, it has a very complex topography. The physical scope of the model is x∈½ � 0; 2:0km , y∈½ � 0; 3:5km , and z∈ ½ � 0; 1:1km . The corresponding 3D mesh is shown in Figure 8b. A coarser version of the mesh is given as the actual mesh in computations is too fine to see clearly in the figure. The model is meshed with 210,701 relative coarse tetrahedral elements. Each coarse tetrahedron is subdivided into Ne ¼ 64 subelements as we adopt P<sup>4</sup> polynomial reconstruction, and thus there are 13,484,864 fine elements totally. The time step Δt is 10�<sup>4</sup> s. The source is

The 3D snapshots of u component (a), v component (b), and w component (c) at propagation time 0:42 s in a

 <sup>¼</sup> ð Þ <sup>750</sup>m; <sup>1300</sup>m; <sup>300</sup><sup>m</sup> with the same time history in Eq. (45). The media velocities of vp and vs are vp ¼ 3000 m=s and vs ¼ 2000 m=s. The MPI parallelization based on spatial domain decomposition is applied. The

A real 3D model with complex topography. (a) model and (b) unstructured tetrahedral meshes.

situated at x0; y0; z<sup>0</sup>

Figure 8.

14

cuboid model. The source is located in the center of the model.

Seismic Waves - Probing Earth System

Figure 7.

3D snapshots of u, v, and w components at propagation time 0:80 s in a real 3D model. The results are obtained by the method in this chapter with P<sup>4</sup> reconstruction. (a) u component, (b) v component, (c) w component.

nonconvolutional splitting PML [32] is adopted. The 3D snapshots of u, v, and w components at propagation time 0:80 s are shown in Figure 9. The CPU time for extrapolation 10,000 time steps is 100, 449 s with 256 processors each with 2.6 GHz main frequency. From Figure 9, we can see clearly the propagation of P wave and S wave.

#### 4. Conclusions

A new efficient high-order finite volume method for the 3D elastic wave simulation on unstructured meshes has been developed. It combines the advantages of the DG method and the traditional FV method. It adapts irregular topography very well. The reconstruction stencil is generated by refining each coarse tetrahedron which can be implemented effectively for all tetrahedrons whether they are internal or boundary elements. The hierarchical orthogonal basis functions are exploited to perform the high-order polynomial reconstruction on the stencil. The resulting reconstruction matrix remains unchanged for all tetrahedrons and can be precomputed and stored before time evolution. The method preserves a very local property like the DG method, while it has high computational efficiency like the FV method. These advantages facilitate 3D large-scale parallel computations. Numerical computations including a 3D real physical model show its good performance. The method also can be expected to solve other linear hyperbolic equations without essential difficulty.

#### Acknowledgements

I appreciate Dr. Y. Zhuang, Prof. Chung, and Dr. L. Zhang very much for their important help and cooperation. This work is supported by the National Natural Science Foundation of China under the grant number 11471328 and 51739007. It is also partially supported by the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.

References

1.1442147

2010.04763.x

cicp.120610.090911a

[1] Minkoff SE. Spatial parallelism of a 3D finite difference velocity-stress elastic wave propagation code. SIAM Journal on Scientific Computing. 2002;24:1-19. DOI:

DOI: http://dx.doi.org/10.5772/intechopen.86400

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

Computing. 2005;26:864-884. DOI: 10.1137/S1064827502407457

[9] Zhang W, Chung E, Wang C. Stability for imposing absorbing boundary conditions in the finite element simulation of acoustic wave propagation. Journal of Computational Mathematics. 2014;32:1-20. DOI:

[10] Dubiner M. Spectral methods on triangles and other domains. Journal of Scientific Computing. 1991;6:345-390.

Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International. 1999;139:806-822. DOI: 10.1046/j.1365-246x.1999.00967.x

[12] Komatitsch D, Martin R, Tromp J, Taylor MA, Wingate BA. Wave

propagation in 2-D elastic media using a spectral element method with triangles

Computational Acoustics. 2001;9:703-718. DOI: 10.1142/S0218396X01000796

[13] Komatitsch D, Tromp J. Spectralelement simulations of global seismic wave propagation—I. Validation. Geophysical Journal International. 2002;

[14] Seriani G. 3-D large-scale wave propagation modeling by spectralelement method on Cray T3E

multiprocessor. Computer Methods in Applied Mechanics and Engineering. 1998;164:235-247. DOI: 10.1016/

[15] Dumbser M, Käser M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-II: The three-dimensional isotropic case.

10.4208/jcm.1310-m3942

DOI: 10.1007/BF01060030

[11] Komatitsch D, Tromp J.

and quadrangles. Journal of

149:390-412. DOI: 10.1046/ j.1365-246X.2002.01653.x

S0045-7825(98)00057-7

[2] Virieux J. P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics. 1986;51:889-901. DOI: 10.1190/

[3] Klin P, Priolo E, Seriani G. Numerical simulation of seismic wave propagation in realistic 3-D geo-models with a Fourier pseudo spectral method.

Geophysical Journal International. 2010; 183:905-922. DOI: 10.1111/j.1365-246X.

[4] Zhang W. Stability conditions for wave simulation in 3-D anisotropic media with the pseudospectral method. Communications in Computational Physics. 2012;12:703-720. DOI: 10.4208/

[5] Bécache E, Joly P, Tsogka C. A new family of mixed finite elements for the linear elastodynamic problem. SIAM Journal on Numerical Analysis. 2002;39: 2109-2132. DOI: 10.2307/4101053

[6] Cohen G, Joly P, Roberts JE, Tordjman N. Higher order triangular finite elements with mass lumping for the wave equations. SIAM Journal on Numerical Analysis. 2001;38:2047-2078. DOI: 10.1137/s0036142997329554

[7] Cohen G, Fauqueux S. Mixed finite elements with mass-lumping for the transient wave equation. Journal of Computational Acoustics. 2000;8:171-188. DOI: 10.1142/

[8] Cohen G, Fauqueux S. Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM Journal on Scientific

s0218396x0000011x

17

10.1137/S1064827501390960

#### Author details

Wensheng Zhang

1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

2 School of Mathematics Sciences, University of Chinese Academy Sciences, Beijing, China

\*Address all correspondence to: zws@lsec.cc.ac.cn

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

#### References

Acknowledgements

Seismic Waves - Probing Earth System

Author details

Wensheng Zhang

China

16

of Sciences, Beijing, China

\*Address all correspondence to: zws@lsec.cc.ac.cn

provided the original work is properly cited.

ary Sciences, Chinese Academy of Sciences.

I appreciate Dr. Y. Zhuang, Prof. Chung, and Dr. L. Zhang very much for their important help and cooperation. This work is supported by the National Natural Science Foundation of China under the grant number 11471328 and 51739007. It is also partially supported by the National Center for Mathematics and Interdisciplin-

1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy

2 School of Mathematics Sciences, University of Chinese Academy Sciences, Beijing,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[1] Minkoff SE. Spatial parallelism of a 3D finite difference velocity-stress elastic wave propagation code. SIAM Journal on Scientific Computing. 2002;24:1-19. DOI: 10.1137/S1064827501390960

[2] Virieux J. P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics. 1986;51:889-901. DOI: 10.1190/ 1.1442147

[3] Klin P, Priolo E, Seriani G. Numerical simulation of seismic wave propagation in realistic 3-D geo-models with a Fourier pseudo spectral method. Geophysical Journal International. 2010; 183:905-922. DOI: 10.1111/j.1365-246X. 2010.04763.x

[4] Zhang W. Stability conditions for wave simulation in 3-D anisotropic media with the pseudospectral method. Communications in Computational Physics. 2012;12:703-720. DOI: 10.4208/ cicp.120610.090911a

[5] Bécache E, Joly P, Tsogka C. A new family of mixed finite elements for the linear elastodynamic problem. SIAM Journal on Numerical Analysis. 2002;39: 2109-2132. DOI: 10.2307/4101053

[6] Cohen G, Joly P, Roberts JE, Tordjman N. Higher order triangular finite elements with mass lumping for the wave equations. SIAM Journal on Numerical Analysis. 2001;38:2047-2078. DOI: 10.1137/s0036142997329554

[7] Cohen G, Fauqueux S. Mixed finite elements with mass-lumping for the transient wave equation. Journal of Computational Acoustics. 2000;8:171-188. DOI: 10.1142/ s0218396x0000011x

[8] Cohen G, Fauqueux S. Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM Journal on Scientific

Computing. 2005;26:864-884. DOI: 10.1137/S1064827502407457

[9] Zhang W, Chung E, Wang C. Stability for imposing absorbing boundary conditions in the finite element simulation of acoustic wave propagation. Journal of Computational Mathematics. 2014;32:1-20. DOI: 10.4208/jcm.1310-m3942

[10] Dubiner M. Spectral methods on triangles and other domains. Journal of Scientific Computing. 1991;6:345-390. DOI: 10.1007/BF01060030

[11] Komatitsch D, Tromp J. Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophysical Journal International. 1999;139:806-822. DOI: 10.1046/j.1365-246x.1999.00967.x

[12] Komatitsch D, Martin R, Tromp J, Taylor MA, Wingate BA. Wave propagation in 2-D elastic media using a spectral element method with triangles and quadrangles. Journal of Computational Acoustics. 2001;9:703-718. DOI: 10.1142/S0218396X01000796

[13] Komatitsch D, Tromp J. Spectralelement simulations of global seismic wave propagation—I. Validation. Geophysical Journal International. 2002; 149:390-412. DOI: 10.1046/ j.1365-246X.2002.01653.x

[14] Seriani G. 3-D large-scale wave propagation modeling by spectralelement method on Cray T3E multiprocessor. Computer Methods in Applied Mechanics and Engineering. 1998;164:235-247. DOI: 10.1016/ S0045-7825(98)00057-7

[15] Dumbser M, Käser M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-II: The three-dimensional isotropic case.

Geophysical Journal International. 2006;167:319-336. DOI: 10.1111/ j.1365-246X.2006.03120.x

[16] Käser M, Dumbser M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes. I: The twodimensional isotropic case with external source terms. Geophysical Journal International. 2006;166:855-877. DOI: 10.1111/j.1365-246X.2006.03051.x

[17] Käser M, Dumbser M, Puente J, Igel H. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes— III: Viscoelastic attenuation. Geophysical Journal International. 2007;168:224-242. DOI: 10.1111/j.1365-246X.2006.03193.x

[18] Ye R, de Hoop MV, Petrovitch CL, Pyrak-Nolte LJ, Wilcox LC. A discontinuous Galerkin method with a modified penalty flux for the propagation and scattering of acoustoelastic waves. Geophysical Journal International. 2016;205:1267-1289. DOI: 10.1093/gji/ggw070

[19] Dumbser M, Käser M, de la Puente J. Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D. Geophysical Journal International. 2007; 171:665-694. DOI: 10.1111/ j.1365-246X.2007.03421.x

[20] Dumbser M, Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics. 2007;221: 693-723. DOI: 10.1016/j.jcp.2006.06.043

[21] Käser M, Iske A. ADER schemes on adaptive triangular meshes for scalar conservation laws. Journal of Computational Physics. 2005;205: 486-508. DOI: 10.1016/j.jcp.2004.11.015

[22] Zhang W, Zhuang Y, Chung ET. A new spectral finite volume method for

elastic modelling on unstructured meshes. Geophysical Journal International. 2016;206:292-307. DOI: 10.1093/gji/ggw148

propagation in poroelastic media. Geophysics. 2008;73:T77-T97. DOI:

[30] Hesthaven JS, Warburton T. Nodal Discontinuous Galerkin Methods. New York: Springer-Verlag; 2008. 502 p. DOI: 10.1007/978-0-387-72067-8

DOI: http://dx.doi.org/10.5772/intechopen.86400

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes

[31] Bérenger JP. A perfectly matched

[32] Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics. 2001;66:294-307.

electromagnetic waves. Journal of Computational Physics. 1994;114: 185-200. DOI: 10.1006/jcph.1996.0181

layer for the absorption of

DOI: 10.1190/1.1444908

19

10.1190/1.2965027

[23] Dumbser M, Käser M, Toro EF. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—V: Local time stepping and p-adaptivity. Geophysical Journal International. 2007;171:695-717. DOI: 10.1111/j.1365-246X.2007.03427.x

[24] Liu Y, Vinokur M, Wang ZJ. Spectral (finite) volume method for conservation laws on unstructured grids. V: Extension to three-dimensional systems. Journal of Computational Physics. 2006;212:454-472. DOI: 10.1016/j.jcp.2003.09.012

[25] Wang ZJ. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation. Journal of Computational Physics. 2002; 178:210-251. DOI: 10.1006/ jcph.2002.7041

[26] Wang ZJ, Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids. II: Extension to two-dimensional scalar equation. Journal of Computational Physics. 2002; 179:665-697. DOI: 10.1006/jcph. 2002.7082

[27] Zhang W, Zhuang Y, Zhang L. A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes. Journal of Computational Physics. 2017;340: 534-555. DOI: 10.1016/j.jcp.2017.03.050

[28] Puente J, Käser M, Dumbser M, Igel H. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes-IV: Anisotropy. Geophysical Journal International. 2007;169:1210-1228. DOI: 10.1111/j.1365-246X.2007.03381.x

[29] Puente J, Dumbser M, Käser M, Igel H. Discontinuous Galerkin method for

A High-Order Finite Volume Method for 3D Elastic Modelling on Unstructured Meshes DOI: http://dx.doi.org/10.5772/intechopen.86400

propagation in poroelastic media. Geophysics. 2008;73:T77-T97. DOI: 10.1190/1.2965027

Geophysical Journal International. 2006;167:319-336. DOI: 10.1111/ j.1365-246X.2006.03120.x

Seismic Waves - Probing Earth System

[16] Käser M, Dumbser M. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes. I: The twodimensional isotropic case with external source terms. Geophysical Journal International. 2006;166:855-877. DOI: 10.1111/j.1365-246X.2006.03051.x

elastic modelling on unstructured meshes. Geophysical Journal

10.1093/gji/ggw148

International. 2016;206:292-307. DOI:

[23] Dumbser M, Käser M, Toro EF. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—V: Local time stepping and p-adaptivity. Geophysical Journal International. 2007;171:695-717. DOI: 10.1111/j.1365-246X.2007.03427.x

[24] Liu Y, Vinokur M, Wang ZJ. Spectral (finite) volume method for conservation laws on unstructured grids. V: Extension to three-dimensional systems. Journal of Computational Physics. 2006;212:454-472. DOI: 10.1016/j.jcp.2003.09.012

[25] Wang ZJ. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation. Journal of Computational Physics. 2002;

[26] Wang ZJ, Liu Y. Spectral (finite) volume method for conservation laws on unstructured grids. II: Extension to two-dimensional scalar equation. Journal of Computational Physics. 2002;

179:665-697. DOI: 10.1006/jcph.

[27] Zhang W, Zhuang Y, Zhang L. A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes. Journal of Computational Physics. 2017;340: 534-555. DOI: 10.1016/j.jcp.2017.03.050

[28] Puente J, Käser M, Dumbser M, Igel

[29] Puente J, Dumbser M, Käser M, Igel H. Discontinuous Galerkin method for

discontinuous Galerkin method for elastic waves on unstructured meshes-IV: Anisotropy. Geophysical Journal International. 2007;169:1210-1228. DOI: 10.1111/j.1365-246X.2007.03381.x

H. An arbitrary high-order

178:210-251. DOI: 10.1006/

jcph.2002.7041

2002.7082

[17] Käser M, Dumbser M, Puente J, Igel

[18] Ye R, de Hoop MV, Petrovitch CL,

discontinuous Galerkin method with a

propagation and scattering of acoustoelastic waves. Geophysical Journal International. 2016;205:1267-1289. DOI:

[19] Dumbser M, Käser M, de la Puente J. Arbitrary high-order finite volume schemes for seismic wave propagation on unstructured meshes in 2D and 3D. Geophysical Journal International. 2007;

[20] Dumbser M, Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. Journal of Computational Physics. 2007;221: 693-723. DOI: 10.1016/j.jcp.2006.06.043

[21] Käser M, Iske A. ADER schemes on adaptive triangular meshes for scalar

[22] Zhang W, Zhuang Y, Chung ET. A new spectral finite volume method for

conservation laws. Journal of Computational Physics. 2005;205: 486-508. DOI: 10.1016/j.jcp.2004.11.015

18

Pyrak-Nolte LJ, Wilcox LC. A

modified penalty flux for the

10.1093/gji/ggw070

171:665-694. DOI: 10.1111/ j.1365-246X.2007.03421.x

discontinuous Galerkin method for elastic waves on unstructured meshes— III: Viscoelastic attenuation. Geophysical Journal International. 2007;168:224-242. DOI: 10.1111/j.1365-246X.2006.03193.x

H. An arbitrary high-order

[30] Hesthaven JS, Warburton T. Nodal Discontinuous Galerkin Methods. New York: Springer-Verlag; 2008. 502 p. DOI: 10.1007/978-0-387-72067-8

[31] Bérenger JP. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics. 1994;114: 185-200. DOI: 10.1006/jcph.1996.0181

[32] Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics. 2001;66:294-307. DOI: 10.1190/1.1444908

Chapter 2

Abstract

1. Introduction

21

Problem

Jeremiah Rushchitsky

Cylindrical Surface Wave:

Revisiting the Classical Biot'

The problem on a surface harmonic elastic wave propagating along the free surface of cylindrical cavity in the direction of cavity axis is considered. In the case of isotropic medium, this is the classical Biot's problem of 1952. First, the Biot pioneer work is revisited: the analytical part of Biot's findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. Then the new direct generalization of Biot's problem on the case of transversely isotropic medium within the framework of linear theory of elasticity is proposed. Transition to the transverse isotropy needs some novelty—necessity of using the more complex representations of displacements through two potentials. Finally, a generalization of Biot's problem on the case of isotropic and transversely isotropic media in the framework of linearized theory of elasticity with allowance for initial stresses is stated. This part repeats briefly the results of A.N. Guz with co-authors of 1974. The main features are using the linearized theory of elasticity and one only potential. All three parts are shown as analytical study up to the level when the numerical methods have to be used.

Keywords: surface harmonic cylindrical wave, classical Biot's problem,

of Biot's Problem on the Case of Transversely Isotropic Media within the

Note first that the seismic waves include mainly the primary and secondary body waves and different kinds of surface waves. This chapter is devoted to one kind of surface waves. The problem is stated as follows: the infinite medium with cylindrical circular cavity having the symmetry axis Oz and constant radius is analyzed. An attenuating in depth of medium surface harmonic wave propagates along the cavity surface in directionOz. In this case, the problem becomes mathematically the axisymmetric one. This problem is solved by Biot in 1952 [1] with assumption that the medium is isotropic. The context of this chapter includes four parts. The subchapter 1 "Introduction" is the standard one. The subchapter 2 is named: "Main Stages of Solving the Classical Biot's Problem on Surface Wave along Cylindrical Cavity." Here, the analytical part of Biot's findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. The subchapter 3 "Direct Generalization

generalization to the case of transversely isotropic medium

s

#### Chapter 2

## Cylindrical Surface Wave: Revisiting the Classical Biot' s Problem

Jeremiah Rushchitsky

#### Abstract

The problem on a surface harmonic elastic wave propagating along the free surface of cylindrical cavity in the direction of cavity axis is considered. In the case of isotropic medium, this is the classical Biot's problem of 1952. First, the Biot pioneer work is revisited: the analytical part of Biot's findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. Then the new direct generalization of Biot's problem on the case of transversely isotropic medium within the framework of linear theory of elasticity is proposed. Transition to the transverse isotropy needs some novelty—necessity of using the more complex representations of displacements through two potentials. Finally, a generalization of Biot's problem on the case of isotropic and transversely isotropic media in the framework of linearized theory of elasticity with allowance for initial stresses is stated. This part repeats briefly the results of A.N. Guz with co-authors of 1974. The main features are using the linearized theory of elasticity and one only potential. All three parts are shown as analytical study up to the level when the numerical methods have to be used.

Keywords: surface harmonic cylindrical wave, classical Biot's problem, generalization to the case of transversely isotropic medium

#### 1. Introduction

Note first that the seismic waves include mainly the primary and secondary body waves and different kinds of surface waves. This chapter is devoted to one kind of surface waves. The problem is stated as follows: the infinite medium with cylindrical circular cavity having the symmetry axis Oz and constant radius is analyzed. An attenuating in depth of medium surface harmonic wave propagates along the cavity surface in directionOz. In this case, the problem becomes mathematically the axisymmetric one. This problem is solved by Biot in 1952 [1] with assumption that the medium is isotropic. The context of this chapter includes four parts. The subchapter 1 "Introduction" is the standard one. The subchapter 2 is named: "Main Stages of Solving the Classical Biot's Problem on Surface Wave along Cylindrical Cavity." Here, the analytical part of Biot's findings is shown in the main fragments. The features are using two potentials and representation of solution by Macdonald functions of different indexes. The subchapter 3 "Direct Generalization of Biot's Problem on the Case of Transversely Isotropic Media within the

Framework of Linear Theory of Elasticity" contains the new approach to the classical Biot's problem and represents the direct generalization of this problem that uses the Biot's scheme of analysis. Transition to the case of transverse isotropy needs some novelty—necessity of using the more complex representations of displacements through two potentials. The subchapter 4 "Genera-lization of Biot's Problem on the case of Isotropic and Transversely Isotropic Media within the framework of Linearized Theory of Elasticity with Allowance for Initial Stresses" repeats briefly the results of A.N. Guz with co-authors (1974). They considered a generalization of the Biot's problem on the case of elastic media with allowance for the initial stresses. The main features are using the linearized theory of elasticity, one only potential, and Macdonald function of one index.

Here the standard notations of Laplace operator Δrz and velocities of longitudi-

The solution of Eqs. (6) and (7) is found in the form of harmonic waves in the

A substitution of representations (8) into the wave Eqs. (6) and (7) gives the

,r � <sup>k</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>v</sup>=vL

,r � <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

These equations correspond to the Bessel equation for Macdonald functions

More exactly, Eqs. (9) and (10) have the solutions in the form of Macdonald

are fulfilled. According to conditions (12), the wave number of cylindrical wave

Further the wave Eqs. (9) and (10) are considered separately. The first equation

<sup>L</sup>Φ<sup>∗</sup> <sup>¼</sup> <sup>0</sup> mL <sup>¼</sup> <sup>k</sup>

of zeroth order and unknown argument x ¼ mLr, which includes the unknown

<sup>2</sup> � � � � <sup>Ψ</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup> mT <sup>¼</sup> <sup>k</sup>

<sup>2</sup> � �>0; <sup>k</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>v</sup>=vT

<sup>2</sup> h i <sup>þ</sup> <sup>1</sup>=<sup>r</sup> <sup>2</sup> n o � � <sup>Ψ</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup> � � (10)

<sup>2</sup> � �Φ<sup>∗</sup> <sup>¼</sup> <sup>0</sup> � �, (9)

> <sup>T</sup> þ 1=r <sup>2</sup> � � � � <sup>Ψ</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>

<sup>=</sup>x<sup>2</sup> � � � � <sup>y</sup> <sup>¼</sup> <sup>0</sup> (11)

<sup>2</sup> � �><sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

<sup>Φ</sup><sup>∗</sup> ð Þ¼ <sup>r</sup> <sup>A</sup>ΦK0ð Þ mLr (14)

2 r� � (13)

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vT

2 r� �: (15)

� � (12)

i kz ð Þ �ω<sup>t</sup> , <sup>Ψ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> <sup>e</sup>

ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>=</sup><sup>ρ</sup> <sup>p</sup> , vT <sup>¼</sup> ffiffiffiffiffiffiffi

i kz ð Þ �ω<sup>t</sup> , (8)

μ=ρ p

nal and transverse waves in isotropic elastic medium vL <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

DOI: http://dx.doi.org/10.5772/intechopen.86910

2.2 Solving the wave equations in the form of Macdonald functions

<sup>Φ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> cos k zð Þ � vt , <sup>Ψ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> sin k zð Þ � vt :

equations relative to the unknown amplitudes <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> , <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup>

L � �Φ<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:

,rr <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>Φ</sup><sup>∗</sup>

,rr � ð Þ <sup>1</sup>=<sup>r</sup> <sup>Ψ</sup><sup>∗</sup>

Kλð Þ x (modified Bessel functions of the second kind [2–4])

,r � <sup>m</sup><sup>2</sup>

The second equation can be written in the form

<sup>T</sup> þ 1=r

,r � <sup>m</sup><sup>2</sup>

,r � <sup>k</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>v</sup>=vT

<sup>y</sup>″ <sup>þ</sup> ð Þ <sup>1</sup>=<sup>x</sup> <sup>y</sup><sup>0</sup> � <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

<sup>T</sup>><sup>0</sup> <sup>k</sup><sup>2</sup> <sup>1</sup> � ð Þ <sup>v</sup>=vL

must be real, and the wave velocity must be less of the velocities of classical

This equation has the solution in the form of Macdonald function:

are used.

Φ<sup>∗</sup>

,rr <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>Φ</sup><sup>∗</sup>

Ψ∗

functions, if the conditions.

<sup>L</sup>>0, k<sup>2</sup> � <sup>k</sup><sup>2</sup>

longitudinal and transverse plane waves.

,rr <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>Φ</sup><sup>∗</sup>

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

is written in the form

Φ<sup>∗</sup>

phase velocity of wave.

,rr � ð Þ <sup>1</sup>=<sup>r</sup> <sup>Ψ</sup><sup>∗</sup>

Ψ∗

23

direction of vertical coordinate:

<sup>Φ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> <sup>e</sup>

,r � <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

Ψ∗

Φ<sup>∗</sup>

,rr � ð Þ <sup>1</sup>=<sup>r</sup> <sup>Ψ</sup><sup>∗</sup>

#### 2. Main stages of solving the classical Biot's problem on surface wave along a cylindrical cavity

#### 2.1 Statement of problem and main equations in potentials

A cylindrical system of coordinates Orϑz is chosen, and a harmonic wave is considered that has the phase variable σ ¼ k zð Þ � vt , unknown wave number k ¼ ð Þ ω=v , unknown phase velocity v, and arbitrary (but given) frequency ω and ampitude A. It is supposed that the wave propagates in an infinite medium with cylindrical cavity of constant radius ro in the direction of vertical coordinate z and possibly attenuates in the direction of radial coordinate r. In this linear statement and in assumption that deformations are small, the problem is axisymmetric, and deformations are described by two displacements urð Þ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> ; <sup>u</sup>φð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>0</sup> and uzð ÞÞ r; z; t and two Lame equations of the form

$$\frac{\mathbf{C}\_{11} - \mathbf{C}\_{12}}{2} \left( \Delta\_{\text{rx}} u\_r - \frac{1}{r^2} u\_r \right) + \frac{\mathbf{C}\_{11} + \mathbf{C}\_{12}}{2} \left( u\_{r,r} + \frac{1}{r} u\_r + u\_{\text{z},\text{z}} \right)\_{,r} = \rho u\_{r,\text{tt}},\tag{1}$$

$$\frac{1}{2}(\mathbf{C}\_{11} - \mathbf{C}\_{12})\Delta\_{\text{rx}}u\_{\text{z}} + \frac{1}{2}(\mathbf{C}\_{11} + \mathbf{C}\_{12})\left(u\_{r,r} + \frac{1}{r}u\_{r} + u\_{\text{z},\text{z}}\right)\_{,\text{z}} = \rho u\_{\text{z},\text{tt}},\tag{2}$$

or

$$(\lambda + 2\mu) \left( u\_{r,rr} + \frac{1}{r} u\_{r,r} - \frac{1}{r^2} u\_r + u\_{\text{z},r\text{z}} \right) + \mu (u\_{r,\text{zz}} - u\_{\text{z},r\text{z}}) = \rho u\_{r,\text{tt}} \tag{3}$$

$$\mu \left(\lambda + 2\mu\right) \left(u\_{r,rx} + \frac{1}{r} u\_{r,x} + u\_{x,xx}\right) - \mu \left[\frac{1}{r} (u\_{r,x} - u\_{x,r}) + (u\_{r,rx} - u\_{x,rr})\right] = \rho u\_{x,tt}. \tag{4}$$

Further the potentials Φð Þ r; z; t , Ψð Þ r; z; t are introduced

$$
\mu\_r = \Phi\_{,r} - \Psi\_{,x}, \ \ u\_x = \Phi\_{,x} + \Psi\_{,r} + (\mathbf{1}/r)\Psi. \tag{5}
$$

When Eq. (5) is substituted into Eqs. (3) and (4), then two uncoupled linear wave equations are obtained:

$$
\Delta\_{\rm rx} \Phi - \left(\mathbf{1}/v\_L\right)^2 \Phi\_{,\rm t} = \mathbf{0},\tag{6}
$$

$$
\Delta\_{rz} \Psi - \left(\mathbf{1}/r^2\right) \Psi - \left(\mathbf{1}/\nu\_T\right)^2 \Psi\_{,\mathfrak{u}} = \mathbf{0}.\tag{7}
$$

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

Framework of Linear Theory of Elasticity" contains the new approach to the classical Biot's problem and represents the direct generalization of this problem that uses the Biot's scheme of analysis. Transition to the case of transverse isotropy needs some novelty—necessity of using the more complex representations of displacements through two potentials. The subchapter 4 "Genera-lization of Biot's Problem on the case of Isotropic and Transversely Isotropic Media within the framework of Linearized Theory of Elasticity with Allowance for Initial Stresses" repeats briefly the results of A.N. Guz with co-authors (1974). They considered a generalization of the Biot's problem on the case of elastic media with allowance for the initial stresses. The main features are using the linearized theory of elasticity, one only potential,

2. Main stages of solving the classical Biot's problem on surface wave

A cylindrical system of coordinates Orϑz is chosen, and a harmonic wave is considered that has the phase variable σ ¼ k zð Þ � vt , unknown wave number k ¼ ð Þ ω=v , unknown phase velocity v, and arbitrary (but given) frequency ω and ampitude A. It is supposed that the wave propagates in an infinite medium with cylindrical cavity of constant radius ro in the direction of vertical coordinate z and possibly attenuates in the direction of radial coordinate r. In this linear statement and in assumption that deformations are small, the problem is axisymmetric, and deformations are described by two displacements urð Þ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> ; <sup>u</sup>φð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>0</sup> and

2.1 Statement of problem and main equations in potentials

and Macdonald function of one index.

Seismic Waves - Probing Earth System

along a cylindrical cavity

uzð ÞÞ r; z; t and two Lame equations of the form

<sup>r</sup><sup>2</sup> ur

1 r

ur, <sup>z</sup> þ uz, zz

1 2

ur,r � <sup>1</sup>

Further the potentials Φð Þ r; z; t , Ψð Þ r; z; t are introduced

ΔrzΨ � 1=r

þ

C<sup>11</sup> þ C<sup>12</sup>

ð Þ C<sup>11</sup> þ C<sup>12</sup> ur,r þ

<sup>r</sup><sup>2</sup> ur <sup>þ</sup> uz,rz

When Eq. (5) is substituted into Eqs. (3) and (4), then two uncoupled linear

<sup>2</sup> <sup>Ψ</sup> � ð Þ <sup>1</sup>=vT

2

ΔrzΦ � ð Þ 1=vL

� μ 1 r

<sup>2</sup> ur,r <sup>þ</sup>

1 r

1 r

ur þ uz, <sup>z</sup> 

ur þ uz, <sup>z</sup> 

ð Þþ ur, <sup>z</sup> � uz,r ð Þ ur,rz � uz,rr 

ur ¼ Φ,r � Ψ, z, uz ¼ Φ, <sup>z</sup> þ Ψ,r þ ð Þ 1=r Ψ: (5)

2

,r

þ μð Þ¼ ur, zz � uz,rz ρur,tt (3)

Φ,tt ¼ 0, (6)

Ψ,tt ¼ 0: (7)

, z

¼ ρur,tt, (1)

¼ ρuz,tt, (2)

¼ ρuz,tt: (4)

<sup>Δ</sup>rzur � <sup>1</sup>

ð Þ C<sup>11</sup> � C<sup>12</sup> Δrzuz þ

ð Þ λ þ 2μ ur,rr þ

wave equations are obtained:

1 r

C<sup>11</sup> � C<sup>12</sup> 2

ð Þ λ þ 2μ ur,rz þ

1 2

or

22

Here the standard notations of Laplace operator Δrz and velocities of longitudinal and transverse waves in isotropic elastic medium vL <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>=</sup><sup>ρ</sup> <sup>p</sup> , vT <sup>¼</sup> ffiffiffiffiffiffiffi μ=ρ p are used.

#### 2.2 Solving the wave equations in the form of Macdonald functions

The solution of Eqs. (6) and (7) is found in the form of harmonic waves in the direction of vertical coordinate:

$$\Phi(r,z,t) = \Phi^\*(r)e^{i(kx-\alpha t)}, \quad \Psi(r,z,t) = \Psi^\*(r)e^{i(kx-\alpha t)}, \tag{8}$$

<sup>Φ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> cos k zð Þ � vt , <sup>Ψ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> sin k zð Þ � vt :

A substitution of representations (8) into the wave Eqs. (6) and (7) gives the equations relative to the unknown amplitudes <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> , <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> Φ<sup>∗</sup> ,rr <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>Φ</sup><sup>∗</sup> ,r � <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> L � �Φ<sup>∗</sup> <sup>¼</sup> <sup>0</sup>:

$$\left(\Phi^\*\right)\_{,rr} + \left(\mathbf{1}/r\right)\Phi^\*\left.\_{,r} - k^2 \left(\mathbf{1} - \left(v/v\_L\right)^2\right)\Phi^\*\left.=\mathbf{0}\right),\tag{9}$$

$$\begin{aligned} \Psi^\* \, \_{,rr} - (\mathbf{1}/r) \Psi^\* \, \_{,r} - \left[k^2 - k\_T^2 + \left(\mathbf{1}/r^2\right)\right] \Psi^\* &= \mathbf{0} \\ \left(\Psi^\* \, \_{,rr} - (\mathbf{1}/r) \Psi^\* \, \_{,r} - \left\{k^2 \left[1 - \left(v/v\_T\right)^2\right] + \left(\mathbf{1}/r^2\right)\right\} \Psi^\* &= \mathbf{0}\right) \end{aligned} \tag{10}$$

These equations correspond to the Bessel equation for Macdonald functions Kλð Þ x (modified Bessel functions of the second kind [2–4])

$$y' + (\mathbf{1}/\mathbf{x})y' - \left[\mathbf{1} + \left(\lambda^2/\mathbf{x}^2\right)\right]y = \mathbf{0} \tag{11}$$

More exactly, Eqs. (9) and (10) have the solutions in the form of Macdonald functions, if the conditions.

$$k^2 - k\_L^2 > 0,\\ k^2 - k\_T^2 > 0\\ \left(k^2 \left(1 - \left(v/v\_L\right)^2\right) > 0,\\ k^2 \left(1 - \left(v/v\_T\right)^2\right) > 0\right) \tag{12}$$

are fulfilled. According to conditions (12), the wave number of cylindrical wave must be real, and the wave velocity must be less of the velocities of classical longitudinal and transverse plane waves.

Further the wave Eqs. (9) and (10) are considered separately. The first equation is written in the form

$$\left(\Phi^\*\right)\_{,rr} + \left(\mathbf{1}/r\right)\Phi^\*\left, -m\_L^2\Phi^\* = 0 \quad m\_L = \ \not\!k\sqrt{\left(\mathbf{1} - \left(v/v\_L\right)^2\right)}\tag{13}$$

This equation has the solution in the form of Macdonald function:

$$\Phi^\*(r) = A\_{\Phi} K\_0(m\_L r) \tag{14}$$

of zeroth order and unknown argument x ¼ mLr, which includes the unknown phase velocity of wave.

The second equation can be written in the form

$$\left\{\Psi^\*\right\}\_{,rr} - \left(\mathbf{1}/r\right)\Psi^\*\right\}\_{,r} - \left\{m\_T^2 + \left(\mathbf{1}/r^2\right)\right\}\Psi^\* = \mathbf{0} \, m\_T = \, k\sqrt{\left(\mathbf{1} - \left(\boldsymbol{\nu}/\boldsymbol{\nu}\_T\right)^2\right)}.\tag{15}$$

Figure 1. Plots of the first five Macdonald functions.

The corresponding solution under conditions (12) is expressed by the Macdonald function K<sup>1</sup> r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> T � � q

$$\Psi^\*\left(r\right) = A\,\Psi K\_1(m\_T r) \tag{16}$$

½ � <sup>2</sup>μð Þþ <sup>Φ</sup>,rr � <sup>Ψ</sup>,rz λΔΦ <sup>r</sup>¼ro <sup>¼</sup> <sup>0</sup>, <sup>μ</sup> <sup>2</sup>ð Þþ <sup>Φ</sup>,rz � <sup>Ψ</sup>, zz ΔΨ � <sup>1</sup>=<sup>r</sup>

<sup>Φ</sup>,tt <sup>¼</sup> 0, ΔΨ � <sup>1</sup>=r<sup>2</sup> ð Þ<sup>Ψ</sup> � ð Þ <sup>1</sup>=vT

r¼ro

� �A<sup>Φ</sup> �

2 r� �A<sup>Φ</sup> <sup>þ</sup> <sup>2</sup> � ð Þ <sup>v</sup>=vT

> þ 1 mLro

� �

K1ð Þ mTro

r� �K1ð Þ mLro <sup>A</sup><sup>Φ</sup> <sup>þ</sup> <sup>2</sup> � ð Þ <sup>v</sup>=vT

<sup>0</sup>ð Þ¼� x K1ð Þ x , K<sup>0</sup>

<sup>0</sup>ð Þ¼ x K0ð Þ x , K″

<sup>2</sup> h i K0ð Þ mLro

1 � ð Þ v=vT

¼ 0.

<sup>2</sup> K0ð Þ mLro

K0ð Þþ mLro K2ð Þ mLro

(23) is consideration of the system relative to quantities K1ð Þ mLro A<sup>Φ</sup> and

þ 1 mTro � �K1ð Þ mTro <sup>A</sup><sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>,

� λ 2μ

Solving of systems (24) and (25) gives two results. First, the solution is found accurate within one amplitude factor. Second, an equation for determination of phase velocity of cylindrical surface wave can be obtained in an explicit form.

The work of Biot (1952) has demonstrated some art in handling the Macdonald functions and has written Eq. (24) through only functions of the zeroth and first

have been used [3]. As a result, the equation for determination of phase velocity

K1ð Þ mLro

2 r� � <sup>K</sup>0ð Þ mTro

<sup>1</sup>ð Þ¼� x K″

þ

<sup>0</sup>ð Þ x ,

<sup>0</sup>ð Þ¼ x ð Þ 1=x K1ð Þþ x K0ð Þ x

1 � ð Þ v=vL <sup>2</sup> � �

9 = ;

¼ 0:

mLro

� �

þ 1 mTro

K1ð Þ mTro

� �K1ð Þ mLro <sup>A</sup><sup>Φ</sup>

ð Þ v=vL

In the work [1], Biot has used the expressions.

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

K1ð Þ mLro

2 r� � <sup>K</sup>0ð Þ mTro

2

<sup>Φ</sup>,tt h i

ΔΦ � ð Þ 1=vL

2ð Þþ Φ,rz � Ψ, zz ð Þ 1=vT

amplitude coefficients

1 � ð Þ v=vL

K1ð Þ mSro A<sup>Ψ</sup>

þ

2

K″

1 � ð Þ v=vL

<sup>2</sup> � � K0ð Þ mLro

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

orders. For that, the known formulas

<sup>0</sup>ð Þþ x ð Þ 1=x K<sup>0</sup>

of cylindrical wave has the form

2 � ð Þ v=vT

� 4

25

K0

<sup>2</sup> � � <sup>2</sup> � ð Þ <sup>v</sup>=vT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

2 r� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8 < :

<sup>Ψ</sup>,tt h i

<sup>2</sup> � <sup>λ</sup> μ ð Þ v=vL

2

2

such a way ð Þþ Φ,rr � Ψ,rz ð Þ λ=2μ ð Þ 1=vL

DOI: http://dx.doi.org/10.5772/intechopen.86910

<sup>2</sup> � �Ψ � �

¼ 0,

2

r¼ro

<sup>2</sup> � �K1ð Þ mTro

K1ð Þ mLro

<sup>2</sup> K0ð Þ mLro K1ð Þ mLro

<sup>2</sup> � �K1ð Þ mTro <sup>A</sup><sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>: (25)

2

Then the substitution of solutions (14) and (16) into the boundary conditions (21) gives two homogeneous algebraic equations relative to the unknown constant

An analysis of these equations that describe the cylindrical surface wave is very similar to the analysis that has been carried out by Rayleigh for the classical wave propagating along the plane surface. Some novelty in analysis of systems (22) and

<sup>r</sup>¼ro <sup>¼</sup> 0 (21)

Ψ,tt ¼ 0 and rewrite Eq. (21) in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

2 r� �A<sup>Ψ</sup> <sup>¼</sup> <sup>0</sup>,

A<sup>Ψ</sup> ¼ 0: (23)

(22)

(24)

(26)

(27)

of the first order and unknown argument x ¼ mTr, which includes the unknown wave velocity. The amplitude coefficient A<sup>Ψ</sup> is assumed to be constant and arbitrary.

Note that the Macdonald functions have the property of attenuation with increasing arguments which is shown in Figure 1. Therefore, the propagation along the vertical coordinate z waves (15) and (16) can be considered as the waves with amplitudes <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> , <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> , which attenuate with increasing the radial coordinate <sup>r</sup>.

This means that amplitudes can decrease essentially with increasing the distance from the surface of cylindrical cavity. In this sense, the waves (15) and (16) are the surface ones. This forms also the sense of conditions (12). The same conditions are used in the analysis of classical Rayleigh surface wave which propagates along the plane surface of isotropic elastic medium [5–9]. But the Rayleigh wave attenuates as an exponential function when being moved from the free surface, whereas the cylindrical surface Biot's wave attenuates as the Macdonald functions. At that, the arguments in exponential function and Macdonald functions are identical and depend on the wave velocity.

#### 2.3 Boundary conditions: equations for unknown wave number

The boundary conditions correspond to the absence of stresses on surface r ¼ ro

$$
\sigma\_{rr}(r = r\_o, z, t) = \mathbf{0}, \quad \sigma\_{rz}(r = r\_o, z, t) = \mathbf{0}. \tag{17}
$$

The stresses

$$
\sigma\_{rr} = 2\mu u\_{r,r} + \lambda((u\_r/r) + u\_{r,r} + u\_{x,x}), \quad \sigma\_{rz} = \mu(u\_{r,x} + u\_{x,r}) \tag{18}
$$

are written through the potentials

$$\sigma\_{rr} = (\lambda + 2\mu)(\Phi\_{,rr} - \Psi\_{,rr}) + \lambda\{(\mathbf{1}/r)(\Phi\_{,r} - \Psi\_{,x}) + \Phi\_{,xx} + \Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,x}\}, \tag{19}$$

$$
\sigma\_{rx} = \mu \left[ (\Phi\_{,rx} - \Psi\_{,xx}) + \Phi\_{,xr} + \Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,r} - \left(\mathbf{1}/r^2\right)\Psi \right]. \tag{20}
$$

Then the boundary conditions (17) can be written in the form.

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

$$\left[\left[\Delta\mu(\Phi\_{,rr}-\Psi\_{,zz})+\lambda\Delta\Phi\right]\_{r=r\_o}=0,\mu\left[\mathcal{Q}(\Phi\_{,rz}-\Psi\_{,zz})+\Delta\Psi-\left(\mathbf{1}/r^2\right)\Psi\right]\_{r=r\_o}=0\tag{21}$$

In the work [1], Biot has used the expressions.

ΔΦ � ð Þ 1=vL 2 <sup>Φ</sup>,tt <sup>¼</sup> 0, ΔΨ � <sup>1</sup>=r<sup>2</sup> ð Þ<sup>Ψ</sup> � ð Þ <sup>1</sup>=vT 2 Ψ,tt ¼ 0 and rewrite Eq. (21) in such a way ð Þþ Φ,rr � Ψ,rz ð Þ λ=2μ ð Þ 1=vL 2 <sup>Φ</sup>,tt h i r¼ro ¼ 0, 2ð Þþ Φ,rz � Ψ, zz ð Þ 1=vT 2 <sup>Ψ</sup>,tt h i r¼ro ¼ 0.

Then the substitution of solutions (14) and (16) into the boundary conditions (21) gives two homogeneous algebraic equations relative to the unknown constant amplitude coefficients

$$
\left[\mathbf{1} - \left(v/v\_L\right)^2 - \frac{\lambda}{\mu} \left(v/v\_L\right)^2 \frac{K\_0(m\_L r\_o)}{K\_0(m\_L r\_o) + K\_2(m\_L r\_o)}\right] A\_\Phi - \sqrt{\left(\mathbf{1} - \left(v/v\_L\right)^2\right)} A\_\Psi = 0,\tag{22}
$$

$$2\sqrt{\left(1-\left(v/v\_L\right)^2\right)}A\_\Phi + \left(2-\left(v/v\_T\right)^2\right)\frac{K\_1(\mathfrak{m}\_Tr\_o)}{K\_1(\mathfrak{m}\_Tr\_o)}A\_\Psi = 0.\tag{23}$$

An analysis of these equations that describe the cylindrical surface wave is very similar to the analysis that has been carried out by Rayleigh for the classical wave propagating along the plane surface. Some novelty in analysis of systems (22) and (23) is consideration of the system relative to quantities K1ð Þ mLro A<sup>Φ</sup> and K1ð Þ mSro A<sup>Ψ</sup>

$$\begin{cases} \left(1-\left(\boldsymbol{v}/\boldsymbol{v}\_{L}\right)^{2}\right) \left[\frac{K\_{0}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)}{K\_{1}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)}+\frac{1}{m\_{L}\boldsymbol{r}\_{o}}\right] - \frac{\lambda}{2\mu}\left(\boldsymbol{v}/\boldsymbol{v}\_{L}\right)^{2}\frac{K\_{0}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)}{K\_{1}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)}\right]K\_{1}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)A\_{\Phi} \\ \quad + \sqrt{\left(1-\left(\boldsymbol{v}/\boldsymbol{v}\_{S}\right)^{2}\right)} \left[\frac{K\_{0}\left(\boldsymbol{m}\_{T}\boldsymbol{r}\_{o}\right)}{K\_{1}\left(\boldsymbol{m}\_{T}\boldsymbol{r}\_{o}\right)}+\frac{1}{m\_{T}\boldsymbol{r}\_{o}}\right]K\_{1}\left(\boldsymbol{m}\_{T}\boldsymbol{r}\_{o}\right)A\_{\Psi} = \boldsymbol{0}, \end{cases} \tag{24}$$
 
$$2\sqrt{\left(1-\left(\boldsymbol{v}/\boldsymbol{v}\_{L}\right)^{2}\right)}K\_{1}\left(\boldsymbol{m}\_{L}\boldsymbol{r}\_{o}\right)A\_{\Phi} + \left(2-\left(\boldsymbol{v}/\boldsymbol{v}\_{T}\right)^{2}\right)K\_{1}\left(\boldsymbol{m}\_{T}\boldsymbol{r}\_{o}\right)A\_{\Psi} = \boldsymbol{0}. \tag{25}$$

Solving of systems (24) and (25) gives two results. First, the solution is found accurate within one amplitude factor. Second, an equation for determination of phase velocity of cylindrical surface wave can be obtained in an explicit form.

The work of Biot (1952) has demonstrated some art in handling the Macdonald functions and has written Eq. (24) through only functions of the zeroth and first orders. For that, the known formulas

$$\begin{aligned} K\_0'(\boldsymbol{\omega}) &= -K\_1(\boldsymbol{\omega}), & K\_1'(\boldsymbol{\omega}) &= -\boldsymbol{K\_0}(\boldsymbol{\omega}),\\ K\_0'(\boldsymbol{\omega}) + (\mathbf{1}/\boldsymbol{\omega})K\_0'(\boldsymbol{\omega}) &= K\_0(\boldsymbol{\omega}), & \boldsymbol{K\_0'}(\boldsymbol{\omega}) &= (\mathbf{1}/\boldsymbol{\omega})K\_1(\boldsymbol{\omega}) + K\_0(\boldsymbol{\omega}) \end{aligned} \tag{26}$$

have been used [3]. As a result, the equation for determination of phase velocity of cylindrical wave has the form

$$\begin{split} & \left( 2 - \left( \boldsymbol{v}/\boldsymbol{v}\_{T} \right)^{2} \right) \left\{ \quad \left[ 2 - \left( \boldsymbol{v}/\boldsymbol{v}\_{T} \right)^{2} \right] \frac{K\_{0} \left( m\_{L} \boldsymbol{r}\_{o} \right)}{K\_{1} \left( m\_{L} \boldsymbol{r}\_{o} \right)} + \frac{\left( \mathbf{1} - \left( \boldsymbol{v}/\boldsymbol{v}\_{L} \right)^{2} \right)}{m\_{L} \boldsymbol{r}\_{o}} \right\} \\ & - 4 \cdot \sqrt{\left( \mathbf{1} - \left( \boldsymbol{v}/\boldsymbol{v}\_{L} \right)^{2} \right)} \sqrt{\left( \mathbf{1} - \left( \boldsymbol{v}/\boldsymbol{v}\_{T} \right)^{2} \right)} \; \; \left[ \frac{K\_{0} \left( m\_{T} \boldsymbol{r}\_{o} \right)}{K\_{1} \left( m\_{T} \boldsymbol{r}\_{o} \right)} + \frac{\mathbf{1}}{m\_{T} \boldsymbol{r}\_{o}} \right]} = \mathbf{0}. \end{split} \tag{27}$$

The corresponding solution under conditions (12) is expressed by the

wave velocity. The amplitude coefficient A<sup>Ψ</sup> is assumed to be constant and

2.3 Boundary conditions: equations for unknown wave number

Note that the Macdonald functions have the property of attenuation with increasing arguments which is shown in Figure 1. Therefore, the propagation along the vertical coordinate z waves (15) and (16) can be considered as the waves with amplitudes <sup>Φ</sup><sup>∗</sup> ð Þ<sup>r</sup> , <sup>Ψ</sup><sup>∗</sup> ð Þ<sup>r</sup> , which attenuate with increasing the radial coordinate <sup>r</sup>. This means that amplitudes can decrease essentially with increasing the distance from the surface of cylindrical cavity. In this sense, the waves (15) and (16) are the surface ones. This forms also the sense of conditions (12). The same conditions are used in the analysis of classical Rayleigh surface wave which propagates along the plane surface of isotropic elastic medium [5–9]. But the Rayleigh wave attenuates as an exponential function when being moved from the free surface, whereas the cylindrical surface Biot's wave attenuates as the Macdonald functions. At that, the arguments in exponential function and Macdonald functions are

of the first order and unknown argument x ¼ mTr, which includes the unknown

The boundary conditions correspond to the absence of stresses on surface r ¼ ro

σrr ¼ 2μur,r þ λð Þ ð Þþ ur=r ur,r þ uz, <sup>z</sup> , σrz ¼ μð Þ ur, <sup>z</sup> þ uz,r (18)

<sup>2</sup> � �Ψ � �: (20)

σrr ¼ ð Þ λ þ 2μ ð Þþ Φ,rr � Ψ,rz λf g ð Þ 1=r ð Þþ Φ,r � Ψ, <sup>z</sup> Φ, zz þ Ψ,rz þ ð Þ 1=r Ψ, <sup>z</sup> , (19)

σrz ¼ μ ð Þþ Φ,rz � Ψ, zz Φ, zr þ Ψ,rr þ ð Þ 1=r Ψ,r � 1=r

Then the boundary conditions (17) can be written in the form.

σrrð Þ¼ r ¼ ro; z; t 0, σrzð Þ¼ r ¼ ro; z; t 0: (17)

<sup>Ψ</sup><sup>∗</sup> ð Þ¼ <sup>r</sup> <sup>A</sup>ΨK1ð Þ mTr (16)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> T

� � q

Macdonald function K<sup>1</sup> r

Plots of the first five Macdonald functions.

Seismic Waves - Probing Earth System

identical and depend on the wave velocity.

are written through the potentials

arbitrary.

Figure 1.

The stresses

24

Let us write the corresponding equation for the Rayleigh wave [5–9] as

$$4\sqrt{1-\left(\upsilon/\upsilon\_L\right)^2}\sqrt{\mathbf{1}-\left(\upsilon/\upsilon\_S\right)^2}-\left[2-\left(\upsilon/\upsilon\_S\right)^2\right]^2=\mathbf{0}.\tag{28}$$

Method 1 (graphical method [10, 11]). Eq. (30) is considered as a sum of two summands <sup>Z</sup><sup>1</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>¼</sup> 0. The first summand Z1 = z3 describes a cubic parabola; the lower branch of which lies in the first quadrant of the plane zOZ1. The second

allowance for the shear modulus μ that is positive. These parabolas are intersected on the interval 0ð Þ ; 1 . More exactly, one of the roots z ¼ zC of Eq. (30) can be

the case when the parabola is tangent to the abscissa axis, and the maximal value corresponds to the case when the parabola is moved partially into the fourth quadrant. Thus, the velocity of Rayleigh wave is close to the velocity of plane transverse

Method 2 (method of finding the interval, on ends of which the equation possesses the different by sign values [2, 11]). This method is based on the analysis of Eq. (30). The value of equation that corresponds to the point cR ¼ cT is positive and equal to 1. The second point is chosen as cR ¼ εcT, where ε is assumed as the small quantity (this point is close to 0). When this value is substituted into Eq. (30), then

Method 3 (another method of finding the interval, on ends of which the equation possesses the different by sign values [5]). This method is based on the analysis of Eq. (31). The right point is chosen as θ ¼ ð Þ ð Þ 1=cT (similar to method 2). Then Eq. (31) possesses the positive value. The left point corresponds to θ ! ∞. Further an expression (31) is expanded into the power series near the point at infinity. This

L

<sup>3</sup> � � <sup>p</sup> . Since the condition <sup>θ</sup> < 1 has been fulfilled, then the corresponding

The main conclusion from the shown above methods is that they really allow to establish an existence of real root of Rayleigh equation (the real value of velocity of harmonic Rayleigh wave). They give the positive answer on the question whether the Rayleigh wave exists. In the case of other surface waves including the cylindrical wave under consideration, the experience of the classical Rayleigh wave analysis

� � � � , which is always negative. So this

T � � � <sup>1</sup>=c<sup>2</sup>

be solved exactly, and the roots possess the values <sup>θ</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>, <sup>θ</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>=</sup> ffiffiffi

3. Cylindrical wave propagating along the surface of the cylindrical cavity in the direction of vertical axis: The case of transversal

that the wave is harmonic in time, and attenuating deep into medium wave

Let us return to the initial statement of problem and consider an infinite medium with cylindrical circular cavity that has the symmetry axis Oz and radius ro. The medium is assumed to be the transversely isotropic elastic one. It is assumed further

equation possesses in the chosen points the different sign values. Thus, at least one

Method 4 (method based on assumption relative to the Poisson ratio [7]). This assumption consists in the choice of value of Poisson ratio that is often used in the analysis of seismic waves in Earth's crust ν ¼ λ=½ �¼ 2ð Þ λ þ μ ð Þ! 1=4 λ ¼ μ. Then cubic Eq. (31) (the zeroth root θ<sup>1</sup> ¼ 0 is ignored from a physical considerations) can

<sup>2</sup> � �≤0:912: Here, the minimal value corresponds to

� � � � is always negative. Hence, at least one root of equa-

T=c<sup>2</sup> L

<sup>L</sup> ≤1=2 with

<sup>3</sup> � � <sup>p</sup> ,

� � � � � � ,

T=c<sup>2</sup>

summand describes a quadratic parabola <sup>Z</sup><sup>2</sup> ¼ �8ð Þ <sup>z</sup> � <sup>1</sup> <sup>z</sup> � 2 1 � <sup>c</sup><sup>2</sup>

� � <sup>¼</sup> ð Þ <sup>μ</sup>=ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> can be estimated from below and top 0 <sup>≤</sup>c<sup>2</sup>

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

DOI: http://dx.doi.org/10.5772/intechopen.86910

wave, but always less of its 0.874 ≤ (c/cT) ≤ 0.955.

T=c<sup>2</sup> L

root of the equation lies in the interval 1 ð Þ ð Þ =cT ; ∞ .

c2 T=c<sup>2</sup> L

estimated 0:764≤ z ¼ ð Þ c=cT

expression �2ε<sup>2</sup> <sup>1</sup> � <sup>c</sup><sup>2</sup>

<sup>θ</sup><sup>4</sup> <sup>¼</sup> <sup>2</sup> � <sup>2</sup><sup>=</sup> ffiffiffi

can be quite useful.

27

root is equal to θ<sup>4</sup> ¼ 0:8453.

isotropy of medium

tion lies in the interval ð Þ εcT;cT .

series starts with the term �2θ<sup>2</sup> <sup>1</sup>=c<sup>2</sup>

which is concave in the direction of coordinate axis OZ2. Further the ratio

Thus, a presence of Macdonald functions in Eq. (27) complicates essentially an analysis of this equation because according to relations mL ¼ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL 2 r� �,

mS ¼ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vT 2 r� � these functions have the unknown velocity in argument.

If the cavity radius is not small, then the Macdonald functions can be represented by the simple formula <sup>K</sup>0ð Þ¼ <sup>r</sup> <sup>K</sup>1ð Þ¼ <sup>r</sup> <sup>e</sup>�<sup>r</sup> ffiffiffiffiffiffiffiffiffi π=2r p , and Eq. (27) is reduced to the Rayleigh Eq. (28).

Strictly speaking, the analytical part of analysis is ended by obtaining Eq. (27). Further analysis can be continued with the aim of the numerical methods. Biot in [1] has shown some comments and conclusions based on resources of the 1950s.

A possibility of analytical approach is still saved in the problem on existence of the appropriate wave velocity. First of all, Eq. (27) depends on the elastic constants, and this dependence can be shown in the form of dependence on the ratio of known velocities ð Þ vL=vT . If the notation <sup>v</sup><sup>2</sup>=v<sup>2</sup> T � � <sup>¼</sup> <sup>z</sup> is used, then Eq. (27) can be written in the form

$$
\begin{split} & \left( 2 - z \left( v\_{L}/v\_{T} \right)^{2} \right) \left\{ \begin{array}{c} K\_{0} \left( r\_{o} k \sqrt{1 - z \left( v\_{L}/v\_{T} \right)^{2}} \right) \\\ K\_{1} \left( r\_{o} k \sqrt{1 - z \left( v\_{L}/v\_{T} \right)^{2}} \right) \end{array} + \frac{\left( 1 - z \left( v\_{L}/v\_{T} \right)^{2} \right)}{r\_{o} k \sqrt{1 - z \left( v\_{L}/v\_{T} \right)^{2}}} \right\} \\\ & - 4 \left. \sqrt{(1 - z)} \sqrt{1 - z \left( v\_{L}/v\_{T} \right)^{2}} \left[ \frac{K\_{0} \left( r\_{o} k \sqrt{1 - z} \right)}{K\_{1} \left( r\_{o} k \sqrt{1 - z} \right)} + \frac{1}{r\_{o} k \sqrt{1 - z}} \right] = 0. \end{split} \tag{29}
$$

It seems appropriate to recall here the most known ways of proving the existence of velocity of the classical Rayleigh wave. An initial equation is always Eq. (28). Two different notations v<sup>2</sup>=v<sup>2</sup> T � � <sup>¼</sup> <sup>z</sup> and <sup>v</sup> <sup>¼</sup> ð Þ <sup>1</sup>=<sup>θ</sup> are used, which generate two different representations of Eq. (28)

$$\{z\{x^3 - 8(z - 1)\left[z - 2\left(1 - \left(v\_T^2/v\_L^2\right)\right)\right]\} = 0,\tag{30}$$

$$\left(2\theta^2 - \left(\mathbf{1}/v\_T^2\right)\right)^2 - 4\theta^2\sqrt{\theta^2 - \left(\mathbf{1}/v\_T^2\right)}\sqrt{\theta^2 - \left(\mathbf{1}/v\_L^2\right)} = \mathbf{0}.\tag{31}$$

Finding the real root of Eq. (30) is the key step in the analysis of the Rayleigh wave [5–9] . For more than 100 years of analysis of this wave, many methods of proving the existence of real velocity of wave were elaborated.

First of all, the sufficiently useful and exact empirical Viktorov's formula [5].

$$\mathbf{r}(v/v\_T) = \sqrt{\mathbf{z}} \approx \frac{\mathbf{0}.87 + \mathbf{1}.12v}{\mathbf{1} + v} \text{ ( $v$  the Poisson ratio)}\tag{32}$$

should be shown.

Let us show further briefly some phenomenological methods. Note that the restriction on the Rayleigh wave velocity is already obtained from a statement of the problem—it is less of the velocity of plane transverse wave. This restriction can be written in the form z < 1 or θ>ð Þ 1=cT .

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

Let us write the corresponding equation for the Rayleigh wave [5–9] as

1 � ð Þ v=vS

If the cavity radius is not small, then the Macdonald functions can be

2

Thus, a presence of Macdonald functions in Eq. (27) complicates essentially an

Strictly speaking, the analytical part of analysis is ended by obtaining Eq. (27). Further analysis can be continued with the aim of the numerical methods. Biot in [1] has shown some comments and conclusions based on resources of the 1950s. A possibility of analytical approach is still saved in the problem on existence of the appropriate wave velocity. First of all, Eq. (27) depends on the elastic constants, and this dependence can be shown in the form of dependence on the ratio of

T

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � z vð Þ <sup>L</sup>=vT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � z vð Þ <sup>L</sup>=vT

<sup>1</sup> � <sup>z</sup> � � <sup>p</sup> K<sup>1</sup> rok ffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>z</sup> � � <sup>p</sup> <sup>þ</sup>

It seems appropriate to recall here the most known ways of proving the existence of velocity of the classical Rayleigh wave. An initial equation is always Eq. (28).

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>θ</sup><sup>2</sup> � <sup>1</sup>=v<sup>2</sup>

Finding the real root of Eq. (30) is the key step in the analysis of the Rayleigh wave [5–9] . For more than 100 years of analysis of this wave, many methods of

First of all, the sufficiently useful and exact empirical Viktorov's formula [5].

Let us show further briefly some phenomenological methods. Note that the restriction on the Rayleigh wave velocity is already obtained from a statement of the problem—it is less of the velocity of plane transverse wave. This restriction can be

� � q þ

� � q

2

2

" #

� � <sup>¼</sup> <sup>z</sup> and <sup>v</sup> <sup>¼</sup> ð Þ <sup>1</sup>=<sup>θ</sup> are used, which generate

T=v<sup>2</sup> L � � � � � � � � <sup>¼</sup> <sup>0</sup>, (30)

<sup>θ</sup><sup>2</sup> � <sup>1</sup>=v<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>υ</sup> ð Þ <sup>υ</sup> the Poisson ratio (32)

L <sup>q</sup> � � <sup>¼</sup> <sup>0</sup>: (31)

T q � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� 2 � ð Þ v=vS <sup>2</sup> h i<sup>2</sup>

these functions have the unknown velocity in argument.

¼ 0: (28)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

r� �

π=2r p , and Eq. (27) is

1 � z vð Þ <sup>L</sup>=vT <sup>2</sup> � �

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � z vð Þ <sup>L</sup>=vT

> > ¼ 0:

2

9 >>=

>>;

(29)

� � <sup>¼</sup> <sup>z</sup> is used, then Eq. (27) can be

rok

1 rok ffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>z</sup> <sup>p</sup>

q

2

,

4

Seismic Waves - Probing Earth System

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vT

reduced to the Rayleigh Eq. (28).

written in the form

2 � z vð Þ <sup>L</sup>=vT <sup>2</sup> � �

� <sup>4</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Two different notations v<sup>2</sup>=v<sup>2</sup>

two different representations of Eq. (28)

<sup>2</sup>θ<sup>2</sup> � <sup>1</sup>=v<sup>2</sup>

ð Þ¼ v=vT

written in the form z < 1 or θ>ð Þ 1=cT .

should be shown.

26

T � � � � <sup>2</sup> � <sup>4</sup>θ<sup>2</sup>

r� �

2

known velocities ð Þ vL=vT . If the notation <sup>v</sup><sup>2</sup>=v<sup>2</sup>

8 >><

>>:

ð Þ <sup>1</sup> � <sup>z</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Þ 2 � z

1 � z vð Þ <sup>L</sup>=vT

T

proving the existence of real velocity of wave were elaborated.

ffiffi

K<sup>0</sup> rok

K<sup>1</sup> rok

2 q K<sup>0</sup> rok ffiffiffiffiffiffiffiffiffiffi

z z<sup>3</sup> � <sup>8</sup>ð Þ <sup>z</sup> � <sup>1</sup> <sup>z</sup> � 2 1 � <sup>v</sup><sup>2</sup>

<sup>z</sup> <sup>p</sup> <sup>≈</sup>0:<sup>87</sup> <sup>þ</sup> <sup>1</sup>:12<sup>υ</sup>

mS ¼ k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ð Þ v=vL

2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

analysis of this equation because according to relations mL ¼ k

represented by the simple formula <sup>K</sup>0ð Þ¼ <sup>r</sup> <sup>K</sup>1ð Þ¼ <sup>r</sup> <sup>e</sup>�<sup>r</sup> ffiffiffiffiffiffiffiffiffi

q

Method 1 (graphical method [10, 11]). Eq. (30) is considered as a sum of two summands <sup>Z</sup><sup>1</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>¼</sup> 0. The first summand Z1 = z3 describes a cubic parabola; the lower branch of which lies in the first quadrant of the plane zOZ1. The second summand describes a quadratic parabola <sup>Z</sup><sup>2</sup> ¼ �8ð Þ <sup>z</sup> � <sup>1</sup> <sup>z</sup> � 2 1 � <sup>c</sup><sup>2</sup> T=c<sup>2</sup> L � � � � � � , which is concave in the direction of coordinate axis OZ2. Further the ratio c2 T=c<sup>2</sup> L � � <sup>¼</sup> ð Þ <sup>μ</sup>=ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> can be estimated from below and top 0 <sup>≤</sup>c<sup>2</sup> T=c<sup>2</sup> <sup>L</sup> ≤1=2 with allowance for the shear modulus μ that is positive. These parabolas are intersected on the interval 0ð Þ ; 1 . More exactly, one of the roots z ¼ zC of Eq. (30) can be estimated 0:764≤ z ¼ ð Þ c=cT <sup>2</sup> � �≤0:912: Here, the minimal value corresponds to the case when the parabola is tangent to the abscissa axis, and the maximal value corresponds to the case when the parabola is moved partially into the fourth quadrant. Thus, the velocity of Rayleigh wave is close to the velocity of plane transverse wave, but always less of its 0.874 ≤ (c/cT) ≤ 0.955.

Method 2 (method of finding the interval, on ends of which the equation possesses the different by sign values [2, 11]). This method is based on the analysis of Eq. (30). The value of equation that corresponds to the point cR ¼ cT is positive and equal to 1. The second point is chosen as cR ¼ εcT, where ε is assumed as the small quantity (this point is close to 0). When this value is substituted into Eq. (30), then expression �2ε<sup>2</sup> <sup>1</sup> � <sup>c</sup><sup>2</sup> T=c<sup>2</sup> L � � � � is always negative. Hence, at least one root of equation lies in the interval ð Þ εcT;cT .

Method 3 (another method of finding the interval, on ends of which the equation possesses the different by sign values [5]). This method is based on the analysis of Eq. (31). The right point is chosen as θ ¼ ð Þ ð Þ 1=cT (similar to method 2). Then Eq. (31) possesses the positive value. The left point corresponds to θ ! ∞. Further an expression (31) is expanded into the power series near the point at infinity. This series starts with the term �2θ<sup>2</sup> <sup>1</sup>=c<sup>2</sup> T � � � <sup>1</sup>=c<sup>2</sup> L � � � � , which is always negative. So this equation possesses in the chosen points the different sign values. Thus, at least one root of the equation lies in the interval 1 ð Þ ð Þ =cT ; ∞ .

Method 4 (method based on assumption relative to the Poisson ratio [7]). This assumption consists in the choice of value of Poisson ratio that is often used in the analysis of seismic waves in Earth's crust ν ¼ λ=½ �¼ 2ð Þ λ þ μ ð Þ! 1=4 λ ¼ μ. Then cubic Eq. (31) (the zeroth root θ<sup>1</sup> ¼ 0 is ignored from a physical considerations) can be solved exactly, and the roots possess the values <sup>θ</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>, <sup>θ</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>=</sup> ffiffiffi <sup>3</sup> � � <sup>p</sup> , <sup>θ</sup><sup>4</sup> <sup>¼</sup> <sup>2</sup> � <sup>2</sup><sup>=</sup> ffiffiffi <sup>3</sup> � � <sup>p</sup> . Since the condition <sup>θ</sup> < 1 has been fulfilled, then the corresponding root is equal to θ<sup>4</sup> ¼ 0:8453.

The main conclusion from the shown above methods is that they really allow to establish an existence of real root of Rayleigh equation (the real value of velocity of harmonic Rayleigh wave). They give the positive answer on the question whether the Rayleigh wave exists. In the case of other surface waves including the cylindrical wave under consideration, the experience of the classical Rayleigh wave analysis can be quite useful.

#### 3. Cylindrical wave propagating along the surface of the cylindrical cavity in the direction of vertical axis: The case of transversal isotropy of medium

Let us return to the initial statement of problem and consider an infinite medium with cylindrical circular cavity that has the symmetry axis Oz and radius ro. The medium is assumed to be the transversely isotropic elastic one. It is assumed further that the wave is harmonic in time, and attenuating deep into medium wave

propagates in the direction of axis Oz along the cavity surface. Such a problem can be considered as some generalization of Biot's [1] problem that is solved in the assumption of isotropy of medium on the case of transversal isotropy of medium. Therefore, it seems expedient to recall some facts from the theory of elasticity of transversally isotropic medium.

The shear modulus that corresponds to the shear along the symmetry axis Oz

The Poisson ratio that corresponds to the shear along the symmetry axis Oz under tension in the isotropy plane and characterizes the shortening in this plane

λxy þ 2μxy ¼ C11, λxy ¼ C12, μxy ¼ ð Þ 1=2 ð Þ C<sup>11</sup> � C<sup>12</sup>

The Poisson ratio (40) is determined by the known formula of isotropic theory

The Poisson ratio υxy that corresponds to the shear along the symmetry axis Oz under tension along the isotropy plane is determined also by the classical formula

The constants C11, C12, C13, C33, C<sup>44</sup> are represented through the technical con-

, C<sup>44</sup> ¼ G<sup>0</sup>

Let us comment briefly some features of transversally isotropic materials. They can be divided on the natural and artificial ones. An example of the classical natural material is the rock. An example of the modern material is a family of fibers

An example of composite materials can be four fibrous composites of micro- and nanolevels, which are described in [15]. The corresponding elastic constants for

The shown above values are typical for the transversally isotropic materials, and

Comment 1. The Young modulus in the direction along the symmetry axis Ez exceeds essentially the Young modulus in the isotropy plane Ex (from 6 to 34 times

:

<sup>1</sup> � <sup>ν</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>ν</sup> <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ E, C<sup>12</sup> <sup>¼</sup> <sup>v</sup> � <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ

E0

Ex ¼ 1:34GPa, Ez ¼ 84:62GPa, Gxz ¼ 24:40GPa, υxy ¼ 0:24, υxz ¼ 0:60.

Ex ¼ 3:59GPa, Ez ¼ 25:22GPa, Gxz ¼ 1:17 GPa, υxy ¼ 0:39, υxz ¼ 0:58.

Ex ¼ 3:69GPa, Ez ¼ 102:4GPa, Gxz ¼ 1:14GPa, υxy ¼ 0:39, υxz ¼ 0:62.

Ex ¼ 3:70GPa, Ez ¼ 67:21GPa, Gxz ¼ 1:14GPa, υxy ¼ 0:39, υxz ¼ 0:62.

Ex ¼ 3:67 GPa, Ez ¼ 126:4GPa, Gxz ¼ 1:14 GPa, υxy ¼ 0:39, υxz ¼ 0:62.

"Kevlar®." Kevlar® KM2 [15] is characterized by elastic constants

<sup>λ</sup>xz <sup>þ</sup> <sup>2</sup>μxz <sup>¼</sup> <sup>C</sup>33, <sup>λ</sup>xz <sup>¼</sup> <sup>C</sup>13, <sup>μ</sup>xz <sup>¼</sup> <sup>C</sup>44: (41)

Sometimes, the corresponding Lame moduli are used.

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

DOI: http://dx.doi.org/10.5772/intechopen.86910

, G<sup>0</sup> by the formulas.

ð Þ 1 � ν <sup>1</sup> � <sup>ν</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>ν</sup> <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ E,

<sup>1</sup> � <sup>ν</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>ν</sup> <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ

some variants of these materials are as follows [15]:

<sup>υ</sup>xz <sup>¼</sup> <sup>λ</sup>xz=<sup>2</sup> <sup>λ</sup>xy <sup>þ</sup> <sup>μ</sup>xy .

<sup>υ</sup>xy <sup>¼</sup> <sup>λ</sup>xy<sup>=</sup> <sup>λ</sup>xy <sup>þ</sup> <sup>μ</sup>xy .

<sup>C</sup><sup>13</sup> <sup>¼</sup> <sup>ν</sup><sup>0</sup>

<sup>C</sup><sup>33</sup> <sup>¼</sup> <sup>1</sup> � <sup>ν</sup><sup>2</sup>

10% of carbon microfibers

10% of graphite microwhiskers

10% of zig-zag carbon nanotubes

10% of chiral carbon nanotubes

therefore they are briefly commented below.

in examples above but can in some cases exceed 100 times).

, ν, ν<sup>0</sup>

<sup>C</sup><sup>11</sup> <sup>¼</sup> <sup>1</sup> � <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ

stants E, E<sup>0</sup>

29

Gxz ¼ C44: (39)

<sup>1</sup> � <sup>ν</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>ν</sup> <sup>ν</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>E</sup>=E<sup>0</sup> ð Þ E,

(42)

υxz ¼ C13=ð Þ C<sup>11</sup> þ C<sup>12</sup> : (40)

#### 3.1 Some information on transversally isotropic medium

Let us consider the case when Ox<sup>3</sup> is the axis of symmetry and Ox1x<sup>2</sup> is the plane of isotropy. This symmetry corresponds to the hexagonal crystalline system. The matrix of elastic properties is characterized by 5 independent elastic constants C11, C12, C13, C33, C<sup>44</sup> and 12 non-zero components [11–13]:

$$\mathbf{C}\_{IK} = \begin{vmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{C}\_{12} & \mathbf{C}\_{11} & \mathbf{C}\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{C}\_{13} & \mathbf{C}\_{13} & \mathbf{C}\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{44} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{C}\_{44} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & (\mathbf{1}/2)(\mathbf{C}\_{11} - \mathbf{C}\_{12}) \end{vmatrix} . \tag{33}$$

Then the constitutive relations σ � ε have the form [12, 14].

$$\begin{aligned} \sigma\_{11} &= \mathbf{C}\_{1kkl}\varepsilon\_{kl} = \mathbf{C}\_{11}\varepsilon\_{11} + \mathbf{C}\_{12}\varepsilon\_{22} + \mathbf{C}\_{13}\varepsilon\_{33}, \\ \sigma\_{22} &= \mathbf{C}\_{22kl}\varepsilon\_{kl} = \mathbf{C}\_{12}\varepsilon\_{11} + \mathbf{C}\_{22}\varepsilon\_{22} + \mathbf{C}\_{13}\varepsilon\_{33}, \\ \sigma\_{33} &= \mathbf{C}\_{33kl}\varepsilon\_{kl} = \mathbf{C}\_{13}\varepsilon\_{11} + \mathbf{C}\_{13}\varepsilon\_{22} + \mathbf{C}\_{33}\varepsilon\_{33}, \\ \sigma\_{12} &= (\mathbf{C}\_{11} - \mathbf{C}\_{12})\varepsilon\_{12}, \sigma\_{13} = 2\mathbf{C}\_{44}\varepsilon\_{13}, \sigma\_{23} = 2\mathbf{C}\_{44}\varepsilon\_{23}. \end{aligned} \tag{34}$$

or in notations σ � u [12, 14].

$$\begin{aligned} \sigma\_{11} &= \mathbf{C}\_{11}\boldsymbol{u}\_{1,1} + \mathbf{C}\_{12}\boldsymbol{u}\_{2,2} + \mathbf{C}\_{13}\boldsymbol{u}\_{3,3}, \sigma\_{22} = \mathbf{C}\_{12}\boldsymbol{u}\_{1,1} + \mathbf{C}\_{11}\boldsymbol{u}\_{2,2} + \mathbf{C}\_{13}\boldsymbol{u}\_{3,3}, \\ \sigma\_{33} &= \mathbf{C}\_{13}\boldsymbol{u}\_{1,1} + \mathbf{C}\_{13}\boldsymbol{u}\_{2,2} + \mathbf{C}\_{11}\boldsymbol{u}\_{3,3}, \sigma\_{12} = (\mathbf{1}/2)(\mathbf{C}\_{11} - \mathbf{C}\_{12})(\boldsymbol{u}\_{1,2} + \boldsymbol{u}\_{2,1}), \\ \sigma\_{13} &= \mathbf{C}\_{44}(\boldsymbol{u}\_{1,3} + \boldsymbol{u}\_{3,1}), \sigma\_{23} = (\mathbf{1}/2)\mathbf{C}\_{44}(\boldsymbol{u}\_{2,3} + \boldsymbol{u}\_{3,2}). \end{aligned} \tag{35}$$

Also, five independent elastic technical constants are often used.

Ex <sup>¼</sup> Ey, Ex <sup>¼</sup> Ey, Ez, Gxy, Gxz <sup>¼</sup> Gyz, <sup>υ</sup>xy, <sup>υ</sup>xz <sup>¼</sup> <sup>υ</sup>yz, Gxy <sup>¼</sup> Ex<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>υxy � �. They are evaluated through CNM by the following formulas:

The longitudinal Young modulus that corresponds to tension along the symmetry axis Oz

$$E\_x = \mathbf{C}\_{33} - \left[ \mathbf{2} (\mathbf{C}\_{13})^2 / (\mathbf{C}\_{11} + \mathbf{C}\_{12}) \right]. \tag{36}$$

The transverse Young modulus that corresponds to tension in the isotropy plane Oxy

$$E\_{\mathbf{x}} = (\mathbf{C}\_{11} - \mathbf{C}\_{12}) \left[ (\mathbf{C}\_{11} + \mathbf{C}\_{12})\mathbf{C}\_{33} - 2(\mathbf{C}\_{13})^2 \right] / \left[ \mathbf{C}\_{11}\mathbf{C}\_{33} + (\mathbf{C}\_{13})^2 \right]. \tag{37}$$

The shear modulus that corresponds to the shear along the isotropy plane Oxy

$$\mathbf{G\_{xy}} = \mathbf{C\_{66}} = (\mathbf{1}/\mathbf{2})(\mathbf{C\_{11}} - \mathbf{C\_{12}}).\tag{38}$$

propagates in the direction of axis Oz along the cavity surface. Such a problem can be considered as some generalization of Biot's [1] problem that is solved in the assumption of isotropy of medium on the case of transversal isotropy of medium. Therefore, it seems expedient to recall some facts from the theory of elasticity of

Let us consider the case when Ox<sup>3</sup> is the axis of symmetry and Ox1x<sup>2</sup> is the plane of isotropy. This symmetry corresponds to the hexagonal crystalline system. The matrix of elastic properties is characterized by 5 independent elastic constants

> C<sup>11</sup> C<sup>12</sup> C<sup>13</sup> 00 0 C<sup>12</sup> C<sup>11</sup> C<sup>13</sup> 00 0 C<sup>13</sup> C<sup>13</sup> C<sup>33</sup> 00 0 000 C<sup>44</sup> 0 0 0000 C<sup>44</sup> 0

0000 01ð Þ =2 ð Þ C<sup>11</sup> � C<sup>12</sup>

� � � � � � � � � � � � � �

: (33)

(34)

(35)

� �. They are

: (37)

: (36)

3.1 Some information on transversally isotropic medium

C11, C12, C13, C33, C<sup>44</sup> and 12 non-zero components [11–13]:

Then the constitutive relations σ � ε have the form [12, 14].

σ<sup>11</sup> ¼ C11klεkl ¼ C11ε<sup>11</sup> þ C12ε<sup>22</sup> þ C13ε33, σ<sup>22</sup> ¼ C22klεkl ¼ C12ε<sup>11</sup> þ C22ε<sup>22</sup> þ C13ε33, σ<sup>33</sup> ¼ C33klεkl ¼ C13ε<sup>11</sup> þ C13ε<sup>22</sup> þ C33ε33,

σ<sup>11</sup> ¼ C11u1, <sup>1</sup> þ C12u2, <sup>2</sup> þ C13u3, <sup>3</sup>, σ<sup>22</sup> ¼ C12u1, <sup>1</sup> þ C11u2,<sup>2</sup> þ C13u3, <sup>3</sup>, σ<sup>33</sup> ¼ C13u1, <sup>1</sup> þ C13u2, <sup>2</sup> þ C11u3,3, σ<sup>12</sup> ¼ ð Þ 1=2 ð Þ C<sup>11</sup> � C<sup>12</sup> ð Þ u1, <sup>2</sup> þ u2, <sup>1</sup> ,

Also, five independent elastic technical constants are often used. Ex ¼ Ey, Ex ¼ Ey, Ez, Gxy, Gxz ¼ Gyz, υxy, υxz ¼ υyz, Gxy ¼ Ex= 1 þ 2υxy

Ez ¼ C<sup>33</sup> � 2ð Þ C<sup>13</sup>

The longitudinal Young modulus that corresponds to tension along the

The transverse Young modulus that corresponds to tension in the isotropy

The shear modulus that corresponds to the shear along the isotropy plane Oxy

<sup>2</sup> h i

2

=ð Þ C<sup>11</sup> þ C<sup>12</sup> h i

> = C11C<sup>33</sup> þ ð Þ C<sup>13</sup> <sup>2</sup> h i

Gxy ¼ C<sup>66</sup> ¼ ð Þ 1=2 ð Þ C<sup>11</sup> � C<sup>12</sup> : (38)

σ<sup>13</sup> ¼ C44ð Þ u1,<sup>3</sup> þ u3, <sup>1</sup> , σ<sup>23</sup> ¼ ð Þ 1=2 C44ð Þ u2, <sup>3</sup> þ u3, <sup>2</sup> :

evaluated through CNM by the following formulas:

Ex ¼ ð Þ C<sup>11</sup> � C<sup>12</sup> ð Þ C<sup>11</sup> þ C<sup>12</sup> C<sup>33</sup> � 2ð Þ C<sup>13</sup>

σ<sup>12</sup> ¼ ð Þ C<sup>11</sup> � C<sup>12</sup> ε12, σ<sup>13</sup> ¼ 2C44ε13, σ<sup>23</sup> ¼ 2C44ε23,

transversally isotropic medium.

Seismic Waves - Probing Earth System

CIK ¼

or in notations σ � u [12, 14].

symmetry axis Oz

plane Oxy

28

� � � � � � � � � � � � � � The shear modulus that corresponds to the shear along the symmetry axis Oz

$$\mathbf{G}\_{\text{xx}} = \mathbf{C}\_{\text{44}}.\tag{39}$$

The Poisson ratio that corresponds to the shear along the symmetry axis Oz under tension in the isotropy plane and characterizes the shortening in this plane

$$
\mu\_{\text{xx}} = \mathbf{C}\_{13}/(\mathbf{C}\_{11} + \mathbf{C}\_{12}).\tag{40}
$$

Sometimes, the corresponding Lame moduli are used.

$$\begin{aligned} \lambda\_{\text{xy}} + 2\mu\_{\text{xy}} &= \mathbf{C\_{11}}, \ \lambda\_{\text{xy}} = \mathbf{C\_{12}}, \ \mu\_{\text{xy}} = (\mathbf{1}/2)(\mathbf{C\_{11}} - \mathbf{C\_{12}})\\ \lambda\_{\text{xx}} + 2\mu\_{\text{xx}} &= \mathbf{C\_{33}}, \ \lambda\_{\text{xx}} = \mathbf{C\_{13}}, \ \mu\_{\text{xx}} = \mathbf{C\_{44}}. \end{aligned} \tag{41}$$

The Poisson ratio (40) is determined by the known formula of isotropic theory <sup>υ</sup>xz <sup>¼</sup> <sup>λ</sup>xz=<sup>2</sup> <sup>λ</sup>xy <sup>þ</sup> <sup>μ</sup>xy .

The Poisson ratio υxy that corresponds to the shear along the symmetry axis Oz under tension along the isotropy plane is determined also by the classical formula <sup>υ</sup>xy <sup>¼</sup> <sup>λ</sup>xy<sup>=</sup> <sup>λ</sup>xy <sup>þ</sup> <sup>μ</sup>xy .

The constants C11, C12, C13, C33, C<sup>44</sup> are represented through the technical constants E, E<sup>0</sup> , ν, ν<sup>0</sup> , G<sup>0</sup> by the formulas.

$$\begin{aligned} \mathbf{C}\_{11} &= \frac{\mathbf{1} - \left(\nu'\right)^2 (\mathbf{E}/E')}{\mathbf{1} - \nu^2 + (\mathbf{1} + 2\nu) \left(\nu'\right)^2 (\mathbf{E}/E')} \mathbf{E}, \mathbf{C}\_{12} = \frac{\nu - \left(\nu'\right)^2 (\mathbf{E}/E')}{\mathbf{1} - \nu^2 + (\mathbf{1} + 2\nu) \left(\nu'\right)^2 (\mathbf{E}/E')} \mathbf{E}, \\ \mathbf{C}\_{13} &= \frac{\nu'(1 - \nu)}{\mathbf{1} - \nu^2 + (\mathbf{1} + 2\nu) \left(\nu'\right)^2 (\mathbf{E}/E')} \mathbf{E}, \\ \mathbf{C}\_{33} &= \frac{\mathbf{1} - \nu^2}{\mathbf{1} - \nu^2 + (\mathbf{1} + 2\nu) \left(\nu'\right)^2 (\mathbf{E}/E')} \mathbf{E}', \mathbf{C}\_{44} = \mathbf{G}'. \end{aligned} \tag{42}$$

Let us comment briefly some features of transversally isotropic materials. They can be divided on the natural and artificial ones. An example of the classical natural material is the rock. An example of the modern material is a family of fibers "Kevlar®." Kevlar® KM2 [15] is characterized by elastic constants

$$E\_{\rm x} = 1.34 \,\text{GPa}, \quad E\_{\rm x} = 84.62 \,\text{GPa}, \quad G\_{\rm xx} = 24.40 \,\text{GPa}, \quad \nu\_{\rm xy} = 0.24, \quad \nu\_{\rm xx} = 0.60.1$$

An example of composite materials can be four fibrous composites of micro- and nanolevels, which are described in [15]. The corresponding elastic constants for some variants of these materials are as follows [15]:

10% of carbon microfibers


Ex ¼ 3:67 GPa, Ez ¼ 126:4GPa, Gxz ¼ 1:14 GPa, υxy ¼ 0:39, υxz ¼ 0:62.

The shown above values are typical for the transversally isotropic materials, and therefore they are briefly commented below.

Comment 1. The Young modulus in the direction along the symmetry axis Ez exceeds essentially the Young modulus in the isotropy plane Ex (from 6 to 34 times in examples above but can in some cases exceed 100 times).

Comment 2. The Lame moduli λ<sup>x</sup> and λ<sup>z</sup> repeat the relations between Ex and Ez. Comment 3. The Poisson ratio υxz along the symmetry axis Oz exceeds the classical red line in 0.5 for values of this ratio.

Δs1Δs2φ ¼ 0, (49)

<sup>2</sup> � �φ, zz <sup>N</sup> <sup>¼</sup> <sup>1</sup>; 2 are some "complicated"

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2d vuut :

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ a þ c

¼ V:

<sup>2</sup> � �φN, zz: (53)

¼ 0: (52)

(54)

<sup>2</sup> � <sup>4</sup><sup>d</sup>

(50)

a þ c �

ur ¼ ϕ1,r þ ϕ2,r, uz ¼ k1ϕ1, <sup>z</sup> þ k2ϕ2, <sup>z</sup>: (51)

where ΔsNφ ¼ φ,rr þ ð Þ 1=r φ,r þ 1=ð Þ sN

DOI: http://dx.doi.org/10.5772/intechopen.86910

<sup>2</sup> <sup>þ</sup> ð Þ¼ <sup>1</sup>=<sup>d</sup> <sup>0</sup>,

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ a þ c

vuut , s2,<sup>4</sup> ¼ �

<sup>2</sup> � <sup>4</sup><sup>d</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2d

q

s

s1, <sup>3</sup> ¼ �

<sup>4</sup> � ½ � ð Þ <sup>a</sup> <sup>þ</sup> <sup>c</sup> <sup>=</sup><sup>d</sup> <sup>s</sup>

a þ c þ

zation of this equation in the form (49).

C<sup>11</sup> ur,rr þ ð Þ 1=r ur,r � 1=r

<sup>V</sup><sup>2</sup> <sup>þ</sup>

<sup>σ</sup>zz <sup>¼</sup> ð Þ <sup>C</sup>33k<sup>1</sup> � <sup>C</sup>13V<sup>1</sup> <sup>ϕ</sup>1, zz <sup>þ</sup> ð Þ <sup>C</sup>33k<sup>2</sup> � <sup>C</sup>13V<sup>2</sup> <sup>ϕ</sup>2, zz � �,

<sup>σ</sup>rz <sup>¼</sup> <sup>C</sup><sup>44</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>1</sup> <sup>ϕ</sup>1,rz <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>ϕ</sup>2,rz � �:

31

k1 2ð Þð Þþ C<sup>13</sup> þ C<sup>44</sup> C<sup>44</sup> C<sup>11</sup>

This expression gives the quadratic equation for k1 2ð Þ and V

C13ð Þ� 2C<sup>44</sup> þ C<sup>33</sup> C11C<sup>33</sup> C11C<sup>44</sup>

ΔrzNφ<sup>N</sup> ¼ φN,rr þ ð Þ 1=r φN,r þ 1=ð Þ VN

The stresses are expressed through new potentials in such a way

<sup>σ</sup>rr ¼ �ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>ϕ</sup>1,rr <sup>þ</sup> <sup>ϕ</sup>2,rr � � � <sup>C</sup><sup>44</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>1</sup> <sup>ϕ</sup>1, zz <sup>þ</sup> ð Þ <sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>ϕ</sup>2, zz � �,

Note that the simple link VN ¼ ð Þ 1=sN exists between constants VN and sN, which makes the ways 1 and 2 very similar. Then the potentials fulfill the equations

σθθ ¼ �ð Þ <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>12</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>ϕ</sup>1,rr <sup>þ</sup> <sup>ϕ</sup>2,rr � � � ð Þ <sup>C</sup>13k<sup>1</sup> � <sup>C</sup>12V<sup>1</sup> <sup>ϕ</sup>1, zz <sup>þ</sup> ð Þ <sup>C</sup>13k<sup>2</sup> � <sup>C</sup>12V<sup>2</sup> <sup>ϕ</sup>2, zz � �,

Way 3 [1, 16]. This way is proposed for equations of motion, but only for the isotropic theory of elasticity. It can be used for the static problems of transversely

comparing some coefficients

copies of classical expressions Δφ ¼ φ,rr þ ð Þ 1=r φ,r þ φ, zz associated with the Laplace operator. Two constants sN are determined from the algebraic equations

Thus, a transition from the isotropic case to the transversally isotropic one complicates the procedure of solving the static problems. Here a necessity of solving the classical biharmonic equation is changed on necessity of solving some generali-

Way 2 [12, 16]. This way is also proposed for the static problems. Here, two potentials are introduced which are linked immediately with displacements

A substitution of representations (51) into equations of equilibrium (45), (46)

� � <sup>þ</sup> <sup>C</sup>44ur, zz <sup>þ</sup> ½ � <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> uz,rz <sup>¼</sup> <sup>0</sup>,

allows to determine the unknown constants k1, k2. An idea consists in that both equations must be transformed in identical equations relative to the potentials by

<sup>¼</sup> kC<sup>33</sup>

k1 2ð ÞC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

V þ C<sup>33</sup> C<sup>11</sup>

C44ðuz,rr þ ð Þ 1=r uz,rÞ þ C33uz, zz þ ½ � C<sup>13</sup> þ C<sup>44</sup> ður,rz þ ð Þ 1=r ur, <sup>z</sup>Þ ¼ 0

<sup>2</sup> � �ur

Comment 4. The shear moduli Gxy and Gxz are differed quite moderately.

#### 3.2 The basic formulas for elastic transversely isotropic medium with axial symmetry

Let us write the basic formulas for the case of symmetry axis Oz. Then displacements are characterized by two components urð Þ r; z; t , uzð Þ r; z; t . The motion equations in stresses have the form.

$$
\sigma\_{\eta r,r} + \sigma\_{\eta z,x} + (\mathbf{1}/r) \left(\sigma\_{rr} - \sigma\_{\eta \eta}\right) = \mathbf{0},\\
\sigma\_{\mathbf{z},r} + (\mathbf{1}/r)\sigma\_{q\mathbf{z},q} + \sigma\_{\mathbf{z}z,x} + (\mathbf{1}/r)\sigma\_{\mathbf{z}} = \mathbf{0}. \tag{43}
$$

The substitution of constitutive equations.

$$\begin{aligned} \sigma\_{rr} &= \mathbf{C}\_{11}\boldsymbol{u}\_{r,r} + \mathbf{C}\_{12}(\mathbf{1}/r)\boldsymbol{u}\_r + \mathbf{C}\_{13}\boldsymbol{u}\_{z,x}, \sigma\_{xx} = \mathbf{C}\_{13}\boldsymbol{u}\_{r,r} + \mathbf{C}\_{13}(\mathbf{1}/r)\boldsymbol{u}\_r + \mathbf{C}\_{33}\boldsymbol{u}\_{z,x}, \\ \sigma\_{\overline{u}} &= (\mathbf{1}/2)\mathbf{C}\_{44}(\boldsymbol{u}\_{z,r} + \boldsymbol{u}\_{r,x}), \sigma\_{\overline{\psi}} = \sigma\_{r\psi} = \mathbf{0} \end{aligned} \tag{44}$$

into the motion Eqs. (43) gives the motion equations in displacements

$$\mathbf{C}\_{11}\left[u\_{r,rr} + (\mathbf{1}/r)u\_{r,r} - \left(\mathbf{1}/r^2\right)u\_r\right] + \mathbf{C}\_{44}u\_{r,\infty} + [\mathbf{C}\_{13} + \mathbf{C}\_{44}]u\_{\mathbf{z},r\mathbf{z}} = \rho u\_{r,\text{th}},\tag{45}$$

$$\mathbf{C}\_{44}(\boldsymbol{u}\_{\boldsymbol{z},\boldsymbol{r}\prime} + (\mathbf{1}/\boldsymbol{r})\boldsymbol{u}\_{\boldsymbol{z},\boldsymbol{r}\prime}) + \mathbf{C}\_{33}\boldsymbol{u}\_{\boldsymbol{z},\boldsymbol{z}\prime} + [\mathbf{C}\_{13} + \mathbf{C}\_{44}](\boldsymbol{u}\_{\boldsymbol{r},\boldsymbol{r}\prime} + (\mathbf{1}/\boldsymbol{r})\boldsymbol{u}\_{\boldsymbol{r},\boldsymbol{z}\prime}) = \rho\boldsymbol{u}\_{\boldsymbol{z},\boldsymbol{u}}.\tag{46}$$

Note that Eqs. (45) and (46) include only four constants (the constant C<sup>12</sup> is not represented in these equations). This means that displacements and strains are described by only four constants. But the stress state is already described by all five constants.

#### 3.3 Three classical ways of introducing the potentials in transversely isotropic elasticity

The basic equations of the theory of transversely isotropic elasticity are frequently analyzed by the use of potentials. The potentials are introduced in theory of elasticity mainly for static problems. Transition to the dynamic problems is associated with complications that are sometimes impassable. Because the problem on waves is related to the dynamic ones, let us show further the possible complications with introducing the potentials.

Way 1 [12]. It is proposed for the axisymmetric problems of equilibrium (not motion) and is based on introducing one only potential φð Þ r; z as the function of stresses. The formulas for stresses include four unknown parameters a, b, c, d, which is characteristic for representations in the transversely isotropic elasticity.

$$\sigma\_{rr} = -\left\{\rho\_{,rr} + b(\mathbf{1}/r)\rho\_{,r} + a\rho\_{,xx}\right\}\_{,x'}\\\sigma\_{\theta\theta} = -\left\{b\rho\_{,rr} + (\mathbf{1}/r)\rho\_{,r} + a\rho\_{,xx}\right\}\_{,x'} \tag{47}$$

$$\sigma\_{xx} = -\left\{ c\rho\_{,rr} + c(\mathbf{1}/r)\rho\_{,r} + d\rho\_{,xx} \right\}\_{,x'}, \\ \sigma\_{rz} = -\left\{ \rho\_{,rr} + (\mathbf{1}/r)\rho\_{,r} + a\rho\_{,xx} \right\}\_{,r}. \tag{48}$$

The next step consists in substitution of formulas (47) and (48) into the first equation of equilibrium and the equations that are obtained from the Cauchy relations and formulas for the strain tensor. This permits to determine the unknown parameters through the elastic constants represented in the equilibrium equations. Further, the second equation of equilibrium gives the biharmonic equation for finding the potentials

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

Comment 2. The Lame moduli λ<sup>x</sup> and λ<sup>z</sup> repeat the relations between Ex and Ez. Comment 3. The Poisson ratio υxz along the symmetry axis Oz exceeds the

Let us write the basic formulas for the case of symmetry axis Oz. Then displacements are characterized by two components urð Þ r; z; t , uzð Þ r; z; t . The motion equa-

<sup>¼</sup> <sup>0</sup>, <sup>σ</sup>rz,r <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> σφz,<sup>φ</sup> <sup>þ</sup> <sup>σ</sup>zz, <sup>z</sup> <sup>þ</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>σ</sup>rz <sup>¼</sup> <sup>0</sup>: (43)

Comment 4. The shear moduli Gxy and Gxz are differed quite moderately.

3.2 The basic formulas for elastic transversely isotropic medium with axial

σrr ¼ C11ur,r þ C12ð Þ 1=r ur þ C13uz, z, σzz ¼ C13ur,r þ C13ð Þ 1=r ur þ C33uz, z, <sup>σ</sup>rz <sup>¼</sup> ð Þ <sup>1</sup>=<sup>2</sup> <sup>C</sup>44ð Þ uz,r <sup>þ</sup> ur, <sup>z</sup> , σφ<sup>z</sup> <sup>¼</sup> <sup>σ</sup>r<sup>φ</sup> <sup>¼</sup> <sup>0</sup> (44)

into the motion Eqs. (43) gives the motion equations in displacements

<sup>þ</sup> <sup>C</sup>44ur, zz <sup>þ</sup> ½ � <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> uz,rz <sup>¼</sup> <sup>ρ</sup>ur,tt, (45)

C44ðuz,rr þ ð Þ 1=r uz,rÞ þ C33uz, zz þ ½ � C<sup>13</sup> þ C<sup>44</sup> ður,rz þ ð Þ 1=r ur, <sup>z</sup>Þ ¼ ρuz,tt: (46)

Note that Eqs. (45) and (46) include only four constants (the constant C<sup>12</sup> is not represented in these equations). This means that displacements and strains are described by only four constants. But the stress state is already described by all five constants.

3.3 Three classical ways of introducing the potentials in transversely isotropic

The basic equations of the theory of transversely isotropic elasticity are frequently analyzed by the use of potentials. The potentials are introduced in theory of elasticity mainly for static problems. Transition to the dynamic problems is associated with complications that are sometimes impassable. Because the problem on waves is related to the dynamic ones, let us show further the possible complications

Way 1 [12]. It is proposed for the axisymmetric problems of equilibrium (not motion) and is based on introducing one only potential φð Þ r; z as the function of stresses. The formulas for stresses include four unknown parameters a, b, c, d, which is characteristic for representations in the transversely isotropic elasticity.

, σθθ ¼ � bφ,rr þ ð Þ 1=r φ,r þ aφ, zz

, σrz ¼ � φ,rr þ ð Þ 1=r φ,r þ aφ, zz

, z

,r

, (47)

: (48)

, z

, z

The next step consists in substitution of formulas (47) and (48) into the first equation of equilibrium and the equations that are obtained from the Cauchy relations and formulas for the strain tensor. This permits to determine the unknown parameters through the elastic constants represented in the equilibrium equations. Further, the second equation of equilibrium gives the biharmonic equation for

<sup>2</sup> ur

classical red line in 0.5 for values of this ratio.

The substitution of constitutive equations.

symmetry

tions in stresses have the form.

Seismic Waves - Probing Earth System

σrr,r þ σrz, <sup>z</sup> þ ð Þ 1=r σrr � σφφ

C<sup>11</sup> ur,rr þ ð Þ 1=r ur,r � 1=r

with introducing the potentials.

σrr ¼ � φ,rr þ bð Þ 1=r φ,r þ aφ, zz 

σzz ¼ � cφ,rr þ cð Þ 1=r φ,r þ dφ, zz 

finding the potentials

30

elasticity

$$
\Delta\_{s1} \Delta\_{s2} \rho = 0,\tag{49}
$$

where ΔsNφ ¼ φ,rr þ ð Þ 1=r φ,r þ 1=ð Þ sN <sup>2</sup> � �φ, zz <sup>N</sup> <sup>¼</sup> <sup>1</sup>; 2 are some "complicated" copies of classical expressions Δφ ¼ φ,rr þ ð Þ 1=r φ,r þ φ, zz associated with the Laplace operator. Two constants sN are determined from the algebraic equations

$$\begin{aligned} s^4 - [(a+c)/d]s^2 + (1/d) &= \mathbf{0}, \\ s\_{1,3} &= \pm \sqrt{\frac{a+c+\sqrt{\left(a+c\right)^2 - 4d}}{2d}} \quad s\_{2,4} = \pm \sqrt{\frac{a+c-\sqrt{\left(a+c\right)^2 - 4d}}{2d}}. \end{aligned} \tag{50}$$

Thus, a transition from the isotropic case to the transversally isotropic one complicates the procedure of solving the static problems. Here a necessity of solving the classical biharmonic equation is changed on necessity of solving some generalization of this equation in the form (49).

Way 2 [12, 16]. This way is also proposed for the static problems. Here, two potentials are introduced which are linked immediately with displacements

$$
\mu\_r = \phi\_{1,r} + \phi\_{2,r}, \quad \mu\_x = k\_1 \phi\_{1,x} + k\_2 \phi\_{2,x}.\tag{51}
$$

A substitution of representations (51) into equations of equilibrium (45), (46)

$$\mathbf{C}\_{11} \left[ u\_{\tau,rr} + (\mathbf{1}/r) u\_{\tau,r} - \left( \mathbf{1}/r^2 \right) u\_r \right] + \mathbf{C}\_{44} u\_{\tau,zz} + \left[ \mathbf{C}\_{13} + \mathbf{C}\_{44} \right] u\_{z,zz} = \mathbf{0},$$

$$\mathbf{C}\_{44} (u\_{z,rr} + (\mathbf{1}/r) u\_{z,r}) + \mathbf{C}\_{33} u\_{z,zz} + \left[ \mathbf{C}\_{13} + \mathbf{C}\_{44} \right] (u\_{\tau,rz} + (\mathbf{1}/r) u\_{\tau,z}) = \mathbf{0}$$

allows to determine the unknown constants k1, k2. An idea consists in that both equations must be transformed in identical equations relative to the potentials by comparing some coefficients

$$\frac{k\_{1(2)}\left(\mathbf{C}\_{13} + \mathbf{C}\_{44}\right) + \mathbf{C}\_{44}}{\mathbf{C}\_{11}} = \frac{k\mathbf{C}\_{33}}{k\_{1(2)}\mathbf{C}\_{44} + \left(\mathbf{C}\_{13} + \mathbf{C}\_{44}\right)} = V.$$

This expression gives the quadratic equation for k1 2ð Þ and V

$$V^2 + \frac{C\_{13}(2C\_{44} + C\_{33}) - C\_{11}C\_{33}}{C\_{11}C\_{44}}V + \frac{C\_{33}}{C\_{11}} = 0.\tag{52}$$

Note that the simple link VN ¼ ð Þ 1=sN exists between constants VN and sN, which makes the ways 1 and 2 very similar. Then the potentials fulfill the equations

$$
\Delta\_{\text{rxN}} \rho\_N = \rho\_{N,rr} + (\mathbf{1}/r)\rho\_{N,r} + \left(\mathbf{1}/(V\_N)^2\right)\rho\_{N,\text{xx}}.\tag{53}
$$

The stresses are expressed through new potentials in such a way

$$\begin{aligned} \sigma\_{rr} &= -(\mathbf{C}\_{11} - \mathbf{C}\_{12})(\mathbf{1}/r)(\boldsymbol{\phi}\_{1,rr} + \boldsymbol{\phi}\_{2,rr}) - \mathbf{C}\_{44}((\mathbf{1}+k\_1)\boldsymbol{\phi}\_{1,zz} + (\mathbf{1}+k\_2)\boldsymbol{\phi}\_{2,zz}), \\ \sigma\_{\theta\theta} &= -(\mathbf{C}\_{11} - \mathbf{C}\_{12})(\mathbf{1}/r)(\boldsymbol{\phi}\_{1,rr} + \boldsymbol{\phi}\_{2,rr}) - \left((\mathbf{C}\_{13}k\_1 - \mathbf{C}\_{12}V\_1)\boldsymbol{\phi}\_{1,zz} + (\mathbf{C}\_{13}k\_2 - \mathbf{C}\_{12}V\_2)\boldsymbol{\phi}\_{2,zz}\right), \\ \sigma\_{zz} &= \left((\mathbf{C}\_{33}k\_1 - \mathbf{C}\_{13}V\_1)\boldsymbol{\phi}\_{1,zz} + (\mathbf{C}\_{33}k\_2 - \mathbf{C}\_{13}V\_2)\boldsymbol{\phi}\_{2,zz}\right), \\ \sigma\_{rz} &= \mathbf{C}\_{44}\left((\mathbf{1}+k\_1)\boldsymbol{\phi}\_{1,zz} + (\mathbf{1}+k\_2)\boldsymbol{\phi}\_{2,zz}\right). \end{aligned} \tag{54}$$

Way 3 [1, 16]. This way is proposed for equations of motion, but only for the isotropic theory of elasticity. It can be used for the static problems of transversely isotropic theory of elasticity. The initial equations here are the equations of motion (43) without inertial summands

$$\mathbf{C}\_{11}\left[u\_{\tau,rr} + (\mathbf{1}/r)u\_{\tau,r} - \left(\mathbf{1}/r^2\right)u\_r\right] + \mathbf{C}\_{44}u\_{\tau,xx} + [\mathbf{C}\_{13} + \mathbf{C}\_{44}]u\_{z,rx} = \mathbf{0},\tag{55}$$

$$\mathbf{C}\_{44}(u\_{\mathbf{z},rr} + (\mathbf{1}/r)u\_{\mathbf{z},r}) + \mathbf{C}\_{33}u\_{\mathbf{z},\mathbf{z}} + [\mathbf{C}\_{13} + \mathbf{C}\_{44}](u\_{r,rz} + (\mathbf{1}/r)u\_{r,\mathbf{z}}) = \mathbf{0}.\tag{56}$$

The potentials are introduced like (51), but the representations are complicated by necessity of introducing two new unknown parameters:

$$u\_r = \Phi\_{,r} - \Psi\_{,x}, \quad u\_x = n\Phi\_{,x} + m\Psi\_{,r} + m(\mathbf{1}/r)\Psi,\tag{57}$$

<sup>m</sup>1,<sup>2</sup> ¼ � ð Þ <sup>C</sup><sup>44</sup>

vuut

DOI: http://dx.doi.org/10.5772/intechopen.86910

1 � 4

C<sup>11</sup> C<sup>33</sup>

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

� 1 �

8 < : <sup>2</sup> <sup>þ</sup> <sup>C</sup>11C<sup>33</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The unknown potentials Φð Þ r; z and Ψð Þ r; z have to be determined from the simple Eqs. (63) and (65) which are the classical Bessel equations of orders 0 and 1 and arguments depending on some rational combination of elastic constants. Thus, three ways of introduction of potentials in the static problems of transversely isotropic theory of elasticity are shown. The different attempts to transfer these ways into the dynamic problems meet some troubles—the presence of inertial summands generates new additional conditions for the unknown constants in representations of potentials. Introducing the new constants does not help—the num-

3.4 Solving the problem on the propagation in the direction of vertical axis surface cylindrical wave for the case of transversal isotropy of medium

Consider now equations of motion (45) and (46) and introduce the potentials by the formula (57). A substitution of formula (57) into equations of motion gives five

> C<sup>44</sup> þ n Cð Þ <sup>13</sup> þ C<sup>44</sup> C<sup>11</sup>

C<sup>44</sup> C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup> <sup>Φ</sup>, zz <sup>¼</sup> <sup>ρ</sup> C<sup>11</sup>

<sup>Φ</sup>, zz <sup>¼</sup> <sup>n</sup><sup>ρ</sup>

C33m � ð Þ C<sup>13</sup> þ C<sup>44</sup> C44m

C33m � ð Þ C<sup>13</sup> þ C<sup>44</sup> C44m

i kz ð Þ �ω<sup>t</sup> , <sup>Ψ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>Ψ</sup>

ð Þr is unknown. They must be found from equations, which are obtained

Two last equations are identical. Also the equations for potential Φ must be identical as well as the equations for potential Ψ must be identical. Let us assume additionally that the problem in hand considering the solution in the form of harmonic in time cylindrical wave with unknown wave number k and known

Note that characterization of an attenuation of wave depth down functions

<sup>Ψ</sup>, zz <sup>¼</sup> <sup>ρ</sup>

nC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

<sup>Ψ</sup>, zzz <sup>¼</sup> <sup>ρ</sup>

<sup>Ψ</sup>, zz <sup>¼</sup> <sup>ρ</sup> C<sup>44</sup>

\_ ð Þr e C<sup>44</sup>

Φ,tt, (67)

Ψ,tt:

(68)

Φ,tt, (69)

Ψ, ztt, (70)

Ψ,tt: (71)

i kz ð Þ �ω<sup>t</sup> : (72)

C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup>

2C33ð Þ C<sup>13</sup> þ C<sup>44</sup>

ð Þ C<sup>44</sup>

ber of conditions is still more than the number of all constants.

equations relative to the potentials. Eq. (57) gives two equations:

nC<sup>33</sup> nC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

> 2 Ψ, <sup>z</sup> þ

> > 2 Ψ þ

Φ,rr þ ð Þ 1=r Φ,r þ

<sup>2</sup> � �<sup>Ψ</sup> <sup>þ</sup>

Ψ,rr þ ð Þ 1=r Ψ,r � 1=r

Φ,rr þ ð Þ 1=r Φ,r þ

frequency ω:

ð Þr , Ψ \_

Φ \_

33

Eq. (46) gives three equations:

Ψ,rrz þ ð Þ 1=r Ψ,rz � ð Þ 1=r

Ψ,rr þ ð Þ 1=r Ψ,r � ð Þ 1=r

Φð Þ¼ r; z; t Φ

\_ ð Þr e

by substitution of representations (72) into Eqs. (67) and (71):

2

C33ð Þ C<sup>13</sup> þ C<sup>44</sup>

" #<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>C</sup>11C<sup>33</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

2

9 = ;: (66)

A substitution of representations (57) into equations of motion (45) and (46) gives equations relative to the potentials. Eq. (45) gives two equations:

$$
\Phi\_{,rr} + (1/r)\Phi\_{,r} + \frac{C\_{44} + n(C\_{13} + C\_{44})}{C\_{11}}\Phi\_{,xx} = 0,\tag{58}
$$

$$
\Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,r} - \left(\mathbf{1}/r^2\right)\Psi + \frac{\mathbf{C}\_{44}}{\mathbf{C}\_{11} - m(\mathbf{C}\_{13} + \mathbf{C}\_{44})}\Psi\_{,xx} = \mathbf{0},\tag{59}
$$

whereas Eq. (46) gives three equations:

$$
\Phi\_{,rr} + (\mathbf{1}/r)\Phi\_{,r} + \frac{nC\_{33}}{nC\_{44} + (C\_{13} + C\_{44})}\Phi\_{,xx} = \mathbf{0},\tag{60}
$$

$$
\Psi\_{,rrz} + (\mathbf{1}/r)\Psi\_{,rz} - (\mathbf{1}/r)^2\Psi\_{,x} + \frac{\mathbf{C}\_{33}m - (\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{44}m}\Psi\_{,xxx} = \mathbf{0},\tag{61}
$$

$$
\Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,r} - (\mathbf{1}/r)^2\Psi + \frac{C\_{33}\mathfrak{m} - (C\_{13} + C\_{44})}{C\_{44}\mathfrak{m}}\Psi\_{,xx} = \mathbf{0}.\tag{62}
$$

The last two equations are identical. Eqs. (58) and (60) and (59) and (62) have to be identical. This means that the coefficients in these equations have to be identical. As a result, two equations can be obtained for the determination of unknown constants n, m.

$$\begin{aligned} \frac{C\_{44} + n(C\_{13} + C\_{44})}{C\_{11}} &= \frac{nC\_{33}}{nC\_{44} + (C\_{13} + C\_{44})} \to \\ n^2 - n \frac{C\_{11}C\_{33} - (C\_{44})^2 - (C\_{13} + C\_{44})^2}{C\_{44}(C\_{13} + C\_{44})} + 1 = 0, \end{aligned} \tag{63}$$

$$m\_{1,2} = \frac{C\_{11}C\_{33} - (C\_{44})^2 - (C\_{13} + C\_{44})^2}{2C\_{44}(C\_{13} + C\_{44})}$$

$$\times \left(1 \pm \sqrt{1 - 4\left[\frac{C\_{44}(C\_{13} + C\_{44})}{C\_{11}C\_{33} - (C\_{44})^2 - (C\_{13} + C\_{44})^2}\right]^2}\right), \tag{64}$$

$$\frac{C\_{44}}{C\_{11} - m(C\_{13} + C\_{44})} = \frac{C\_{33}m - (C\_{13} + C\_{44})}{C\_{44}m} \to m^2$$

$$+ m\left[\frac{(C\_{44})^2 + C\_{31}C\_{33} + (C\_{13} + C\_{44})^2}{C\_{33}(C\_{13} + C\_{44})}\right] + \frac{C\_{11}}{C\_{33}} = 0. \tag{65}$$

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

isotropic theory of elasticity. The initial equations here are the equations of motion

� � <sup>þ</sup> <sup>C</sup>44ur, zz <sup>þ</sup> ½ � <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> uz,rz <sup>¼</sup> <sup>0</sup>, (55)

ur ¼ Φ,r � Ψ, z, uz ¼ nΦ, <sup>z</sup> þ mΨ,r þ mð Þ 1=r Ψ, (57)

Φ, zz ¼ 0, (58)

Ψ, zz ¼ 0, (59)

Φ, zz ¼ 0, (60)

Ψ, zzz ¼ 0, (61)

Ψ, zz ¼ 0: (62)

(63)

(64)

(65)

C44ðuz,rr þ ð Þ 1=r uz,rÞ þ C33uz, zz þ ½ � C<sup>13</sup> þ C<sup>44</sup> ður,rz þ ð Þ 1=r ur, <sup>z</sup>Þ ¼ 0: (56)

The potentials are introduced like (51), but the representations are complicated

A substitution of representations (57) into equations of motion (45) and (46)

C<sup>44</sup> þ n Cð Þ <sup>13</sup> þ C<sup>44</sup> C<sup>11</sup>

nC<sup>33</sup> nC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

> C33m � ð Þ C<sup>13</sup> þ C<sup>44</sup> C44m

C33m � ð Þ C<sup>13</sup> þ C<sup>44</sup> C44m

The last two equations are identical. Eqs. (58) and (60) and (59) and (62) have

<sup>¼</sup> nC<sup>33</sup>

<sup>2</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

nC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

2

2

<sup>2</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

<sup>C</sup>44<sup>m</sup> ! <sup>m</sup><sup>2</sup>

þ C<sup>11</sup> C<sup>33</sup>

2

!

þ 1 ¼ 0,

2

¼ 0:

1

CA,

to be identical. This means that the coefficients in these equations have to be identical. As a result, two equations can be obtained for the determination of

C44ð Þ C<sup>13</sup> þ C<sup>44</sup>

<sup>1</sup> � <sup>4</sup> <sup>C</sup>44ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> C11C<sup>33</sup> � ð Þ C<sup>44</sup>

<sup>2</sup> <sup>þ</sup> <sup>C</sup>11C<sup>33</sup> <sup>þ</sup> ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

C33ð Þ C<sup>13</sup> þ C<sup>44</sup> " #

2C44ð Þ C<sup>13</sup> þ C<sup>44</sup>

<sup>2</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

" #<sup>2</sup>

<sup>¼</sup> <sup>C</sup>33<sup>m</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup>

C<sup>44</sup> C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup>

gives equations relative to the potentials. Eq. (45) gives two equations:

<sup>2</sup> � �<sup>Ψ</sup> <sup>þ</sup>

2 Ψ, <sup>z</sup> þ

2 Ψ þ

<sup>2</sup> � �ur

by necessity of introducing two new unknown parameters:

Φ,rr þ ð Þ 1=r Φ,r þ

Φ,rr þ ð Þ 1=r Φ,r þ

C<sup>44</sup> þ n Cð Þ <sup>13</sup> þ C<sup>44</sup> C<sup>11</sup>

C11C<sup>33</sup> � ð Þ C<sup>44</sup>

Ψ,rr þ ð Þ 1=r Ψ,r � 1=r

Ψ,rrz þ ð Þ 1=r Ψ,rz � ð Þ 1=r

Ψ,rr þ ð Þ 1=r Ψ,r � ð Þ 1=r

<sup>n</sup><sup>2</sup> � <sup>n</sup>

� 1 �

B@

32

0

<sup>n</sup>1,<sup>2</sup> <sup>¼</sup> <sup>C</sup>11C<sup>33</sup> � ð Þ <sup>C</sup><sup>44</sup>

vuut

<sup>þ</sup> <sup>m</sup> ð Þ <sup>C</sup><sup>44</sup>

C<sup>44</sup> C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup>

unknown constants n, m.

whereas Eq. (46) gives three equations:

(43) without inertial summands

Seismic Waves - Probing Earth System

C<sup>11</sup> ur,rr þ ð Þ 1=r ur,r � 1=r

$$\begin{split} m\_{1,2} &= -\frac{\left(\mathbf{C\_{44}}\right)^2 + \mathbf{C\_{11}}\mathbf{C\_{33}} + \left(\mathbf{C\_{13}} + \mathbf{C\_{44}}\right)^2}{2\mathbf{C\_{33}}(\mathbf{C\_{13}} + \mathbf{C\_{44}})} \\ &\times \left\{ 1 \pm \sqrt{1 - 4\frac{\mathbf{C\_{11}}}{\mathbf{C\_{33}}} \left[\frac{\mathbf{C\_{33}}(\mathbf{C\_{13}} + \mathbf{C\_{44}})}{\left(\mathbf{C\_{44}}\right)^2 + \mathbf{C\_{11}}\mathbf{C\_{33}} + \left(\mathbf{C\_{13}} + \mathbf{C\_{44}}\right)^2} \right]^2} \right\}. \end{split} \tag{66}$$

The unknown potentials Φð Þ r; z and Ψð Þ r; z have to be determined from the simple Eqs. (63) and (65) which are the classical Bessel equations of orders 0 and 1 and arguments depending on some rational combination of elastic constants.

Thus, three ways of introduction of potentials in the static problems of transversely isotropic theory of elasticity are shown. The different attempts to transfer these ways into the dynamic problems meet some troubles—the presence of inertial summands generates new additional conditions for the unknown constants in representations of potentials. Introducing the new constants does not help—the number of conditions is still more than the number of all constants.

#### 3.4 Solving the problem on the propagation in the direction of vertical axis surface cylindrical wave for the case of transversal isotropy of medium

Consider now equations of motion (45) and (46) and introduce the potentials by the formula (57). A substitution of formula (57) into equations of motion gives five equations relative to the potentials. Eq. (57) gives two equations:

$$n\Phi\_{,rr} + (1/r)\Phi\_{,r} + \frac{\mathbf{C}\_{44} + n(\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{11}}\Phi\_{,xx} = \frac{\rho}{\mathbf{C}\_{11}}\Phi\_{,tt} \tag{67}$$

$$
\Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,r} - \left(\mathbf{1}/r^2\right)\Psi + \frac{\mathbf{C\_{44}}}{\mathbf{C\_{11}} - m(\mathbf{C\_{13}} + \mathbf{C\_{44}})}\Psi\_{,xx} = \frac{\rho}{\mathbf{C\_{11}} - m(\mathbf{C\_{13}} + \mathbf{C\_{44}})}\Psi\_{,\mu}.\tag{68}
$$

Eq. (46) gives three equations:

$$\Phi\_{,rr} + (\mathbf{1}/r)\Phi\_{,r} + \frac{n\mathbf{C}\_{33}}{n\mathbf{C}\_{44} + (\mathbf{C}\_{13} + \mathbf{C}\_{44})}\Phi\_{,xx} = \frac{n\rho}{n\mathbf{C}\_{44} + (\mathbf{C}\_{13} + \mathbf{C}\_{44})}\Phi\_{,tt},\tag{69}$$

$$
\Psi\_{,rrz} + (\mathbf{1}/r)\Psi\_{,rz} - (\mathbf{1}/r)^2\Psi\_{,z} + \frac{\mathbf{C}\_{33}m - (\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{44}m}\Psi\_{,zzz} = \frac{\rho}{\mathbf{C}\_{44}}\Psi\_{,z\text{tt}}\tag{70}
$$

$$
\Psi\_{,rr} + (\mathbf{1}/r)\Psi\_{,r} - (\mathbf{1}/r)^2\Psi + \frac{\mathbf{C}\_{33}m - (\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{44}m}\Psi\_{,xx} = \frac{\rho}{\mathbf{C}\_{44}}\Psi\_{,tt}.\tag{71}
$$

Two last equations are identical. Also the equations for potential Φ must be identical as well as the equations for potential Ψ must be identical. Let us assume additionally that the problem in hand considering the solution in the form of harmonic in time cylindrical wave with unknown wave number k and known frequency ω:

$$\Phi(r,z,t) = \stackrel{\frown}{\Phi}(r)e^{i(kx-\alpha t)}, \quad \Psi(r,z,t) = \stackrel{\frown}{\Psi}(r)e^{i(kx-\alpha t)}.\tag{72}$$

Note that characterization of an attenuation of wave depth down functions Φ \_ ð Þr , Ψ \_ ð Þr is unknown. They must be found from equations, which are obtained by substitution of representations (72) into Eqs. (67) and (71):

$$
\stackrel{\frown}{\Phi}\_{,rr} + (\mathbf{1}/r)\stackrel{\frown}{\Phi}\_{,r} - \left[\frac{\mathbf{C}\_{44} + \mathfrak{n}(\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{11}} k^2 - k\_{L(11)}^2\right] \stackrel{\frown}{\Phi} = \mathbf{0},\tag{73}
$$

A success in the determination of transformed potentials is accompanied by a complication of conditions which provide the wave attenuation. They have the

Let us recall that the similar conditions for the case of isotropic medium

conditions of classical Rayleigh surface wave [5–9, 17]. A complexity of conditions (83) is increased by the complex form of dependence of constants n, m on the

If the conditions (83) are fulfilled, then the solution of wave equations for

\_

With allowance for formulas (84), the representations of potentials becomes

i kz ð Þ �ω<sup>t</sup> , <sup>Ψ</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> <sup>A</sup>

The formula (85) completes the first analytical part of solving the problem on

<sup>Φ</sup>K<sup>0</sup> MLð Þ <sup>11</sup> <sup>r</sup> , <sup>Ψ</sup>

3.5 Boundary conditions: equations for the unknown wave number

<sup>σ</sup>rr <sup>¼</sup> ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> ð Þþ <sup>Φ</sup>,rr � <sup>Ψ</sup>,rz <sup>λ</sup> ð Þ <sup>1</sup>=<sup>r</sup> ð Þþ <sup>Φ</sup>,r � <sup>Ψ</sup>, <sup>z</sup>

σrz ¼ μ ð Þþ Φ,rz � Ψ, zz nΦ, zr þ mΨ,rr þ mð Þ 1=r Ψ,r � m 1=r

<sup>K</sup><sup>0</sup> MLð Þ <sup>11</sup> <sup>r</sup> <sup>=</sup>dr<sup>2</sup> <sup>¼</sup> MLð Þ <sup>11</sup> ð Þ <sup>1</sup>=<sup>r</sup> <sup>K</sup><sup>1</sup> MLð Þ <sup>11</sup> <sup>r</sup> <sup>þ</sup> MLð Þ <sup>11</sup> <sup>2</sup>

\_

A

Further, the representations (86) and (87) should be substituted into the boundary conditions, and the formulas on differentiation of Macdonald functions

dK<sup>0</sup> MLð Þ <sup>11</sup> rx <sup>=</sup>dr ¼ �MLð Þ <sup>11</sup> <sup>K</sup><sup>1</sup> MLð Þ <sup>11</sup> rx ,

dK<sup>1</sup> MTð Þ <sup>44</sup> <sup>r</sup> <sup>=</sup>dr ¼ �ð Þ <sup>1</sup>=<sup>r</sup> <sup>K</sup><sup>1</sup> MTð Þ <sup>44</sup> <sup>r</sup> � MTð Þ <sup>44</sup> <sup>K</sup><sup>0</sup> MTð Þ <sup>44</sup> <sup>r</sup> :

Then the boundary conditions are transformed into the algebraic equations

<sup>Φ</sup>, K<sup>1</sup> MTð Þ <sup>44</sup> ro A

This part of analysis can be treated as the second analytical part. The boundary conditions have the form identical for all kinds of symmetry of properties. That is, they have the form (17) or (21). The formulas for stresses depend already on the symmetry of medium. The expressions for stresses through the potential reflect the features of introducing the potentials. In this case, they have

C33m � ð Þ C<sup>13</sup> þ C<sup>44</sup> C44m

<sup>T</sup>>0 are slightly simpler and coincide with the corresponding

ð Þ¼ r A \_

\_

þnΦ, zz þ mΨ,rz þ mð Þ 1=r Ψ, <sup>z</sup>

\_ Ψ

<sup>2</sup> Ψ : (87)

, (86)

<sup>K</sup><sup>0</sup> MLð Þ <sup>11</sup> <sup>r</sup> ,

<sup>Ψ</sup>K<sup>1</sup> MTð Þ <sup>44</sup> <sup>r</sup> <sup>e</sup>

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

<sup>Ψ</sup>K<sup>1</sup> MTð Þ <sup>44</sup> <sup>r</sup> : (84)

<sup>T</sup>ð Þ <sup>44</sup> <sup>&</sup>gt;0: (83)

i kz ð Þ �ω<sup>t</sup> : (85)

form.

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

C<sup>44</sup> þ n Cð Þ <sup>13</sup> þ C<sup>44</sup> C<sup>11</sup>

<sup>L</sup>>0, <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

potentials can be written in the form.

Φ \_

\_

[3] should be taken into account:

relative to quantities K<sup>1</sup> MLð Þ <sup>11</sup> ro

d2

35

ð Þ¼ r A \_

<sup>Φ</sup>K<sup>0</sup> MLð Þ <sup>11</sup> <sup>r</sup> <sup>e</sup>

wave number k.

more definite

the form

Φð Þ¼ r; z; t A

cylindrical surface wave.

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

DOI: http://dx.doi.org/10.5772/intechopen.86910

<sup>L</sup>ð Þ <sup>11</sup> <sup>&</sup>gt;0,

$$k\_{L(11)} = \left(w/\nu\_{L(11)}\right), \nu\_{L(11)} = \sqrt{\mathcal{C}\_{11}/\rho},$$

$$\stackrel{\frown}{\Phi}\_{,rr} + (1/r)\stackrel{\frown}{\Phi}\_{,r} - \frac{n}{n\mathcal{C}\_{44} + (\mathcal{C}\_{13} + \mathcal{C}\_{44})} \left(\mathcal{C}\_{33}k^2 - \mathcal{C}\_{11}k\_{L(11)}^2\right) \stackrel{\frown}{\Phi} = \mathbf{0},\tag{74}$$

$$
\widehat{\Psi}\_{,rr} + (\mathbf{1}/r)\widehat{\Psi}\_{,r} - \left(\mathbf{1}/r^2\right)\widehat{\Psi} - \frac{\mathbf{C}\_{44}}{\mathbf{C}\_{11} - m(\mathbf{C}\_{13} + \mathbf{C}\_{44})} \left(k^2 - k\_{T(44)}^2\right) \widehat{\Psi} = \mathbf{0},\tag{75}
$$

$$\begin{aligned} k\_{T(44)} &= \left(\alpha/v\_{L(44)}\right), v\_{L(44)} = \sqrt{\mathcal{C}\_{44}/\rho}, \\ \widehat{\Psi}\_{,rr} &+ \left(\mathbf{1}/r\right)\widehat{\Psi}\_{,r} - \left(\mathbf{1}/r\right)^{2}\widehat{\Psi} - \left[\frac{\mathcal{C}\_{33}m - \left(\mathcal{C}\_{13} + \mathcal{C}\_{44}\right)}{\mathcal{C}\_{44}m}k^{2} - k\_{T(44)}^{2}\right]\widehat{\Psi} = 0. \end{aligned} \tag{76}$$

As a result, two equations can be obtained that permit to determine the constants n, m

$$m^2 - 2N\_1 n + N\_2 = 0,\\ m^2 + 2M\_1 m + M\_2 = 0,\tag{77}$$

$$N\_{\pm}(M\_{\pm}) = N\_1(M\_1) \pm \sqrt{\left[N\_1(M\_1)\right]^2 - N\_2(M\_2)},\tag{78}$$

$$2N\_1 = \frac{\left[\mathbf{C}\_{11}\mathbf{C}\_{33} - \left(\mathbf{C}\_{13} + \mathbf{C}\_{44}\right)^2\right]\mathbf{k}^2 - \mathbf{C}\_{11}[\mathbf{C}\_{11} - \mathbf{C}\_{44}]\mathbf{k}\_{L(11)}^2 - \left(\mathbf{C}\_{44}\right)^2}{\mathbf{C}\_{44}(\mathbf{C}\_{13} + \mathbf{C}\_{44})\mathbf{k}^2},\tag{79}$$

$$\begin{aligned} N\_2 &= \frac{\mathbf{C}\_{44} - \mathbf{C}\_{11}k\_{L(11)}^2}{\mathbf{C}\_{44}k^2} = \mathbf{0} \\\\ 2M\_1 &= \frac{\left[\left(\mathbf{C}\_{44}\right)^2 - \mathbf{C}\_{11}\mathbf{C}\_{33} - \left(\mathbf{C}\_{13} + \mathbf{C}\_{44}\right)^2\right]k^2 - \left[\left(\mathbf{C}\_{44}\right)^2 - \mathbf{C}\_{11}\mathbf{C}\_{44}\right]k\_{T(44)}^2}{\left(\mathbf{C}\_{13} + \mathbf{C}\_{44}\right)\left(\mathbf{C}\_{33}k^2 - \mathbf{C}\_{44}k\_{T(44)}^2\right)}, \\\\ M\_2 &= \frac{\mathbf{C}\_{11}}{\left(\mathbf{C}\_{33}k^2 - \mathbf{C}\_{44}k\_{T(44)}^2\right)}k^2. \end{aligned} \tag{80}$$

Note that restriction on the kind of solution (it has to be a wave) allows to unite two different conditions into one—conditions for equaling coefficients in summands with the second derivative by time t and vertical coordinate z. In this case, the number of unknown constants coincides with the number of conditions which are necessary for the determination of potentials. As a result, the wave attenuationtransformed potentials can be determined from the equations of Bessel type:

$$
\widehat{\Phi}\_{,rr} + (\mathbf{1}/r)\widehat{\Phi}\_{,r} - M\_{L(11)}^2 \overset{\frown}{\Phi} = \mathbf{0}, \quad M\_{L(11)} = \sqrt{\frac{\mathbf{C}\_{44} + n(\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{11}}} k^2 - k\_{L(11)}^2. \tag{81}
$$

$$\begin{aligned} \widehat{\Psi}\_{,rr} + (\mathbf{1}/r)\widehat{\Psi}\_{,r} - \left[ (\mathbf{1}/r^2) + M\_{T(44)}^2 \right] \widehat{\Psi} &= \mathbf{0}, \\\ M\_{T(44)} = \sqrt{\frac{\mathbf{C}\_{44}}{\mathbf{C}\_{11} - m(\mathbf{C}\_{13} + \mathbf{C}\_{44})}} \left( k^2 - k\_{T(44)}^2 \right), \end{aligned} \tag{82}$$

34

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

Φ \_

Seismic Waves - Probing Earth System

,rr þ ð Þ 1=r Φ

\_

\_

,rr þ ð Þ 1=r Ψ

kTð Þ <sup>44</sup> ¼ ω=vLð Þ <sup>44</sup>

,rr þ ð Þ 1=r Ψ

Φ \_

Ψ \_

> Ψ \_

constants n, m

2N<sup>1</sup> ¼

2M<sup>1</sup> ¼

Φ \_

34

<sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup><sup>44</sup> � <sup>C</sup>11k<sup>2</sup>

ð Þ C<sup>44</sup>

<sup>C</sup>33k<sup>2</sup> � <sup>C</sup>44k<sup>2</sup>

� � <sup>k</sup><sup>2</sup>

<sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>C</sup><sup>11</sup>

,rr þ ð Þ 1=r Φ

\_

,r � <sup>M</sup><sup>2</sup>

Ψ \_ <sup>L</sup>ð Þ <sup>11</sup> Φ \_

,rr þ ð Þ 1=r Ψ

MTð Þ <sup>44</sup> ¼

\_

,rr þ ð Þ 1=r Φ

\_

,r � 1=r <sup>2</sup> � � Ψ \_

,r � ð Þ 1=r

C11C<sup>33</sup> � ð Þ C<sup>13</sup> þ C<sup>44</sup> <sup>2</sup> h i

> Lð Þ 11 <sup>C</sup>44k<sup>2</sup> <sup>¼</sup> <sup>0</sup>

\_

,r � <sup>n</sup>

� �, vLð Þ <sup>44</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> Ψ \_

N�ð Þ¼ M� N1ð Þ� M<sup>1</sup>

<sup>2</sup> � <sup>C</sup>11C<sup>33</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> <sup>2</sup> h i

:

Tð Þ 44

,r � <sup>C</sup><sup>44</sup> <sup>þ</sup> n Cð Þ <sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> C<sup>11</sup>

kLð Þ <sup>11</sup> ¼ ω=vLð Þ <sup>11</sup>

nC<sup>44</sup> þ ð Þ C<sup>13</sup> þ C<sup>44</sup>

� <sup>C</sup><sup>44</sup>

C44=ρ p ,

C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup>

� <sup>C</sup>33<sup>m</sup> � ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> C44m

q

As a result, two equations can be obtained that permit to determine the

ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> <sup>C</sup>33k<sup>2</sup> � <sup>C</sup>44k<sup>2</sup>

Note that restriction on the kind of solution (it has to be a wave) allows to unite

two different conditions into one—conditions for equaling coefficients in summands with the second derivative by time t and vertical coordinate z. In this case, the number of unknown constants coincides with the number of conditions which are necessary for the determination of potentials. As a result, the wave attenuationtransformed potentials can be determined from the equations of Bessel type:

¼ 0, MLð Þ <sup>11</sup> ¼

,r � <sup>1</sup>=r<sup>2</sup> ð Þþ <sup>M</sup><sup>2</sup>

C<sup>11</sup> � m Cð Þ <sup>13</sup> þ C<sup>44</sup>

h i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C<sup>44</sup>

r � �

� �

� �, vLð Þ <sup>11</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

<sup>C</sup>33k<sup>2</sup> � <sup>C</sup>11k<sup>2</sup>

� �

<sup>n</sup><sup>2</sup> � <sup>2</sup>N1<sup>n</sup> <sup>þ</sup> <sup>N</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, m<sup>2</sup> <sup>þ</sup> <sup>2</sup>M1<sup>m</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, (77)

<sup>C</sup>44ð Þ <sup>C</sup><sup>13</sup> <sup>þ</sup> <sup>C</sup><sup>44</sup> <sup>k</sup><sup>2</sup> ,

<sup>k</sup><sup>2</sup> � ð Þ <sup>C</sup><sup>44</sup>

s

Tð Þ 44

Ψ \_ ¼ 0,

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

Tð Þ 44

,

Tð Þ 44 � � ,

½ � N1ð Þ M<sup>1</sup>

<sup>k</sup><sup>2</sup> � <sup>C</sup>11½ � <sup>C</sup><sup>11</sup> � <sup>C</sup><sup>44</sup> <sup>k</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> � <sup>N</sup>2ð Þ <sup>M</sup><sup>2</sup>

<sup>L</sup>ð Þ <sup>11</sup> � ð Þ C<sup>44</sup>

<sup>2</sup> � <sup>C</sup>11C<sup>44</sup> h i

� �

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> Lð Þ 11

C11=ρ p ,

Lð Þ 11

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup>

Φ \_

Φ \_

> Ψ \_

Ψ \_

Tð Þ 44 � �

Tð Þ 44

¼ 0, (73)

¼ 0, (74)

¼ 0, (75)

¼ 0: (76)

, (78)

(79)

(80)

2

k2 Tð Þ 44

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> Lð Þ 11

,

(81)

(82)

C<sup>44</sup> þ n Cð Þ <sup>13</sup> þ C<sup>44</sup> C<sup>11</sup>

A success in the determination of transformed potentials is accompanied by a complication of conditions which provide the wave attenuation. They have the form.

$$\frac{\mathbf{C}\_{44} + n(\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{11}} k^2 - k\_{L(11)}^2 > 0,\\ \frac{\mathbf{C}\_{33}m - (\mathbf{C}\_{13} + \mathbf{C}\_{44})}{\mathbf{C}\_{44}m} k^2 - k\_{T(44)}^2 > 0. \tag{83}$$

Let us recall that the similar conditions for the case of isotropic medium <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> <sup>L</sup>>0, <sup>k</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> <sup>T</sup>>0 are slightly simpler and coincide with the corresponding conditions of classical Rayleigh surface wave [5–9, 17]. A complexity of conditions (83) is increased by the complex form of dependence of constants n, m on the wave number k.

If the conditions (83) are fulfilled, then the solution of wave equations for potentials can be written in the form.

$$
\stackrel{\frown}{\Phi}(r) = \stackrel{\frown}{A}\_{\Phi} \text{K}\_{0}(\mathcal{M}\_{L(11)}r), \; \stackrel{\frown}{\Psi}(r) = \stackrel{\frown}{A}\_{\Psi} \text{K}\_{1}(\mathcal{M}\_{T(44)}r). \tag{84}
$$

With allowance for formulas (84), the representations of potentials becomes more definite

$$\Phi(r,z,t) = \stackrel{\frown}{A}\_{\Phi} \mathcal{K}\_0(\mathcal{M}\_{L(11)}r)e^{i(kx-\alpha t)}, \quad \Psi(r,z,t) = \stackrel{\frown}{A}\_{\Psi} \mathcal{K}\_1(\mathcal{M}\_{T(44)}r)e^{i(kx-\alpha t)}.\tag{85}$$

The formula (85) completes the first analytical part of solving the problem on cylindrical surface wave.

#### 3.5 Boundary conditions: equations for the unknown wave number

This part of analysis can be treated as the second analytical part. The boundary conditions have the form identical for all kinds of symmetry of properties. That is, they have the form (17) or (21). The formulas for stresses depend already on the symmetry of medium. The expressions for stresses through the potential reflect the features of introducing the potentials. In this case, they have the form

$$\sigma\_{rr} = (\lambda + 2\mu)(\Phi\_{,rr} - \Psi\_{,rr}) + \lambda \begin{Bmatrix} (\mathbf{1}/r)(\Phi\_{,r} - \Psi\_{,x}) + \\ + n\Phi\_{,xx} + m\Psi\_{,rr} + m(\mathbf{1}/r)\Psi\_{,x} \end{Bmatrix},\tag{86}$$

$$
\sigma\_{rz} = \mu \left[ (\Phi\_{,rr} - \Psi\_{,xx}) + n\Phi\_{,rr} + m\Psi\_{,rr} + m(\mathbf{1}/r)\Psi\_{,r} - m\left(\mathbf{1}/r^2\right)\Psi \right].\tag{87}
$$

Further, the representations (86) and (87) should be substituted into the boundary conditions, and the formulas on differentiation of Macdonald functions [3] should be taken into account:

$$\left[d\mathcal{K}\_0\left(\mathcal{M}\_{L(11)}r\infty\right)/dr\right] = -\mathcal{M}\_{L(11)}\mathcal{K}\_1\left(\mathcal{M}\_{L(11)}r\infty\right),$$

$$\left[d^2\mathcal{K}\_0\left(\mathcal{M}\_{L(11)}r\right)/dr^2\right] = \mathcal{M}\_{L(11)}\left(1/r\right)\mathcal{K}\_1\left(\mathcal{M}\_{L(11)}r\right) + \left(\mathcal{M}\_{L(11)}\right)^2\mathcal{K}\_0\left(\mathcal{M}\_{L(11)}r\right),$$

$$\left[d\mathcal{K}\_1\left(\mathcal{M}\_{T(44)}r\right)/dr\right] = -\left(1/r\right)\mathcal{K}\_1\left(\mathcal{M}\_{T(44)}r\right) - \mathcal{M}\_{T(44)}\mathcal{K}\_0\left(\mathcal{M}\_{T(44)}r\right).$$

Then the boundary conditions are transformed into the algebraic equations relative to quantities K<sup>1</sup> MLð Þ <sup>11</sup> ro A \_ <sup>Φ</sup>, K<sup>1</sup> MTð Þ <sup>44</sup> ro A \_ Ψ

$$\begin{aligned} \left[\mathbf{M}\_{L(11)}\frac{1}{r\_o} + \frac{v\_L^2}{v\_T^2} \begin{pmatrix} \left(\boldsymbol{M}\_{L(11)}\right)^2 -\\ -\frac{v\_L^2}{v\_L^2} - v\_{T0}^2 \end{pmatrix} \frac{\mathbf{K}\_0\left(\boldsymbol{M}\_{L(11)}r\_o\right)}{\mathbf{K}\_1\left(\boldsymbol{M}\_{L(11)}r\_o\right)}\right] \hat{\boldsymbol{A}}\_{\Phi}\mathbf{K}\_1\left(\boldsymbol{M}\_{L(11)}r\_o\right) - ik \frac{v\_L^2 - v\_T^2}{v\_T^2} \\ \times \begin{bmatrix} \left(2(\mathbf{1} - \boldsymbol{m}) + \frac{v\_T^2}{v\_L^2 - v\_T^2}\right) \frac{1}{r\_o} + \\ + \left((\mathbf{1} - \boldsymbol{m}) + \frac{v\_T^2}{v\_L^2 - v\_T^2}\right) \boldsymbol{M}\_{T(44)} \frac{\boldsymbol{K}\_0\left(\boldsymbol{M}\_{T(44)}r\_o\right)}{\boldsymbol{K}\_1\left(\boldsymbol{M}\_{T(44)}r\_o\right)} \end{bmatrix} \hat{\boldsymbol{A}}\_{\Phi}\mathbf{K}\_1\left(\boldsymbol{M}\_{T(44)}r\_o\right) = 0, \end{aligned} \tag{88}$$

Let us show below an analysis of the problem in hand that is carried out in Subchapter "Longitudinal Waves" of Chapter 4 "Waves in Cylindrical Media" of volume 2 of edition [19]. Here, the cylinder of circular cross-section is considered, and the longitudinal wave is defined as the wave propagating in the direction of cylinder axis Oy3. The problem is assumed to be axisymmetric and is described within the framework of linearized theory of elasticity for bodies with initial

; θ; y<sup>3</sup>

; <sup>y</sup>3; <sup>t</sup> � �, u<sup>θ</sup> <sup>¼</sup> <sup>0</sup>, uy<sup>3</sup> <sup>¼</sup> uy<sup>3</sup> <sup>r</sup>

The medium is assumed isotropic or transversally isotropic. The main relations

Note that as shown in (92), eight constants are necessary in the linearized theory, but in the framework of linear theory, they have the form (33), and their

where only eight independent constants (92) must be taken into account. The corresponding equations of linear theory of elasticity for the case of transversally isotropic medium without of initial stresses are written above as

The general solutions for the case of axial symmetry are expressed through one

=∂r 0 ∂y3 � �X<sup>0</sup>

<sup>1</sup> þ ω<sup>0</sup>

Note that in Section 3 of this chapter, two potentials Φ, Ψ are introduced by formula (57), which corresponds and generalizes the procedure used in Biot's

The longitudinal harmonic wave is described analytically through the potential

ð Þ<sup>1</sup> r <sup>0</sup> ð Þe

<sup>1111</sup> þ ω<sup>0</sup>

<sup>3113</sup> <sup>∂</sup><sup>2</sup>

=∂y<sup>2</sup> 3 � � � <sup>ρ</sup><sup>0</sup> <sup>∂</sup><sup>2</sup> =∂τ<sup>2</sup> � � � � X<sup>0</sup>

uα=∂xk∂x<sup>β</sup> � � <sup>¼</sup> ρδm<sup>α</sup> <sup>∂</sup><sup>2</sup>

ur<sup>0</sup> ¼ � <sup>∂</sup><sup>2</sup>

; <sup>y</sup>3; <sup>τ</sup> � � <sup>¼</sup> <sup>X</sup><sup>0</sup>

solution (4.13) [19] into the second Eq. (3.362) [19] (for potential X<sup>0</sup>

<sup>1</sup> � <sup>k</sup><sup>2</sup> ξ0 32 � �

<sup>1</sup> � <sup>k</sup><sup>2</sup> <sup>ω</sup><sup>0</sup>

<sup>3113</sup> � � � � <sup>þ</sup> <sup>ρ</sup>0<sup>2</sup>C<sup>2</sup>

<sup>0</sup> ð Þ <sup>∂</sup>=∂<sup>r</sup> <sup>0</sup> ð Þ:

X<sup>0</sup> r 0 1111Δ<sup>0</sup>

Further, the general solutions of basic equations in displacements are utilized.

ω1111, ω1122, ω1133,ω1221,ω1313, ω1331,ω3113, ω3333: (92)

� � are introduced, and displacements are

; <sup>y</sup>3; <sup>t</sup> � � (91)

uα=∂τ<sup>2</sup> � � (93)

, (94)

i ky ð Þ <sup>3</sup>�ωτ , (95)

ð Þ<sup>1</sup> <sup>r</sup><sup>0</sup> ð Þ has to be determined by substitution of

cpo X0 ,

). This gives

ð Þ<sup>1</sup> <sup>¼</sup> <sup>0</sup>, (96)

0

stresses. The cylindrical coordinates r<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86910

ur<sup>0</sup> ¼ ur<sup>0</sup> r

These equations have the form (3.174) [19]

ωlmαβ ∂<sup>2</sup>

0

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

for transversal isotropy are described by independent constant

taken in the form

number is five.

Eqs. (45) and (46).

analysis [1].

Eq. (4.16) [19]:

ω0 1111ω<sup>0</sup> <sup>1331</sup> �� � <sup>Δ</sup><sup>0</sup>

�k<sup>2</sup> ρ0 C2 cp ω<sup>0</sup>

37

in the form (101) [19]

potential in the form (4.13) [19]

u<sup>3</sup> ¼ ω<sup>0</sup>

<sup>Δ</sup><sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

<sup>1111</sup> þ ω<sup>0</sup> <sup>1313</sup> � ��<sup>1</sup> <sup>ω</sup><sup>0</sup>

=∂r <sup>0</sup><sup>2</sup> � � <sup>þ</sup> <sup>1</sup>=<sup>r</sup>

where the unknown amplitude X<sup>0</sup>

<sup>1</sup> � <sup>k</sup><sup>2</sup> ξ0 22 � � <sup>Δ</sup><sup>0</sup>

<sup>1111</sup> þ ω<sup>0</sup> <sup>1331</sup> � �Δ<sup>0</sup>

$$\frac{1}{2}(1+n)\mathrm{i}k\frac{K\_0\left(M\_{L(11)}r\_o\right)}{K\_1\left(M\_{L(11)}r\_0\right)}K\_1\left(M\_{L(11)}r\_o\right)\widehat{A}\_\Phi + \left[m\left(M\_{T(44)}\right)^2 + k^2\right]K\_1\left(M\_{T(44)}r\right)\widehat{A}\_\Psi = 0. \tag{89}$$

When the determinant of linear homogeneous system of Eqs. (88) and (89) is equaled to zero, then the equations for the unknown wave number can be obtained:

$$\begin{aligned} &(1+n)k^2 \frac{v\_L^2 - v\_T^2}{v\_T^2} \frac{K\_0\left(M\_{L(11)}r\_o\right)}{K\_1\left(M\_{L(11)}r\_o\right)} \Bigg[ \begin{aligned} &\left(2(1-m) + \frac{v\_T^2}{v\_L^2 - v\_T^2}\right)(1/r\_o) \\ &+\left(\left(1-m\right) + \frac{v\_T^2}{v\_L^2 - v\_T^2}\right)M\_{T(44)}\frac{K\_0\left(M\_{T(44)}r\_o\right)}{K\_1\left(M\_{T(44)}r\_o\right)} \Bigg] \\ &-\left[m\left(M\_{T(44)}\right)^2 + k^2\right] \Bigg[ \begin{aligned} &\frac{1}{H\_0} +\\ &\frac{v\_L^2}{v\_T^2 - v\_L^2}\left(\left(M\_{L(11)}\right)^2 - \frac{v\_L^2 - v\_T^2}{v\_L^2}mk^2\right) \frac{K\_0\left(M\_{L(11)}r\_o\right)}{K\_1\left(M\_{L(11)}r\_o\right)} \end{aligned} \end{aligned} \tag{90}$$

Note that the sufficiently complex expression relative to the wave number is hidden coefficients MLð Þ <sup>11</sup> , MTð Þ <sup>44</sup> of Macdonald's functions <sup>K</sup>0ð Þ MLð Þ <sup>11</sup> ro <sup>K</sup>1ð Þ MLð Þ <sup>11</sup> <sup>r</sup><sup>0</sup> , <sup>K</sup>0ð Þ MTð Þ <sup>44</sup> ro <sup>K</sup>1ð Þ MTð Þ <sup>44</sup> ro . Therefore, the analytical part of analysis is finished on these formulas. Further, the numerical approaches have to be utilized.

Note also that the simple and convenient condition from analysis of classical surface Rayleigh wave [6–10, 17], when the wave number depends only on ratio v2 L=v<sup>2</sup> T � �, does not exist in the analysis of cylindrical surface wave. Here, the parameters MLð Þ <sup>11</sup> , MTð Þ <sup>44</sup> depend on the complicated form on all elastic constants. Of course, the Macdonald functions can be represented approximately through their arguments. But only the numerical methods can give the final result—the value of wave number or phase velocity.

#### 4. Solving the problem on propagating in the direction of symmetry axis surface wave within the framework of linearized theory of elasticity with allowance for initial stresses

Note that analysis of cylindrical surface wave in isotropic medium was first carried out by Biot [1] in 1952 and the transversally isotropic medium with initial stresses was first carried out by Guz et al. in 1974 [18].

#### Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

MLð Þ <sup>11</sup>

�

ð Þ 1 þ n ik

ð Þ <sup>1</sup> <sup>þ</sup> <sup>n</sup> <sup>k</sup><sup>2</sup> <sup>v</sup><sup>2</sup>

� m MTð Þ <sup>44</sup>

v2 L=v<sup>2</sup> T

36

1 ro þ v2 L v2 T

MLð Þ <sup>11</sup> � �<sup>2</sup>

� � 1

� �

� �K<sup>1</sup> MLð Þ <sup>11</sup> ro

K<sup>0</sup> MLð Þ <sup>11</sup> ro � �

K<sup>1</sup> MLð Þ <sup>11</sup> r<sup>0</sup> � �

MLð Þ <sup>11</sup>

v2 L v2 T

� v2 <sup>L</sup> � <sup>v</sup><sup>2</sup> T v2 L

0

Seismic Waves - Probing Earth System

BB@

2 1ð Þþ � m

þ ð Þþ 1 � m

K<sup>0</sup> MLð Þ <sup>11</sup> ro � �

K<sup>1</sup> MLð Þ <sup>11</sup> r<sup>0</sup>

<sup>L</sup> � <sup>v</sup><sup>2</sup> T v2 T

� �<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> h i

numerical approaches have to be utilized.

with allowance for initial stresses

stresses was first carried out by Guz et al. in 1974 [18].

wave number or phase velocity.

�

v2 T v2 <sup>L</sup> � <sup>v</sup><sup>2</sup> T

v2 T v2 <sup>L</sup> � <sup>v</sup><sup>2</sup> T

� �A

1 ro þ

MLð Þ <sup>11</sup> � �<sup>2</sup> � <sup>v</sup><sup>2</sup>

hidden coefficients MLð Þ <sup>11</sup> , MTð Þ <sup>44</sup> of Macdonald's functions <sup>K</sup>0ð Þ MLð Þ <sup>11</sup> ro

nk<sup>2</sup>

1

CCA

ro þ

MTð Þ <sup>44</sup>

\_

K<sup>0</sup> MLð Þ <sup>11</sup> ro � �

K<sup>0</sup> MTð Þ <sup>44</sup> ro � �

K<sup>1</sup> MTð Þ <sup>44</sup> ro � �

� �<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> h i

v2 T v2 <sup>L</sup> � <sup>v</sup><sup>2</sup> T

v2 T v2 <sup>L</sup> � <sup>v</sup><sup>2</sup> T

nk<sup>2</sup> � � <sup>K</sup><sup>0</sup> MLð Þ <sup>11</sup> ro

ð Þ 1=ro

MTð Þ <sup>44</sup>

� �

<sup>K</sup>1ð Þ MLð Þ <sup>11</sup> <sup>r</sup><sup>0</sup>

K<sup>1</sup> MLð Þ <sup>11</sup> ro � �

<sup>Φ</sup> þ m MTð Þ <sup>44</sup>

� �

� �

<sup>L</sup> � <sup>v</sup><sup>2</sup> T v2 L

When the determinant of linear homogeneous system of Eqs. (88) and (89) is equaled to zero, then the equations for the unknown wave number can be obtained:

2 1ð Þþ � m

þ ð Þþ 1 � m

Note that the sufficiently complex expression relative to the wave number is

Therefore, the analytical part of analysis is finished on these formulas. Further, the

Note also that the simple and convenient condition from analysis of classical surface Rayleigh wave [6–10, 17], when the wave number depends only on ratio

� �, does not exist in the analysis of cylindrical surface wave. Here, the parameters MLð Þ <sup>11</sup> , MTð Þ <sup>44</sup> depend on the complicated form on all elastic constants. Of course, the Macdonald functions can be represented approximately through their arguments. But only the numerical methods can give the final result—the value of

4. Solving the problem on propagating in the direction of symmetry axis surface wave within the framework of linearized theory of elasticity

Note that analysis of cylindrical surface wave in isotropic medium was first carried out by Biot [1] in 1952 and the transversally isotropic medium with initial

<sup>Φ</sup>K<sup>1</sup> MLð Þ <sup>11</sup> ro

� � � ik <sup>v</sup><sup>2</sup>

<sup>Ψ</sup>K<sup>1</sup> MTð Þ <sup>44</sup> ro

<sup>K</sup><sup>1</sup> MTð Þ <sup>44</sup> <sup>r</sup> � �<sup>A</sup>

K<sup>0</sup> MTð Þ <sup>44</sup> ro � �

K<sup>1</sup> MTð Þ <sup>44</sup> ro � �

,

� � <sup>¼</sup> <sup>0</sup>,

\_ <sup>Ψ</sup> ¼ 0:

<sup>L</sup> � <sup>v</sup><sup>2</sup> T v2 T

(88)

(89)

(90)

.

<sup>K</sup>0ð Þ MTð Þ <sup>44</sup> ro <sup>K</sup>1ð Þ MTð Þ <sup>44</sup> ro

K<sup>1</sup> MLð Þ <sup>11</sup> ro � �

Let us show below an analysis of the problem in hand that is carried out in Subchapter "Longitudinal Waves" of Chapter 4 "Waves in Cylindrical Media" of volume 2 of edition [19]. Here, the cylinder of circular cross-section is considered, and the longitudinal wave is defined as the wave propagating in the direction of cylinder axis Oy3. The problem is assumed to be axisymmetric and is described within the framework of linearized theory of elasticity for bodies with initial stresses. The cylindrical coordinates r<sup>0</sup> ; θ; y<sup>3</sup> � � are introduced, and displacements are taken in the form

$$u\_{r'} = u\_{r'}(r', \boldsymbol{y}\_3, t), \ u\_{\theta} = 0, \ u\_{\boldsymbol{\gamma}\_3} = u\_{\boldsymbol{\gamma}\_3}(r', \boldsymbol{y}\_3, t) \tag{91}$$

The medium is assumed isotropic or transversally isotropic. The main relations for transversal isotropy are described by independent constant

$$
\begin{pmatrix}
\;\_{1111}\;\;\;0\;\_{1122}\;\;\;0\;\_{1133}\;\;\;0\;\_{1213}\;\;\;0\;\_{1313}\;\;\;0\;\_{1331}\;\;\;0\;\_{3113}\;\;\;0\;\_{333}\
\end{pmatrix} \tag{92}
$$

Note that as shown in (92), eight constants are necessary in the linearized theory, but in the framework of linear theory, they have the form (33), and their number is five.

Further, the general solutions of basic equations in displacements are utilized. These equations have the form (3.174) [19]

$$
\rho\_{lma\beta} \left( \partial^2 \mathfrak{u}\_a / \partial \mathfrak{x}\_k \partial \mathfrak{x}\_\beta \right) = \rho \delta\_{ma} \left( \partial^2 \mathfrak{u}\_a / \partial \mathfrak{x}^2 \right) \tag{93}
$$

where only eight independent constants (92) must be taken into account.

The corresponding equations of linear theory of elasticity for the case of transversally isotropic medium without of initial stresses are written above as Eqs. (45) and (46).

The general solutions for the case of axial symmetry are expressed through one potential in the form (4.13) [19]

$$u\_{r'} = -\left(\partial^2 / \partial r' \partial \eta\_3\right) \mathbf{X}',\tag{94}$$

$$\begin{split} u\_3 &= \left(\boldsymbol{\alpha}\_{1111}' + \boldsymbol{\alpha}\_{1313}'\right)^{-1} \left[\boldsymbol{\alpha}\_{1111}' \Delta\_1' + \boldsymbol{\alpha}\_{3113}' \left(\boldsymbol{\partial}^2 / \partial \eta\_3^2\right) - \boldsymbol{\rho}' \left(\boldsymbol{\sigma}^2 / \partial \mathbf{r}^2\right)\right] \mathbf{X}',\\ \Delta' &= \left(\boldsymbol{\sigma}^2 / \partial r'^2\right) + (\mathbf{1}/r') (\boldsymbol{\partial} / \partial r').\end{split}$$

Note that in Section 3 of this chapter, two potentials Φ, Ψ are introduced by formula (57), which corresponds and generalizes the procedure used in Biot's analysis [1].

The longitudinal harmonic wave is described analytically through the potential in the form (101) [19]

$$X'(r', \boldsymbol{y}\_{\boldsymbol{\beta}}, \boldsymbol{\tau}) = X'\_{(1)}(r')e^{i\left(k\boldsymbol{y}\_{\boldsymbol{\beta}} - a\boldsymbol{\tau}\right)},\tag{95}$$

where the unknown amplitude X<sup>0</sup> ð Þ<sup>1</sup> <sup>r</sup><sup>0</sup> ð Þ has to be determined by substitution of solution (4.13) [19] into the second Eq. (3.362) [19] (for potential X<sup>0</sup> ). This gives Eq. (4.16) [19]:

$$\begin{cases} \left( \boldsymbol{\alpha}\_{1111}' \boldsymbol{\alpha}\_{1331}' \right) \left( \boldsymbol{\Delta}\_1' - \boldsymbol{k}^2 \boldsymbol{\xi}\_2' \mathbf{2} \right) \left( \boldsymbol{\Delta}\_1' - \boldsymbol{k}^2 \boldsymbol{\xi}\_3' \mathbf{2} \right) \\ - \boldsymbol{k}^2 \rho' \mathbf{C}\_{cp}^2 \left[ \left( \boldsymbol{\alpha}\_{111}' + \boldsymbol{\alpha}\_{1331}' \right) \boldsymbol{\Delta}\_1' - \boldsymbol{k}^2 \left( \boldsymbol{\alpha}\_{111}' + \boldsymbol{\alpha}\_{313}' \right) \right] + \rho'^2 \mathbf{C}\_{cp}^2 \right] \mathbf{X}\_{(1)}' = \mathbf{0}, \end{cases} \tag{96}$$

$$\begin{split} \mathbf{C}\_{cp} &= \boldsymbol{\alpha}/\boldsymbol{k}, \boldsymbol{\xi}\_{2,3}^{\prime 2} = \boldsymbol{c}^{\prime} \pm \sqrt{\boldsymbol{c}^{\prime 2} - \left(\boldsymbol{o}\_{3333}^{\prime} \boldsymbol{o}\_{313}^{\prime}/\boldsymbol{o}\_{1111}^{\prime} \boldsymbol{o}\_{1331}^{\prime}\right)}, \\ \boldsymbol{c}^{\prime} &= (\mathbf{1}/\boldsymbol{2}) \left[ \begin{pmatrix} \left(\boldsymbol{o}\_{3333}^{\prime}/\boldsymbol{o}\_{1331}^{\prime}\right) + \left(\boldsymbol{o}\_{313}^{\prime}/\boldsymbol{o}\_{1111}^{\prime}\right) \\ - \left(\left(\boldsymbol{o}\_{1111}^{\prime} + \boldsymbol{o}\_{1331}^{\prime}\right)^{2}/\boldsymbol{o}\_{1111}^{\prime}\boldsymbol{o}\_{1331}^{\prime}\right) \end{pmatrix} \right], \end{split}$$

which further is written in the form

$$
\left(\Delta\_1' - \zeta\_2'^2\right)\left(\Delta\_1' - \zeta\_3'^2\right) = \mathbf{0} \tag{97}
$$

first fragment shows the analytical part of pioneer work of Biot. It represents the classicism of mathematical procedures and physical comments of Biot. Properly speaking, the clear and understandable Rayleigh's scheme is saved, but it is complemented by some findings reflecting the features of cylindrical waves. Two next fragments show the more late development of the Biot's problem. They are different by influence of the Biot's procedure. The approach shown in Section 3 is more close to the Biot's analytical scheme, whereas Section 4 proposes as an independent scheme that is more close to the Rayleigh scheme. Nevertheless, all fragments testify the mathematical complexity in solving the problem on the cylindrical surface waves. Thus, revisiting the old Biot's problem shows that it still generates

Cylindrical Surface Wave: Revisiting the Classical Biot's Problem

new scientific and practical problems.

DOI: http://dx.doi.org/10.5772/intechopen.86910

Author details

39

Jeremiah Rushchitsky

S.P. Timoshenko Institute of Mechanics, Kyiv, Ukraine

\*Address all correspondence to: rushch@inmech.kiev.ua

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The unknown quantities ζ<sup>0</sup> <sup>2</sup>, <sup>3</sup> must be found from the linear algebraic equation of the fourth degree (4.20) [19].

$$\begin{aligned} & \left( \boldsymbol{w}\_{1111}' \boldsymbol{w}\_{131}' (\boldsymbol{\zeta}')^4 + k^4 \left( \boldsymbol{\rho}' \mathbf{C}\_{cp}^2 - \boldsymbol{w}\_{333}' \right) \left( \boldsymbol{\rho}' \mathbf{C}\_{cp}^2 - \boldsymbol{w}\_{3113}' \right) \\ &+ k^2 \left[ \begin{aligned} & \boldsymbol{w}\_{1111}' \left( \boldsymbol{\rho}' \mathbf{C}\_{cp}^2 - \boldsymbol{w}\_{333}' \right) + \boldsymbol{w}\_{131}' \left( \boldsymbol{\rho}' \mathbf{C}\_{cp}^2 - \boldsymbol{w}\_{3113}' \right) \\ &+ \left( \boldsymbol{w}\_{1111}' + \boldsymbol{w}\_{3113}' \right)^2 \end{aligned} \right] (\boldsymbol{\zeta}')^2 = \mathbf{0}, \end{aligned} \tag{98}$$

The solution (95) describes the surface wave, if amplitude X<sup>0</sup> ð Þ<sup>1</sup> <sup>r</sup><sup>0</sup> ð Þ attenuates with increasing the radius. This is provided by the condition that quantities ζ<sup>0</sup> <sup>2</sup>,<sup>3</sup> is unequal and pure imaginary. Then the potential gains the form (4.22) [19].

$$X\_{(1)}'(r') = B\_{10} J\_0\left( |\zeta\_2'|r'\right) + B\_{20} K\_0\left( |\zeta\_2'|r'\right) + B\_{30} J\_0\left( |\zeta\_3'|r'\right) + B\_{40} K\_0\left( |\zeta\_3'|r'\right), \tag{99}$$

The shown part of analysis from introducing the potential by formula (94) to representation of solution by formula (99) inclusive can be compared with analogous part of analysis from Section 3 of this chapter (from introducing the potentials by formula (57) to the solution in the form of (85)). It is easy to see a difference in representations (99) and (85): formula (99) uses the Bessel functions and in particular the Macdonald function of zero index, whereas formula (85) uses (like the Biot's solution (14)) the Macdonald functions (16) of the zero and first indexes.

The next part of analysis of cylindrical wave consists in substitution of solution into boundary conditions of the form (99) [19]

$$Q'\_{r'r'} = \mathbf{0}, \ Q'\_{r'3} = \mathbf{0} \text{ when } r' = R'\_1, R'\_2. \tag{100}$$

The case of oscillatory behavior of wave in the direction of radius is considered with pointing that the case of surface wave is the same type. A substitution of solution (99) into conditions (4.79) [19] gives the dependence of velocity of surface wave or its wave number on frequency—a dispersion equation in the form of determinant of the fourth order in the form (4.26) [19].

$$\det|a\_{\vec{\eta}}| \equiv \Delta(o,k) = 0; \quad i,j = 1,2,3,4. \tag{101}$$

This finishes the analytical part of analysis shown in [19]. It corresponds to the part of Section 3.5 of this chapter, where the explicit form of dispersive equations is proposed in the form (90) that includes the Macdonald functions of the zero and first orders which represent some generalization of dispersion Eq. (27) obtained by Biot.

#### 5. Conclusions

This chapter proposes three fragments of analytical analysis of the cylindrical surface wave propagating in the vertical direction of circular cylindrical cavity. The

#### Cylindrical Surface Wave: Revisiting the Classical Biot's Problem DOI: http://dx.doi.org/10.5772/intechopen.86910

Ccp ¼ ω=k, ξ

<sup>0</sup> ¼ ð Þ 1=2

which further is written in the form

<sup>1331</sup> <sup>ζ</sup><sup>0</sup> ð Þ<sup>4</sup> <sup>þ</sup> <sup>k</sup><sup>4</sup>

<sup>1111</sup> þ ω<sup>0</sup>

2 � � � �r

into boundary conditions of the form (99) [19]

Q<sup>0</sup> r0 <sup>r</sup><sup>0</sup> ¼ 0, Q<sup>0</sup>

determinant of the fourth order in the form (4.26) [19].

det αij � � � r0

with pointing that the case of surface wave is the same type. A substitution of solution (99) into conditions (4.79) [19] gives the dependence of velocity of surface wave or its wave number on frequency—a dispersion equation in the form of

<sup>1111</sup> ρ<sup>0</sup> C2 cp � ω<sup>0</sup> 3333

þ ω<sup>0</sup>

c

Seismic Waves - Probing Earth System

The unknown quantities ζ<sup>0</sup>

the fourth degree (4.20) [19].

ω0 1111ω<sup>0</sup>

X0 ð Þ<sup>1</sup> r

5. Conclusions

38

<sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>ω</sup><sup>0</sup>

2 4

<sup>0</sup> ð Þ¼ B10J<sup>0</sup> ζ<sup>0</sup>

0 2 <sup>2</sup>, <sup>3</sup> ¼ c 0 �

2 4

ω0 3333=ω<sup>0</sup> 1331 � � <sup>þ</sup> <sup>ω</sup><sup>0</sup>

� ω<sup>0</sup>

Δ0 <sup>1</sup> � ζ 0 2 2 � �

� �

The solution (95) describes the surface wave, if amplitude X<sup>0</sup>

þ ω<sup>0</sup>

with increasing the radius. This is provided by the condition that quantities ζ<sup>0</sup>

unequal and pure imaginary. Then the potential gains the form (4.22) [19].

2 � � � �r <sup>0</sup> � � <sup>þ</sup> <sup>B</sup>30J<sup>0</sup> <sup>ζ</sup><sup>0</sup>

The shown part of analysis from introducing the potential by formula (94) to representation of solution by formula (99) inclusive can be compared with analogous part of analysis from Section 3 of this chapter (from introducing the potentials by formula (57) to the solution in the form of (85)). It is easy to see a difference in representations (99) and (85): formula (99) uses the Bessel functions and in particular the Macdonald function of zero index, whereas formula (85) uses (like the Biot's solution (14)) the Macdonald functions (16) of the zero and first indexes. The next part of analysis of cylindrical wave consists in substitution of solution

<sup>3</sup> ¼ 0 when r

The case of oscillatory behavior of wave in the direction of radius is considered

This finishes the analytical part of analysis shown in [19]. It corresponds to the part of Section 3.5 of this chapter, where the explicit form of dispersive equations is proposed in the form (90) that includes the Macdonald functions of the zero and first orders which represent some generalization of dispersion Eq. (27) obtained by Biot.

This chapter proposes three fragments of analytical analysis of the cylindrical surface wave propagating in the vertical direction of circular cylindrical cavity. The

ρ0 C2 cp � ω<sup>0</sup> 3333

� �

3113 � �<sup>2</sup>

<sup>0</sup> � � <sup>þ</sup> <sup>B</sup>20K<sup>0</sup> <sup>ζ</sup><sup>0</sup>

<sup>1111</sup> þ ω<sup>0</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>q</sup> � �,

3113=ω<sup>0</sup>

<sup>2</sup>, <sup>3</sup> must be found from the linear algebraic equation of

3

<sup>0</sup> � � <sup>þ</sup> <sup>B</sup>40K<sup>0</sup> <sup>ζ</sup><sup>0</sup>

<sup>5</sup> <sup>ζ</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>¼</sup> <sup>0</sup>,

(98)

<sup>2</sup>,<sup>3</sup> is

ð Þ<sup>1</sup> <sup>r</sup><sup>0</sup> ð Þ attenuates

3 � � � �r <sup>0</sup> � �, (99)

<sup>2</sup>: (100)

1111ω<sup>0</sup> 1331

¼ 0 (97)

3 5,

3333ω<sup>0</sup>

3113=ω<sup>0</sup> 1111 � �

> =ω<sup>0</sup> 1111ω<sup>0</sup> 1331

c0<sup>2</sup> � ω<sup>0</sup>

1331 � �<sup>2</sup>

> Δ0 <sup>1</sup> � ζ 0 2 3 � �

> > ρ0 C2 cp � ω<sup>0</sup> 3113

<sup>1331</sup> ρ<sup>0</sup> C2 cp � ω<sup>0</sup> 3113

� �

� �

3 � � � �r

<sup>0</sup> ¼ R<sup>0</sup> 1, R<sup>0</sup>

� � Δð Þ¼ ω; k 0; i, j ¼ 1, 2, 3, 4: (101)

� �

first fragment shows the analytical part of pioneer work of Biot. It represents the classicism of mathematical procedures and physical comments of Biot. Properly speaking, the clear and understandable Rayleigh's scheme is saved, but it is complemented by some findings reflecting the features of cylindrical waves. Two next fragments show the more late development of the Biot's problem. They are different by influence of the Biot's procedure. The approach shown in Section 3 is more close to the Biot's analytical scheme, whereas Section 4 proposes as an independent scheme that is more close to the Rayleigh scheme. Nevertheless, all fragments testify the mathematical complexity in solving the problem on the cylindrical surface waves. Thus, revisiting the old Biot's problem shows that it still generates new scientific and practical problems.

### Author details

Jeremiah Rushchitsky S.P. Timoshenko Institute of Mechanics, Kyiv, Ukraine

\*Address all correspondence to: rushch@inmech.kiev.ua

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Biot MA. Propagation of elastic waves in a cylindrical bore containing a fluid. Journal of Applied Physics. 1952; 23(9):997-1005. DOI: 10.1063/1.1702365

[2] Achenbach JD. Wave Propagation in Elastic Solids. Amsterdam: North-Holland; 1973. 425 p

[3] FWJ O, Lozier DW, Bousvert RF, Clark CW, editors. NIST (National Institute of Standards and Technology). Handbook of Mathematical Functions. Cambridge: Cambridge University Press; 2010. 968 p

[4] Rushchitsky JJ. Nonlinear Elastic Waves in Materials. Series: Foundations of Engineering Mechanics. Heidelberg: Springer; 2014. 455 p. DOI: 10.1007/ 978-3-319-00464-8

[5] Sedov LI. A Course in Continuum Mechanics. Vol. I-IV. Amsterdam: Wolters Noordhoff Publishing; 1971

[6] Viktorov IA. Rayleigh and Lamb Waves. NY: Plenum Press; 1967. 168 p

[7] Nowacki W. Theoria sprazystosci. Warszawa: PWN; 1970. 769 p (in Polish)

[8] Rushchitsky JJ, Tsurpal SI. Waves in Materials with Microstructure. Kiev: S. P. Timoshenko Institute of Mechanics; 1998. 377 p (in Ukrainian)

[9] Fedorov FI. Theory of Elastic Waves in Crystals. NY: Academic Press; 1975. 388 p

[10] Dieulesaint E, Royer D. Ondes elastiques dans les solides. Application au traitement du signal. Paris: Masson et Cie; 1974. 424 p (in French)

[11] Royer D, Dieulesaint E. Elastic Waves in Solids. Vols. I and II. Advanced Texts in Physics. Berlin: Springer; 2000

[12] Lekhnitsky SG. Theory of Elasticity of Anisotropic Elastic Body. San Francisco: Golden Day Inc; 1963. 404 p

Chapter 3

Media

Abstract

1. Introduction

41

Boris Sibiryakov

Appearance of Catastrophes and

Plasticity in Porous and Cracked

This chapter is devoted to study the properties of structured continuum, with specific surface and characteristic size of structure. This linear dimension means the absence of automatic transforming difference relations into differential equations. It is impossible to apply conservation laws at any point of the real structural body, because any closed points in vicinity of inner surface can represent both solid and liquid (gas) phases. We need use some representative minimal volume, which characterized the complicate body at hole. This approach leads to differential equations of motion of the infinite order. Solutions of them, along usual P and S waves, contain many waves with abnormally low velocities, which are not bounded below. It is shown that in such media, weak perturbations can increase or decrease without limit. The reason of the infinite order of differential equations is many degrees of freedom in such media. Catastrophes correspond to unstable solutions equations of motion. Plasticity begins in elastic state like continuous phenomenon, and there is a finite distance between the sliding lines on the contrary with classic

plasticity, where distances between sliding lines are infinitely small.

Keywords: structure of pore space, porous and cracked media, instability, plasticity

The main idea of continuous mechanics is that any volume is the representative one. It means that the integral of loadings, which concentrates on the surface and bounds mentioned volume, is equal to zero in statics or to inertial forces in dynamics. The evident disagreement that the surface forces and inertial ones apply to different points (inertial forces apply to center of gravity of volume) overcomes due to an assumption about infinite small sizes of the mentioned volume. This assumption gives us a possibility to equal the volume forces (divergence of the stress tensor), which was created by the internal stresses, and the inertial forces,

according to the second Newton law. Mathematical technique is based on the Gauss theorem about relation between the field flux across surface and divergence of this field in the volume, which is bounded by closed surface. However, in the structured bodies, there is a fundamentally different situation. The representative volume must contain some set of elementary structures. Otherwise, a small volume will contain only one of the phases, for example, liquid in the pores or the solid skeleton without liquid, and will not characterize the properties of the structured body. The

[13] Kiselev AP. Rayleigh wave with a transverse structure. Proceedings of the Royal Society of London. Series A. 2004;460(2050):3059-3064. DOI: 10.1098/rspa.2004.1353

[14] Guz AN, Rushchitsky JJ. Short Introduction to Mechanics of Nanocomposites. Rosemead, CA: Scientific & Academic Publishing; 2013. 280 p

[15] Cheng M, Chen W. Weerasoorlya Mechanical properties of Kevlar ® KM2. Journal of Engineering Materials and Technology. 2005;127(2):197-203. DOI: 10.1115/1.1857937

[16] Elliot HA. Three-dimensional stress distribution in hexagonal aelotropic crystals. Mathematical Proceedings of the Cambridge Philosophical Society. 1948;44(4):522-533. DOI: 10.1017/ 50305004100024531

[17] Rushchitsky JJ. Theory of Waves in Materials. Copenhagen: Ventus Publishing ApS; 2011. 270 p. DOI: 10.13140/RG.2.1.3162.8647

[18] Guz AN, Kushnir VP, Makhort FG. On propagation of waves in cylinders with initial stresses. Izvestiya, Academy of Sciences, USSR. Seriya Mekhanika Tverdogo Tiela. 1974;5:67-74 (In Russian)

[19] Guz AN. Elastic Waves in Bodies with Initial Stresses. 2 Vols. Naukova Dumka: Kiev; 1986. 376 and 536 p (in Russian)

#### Chapter 3

References

Holland; 1973. 425 p

Press; 2010. 968 p

978-3-319-00464-8

Polish)

388 p

[1] Biot MA. Propagation of elastic waves in a cylindrical bore containing a fluid. Journal of Applied Physics. 1952; 23(9):997-1005. DOI: 10.1063/1.1702365

Seismic Waves - Probing Earth System

[12] Lekhnitsky SG. Theory of Elasticity

[13] Kiselev AP. Rayleigh wave with a transverse structure. Proceedings of the Royal Society of London. Series A. 2004;460(2050):3059-3064. DOI:

[14] Guz AN, Rushchitsky JJ. Short Introduction to Mechanics of Nanocomposites. Rosemead, CA: Scientific & Academic Publishing; 2013.

[15] Cheng M, Chen W. Weerasoorlya Mechanical properties of Kevlar ® KM2. Journal of Engineering Materials and Technology. 2005;127(2):197-203. DOI:

[16] Elliot HA. Three-dimensional stress distribution in hexagonal aelotropic crystals. Mathematical Proceedings of the Cambridge Philosophical Society. 1948;44(4):522-533. DOI: 10.1017/

[17] Rushchitsky JJ. Theory of Waves in

[18] Guz AN, Kushnir VP, Makhort FG. On propagation of waves in cylinders with initial stresses. Izvestiya, Academy of Sciences, USSR. Seriya Mekhanika Tverdogo Tiela. 1974;5:67-74 (In

[19] Guz AN. Elastic Waves in Bodies with Initial Stresses. 2 Vols. Naukova Dumka: Kiev; 1986. 376 and 536 p (in

Materials. Copenhagen: Ventus Publishing ApS; 2011. 270 p. DOI: 10.13140/RG.2.1.3162.8647

of Anisotropic Elastic Body. San Francisco: Golden Day Inc; 1963. 404 p

10.1098/rspa.2004.1353

280 p

10.1115/1.1857937

50305004100024531

Russian)

Russian)

[2] Achenbach JD. Wave Propagation in Elastic Solids. Amsterdam: North-

[3] FWJ O, Lozier DW, Bousvert RF, Clark CW, editors. NIST (National Institute of Standards and Technology). Handbook of Mathematical Functions. Cambridge: Cambridge University

[4] Rushchitsky JJ. Nonlinear Elastic Waves in Materials. Series: Foundations of Engineering Mechanics. Heidelberg: Springer; 2014. 455 p. DOI: 10.1007/

[5] Sedov LI. A Course in Continuum Mechanics. Vol. I-IV. Amsterdam: Wolters Noordhoff Publishing; 1971

[6] Viktorov IA. Rayleigh and Lamb Waves. NY: Plenum Press; 1967. 168 p

[7] Nowacki W. Theoria sprazystosci. Warszawa: PWN; 1970. 769 p (in

[8] Rushchitsky JJ, Tsurpal SI. Waves in Materials with Microstructure. Kiev: S. P. Timoshenko Institute of Mechanics;

[9] Fedorov FI. Theory of Elastic Waves in Crystals. NY: Academic Press; 1975.

[10] Dieulesaint E, Royer D. Ondes elastiques dans les solides. Application au traitement du signal. Paris: Masson et

[11] Royer D, Dieulesaint E. Elastic Waves in Solids. Vols. I and II. Advanced Texts in Physics. Berlin:

Cie; 1974. 424 p (in French)

Springer; 2000

40

1998. 377 p (in Ukrainian)

## Appearance of Catastrophes and Plasticity in Porous and Cracked Media

Boris Sibiryakov

#### Abstract

This chapter is devoted to study the properties of structured continuum, with specific surface and characteristic size of structure. This linear dimension means the absence of automatic transforming difference relations into differential equations. It is impossible to apply conservation laws at any point of the real structural body, because any closed points in vicinity of inner surface can represent both solid and liquid (gas) phases. We need use some representative minimal volume, which characterized the complicate body at hole. This approach leads to differential equations of motion of the infinite order. Solutions of them, along usual P and S waves, contain many waves with abnormally low velocities, which are not bounded below. It is shown that in such media, weak perturbations can increase or decrease without limit. The reason of the infinite order of differential equations is many degrees of freedom in such media. Catastrophes correspond to unstable solutions equations of motion. Plasticity begins in elastic state like continuous phenomenon, and there is a finite distance between the sliding lines on the contrary with classic plasticity, where distances between sliding lines are infinitely small.

Keywords: structure of pore space, porous and cracked media, instability, plasticity

#### 1. Introduction

The main idea of continuous mechanics is that any volume is the representative one. It means that the integral of loadings, which concentrates on the surface and bounds mentioned volume, is equal to zero in statics or to inertial forces in dynamics. The evident disagreement that the surface forces and inertial ones apply to different points (inertial forces apply to center of gravity of volume) overcomes due to an assumption about infinite small sizes of the mentioned volume. This assumption gives us a possibility to equal the volume forces (divergence of the stress tensor), which was created by the internal stresses, and the inertial forces, according to the second Newton law. Mathematical technique is based on the Gauss theorem about relation between the field flux across surface and divergence of this field in the volume, which is bounded by closed surface. However, in the structured bodies, there is a fundamentally different situation. The representative volume must contain some set of elementary structures. Otherwise, a small volume will contain only one of the phases, for example, liquid in the pores or the solid skeleton without liquid, and will not characterize the properties of the structured body. The

characteristic size of the structure leads to fact that the average distance is between one of the cracks to another and one pore to another given by the specific surface of the sample. It is necessary to connect the integral geometric properties of a medium with physical processes of such bodies deforming. On the contrary with a classic continuum of Cauchy and Poisson, the new continuum for structured or blocked media must contain many degrees of freedom. It is evident because elementary blocks may translate the motion by contact interactions, by rotations, and by group of particle's motion. It means that the energy contents not in first derivatives (strains) only. The potential energy contents in the second derivatives (curvatures) and other orders of ones. It means that the equation of motion of a blocked medium should contain many derivatives; in other words, the equation of motion may have been very high, probably, the infinite order. The static and dynamic processes in the classic continuum are divided by the Great Wall of China from each other. The equation of equilibrium never will pass in the equation of motion. However, it is evident that the dynamic processes often arise very slow and are quasi-static motions. It would be nice to destroy this mentioned wall by a newly structured continuum. It would be a good idea to destroy the abovementioned wall by means of justification of the newly structured continuum. The seismic emission, which causes due to static loading, maybe not a bad example of such phenomena, which are existed between statics and dynamics.

#### 2. Equations of motion for structured media

In Figure 1, an element of the volume of structured body is shown, in which l<sup>0</sup> is the average distance between one pore and another. Earlier presented was the result about the relation between the specific surface and the average length between cracks and pores. There is a theorem of integral geometry, which relates the specific surface σ<sup>0</sup> and l0, namely [1]

$$
\sigma\_0 l\_0 = 4(1 - f) \tag{1}
$$

side of grain, we have forces; and on the other part of boundary surface D, we have

is no way for the volume element to tend to zero and to match the points of application of surface forces and inertial forces, as in the classical continuum. Therefore, since we must consider the representative finite volume, we have a

operator, and after this, it is possible to apply the law of conservation for some structural image continuum and to act as in a typical classical model of space. The main feature of this approach is to fill all the space, including the pores and cracks by field force. Because of it, we have a continuous image of a very complex media

and a possibility to apply the physical laws into an image of the media.

u x þ

� � � �

difference may be represented as quadrate of the first difference,

l0 2 � �

� exp � <sup>l</sup><sup>0</sup>

This is a first difference for finite distance between two points. The second

<sup>Δ</sup><sup>2</sup> <sup>¼</sup> u xð Þsinh <sup>l</sup><sup>0</sup>

� � � �

2 Dx

problem of different positions of surface and inertial forces.

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

DOI: http://dx.doi.org/10.5772/intechopen.87014

<sup>Δ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> l0

l0 2 Dx � �

<sup>6</sup> cosh <sup>l</sup><sup>0</sup>

4π

Because there is a Poisson formula [2]

2 ðπ ð π

0

0

2 Dx � �

is given by the symbolic formula [1]

The operator is Dx <sup>¼</sup> <sup>∂</sup>

<sup>¼</sup> u xð Þ l0

Pu x ½ �¼ ð Þ u xð Þ

follows:

43

P l0Dx; l0Dy; l0Dz � � <sup>¼</sup> <sup>1</sup>

exp

two translation operators

The idea of creation of the new model of space is as follows: consider some finite volume of the body (a sphere on a figure with radius l0). Surface forces act on a sphere of radius l0, while inertial forces applied at the center of the structure. There

We need to translate the surface forces to the center of the structure by a special

The one-dimensional operator of field translation from point x into point x � l<sup>0</sup>

u xð Þ¼ � l<sup>0</sup> exp ð Þ l0Dx (2)

<sup>∂</sup>x. The difference operator Δ1ð Þ x is a difference between

2

<sup>¼</sup> u xð Þsinh <sup>l</sup><sup>0</sup>

� �<sup>2</sup> (4)

<sup>þ</sup> cosh <sup>l</sup><sup>0</sup>

2 Dz

(6)

<sup>2</sup> Dx � � l0 2

� � (3)

� u x � <sup>l</sup><sup>0</sup>

<sup>2</sup> Dx � �<sup>2</sup>

l0 2

<sup>þ</sup> cosh <sup>l</sup><sup>0</sup>

The analogous operator of translation for some spheres is given by expression

2 Dy � �

� � � � (5)

exp <sup>l</sup><sup>0</sup> Dxsinθcos<sup>φ</sup> <sup>þ</sup> Dysinθsin<sup>φ</sup> <sup>þ</sup> Dzcos<sup>θ</sup> � � � � sinθdθd<sup>φ</sup>

The formally expansion in Taylor's series gives a finite increment of field. This expansion contains the infinite number of derivatives with different powers of l0. The factor l<sup>0</sup> relates with the specific surface of the sample. The three-dimensional operator of field's translation for some cube with length of l<sup>0</sup> may be constructed as

no forces).

where f is the porosity. Hence, if there is a specific surface of sample, there is automatically the average range of microstructure l0.

The distinction between classic and structured continuums is clear, see Figure 1. In the volume, which is inside into surface C, there is equation of equilibrium, because all forces delete to each other. In the volume, which is inside into surface D, there is equilibrium, because forces do not compensate to each other (on the one

#### Figure 1.

Representative element of structured body for granular medium (left) and average distance l0 from one crack to another (right). On surface C, the equation of equilibrium is complied, and on surface D, it is not satisfied.

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

side of grain, we have forces; and on the other part of boundary surface D, we have no forces).

The idea of creation of the new model of space is as follows: consider some finite volume of the body (a sphere on a figure with radius l0). Surface forces act on a sphere of radius l0, while inertial forces applied at the center of the structure. There is no way for the volume element to tend to zero and to match the points of application of surface forces and inertial forces, as in the classical continuum. Therefore, since we must consider the representative finite volume, we have a problem of different positions of surface and inertial forces.

We need to translate the surface forces to the center of the structure by a special operator, and after this, it is possible to apply the law of conservation for some structural image continuum and to act as in a typical classical model of space. The main feature of this approach is to fill all the space, including the pores and cracks by field force. Because of it, we have a continuous image of a very complex media and a possibility to apply the physical laws into an image of the media.

The one-dimensional operator of field translation from point x into point x � l<sup>0</sup> is given by the symbolic formula [1]

$$u(\mathbf{x} \pm l\_0) = \exp\left(l\_0 D\_\mathbf{x}\right) \tag{2}$$

The operator is Dx <sup>¼</sup> <sup>∂</sup> <sup>∂</sup>x. The difference operator Δ1ð Þ x is a difference between two translation operators

$$\Delta\_1 = \frac{1}{l\_0} \left[ u \left( \varkappa + \frac{l\_0}{2} \right) - u \left( \varkappa - \frac{l\_0}{2} \right) \right]$$

$$= \frac{u(\varkappa)}{l\_0} \left[ \exp\left(\frac{l\_0}{2} D\_\varkappa \right) - \exp\left(-\frac{l\_0}{2} D\_\varkappa \right) \right] = u(\varkappa) \frac{\sinh\left(\frac{l\_0}{2} D\_\varkappa \right)}{\left(\frac{l\_0}{2}\right)}\tag{3}$$

This is a first difference for finite distance between two points. The second difference may be represented as quadrate of the first difference,

$$\Delta\_2 = \mathfrak{u}(\mathfrak{x}) \frac{\sinh\left(\frac{l\_0}{2} D\_\mathbf{x}\right)^2}{\left(\frac{l\_0}{2}\right)^2} \tag{4}$$

The formally expansion in Taylor's series gives a finite increment of field. This expansion contains the infinite number of derivatives with different powers of l0. The factor l<sup>0</sup> relates with the specific surface of the sample. The three-dimensional operator of field's translation for some cube with length of l<sup>0</sup> may be constructed as follows:

$$P[\boldsymbol{\mu}(\mathbf{x})] = \frac{\boldsymbol{\mu}(\mathbf{x})}{6} \left[ \cosh\left(\frac{l\_0}{2} D\_\mathbf{x}\right) + \cosh\left(\frac{l\_0}{2} D\_\mathbf{y}\right) + \cosh\left(\frac{l\_0}{2} D\_\mathbf{z}\right) \right] \tag{5}$$

The analogous operator of translation for some spheres is given by expression

$$P\left(l\_0 D\_{\mathbf{x}}; l\_0 D\_{\mathbf{y}}; l\_0 D\_{\mathbf{z}}\right) = \frac{1}{4\pi} \int\_0^{2\pi} \left[ \int\_0^{\pi} \exp\left[l\_0 \left(D\_{\mathbf{x}} \sin\theta \cos\varphi + D\_{\mathbf{y}} \sin\theta \sin\varphi + D\_{\mathbf{z}} \cos\theta\right)\right] \sin\theta d\theta d\varphi \right] \tag{6}$$

Because there is a Poisson formula [2]

characteristic size of the structure leads to fact that the average distance is between one of the cracks to another and one pore to another given by the specific surface of the sample. It is necessary to connect the integral geometric properties of a medium with physical processes of such bodies deforming. On the contrary with a classic continuum of Cauchy and Poisson, the new continuum for structured or blocked media must contain many degrees of freedom. It is evident because elementary blocks may translate the motion by contact interactions, by rotations, and by group of particle's motion. It means that the energy contents not in first derivatives (strains) only. The potential energy contents in the second derivatives (curvatures) and other orders of ones. It means that the equation of motion of a blocked medium should contain many derivatives; in other words, the equation of motion may have been very high, probably, the infinite order. The static and dynamic processes in the classic continuum are divided by the Great Wall of China from each other. The equation of equilibrium never will pass in the equation of motion. However, it is evident that the dynamic processes often arise very slow and are quasi-static motions. It would be nice to destroy this mentioned wall by a newly structured continuum. It would be a good idea to destroy the abovementioned wall by means of justification of the newly structured continuum. The seismic emission, which causes due to static loading, maybe not a bad example of such phenomena, which

In Figure 1, an element of the volume of structured body is shown, in which l<sup>0</sup> is the average distance between one pore and another. Earlier presented was the result about the relation between the specific surface and the average length between cracks and pores. There is a theorem of integral geometry, which relates the specific

where f is the porosity. Hence, if there is a specific surface of sample, there is

Representative element of structured body for granular medium (left) and average distance l0 from one crack to another (right). On surface C, the equation of equilibrium is complied, and on surface D, it is not satisfied.

In the volume, which is inside into surface C, there is equation of equilibrium, because all forces delete to each other. In the volume, which is inside into surface D, there is equilibrium, because forces do not compensate to each other (on the one

The distinction between classic and structured continuums is clear, see Figure 1.

σ0l<sup>0</sup> ¼ 4 1ð Þ � f (1)

are existed between statics and dynamics.

Seismic Waves - Probing Earth System

surface σ<sup>0</sup> and l0, namely [1]

Figure 1.

42

2. Equations of motion for structured media

automatically the average range of microstructure l0.

Seismic Waves - Probing Earth System

$$\int\_{0}^{2\pi} \int\_{0}^{\pi} [f(\alpha \sin \theta \cos \varphi + \beta \sin \theta \sin \varphi + \gamma \cos \theta)] \sin \theta d\theta d\varphi = 2\pi \int\_{0}^{\pi} f(R \cos p) \sin p dp = 2\pi \int\_{-1}^{1} f(Rt) dt. \tag{7}$$

Puið Þ¼ x; y; z

the system Eq. (9):

Gij <sup>¼</sup> <sup>1</sup>

in the simple poles, <sup>n</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

solids as

variables

Figure 2.

45

number ratio ks(ω)/ks(0) of P waves.

At very small l0, the ratio <sup>l</sup>0<sup>n</sup>

we have the transcendental equations

tan x<sup>∗</sup>

<sup>x</sup><sup>∗</sup> ¼ � tanh<sup>y</sup> <sup>∗</sup>

1 ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup> <sup>∭</sup> ∞

DOI: http://dx.doi.org/10.5772/intechopen.87014

�∞

<sup>μ</sup>n<sup>2</sup> � ρω<sup>2</sup> <sup>l</sup>0<sup>n</sup>

sin ð Þ l0n

Sl0n, where <sup>k</sup><sup>2</sup>

sinh ð Þ l0n

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

<sup>l</sup>0<sup>n</sup> exp i nxx <sup>þ</sup> nyy <sup>þ</sup> nzz � � � � Ui nx; ny; nz

This allows us to calculate the Fourier transform for the fundamental solution of

<sup>δ</sup>ij � ð Þ <sup>λ</sup> <sup>þ</sup> <sup>μ</sup> ninj

At very small values, l0n, the sine and argument ratio approaches unity, and the Fourier transform becomes an ordinary equation for Green's tensor in an elastic continuum. The inverse Fourier transform is obtained by integration of Eq. (13) which includes simple poles corresponding to P and S waves and a set of simple poles where the sine in the denominator of Eq. (13) becomes zero. The residuals are

classical equations that define the poles corresponding to compression and shear waves velocities (Figure 2). Assuming n/ks = m and ksl<sup>0</sup> = ε, we obtain the equation for complex roots that describe waves from a focused source in porous and cracked

If m = x + iy is assumed to be a complex value, for the real and imaginary parts,

xsinεxcoshε<sup>y</sup> � ysinhεycosε<sup>x</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup>

We can rewrite Eq. (15) in a different form with x\* = εx and y\* = εy as new

Wave number ratio as a function of dimensionless ratio ε = 2πl0/λs. Curves: 1—wave number ratio ks(ω)/ ks(0), i.e., S-wave velocity decreasing with frequency; 2—γ = Vs/Vp increasing with frequency; and 3—wave

<sup>x</sup><sup>∗</sup> <sup>þ</sup> sinh <sup>2</sup>

<sup>y</sup> <sup>∗</sup> ; sin <sup>2</sup>

ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>n</sup><sup>2</sup> � ρω<sup>2</sup> <sup>l</sup>0<sup>n</sup>

" # <sup>l</sup>0<sup>n</sup>

sin ð Þ l0n

sin ð Þ <sup>l</sup>0<sup>n</sup> ! 1 and denominators in Eq. (13) become the

ysinεxcoshεy þ xsinhεycosεx ¼ 0 (15)

sin ð Þ l0n

<sup>S</sup> is the wave number of both P and S waves.

msinð Þ¼ εm ε (14)

<sup>y</sup> <sup>∗</sup> <sup>¼</sup> <sup>ε</sup><sup>4</sup>

<sup>x</sup><sup>∗</sup> <sup>2</sup> <sup>þ</sup> <sup>y</sup> <sup>∗</sup> <sup>2</sup> (16)

� �dn (12)

(13)

In the formula (7), parameters α, β, and γ are some quantities. However, in Eqs. (6) and (7), parameters play the role of differential operators. The relation between quantities and operators is established by Maslov [3]. Hence, P operator maybe rewritten as follows [4]

$$\begin{split} P(l\_0 D\_x; l\_0 D\_y; l\_0 D\_x) &= \frac{1}{2} \int\_{-1}^{1} \exp\left(l\_0 \sqrt{\Delta}, t\right) dt = \int\_{0}^{1} \cosh\left(l\_0 \sqrt{\Delta}, t\right) dt\\ &= \frac{\sinh\left(l\_0 \sqrt{\Delta}\right)}{l\_0 \sqrt{\Delta}} = E + \frac{l\_0^2}{3!} \Delta + \frac{l\_0^4}{5!} \Delta \Delta + \dots \end{split} \tag{8}$$

In the classic continuum, we apply the impulse conservation law to any element of the medium. In this situation, we need to fill all pores over space by a force field. Instead of real stresses, which are changing very fast from one point to another, we can construct the continual image of real stresses. Namely, we use a continuous field, which is constructed by the application of the operator P to the real complicated force field. For this continuous image of real stress, Pð Þ σik , we can apply the impulse conservation law. In the classic continuous model, this operation is made by nature itself. This model of a continuum requires some mathematical operations in order to create the continuum medium. Using operator P, we can write the equation of motion of micro-inhomogeneous body, because for an average stresses in structure, the law of impulse conservation takes the usual form, namely [4]

$$\frac{\partial}{\partial \mathbf{x}\_k} [P(\sigma\_{ik})] = \rho \ddot{u}\_i \tag{9}$$

In a more detailed form Eq. (9) can be rewritten as follows

$$\frac{\partial}{\partial \mathbf{x}\_k} \left[ \left( E + \frac{l\_0^2}{\mathbf{3!}!} \Delta + \frac{l\_0^4}{\mathbf{5!}!} \Delta \Delta + \dots \right) \sigma\_{ik} \right] = \rho \ddot{u}\_i \tag{10}$$

No wonder that Eq. (9) contains derivatives of the infinite order. This circumstance is due to many degrees of freedom for structured bodies. At l<sup>0</sup> ! 0, we have the usual equations of motion for classic continuous model of space.

#### 3. Fundamental solutions

We can pass to the image space, following Hooke's law and applying the Fourier transform along three coordinates, as [5]

$$u\_i(\mathbf{x}, y, \mathbf{z}) = \frac{1}{\left(2\pi\right)^3} \iint\limits\_{-\infty}^{\infty} \exp\left[i\left(n\_\mathbf{x}\mathbf{x} + n\_\mathbf{y}\mathbf{y} + n\_\mathbf{z}\mathbf{z}\right)\right] U\_i\left(n\_\mathbf{x}, n\_\mathbf{y}, n\_\mathbf{z}\right) dn\tag{11}$$

$$\text{where } n^2 = n\_x^2 + n\_y^2 + n\_z^2; dn = dn\_x dn\_y dn\_z. \text{ The operator } P \text{ leads to}$$

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

$$Pu\_i(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \frac{1}{\left(2\pi\right)^3} \iint\limits\_{-\infty}^{\infty} \frac{\sinh\left(l\_0 n\right)}{l\_0 n} \exp\left[i\left(n\_\mathbf{x}\mathbf{x} + n\_\mathbf{y}\mathbf{y} + n\_\mathbf{z}\mathbf{z}\right)\right] U\_i\left(n\_\mathbf{x}, n\_\mathbf{y}, n\_\mathbf{z}\right) dn \tag{12}$$

This allows us to calculate the Fourier transform for the fundamental solution of the system Eq. (9):

$$G\_{\vec{\eta}} = \frac{1}{\mu n^2 - \rho o^2 \frac{l\_0 n}{\sin\left(l\_0 n\right)}} \left[ \delta\_{\vec{\eta}} - \frac{(\lambda + \mu) n\_i n\_j}{(\lambda + 2\mu) n^2 - \rho o^2 \frac{l\_0 n}{\sin\left(l\_0 n\right)}} \right] \frac{l\_0 n}{\sin\left(l\_0 n\right)}\tag{13}$$

At very small values, l0n, the sine and argument ratio approaches unity, and the Fourier transform becomes an ordinary equation for Green's tensor in an elastic continuum. The inverse Fourier transform is obtained by integration of Eq. (13) which includes simple poles corresponding to P and S waves and a set of simple poles where the sine in the denominator of Eq. (13) becomes zero. The residuals are in the simple poles, <sup>n</sup><sup>2</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup> Sl0n, where <sup>k</sup><sup>2</sup> <sup>S</sup> is the wave number of both P and S waves. At very small l0, the ratio <sup>l</sup>0<sup>n</sup> sin ð Þ <sup>l</sup>0<sup>n</sup> ! 1 and denominators in Eq. (13) become the classical equations that define the poles corresponding to compression and shear waves velocities (Figure 2). Assuming n/ks = m and ksl<sup>0</sup> = ε, we obtain the equation for complex roots that describe waves from a focused source in porous and cracked solids as

$$
\begin{aligned}
m \sin(\varepsilon m) &= \varepsilon \end{aligned} \tag{14}$$

If m = x + iy is assumed to be a complex value, for the real and imaginary parts, we have the transcendental equations

$$
\varepsilon\kappa\sin\alpha\varepsilon\alpha\kappa\epsilon\nu\jmath - \jmath\sin\alpha\jmath\cos\varepsilon\varkappa = \varepsilon^2
$$

$$
\varepsilon\jmath\sin\alpha\varepsilon\alpha\kappa\epsilon\nu\jmath + \varkappa\sin\alpha\jmath\cos\varepsilon\varkappa = 0\tag{15}
$$

We can rewrite Eq. (15) in a different form with x\* = εx and y\* = εy as new variables

#### Figure 2.

Wave number ratio as a function of dimensionless ratio ε = 2πl0/λs. Curves: 1—wave number ratio ks(ω)/ ks(0), i.e., S-wave velocity decreasing with frequency; 2—γ = Vs/Vp increasing with frequency; and 3—wave number ratio ks(ω)/ks(0) of P waves.

2 ðπ ð π

f½ � ð Þ αsinθcosφ þ βsinθsinφ þ γcosθ sinθdθdφ ¼ 2π

2 ð 1

�1

<sup>¼</sup> sinh <sup>l</sup><sup>0</sup>

l0 ffiffiffi Δ p ¼ E þ

∂ ∂xk

!

" #

No wonder that Eq. (9) contains derivatives of the infinite order. This circumstance is due to many degrees of freedom for structured bodies. At l<sup>0</sup> ! 0, we have

We can pass to the image space, following Hooke's law and applying the Fourier

exp i nxx <sup>þ</sup> nyy <sup>þ</sup> nzz � � � � Ui nx; ny; nz

z; dn ¼ dnxdnydnz. The operator P leads to

ΔΔ þ …

σik

In a more detailed form Eq. (9) can be rewritten as follows

the usual equations of motion for classic continuous model of space.

E þ l 2 0 3! Δ þ l 4 0 5!

∂ ∂xk

3. Fundamental solutions

uið Þ¼ x; y; z

where <sup>n</sup><sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

44

transform along three coordinates, as [5]

<sup>x</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

1 ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup> <sup>∭</sup> ∞

<sup>y</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup>

�∞

exp l<sup>0</sup>

ffiffiffi <sup>Δ</sup> � � <sup>p</sup>

In the classic continuum, we apply the impulse conservation law to any element of the medium. In this situation, we need to fill all pores over space by a force field. Instead of real stresses, which are changing very fast from one point to another, we can construct the continual image of real stresses. Namely, we use a continuous field, which is constructed by the application of the operator P to the real complicated force field. For this continuous image of real stress, Pð Þ σik , we can apply the impulse conservation law. In the classic continuous model, this operation is made by nature itself. This model of a continuum requires some mathematical operations in order to create the continuum medium. Using operator P, we can write the equation of motion of micro-inhomogeneous body, because for an average stresses in structure, the law of impulse conservation takes the usual

ð π

f Rcosp ð Þsinpdp ¼ 2π

ffiffiffi Δ <sup>p</sup> ; <sup>t</sup> � �

dt

ð 1

f Rt ð Þdt

(7)

(8)

�1

0

In the formula (7), parameters α, β, and γ are some quantities. However, in Eqs. (6) and (7), parameters play the role of differential operators. The relation between quantities and operators is established by Maslov [3]. Hence, P operator

> ffiffiffi Δ <sup>p</sup> ; <sup>t</sup> � �

dt ¼ ð 1

> l 2 0 3! Δ þ l 4 0 5!

0

cosh l<sup>0</sup>

ΔΔ þ …

½ �¼ Pð Þ σik ρu€<sup>i</sup> (9)

¼ ρu€<sup>i</sup> (10)

� �dn (11)

0

maybe rewritten as follows [4]

Seismic Waves - Probing Earth System

P l0Dx; l0Dy; l0Dz � � <sup>¼</sup> <sup>1</sup>

form, namely [4]

0

Equation (15) obviously has many real roots corresponding to у = 0. Indeed, at small ε, Eq. (15) gives the solution m = 1, which corresponds to the ordinary P- or S-wave velocity. At large values of m, Eq. (15) is satisfied only if εm approaches a value divisible by n, i.e., at near-zero sine that defines the characteristic anomalous velocity. The unbounded value of the wave number means that normal P and S waves coexist with arbitrarily small P and S velocity anomalies. The existence of these anomalies in a micro-heterogeneous medium has its physical explanation: energy is stored in strain (first derivatives of displacement) as well as in the curvature of higher derivatives. Therefore, there appear as velocities related to flexural and torsion waves and to numerous waves associated with oscillation of groups of particles (blocks) (Figure 3).

The growing of ratio γ = VS/VP causes a very interesting phenomenon, namely an apparent negative Poisson value, for waves with the length not very small compared to size of a grain. The growing value of γ = VS/VP means that the Poisson ratio is decreasing up to negative values [6] (Figure 4).

At the same time, Eqs. (14) and (15) likewise have complex roots. The first Eq. (15) shows that complex roots arise only at some values of e, which are not so small, as they satisfy the inequality εx > π/2. Table 1 lists complex roots corresponding to some relatively small ε. Note that the parameter ε can be expressed via the linear size-to-wavelength ratio (l0/λs).

Complex roots can mean either damping or unlimited growth of wave amplitude, of course, in the presence of an energy-unbounded source. The minimum damping (growth) corresponds to (2.0288)<sup>1</sup> or about a half of the normal velocity. The same process can be expected to cause both excitation and damping in porous and cracked media, depending on the phase of stationary oscillations.

4. One-dimensional case: plane wave and instabilities

u <sup>00</sup> <sup>E</sup> <sup>þ</sup> l 2 0 3! Δ þ l 4 0 5!

Figure 4.

Table 1.

47

Poisson ratios are possible.

Value y is the imaginary part of it.

size of structure l<sup>0</sup> or specific surface of sample σ0:

velocities, which are not bound below.

In one-dimensional case, the Eq. (10) takes more simple expression

The value epsilon means dimensionless product of structure size into wavenumber of usual S waves in continuous medium. Value x means the real value of product of structural wavenumber into structure size.

Gregory experimental data. Poison ratio (the vertical axis) versus pressure. Black color corresponds to water saturated porous shales and gray color corresponds to dry shales with the same porosity. In this case, negative

ε x y 0.2147 2.0288 0.0548 0.2507 2.0645 0.5838 0.2771 2.1064 0.8880 0.3253 2.1560 1.1838 0.3918 2.2157 1.5122

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

DOI: http://dx.doi.org/10.5772/intechopen.87014

!

ΔΔ þ …

This equation by substitution u ¼ exp ð Þ ikx gives us the dispersion equation for an unknown wave number k, or for unknown wave velocity, which depends on the

It is evident that by l<sup>0</sup> ! 0, the wave number k ! kS, i.e., the wave velocity is equal to VP or VS, elastic wave velocity. However, if l<sup>0</sup> is not a very small value, the wave velocity decreases up to zero by kl<sup>0</sup> ! mπ, if m is the integer number. Hence, this model describes along with usual seismic waves many waves of very small

¼ k2 S

sin ð Þ kl<sup>0</sup> kl<sup>0</sup>

<sup>þ</sup> <sup>k</sup><sup>2</sup>

Su ¼ 0 (17)

<sup>k</sup><sup>2</sup> (18)

#### Figure 3.

The decreasing P-wave velocity (the upper line) and S-wave velocity (the middle curve) and the growth of their ratio γ = VS/VP (the lower line) due to increasing size of microstructure. The ratio γ = VS/VP more than 0.705 corresponds to the negative Poisson ratio [6]. The vertical scale is the wave velocities (km/s) and a horizontal scale is the ratio between the size of the microstructure and wavelength.

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

#### Figure 4.

Equation (15) obviously has many real roots corresponding to у = 0. Indeed, at small ε, Eq. (15) gives the solution m = 1, which corresponds to the ordinary P- or S-wave velocity. At large values of m, Eq. (15) is satisfied only if εm approaches a value divisible by n, i.e., at near-zero sine that defines the characteristic anomalous velocity. The unbounded value of the wave number means that normal P and S waves coexist with arbitrarily small P and S velocity anomalies. The existence of these anomalies in a micro-heterogeneous medium has its physical explanation: energy is stored in strain (first derivatives of displacement) as well as in the curvature of higher derivatives. Therefore, there appear as velocities related to flexural and torsion waves and to numerous waves associated with oscillation of groups of

The growing of ratio γ = VS/VP causes a very interesting phenomenon, namely an apparent negative Poisson value, for waves with the length not very small compared to size of a grain. The growing value of γ = VS/VP means that the Poisson ratio is

At the same time, Eqs. (14) and (15) likewise have complex roots. The first Eq. (15) shows that complex roots arise only at some values of e, which are not so

Complex roots can mean either damping or unlimited growth of wave amplitude, of course, in the presence of an energy-unbounded source. The minimum damping (growth) corresponds to (2.0288)<sup>1</sup> or about a half of the normal velocity. The same process can be expected to cause both excitation and damping in porous

The decreasing P-wave velocity (the upper line) and S-wave velocity (the middle curve) and the growth of their ratio γ = VS/VP (the lower line) due to increasing size of microstructure. The ratio γ = VS/VP more than 0.705 corresponds to the negative Poisson ratio [6]. The vertical scale is the wave velocities (km/s) and a horizontal

scale is the ratio between the size of the microstructure and wavelength.

small, as they satisfy the inequality εx > π/2. Table 1 lists complex roots corresponding to some relatively small ε. Note that the parameter ε can be

and cracked media, depending on the phase of stationary oscillations.

particles (blocks) (Figure 3).

Seismic Waves - Probing Earth System

Figure 3.

46

decreasing up to negative values [6] (Figure 4).

expressed via the linear size-to-wavelength ratio (l0/λs).

Gregory experimental data. Poison ratio (the vertical axis) versus pressure. Black color corresponds to water saturated porous shales and gray color corresponds to dry shales with the same porosity. In this case, negative Poisson ratios are possible.


#### Table 1.

The value epsilon means dimensionless product of structure size into wavenumber of usual S waves in continuous medium. Value x means the real value of product of structural wavenumber into structure size. Value y is the imaginary part of it.

#### 4. One-dimensional case: plane wave and instabilities

In one-dimensional case, the Eq. (10) takes more simple expression

$$
\mu^{''}\left(E + \frac{l\_0^2}{3!}\Delta + \frac{l\_0^4}{5!}\Delta\Delta + \dots\right) + k\_S^2 \mu = 0\tag{17}
$$

This equation by substitution u ¼ exp ð Þ ikx gives us the dispersion equation for an unknown wave number k, or for unknown wave velocity, which depends on the size of structure l<sup>0</sup> or specific surface of sample σ0:

$$\frac{\sin\left(kl\_0\right)}{kl\_0} = \frac{k\_\text{S}^2}{k^2} \tag{18}$$

It is evident that by l<sup>0</sup> ! 0, the wave number k ! kS, i.e., the wave velocity is equal to VP or VS, elastic wave velocity. However, if l<sup>0</sup> is not a very small value, the wave velocity decreases up to zero by kl<sup>0</sup> ! mπ, if m is the integer number. Hence, this model describes along with usual seismic waves many waves of very small velocities, which are not bound below.

This effect is more for P waves than for S ones. Eq. (14) shows that if the Poisson ratio is measured on samples by velocities VP and VS, their ratio VS/VP grows by growing l0, and this effect can produce abnormally small Poisson's ratio, up to negative volume of it.

It is evident that at kl ¼ mπ, m is the integer number and the value k ! ∞. It means that there are waves with arbitrary small velocities not bounded below. Beside it, Eq. (15) has complex roots too, because sin ð Þ kl<sup>0</sup> may be negative, while the second term in Eq. (18) contains ð Þ kS=<sup>k</sup> <sup>2</sup> . Eq. (18) means that the complex roots do not by small values of x, because the right-hand expression is a negative value. In order to be complex roots, an evident condition is necessary, i.e., tanx>π=2. The physical sense of it means that the complex roots are possible, if the wavelength is four times (or more than four times) more than the size of the structure. These complex roots mean that amplitude of oscillations may be increasing or decreasing up to infinity or, may be, to zero. These roots are responsible for catastrophe's behavior of structured bodies.

Hence, if there is a source of sufficient energy, even some small oscillations can produce catastrophes. It is interesting that nonlinear deforming of samples decreases this effect, because a wave velocity for rocks is decreasing, by growing amplitude of wave. It means that the wave number is growing by the same frequency in the pure elastic process. In Figure 6, the real roots of dispersion, Eq. (18) are shown. The vertical axis shows a dimensionless frequency, namely ε, while horizontal axis shows us the real and imaginary parts of wave numbers. In Figures 5–7 [7], complex roots as a function of dimensionless frequency ε are shown. Every point is a position of some root, namely a real part, an imaginary one, and a dimensionless frequency. The more is the spreading of ε values, the greater is the number of complex roots.

#### Figure 5.

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The first value of ε corresponds to values of first row from Table 1.

5. Pointing vector and equation of equilibrium for blocked media

value of ε corresponds to values of third row from Table 1.

Eq. (9) as

49

Figure 7.

Figure 6.

The equation of equilibrium for micro-structured media can be written from

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The third

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The

second value of ε corresponds to values of second row from Table 1.

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

DOI: http://dx.doi.org/10.5772/intechopen.87014

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

#### Figure 6.

This effect is more for P waves than for S ones. Eq. (14) shows that if the Poisson ratio is measured on samples by velocities VP and VS, their ratio VS/VP grows by growing l0, and this effect can produce abnormally small Poisson's ratio, up to

It is evident that at kl ¼ mπ, m is the integer number and the value k ! ∞. It means that there are waves with arbitrary small velocities not bounded below. Beside it, Eq. (15) has complex roots too, because sin ð Þ kl<sup>0</sup> may be negative, while

do not by small values of x, because the right-hand expression is a negative value. In order to be complex roots, an evident condition is necessary, i.e., tanx>π=2. The physical sense of it means that the complex roots are possible, if the wavelength is four times (or more than four times) more than the size of the structure. These complex roots mean that amplitude of oscillations may be increasing or decreasing up to infinity or, may be, to zero. These roots are responsible for catastrophe's

Hence, if there is a source of sufficient energy, even some small oscillations can

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The first

produce catastrophes. It is interesting that nonlinear deforming of samples decreases this effect, because a wave velocity for rocks is decreasing, by growing amplitude of wave. It means that the wave number is growing by the same frequency in the pure elastic process. In Figure 6, the real roots of dispersion, Eq. (18) are shown. The vertical axis shows a dimensionless frequency, namely ε, while horizontal axis shows us the real and imaginary parts of wave numbers. In Figures 5–7 [7], complex roots as a function of dimensionless frequency ε are shown. Every point is a position of some root, namely a real part, an imaginary one, and a dimensionless frequency. The more is the spreading of ε values, the greater is

. Eq. (18) means that the complex roots

negative volume of it.

Seismic Waves - Probing Earth System

behavior of structured bodies.

the number of complex roots.

Figure 5.

48

value of ε corresponds to values of first row from Table 1.

the second term in Eq. (18) contains ð Þ kS=<sup>k</sup> <sup>2</sup>

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The second value of ε corresponds to values of second row from Table 1.

#### Figure 7.

The position of complex roots depends on the value ε. The more the value ε, the more numbers of roots. The third value of ε corresponds to values of third row from Table 1.

#### 5. Pointing vector and equation of equilibrium for blocked media

The equation of equilibrium for micro-structured media can be written from Eq. (9) as

$$\frac{\partial}{\partial \mathbf{x}\_k} \left[ P(\sigma\_{ik}) \right] = P \left[ \left. \frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_k} \right| \right] = \mathbf{0}; \frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_k} = P^{-1}(\mathbf{0}) \tag{19}$$

If these indexes coincide, i ¼ m, we get

DOI: http://dx.doi.org/10.5772/intechopen.87014

μ Ui<sup>0</sup>

4 1ð Þ � f nπ � �<sup>2</sup>

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

module is order to strain, multiplied to size of structure l0.

2μ σ0 mk u0 <sup>k</sup> ð Þ x l0

U0ikj þ U0jki

3πμ 1 l0

According to Eq. (9) the additional dilatation is

<sup>θ</sup>ð Þ <sup>n</sup> <sup>¼</sup> <sup>ϕ</sup> <sup>16</sup>γ<sup>2</sup>

(divergence of Pointing vector) namely,

nπð Þ klxl l0

value of these forces is zero) may be written as

� �<sup>&</sup>gt; <sup>¼</sup> 4 1ð Þ � <sup>f</sup>

<sup>θ</sup> <sup>¼</sup> <sup>ϕ</sup> <sup>16</sup>γ<sup>2</sup> 3πμ

� �ð Þ <sup>1</sup> � <sup>f</sup> <sup>2</sup> <sup>1</sup>

<sup>&</sup>lt; <sup>U</sup>0<sup>n</sup><sup>&</sup>gt; <sup>¼</sup> <sup>E</sup>

πn<sup>2</sup>

<sup>þ</sup> <sup>δ</sup>mj � <sup>1</sup> � <sup>γ</sup><sup>2</sup> � �kmkj � �ki<sup>g</sup> <sup>1</sup>

<sup>1</sup> � <sup>1</sup> � <sup>γ</sup><sup>2</sup> 3 � � exp

Take into account that the average value of a quadrat of cosine is <sup>&</sup>lt;kmki<sup>&</sup>gt; <sup>¼</sup> <sup>δ</sup>km

Strains. By differentiating of an integral Eq. (23) take into account that the main part of the field contains in fast changing exponent, not in Green tensor itself, i.е.,

<sup>k</sup> ð Þ x ikjIm∭ Γmið Þ x; y exp ikl xl � yl

4

<sup>n</sup> exp <sup>i</sup>

½ � <sup>U</sup>0<sup>n</sup> ð Þ <sup>1</sup> � <sup>f</sup> <sup>2</sup> <sup>1</sup>

Let us integrate the normal component of the Pointing vector on the small sphere with radius r. This integral must be equal to density of potential energy E

σ0

The average value of fast-changing exponent in Eqs. (28) and (29) on spherical

ð<sup>n</sup><sup>π</sup> 0

The additional dilatation due to randomly oriented volume forces (an average

4 1ð Þ � <sup>f</sup> <sup>3</sup>

π

In Eq. (32) the symbol Si nð Þ π means an integral sine of argument ð Þ nπ . The left hand in Eq. (32) is an additional expansion or compression, so called as dilatancy. It depends on the potential energy of the continuous body E, which may contain shear energy only, but it produces additional expansion or compression. It is a quadrat

sinx x

E

X∞ n¼1

<sup>n</sup> exp <sup>i</sup>

<sup>n</sup> exp <sup>i</sup>

There is a summation with respect to n, and Ui<sup>0</sup> is a Pointing vector for usual continuous model of the medium. This value is a small one of the second order compared to usual displacement, because a Pointing vector, divided on the shear

inπð ÞÞ klxl l0

� � (26)

� � � � dVy (27)

� �kj

� � (28)

� � (29)

<sup>π</sup>n<sup>2</sup> Si nð Þ <sup>π</sup> (31)

<sup>n</sup><sup>2</sup> (32)

<sup>π</sup> 4 1ð Þ � <sup>f</sup> <sup>2</sup> <sup>δ</sup>mi � <sup>1</sup> � <sup>γ</sup><sup>2</sup> � �kmki

nπð Þ k1x þ k2y þ k3z l0 � �

> nπð Þ klxl l0

> > nπð Þ klxl l0

¼ 4 1ð Þ � f El<sup>0</sup> (30)

dx <sup>¼</sup> 4 1ð Þ � <sup>f</sup>

Si nð Þ π

�

3 .

<u1<sup>n</sup>

u1 i,j ð Þ <sup>x</sup> <sup>≈</sup> <sup>ϕ</sup> μ σ2 σ0 mku<sup>0</sup>

<sup>¼</sup> <sup>ϕ</sup> <sup>8</sup>γ<sup>2</sup> 3πμ 1 l0

e 1n ij <sup>¼</sup> <sup>i</sup> 2 u1 i,j <sup>þ</sup> <sup>u</sup><sup>1</sup> j,i � � <sup>¼</sup> <sup>ϕ</sup>

angles is

effect too.

51

< exp 1 n i

<sup>i</sup> ð Þ <sup>x</sup> <sup>&</sup>gt; <sup>¼</sup> <sup>ϕ</sup>

The inverse operator P�<sup>1</sup> ð Þ 0 contains zero, but not zero only. It contains some periodic functions and the average value equal to zero. For example, such construction satisfies to Eq. (19)

$$P\left\{\sum \text{Im}\left\{\exp\left[in\frac{\pi}{l\_0}(k\_1\mathbf{x}+k\_2\mathbf{y}+k\_3\mathbf{z})\right]\right\}=\mathbf{0}\right\}\tag{20}$$

If n is integer number, k<sup>1</sup> ¼ sinθcosφ; k<sup>2</sup> ¼ sinθcosφ; k<sup>3</sup> ¼ cosθ: Physical sense of it means that volume forces are equal to zero in average sense, not at any point. Using mentioned inverse operator, we can write the equilibrium equation for blocked media in the form

$$\frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_k} = \phi \sigma\_0^2 \sigma\_{ik}^0 u\_k^0 \sum\_{n=1}^{\infty} \text{Im} \left\{ \exp \left[ i n \frac{\pi}{l\_0} (k\_1 \mathbf{x} + k\_2 \mathbf{y} + k\_3 \mathbf{z}) \right] \right\} \tag{21}$$

In Eq. (21) σ<sup>2</sup> <sup>0</sup> is a quadrat of specific surface; σ<sup>0</sup> iku<sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup> <sup>i</sup> is the pointing vector of usual continuous body, and ϕ is a dimensionless constant, which must be obtained. These values we can put as constants in small structure volume. The integration with respect to spherical angles gives us a result that the imaginary part of exponent is zero in average sense, namely

$$\frac{1}{4\pi} \int\_{0}^{2\pi} \int\_{0}^{\pi} \exp\left[\frac{m}{l\_{0}}i(\infty in\theta cos\rho + y\sin\theta sin\rho + z\cos\theta)\right] \sin\theta d\theta d\rho$$

$$\int\_{0}^{\pi} \exp\left(ir\frac{n\pi}{l\_{0}}cosp\right) \sin pdp = \frac{1}{2} \int\_{-1}^{1} \exp\left(ir\frac{n\pi}{l\_{0}}t\right) dt = \frac{l\_{0}}{n\pi r} \sin\left(\frac{rn\pi}{l\_{0}}\right) + i0 \tag{22}$$

Partial solution of Eq. (22) is a convolution of Green tensor with right hand of Eq. (21), that is,

$$u\_i^1(\mathbf{x}) = \phi \frac{1}{\mu} \sigma\_0^2 \sigma\_{mk}^0 u\_k^0(\mathbf{x}) \text{Im} \left[ \iint \Gamma\_{mi}(\mathbf{x}, \boldsymbol{y}) \exp\left[ik\_m(\boldsymbol{x}\_m - \boldsymbol{y}\_m)\right] d\boldsymbol{V}\_{\boldsymbol{\mathcal{Y}}} \right] \tag{23}$$

Taking into account that the sizes of area much more, than sizes of structure, the area of integration is the infinite large one. In this case, integral Eq. (13) practically is the Fourier transform of fundamental solution of usual elastic equilibrium equations

$$u\_i^{\rm in}(\mathbf{x}) = \phi \frac{\mathbf{1}}{\mu} \sigma^2 \sigma\_{mk}^0 u\_k^0(\mathbf{x}) \left(\frac{l\_0}{n\pi}\right)^2 \left[\delta\_{mi} - (\mathbf{1} - \boldsymbol{\chi}^2) k\_m k\_i\right] \exp\left[\frac{in\pi(k\_l \chi\_l)}{l\_0}\right] \tag{24}$$

In Eq. (24) the imaginary part of the exponent is used. Hence, the additional value in average sense is equal to zero. Using relation Eq. (1) σ0l<sup>0</sup> ¼ 4 1ð Þ � f , we get a partial solution, which depends on porosity only

$$u\_i^{1n}(\mathbf{x}) = \phi \frac{1}{\mu} \sigma\_{mk}^0 u\_k^0(\mathbf{x}) \left(\frac{4(1-f)}{n\pi}\right)^2 \left[\delta\_{mi} - (1-\chi^2)k\_m k\_i\right] \exp\left[i\frac{n\pi(k\_l \chi\_l)}{l\_0}\right] \tag{25}$$

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

If these indexes coincide, i ¼ m, we get

∂ ∂xk

�

The inverse operator P�<sup>1</sup>

Seismic Waves - Probing Earth System

∂σik ∂xk

is zero in average sense, namely

2 ðπ ð π

exp <sup>π</sup><sup>n</sup> l0

sinpdp <sup>¼</sup> <sup>1</sup>

2 ð1 �1

l0 nπ � �<sup>2</sup>

4 1ð Þ � f nπ � �<sup>2</sup>

a partial solution, which depends on porosity only

0

0

1 4π <sup>¼</sup> ϕσ<sup>2</sup> 0σ<sup>0</sup> iku<sup>0</sup> k X∞ n¼1

tion satisfies to Eq. (19)

media in the form

In Eq. (21) σ<sup>2</sup>

exp ir <sup>n</sup><sup>π</sup> l0 cosp � �

Eq. (21), that is,

u1

u<sup>1</sup><sup>n</sup>

u<sup>1</sup><sup>n</sup>

50

<sup>i</sup> ð Þ¼ <sup>x</sup> <sup>ϕ</sup> <sup>1</sup>

<sup>i</sup> ð Þ¼ <sup>x</sup> <sup>ϕ</sup> <sup>1</sup>

μ σ2 σ0 mku<sup>0</sup> <sup>k</sup> ð Þ x

μ σ0 mku<sup>0</sup> <sup>k</sup> ð Þ x

<sup>i</sup>ð Þ¼ <sup>x</sup> <sup>ϕ</sup> <sup>1</sup>

μ σ2 0σ<sup>0</sup> mku<sup>0</sup>

ð π

0

½ �¼ <sup>P</sup>ð Þ <sup>σ</sup>ik <sup>P</sup> <sup>∂</sup>σik

<sup>P</sup> <sup>X</sup>Im exp in <sup>π</sup>

<sup>0</sup> is a quadrat of specific surface; σ<sup>0</sup>

∂xk � �

periodic functions and the average value equal to zero. For example, such construc-

� � � �

If n is integer number, k<sup>1</sup> ¼ sinθcosφ; k<sup>2</sup> ¼ sinθcosφ; k<sup>3</sup> ¼ cosθ: Physical sense of it means that volume forces are equal to zero in average sense, not at any point. Using mentioned inverse operator, we can write the equilibrium equation for blocked

Im exp in <sup>π</sup>

usual continuous body, and ϕ is a dimensionless constant, which must be obtained. These values we can put as constants in small structure volume. The integration with respect to spherical angles gives us a result that the imaginary part of exponent

> i xsin ð Þ θcosφ þ ysinθsinφ þ zcosθ � �

> > exp ir <sup>n</sup><sup>π</sup> l0 t � �

Partial solution of Eq. (22) is a convolution of Green tensor with right hand of

<sup>k</sup> ð Þ x Im∭ Γmið Þ x; y exp ikm xm � ym

<sup>δ</sup>mi � <sup>1</sup> � <sup>γ</sup><sup>2</sup> � �kmki � � exp

<sup>δ</sup>mi � <sup>1</sup> � <sup>γ</sup><sup>2</sup> � �kmki � � exp i

In Eq. (24) the imaginary part of the exponent is used. Hence, the additional value in average sense is equal to zero. Using relation Eq. (1) σ0l<sup>0</sup> ¼ 4 1ð Þ � f , we get

Taking into account that the sizes of area much more, than sizes of structure, the area of integration is the infinite large one. In this case, integral Eq. (13) practically is the Fourier transform of fundamental solution of usual elastic equilibrium equations

l0

¼ 0;

∂σik ∂xk

ð Þ k1x þ k2y þ k3z

l0

<sup>¼</sup> <sup>P</sup>�<sup>1</sup>

¼ 0

ð Þ k1x þ k2y þ k3z

� � � �

iku<sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup>

dt <sup>¼</sup> <sup>l</sup><sup>0</sup> nπr

ð Þ 0 contains zero, but not zero only. It contains some

ð Þ 0 (19)

(20)

(21)

<sup>i</sup> is the pointing vector of

þ i0 (22)

(24)

(25)

sinθdθdφ

� � � � dVy (23)

inπð ÞÞ klxl l0 � �

> nπð Þ klxl l0 � �

sin rn<sup>π</sup> l0 � �

$$ = \frac{\phi}{\mu} U\_{i0} \left(\frac{4(\mathbf{1}-f)}{n\pi}\right)^2 \left[\mathbf{1} - \frac{\mathbf{1}-\boldsymbol{\gamma}^2}{\mathbf{3}}\right] \exp\left[\frac{in\pi(k\chi\_l)}{l\_0}\right] \tag{26}$$

Take into account that the average value of a quadrat of cosine is <sup>&</sup>lt;kmki<sup>&</sup>gt; <sup>¼</sup> <sup>δ</sup>km 3 . There is a summation with respect to n, and Ui<sup>0</sup> is a Pointing vector for usual continuous model of the medium. This value is a small one of the second order compared to usual displacement, because a Pointing vector, divided on the shear module is order to strain, multiplied to size of structure l0.

Strains. By differentiating of an integral Eq. (23) take into account that the main part of the field contains in fast changing exponent, not in Green tensor itself, i.е.,

$$u\_{i,j}^1(\mathbf{x}) \approx \frac{\phi}{\mu} \sigma^2 \sigma\_{mk}^0 u\_k^0(\mathbf{x}) ik\_j \text{Im} \left[ \iint \Gamma\_{mi}(\mathbf{x}, y) \exp\left[ik\_l(\mathbf{x}\_l - y\_l)\right] d\mathbf{V}\_j \right] \tag{27}$$

$$e\_{ij}^{1n} = \frac{i}{2} \left( u\_{i,j}^1 + u\_{j,i}^1 \right) = \frac{\phi}{2\mu} \sigma\_{mk}^0 \frac{u\_k^0(\mathbf{x})}{l\_0} \frac{4}{\pi} 4(\mathbf{1} - f)^2 \left\{ \left[ \delta\_{mi} - (\mathbf{1} - \boldsymbol{\gamma}^2) k\_m k\_i \right] k\_j \right\}$$

$$+ \left[ \delta\_{mj} - (\mathbf{1} - \boldsymbol{\gamma}^2) k\_m k\_j \right] k\_i \frac{1}{n} \exp\left[ i \frac{n\pi (k\_1 \mathbf{x} + k\_2 \mathbf{y} + k\_3 \mathbf{z})}{l\_0} \right] \tag{28}$$

$$= \phi \frac{8\eta^2}{3\pi\mu} \frac{1}{l\_0} \left[ U\_{0i} k\_j + U\_{0j} k\_i \right] (\mathbf{1} - f)^2 \frac{1}{n} \exp\left[ i \frac{n\pi (k\_l \mathbf{x}\_l)}{l\_0} \right]$$

According to Eq. (9) the additional dilatation is

$$\theta^{(n)} = \phi \frac{\mathbf{1} \mathsf{6} \gamma^2}{\mathfrak{3} \pi \mu} \frac{\mathbf{1}}{l\_0} [U\_{0n}] (\mathbf{1} - f)^2 \frac{\mathbf{1}}{n} \exp\left[i \frac{n \pi (k \!\!/ k \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!/ \!\!\/) \tag{29}$$

Let us integrate the normal component of the Pointing vector on the small sphere with radius r. This integral must be equal to density of potential energy E (divergence of Pointing vector) namely,

$$ = \frac{E}{\sigma\_0} = 4(1-f)El\_0\tag{30}$$

The average value of fast-changing exponent in Eqs. (28) and (29) on spherical angles is

$$<\exp\frac{1}{n}\left[i\frac{n\pi(k\_l\chi\_l)}{l\_0}\right]> = \frac{4(1-f)}{\pi n^2}\int\_0^{n\pi} \frac{\sin\chi}{\chi}d\chi = \frac{4(1-f)}{\pi n^2}Si(n\pi) \tag{31}$$

The additional dilatation due to randomly oriented volume forces (an average value of these forces is zero) may be written as

$$\theta = \phi \frac{\mathbf{1} \mathsf{6} \gamma^2}{3\pi\mu} \frac{\mathbf{4} \left(\mathbf{1} - f\right)^3 E}{\pi} \sum\_{n=1}^{\infty} \frac{\mathrm{Si}(n\pi)}{n^2} \tag{32}$$

In Eq. (32) the symbol Si nð Þ π means an integral sine of argument ð Þ nπ . The left hand in Eq. (32) is an additional expansion or compression, so called as dilatancy. It depends on the potential energy of the continuous body E, which may contain shear energy only, but it produces additional expansion or compression. It is a quadrat effect too.

#### Seismic Waves - Probing Earth System

More strong effect is related with product of high-changing volume force (equal to zero in average) into displacement. This product in not equal to zero in average, because it contains a quadrat of high-changing sine, which is equal to number one third in three dimension space.

$$E\_n = \phi \frac{1}{\mu} \sigma\_{mk}^0 u\_k^0(\mathbf{x}) \left(\frac{4(\mathbf{1} - f)}{n\pi}\right)^2 \left[\delta\_{mi} - (\mathbf{1} - \boldsymbol{\gamma}^2) k\_m k\_i\right] \frac{1}{3} \,\phi \sigma\_0^2 U\_{0i} \tag{33}$$

If indexes coincide, m ¼ i, we get the additional potential energy, due to fluctuations

$$E\_n = \frac{\phi^2}{3(\lambda + 2\mu)} \left[ U\_{01}^2 + U\_{02}^2 + U\_{03}^2 \right] \sigma\_0^2 \left( \frac{4(1 - f)}{n\pi} \right)^2 \tag{34}$$

1 μ

k1x � k2y ¼ 2l0q.

Figures 8.

53

and increasing them near lines itself.

j j E<sup>1</sup> � E<sup>0</sup>

6. The arriving of plasticity

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fð Þ 1 � f

not a highly changed value, because it is equal to1 or �1.

<sup>¼</sup> <sup>8</sup>ϕ<sup>2</sup> <sup>9</sup>ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>σ</sup><sup>2</sup>

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

<sup>0</sup> U<sup>2</sup>

In spite of that, the additional average strains is small, does not means, that these

strains are small in the any point of the volume. Equations (28) and (29) show that on the planes k1x � k2y � k3z ¼ 2l0q (q is an integer number), the exponent is

Successive process of strains localization due to decreasing strains inside of quadrats, making orthogonal lines

In plane situation, the role of these planes plays orthogonal lines

<sup>01</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup>

<sup>02</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup>

<sup>03</sup> � �ð Þ <sup>1</sup> � <sup>f</sup> <sup>2</sup> (40)

q

DOI: http://dx.doi.org/10.5772/intechopen.87014

The summation with respect to index n from unit up to infinity gives

$$E = \frac{8\phi^2}{\Re(\lambda + 2\mu)}\sigma\_0^2 \left[U\_{01}^2 + U\_{02}^2 + U\_{03}^2\right] \left(1 - f\right)^2\tag{35}$$

In spite of a fact that the Pointing vector is the small value of more high order, than stresses, the high value σ<sup>2</sup> <sup>0</sup> (quadrat of specific surface) in Eq. (35) can produce not small common effect. The indefinite factor ϕ depends on the real structure of pore space and macro-stress-strain state. However, in some simple situations, it can calculate elementary. For example, at rigid pressing of globe by spherical force (radial displacements are constants), the stress-strain state is a hydrostatic state in average, but not such state at any point. The compressional energy is proportional to compress module of skeleton and its volume plus the incompressibility of fluid and its volume, namely [8–11]

$$E = \left(\lambda + \frac{2\mu}{3}\right) \frac{\theta\_1^2}{2} (1 - f) + \rho c^2 \frac{\theta\_0^2}{2} f \tag{36}$$

Indexes unit and zero in Eq. (36) mean solid and liquid parameters. The dilatation of two-phase body gives by the formula

$$
\theta = (1 - f)\theta\_1 + f\theta\_0; \theta\_0 = \theta\_1 \tag{37}
$$

If we have uniform random distribution of phases, the average energy is

$$E = E\_1(\mathbf{1} - f) + E\_0 f \tag{38}$$

In Eq. (38) f is the porosity and E<sup>1</sup> and E<sup>0</sup> are the energies of solid and liquid. The dispersion of random value relates with random volume forces, i.e.,

$$\left[\left(E\_1 - \left(E\_1(\mathbf{1} - f) + E\_0 f\right)\right)^2 (\mathbf{1} - f) + f[E\_0 - \left(E\_1(\mathbf{1} - f) + E\_0 f\right)]^2 = (E\_1 - E\_0)^2 f (\mathbf{1} - f) \tag{39}$$

Equation (39) gives the additional energy for very simple macro-hydrostatic state in average. This is the additional of interphase acting. It is equal to additional energy, which is given by Eq. (15). It is reasonable that at unit or zero porosity, an additional energy is equal to zero. The second result is, if the phase energy is equal, the mentioned additional one is equal to zero too. Hence, the indefinite factor ϕ<sup>2</sup> given by the simple equation is

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

$$\frac{1}{\mu} |E\_1 - E\_0| \sqrt{f(1 - f)} = \frac{8\phi^2}{9(\lambda + 2\mu)} \sigma\_0^2 \left[ U\_{01}^2 + U\_{02}^2 + U\_{03}^2 \right] (1 - f)^2 \tag{40}$$

#### 6. The arriving of plasticity

More strong effect is related with product of high-changing volume force (equal to zero in average) into displacement. This product in not equal to zero in average, because it contains a quadrat of high-changing sine, which is equal to number one

If indexes coincide, m ¼ i, we get the additional potential energy, due to

<sup>δ</sup>mi � <sup>1</sup> � <sup>γ</sup><sup>2</sup> kmki 1

> <sup>02</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup> 03

0

<sup>02</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup> 03

σ<sup>2</sup>

<sup>01</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup>

In spite of a fact that the Pointing vector is the small value of more high order,

not small common effect. The indefinite factor ϕ depends on the real structure of pore space and macro-stress-strain state. However, in some simple situations, it can calculate elementary. For example, at rigid pressing of globe by spherical force (radial displacements are constants), the stress-strain state is a hydrostatic state in average, but not such state at any point. The compressional energy is proportional to compress module of skeleton and its volume plus the incompressibility of fluid

1

If we have uniform random distribution of phases, the average energy is

The dispersion of random value relates with random volume forces, i.e.,

In Eq. (38) f is the porosity and E<sup>1</sup> and E<sup>0</sup> are the energies of solid and liquid.

Equation (39) gives the additional energy for very simple macro-hydrostatic state in average. This is the additional of interphase acting. It is equal to additional energy, which is given by Eq. (15). It is reasonable that at unit or zero porosity, an additional energy is equal to zero. The second result is, if the phase energy is equal, the mentioned additional one is equal to zero too. Hence, the indefinite factor ϕ<sup>2</sup>

ð Þþ <sup>1</sup> � <sup>f</sup> f E½ � <sup>0</sup> � ð Þ <sup>E</sup>1ð Þþ <sup>1</sup> � <sup>f</sup> <sup>E</sup>0<sup>f</sup> <sup>2</sup> <sup>¼</sup> ð Þ <sup>E</sup><sup>1</sup> � <sup>E</sup><sup>0</sup>

Indexes unit and zero in Eq. (36) mean solid and liquid parameters. The dilata-

<sup>2</sup> ð Þþ <sup>1</sup> � <sup>f</sup> <sup>ρ</sup><sup>c</sup>

<sup>3</sup> ϕσ<sup>2</sup>

4 1ð Þ � f nπ <sup>2</sup>

ð Þ <sup>1</sup> � <sup>f</sup> <sup>2</sup> (35)

<sup>0</sup> (quadrat of specific surface) in Eq. (35) can produce

<sup>2</sup> θ<sup>2</sup> 0

θ ¼ ð Þ 1 � f θ<sup>1</sup> þ f θ0; θ<sup>0</sup> ¼ θ<sup>1</sup> (37)

E ¼ E1ð Þþ 1 � f E0f (38)

<sup>2</sup> <sup>f</sup> (36)

2 fð Þ 1 � f (39)

<sup>0</sup>U0<sup>i</sup> (33)

(34)

4 1ð Þ � f nπ <sup>2</sup>

> U2 <sup>01</sup> <sup>þ</sup> <sup>U</sup><sup>2</sup>

The summation with respect to index n from unit up to infinity gives

<sup>0</sup> U<sup>2</sup>

En <sup>¼</sup> <sup>ϕ</sup><sup>2</sup>

3ð Þ λ þ 2μ

<sup>E</sup> <sup>¼</sup> <sup>8</sup>ϕ<sup>2</sup>

<sup>9</sup>ð Þ <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> <sup>σ</sup><sup>2</sup>

E ¼ λ þ

tion of two-phase body gives by the formula

2μ 3 θ<sup>2</sup>

third in three dimension space.

Seismic Waves - Probing Earth System

than stresses, the high value σ<sup>2</sup>

and its volume, namely [8–11]

½ � <sup>E</sup><sup>1</sup> � ð Þ <sup>E</sup>1ð Þþ <sup>1</sup> � <sup>f</sup> <sup>E</sup>0<sup>f</sup> <sup>2</sup>

given by the simple equation is

52

En <sup>¼</sup> <sup>ϕ</sup> <sup>1</sup> μ σ0 mku<sup>0</sup> <sup>k</sup> ð Þ x

fluctuations

In spite of that, the additional average strains is small, does not means, that these strains are small in the any point of the volume. Equations (28) and (29) show that on the planes k1x � k2y � k3z ¼ 2l0q (q is an integer number), the exponent is not a highly changed value, because it is equal to1 or �1.

In plane situation, the role of these planes plays orthogonal lines k1x � k2y ¼ 2l0q.

Figures 8.

Successive process of strains localization due to decreasing strains inside of quadrats, making orthogonal lines and increasing them near lines itself.

The series Eqs. (28) and (29) with respect to n in vicinity of mentioned planes are divergent (harmonic) series. It means that the field is decreasing inside of quadrats, making planes, and concentrating in vicinity of planes. Mentioned planes are analogs of slipping lines (lines of Luders) [12] in classic plasticity of the compressible medium. In practical, the number n in Eqs. (28) and (29) is bounded by the elastic limit of the second strain invariant. The field of strains is growing into planes (lines) and decreasing inside of them. This process is called as localization of strains. This localization begins in elasticity, with contrary of classic plasticity and elasticity. The other specific feature of this process is the finite distance between planes (lines). This distance is equal to l<sup>0</sup> (the inverse value of specific surface of sample), while in classic plasticity, this distance is infinitely small. The geological sense of it is interesting. In order to transform the matter from elasticity to plasticity, there is no necessary to have the plastic state at any point of the medium. Plasticity may concentrated near planes, and the other volume can be in elastic state. Rock may flow comparatively light, if they have pores and cracks. On the Figure 8 shown successive process of localization of strains due to decreasing field inside of quadrats, making by orthogonal lines and increasing them near lines itself.

#### 7. Conclusions


Author details

Boris Sibiryakov

55

Trofimuk Institute of Oil and Gas Geology and Geophysics SB RAS,

Appearance of Catastrophes and Plasticity in Porous and Cracked Media

DOI: http://dx.doi.org/10.5772/intechopen.87014

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: sibiryakovbp@ipgg.sbras.ru

Novosibirsk State University, Novosibirsk, Russia

provided the original work is properly cited.

Appearance of Catastrophes and Plasticity in Porous and Cracked Media DOI: http://dx.doi.org/10.5772/intechopen.87014

### Author details

The series Eqs. (28) and (29) with respect to n in vicinity of mentioned planes are divergent (harmonic) series. It means that the field is decreasing inside of quadrats, making planes, and concentrating in vicinity of planes. Mentioned planes are analogs of slipping lines (lines of Luders) [12] in classic plasticity of the compressible medium. In practical, the number n in Eqs. (28) and (29) is bounded by the elastic limit of the second strain invariant. The field of strains is growing into planes (lines) and decreasing inside of them. This process is called as localization of strains. This localization begins in elasticity, with contrary of classic plasticity and elasticity. The other specific feature of this process is the finite distance between planes (lines). This distance is equal to l<sup>0</sup> (the inverse value of specific surface of sample), while in classic plasticity, this distance is infinitely small. The geological sense of it is interesting. In order to transform the matter from elasticity to

plasticity, there is no necessary to have the plastic state at any point of the medium. Plasticity may concentrated near planes, and the other volume can be in elastic state. Rock may flow comparatively light, if they have pores and cracks. On the Figure 8 shown successive process of localization of strains due to decreasing field inside of quadrats, making by orthogonal lines and increasing them near lines itself.

1.The model of the structured continuum with specific surface of the blocked medium or average size of structure, gives us the differential equations of motion of the infinite order. This model includes collective properties of pore space like the porosity and specific surface and predicts besides usual elastic

2.This model predicts the decreasing of the Poisson ratio (up to negative values) due to finite size of microstructure. The reason for this is the decreasing of

3.The localization of stresses and strains in structured media begins in elastic

4.The small areas of a stress-strain concentration looks like usual orthogonal sliding lines in classic plasticity. However, they have a finite effective thickness, which depends on the average size of the structure and the elastic strain limit. Besides, there is a finite distance between analogs of sliding lines, which is equal to the average distance from one pore to another one, or

waves many unusual waves with very small velocities.

wave velocity with finite specific surface of the rock.

7. Conclusions

Seismic Waves - Probing Earth System

state of deforming.

between cracks.

54

Boris Sibiryakov Trofimuk Institute of Oil and Gas Geology and Geophysics SB RAS, Novosibirsk State University, Novosibirsk, Russia

\*Address all correspondence to: sibiryakovbp@ipgg.sbras.ru

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### References

[1] Santalo L. Integral Geometry and Geometrical Probability. 2nd ed. Cambridge University Press; 2004. 405 p

[2] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. In: Zwillinger D, Moll V, editors. Academic Press; 2014. 1184 p

[3] Maslov VP. Operator Methods. Mir. 1976. 559 p

[4] Sibiryakov BP, Prilous BI. The unusual small wave velocities in structural bodies and instability of pore or cracked media by small vibration. WSEAS Transactions on Applied and Theoretical Mechanics. 2007;7:139-144

[5] Fokin AG, Shermergor TD. Theory of propagation of elastic waves in nonhomogeneous media. Springer Link. 1990;25(5):600-609

[6] Gregory AR. Fluid saturation effect on dynamic elastic properties of sedimentary rocks. Geophysics. 1976;41 (5):895-921

[7] Sibiryakov BP, Prilous BI, Kopeykin AV. The nature of instability of Blocked Media and Distribution Law of Unstable States. Physical Mesomechanics. 2013; 16:2:141-151. ISSN: 1029-9599

[8] Biot MA, Willis DJJ. Journal of Applied Mechanics. 1957;24:594-601

[9] Biot MA. General solution of the equations of elasticity and consolidation for a porous material. Journal of Applied Mechanics. 1941;12:155-164

[10] Gassman F. Uber die Elastizitat Poroser Medien: Vier. der Natur. Gesellschaft in Zurich. 1951;96:1-23

[11] Biot MA. Theory of propagation of the elastic waves in a fluid saturated porous solid. 1. Low-frequency range.

The Journal of the Acoustical Society of America. 1956;28:168-178

Chapter 4

Abstract

1. Introduction

57

Efficient Simulation of Fluids

Fluid simulation is based on Navier-Stokes equations. Efficient simulation codes may rely on the smooth particle hydrodynamic toolbox (SPH), a method that uses kernel density estimation. Many variants of SPH have been proposed to optimize the simulation, like implicit incompressible SPH (IISPH) or predictive-corrective incompressible SPH (PC-ISPH). This chapter recalls the formulation of SPH and focuses on its effective parallel implementation using the Nvidia common unified device architecture (CUDA), while message passing interface (MPI) is another option. The key to effective implementation is a dedicated accelerating structure, and therefore some well-chosen parallel design patterns are detailed. Using a rough model of the ocean, this type of simulation can be used directly to simulate a

Pierre Thuillier Le Gac, Emmanuelle Darles,

Pierre-Yves Louis and Lilian Aveneau

tsunami resulting from an underwater earthquake.

and results in a high wave when it arrives on the coast.

Keywords: fluid simulation, SPH, CUDA, MPI, Navier-Stokes, tsunami

Submarine earthquakes may generate tremendous disasters for human, like what occurred during the Tohoku earthquake in 2011. Even if their seismic waves may damage buildings and structures when they occur close to the coast, the tsunami they generally cause are a massive risk for humans. Indeed, the energy produced by a massive undersea quake is transmitted into the water at high speed

To avoid human losses, tsunami's simulations can help to inform the governments and society about the risks before and after a submarine earthquake. This chapter presents solutions for implementing such simulations. The main objective is to be able to calculate the propagation of the tsunami wave into the ocean and then to simulate efficiently its effects when the wave reaches the coast. These kinds of simulation can be done in two dimensions considering only the profile of the coast or in three dimensions when all the topography is considered. In both cases, the

Viscosity is the measure depicting how a fluid resists deformations. Even water is considered having a non-nil viscosity: so, this parameter must be considered carefully for tsunami simulations. Water simulation relies on Navier-Stokes equations that describe the motion of a viscous fluid. Unfortunately, Navier-Stokes equations cannot be solved directly like it is the case for many differential equations. The only way to obtain a solution at a given time consists of approximating it through simulation. In practice, two family of methods may be used. The first one

simulation must handle how water is affected by the earthquake wave.

consists in discretizing the simulation space into small parts and to do the

[12] Kachanov LM. Fundamentals of the Theory of Plasticity. North-Holland Publishing Company, 1971. XIII, 482 p

## Chapter 4 Efficient Simulation of Fluids

Pierre Thuillier Le Gac, Emmanuelle Darles, Pierre-Yves Louis and Lilian Aveneau

#### Abstract

References

Press; 2014. 1184 p

1976. 559 p

405 p

[1] Santalo L. Integral Geometry and Geometrical Probability. 2nd ed. Cambridge University Press; 2004.

Seismic Waves - Probing Earth System

The Journal of the Acoustical Society of

[12] Kachanov LM. Fundamentals of the Theory of Plasticity. North-Holland Publishing Company, 1971. XIII, 482 p

America. 1956;28:168-178

[2] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. In: Zwillinger D, Moll V, editors. Academic

[3] Maslov VP. Operator Methods. Mir.

[5] Fokin AG, Shermergor TD. Theory of

nonhomogeneous media. Springer Link.

[6] Gregory AR. Fluid saturation effect on dynamic elastic properties of

sedimentary rocks. Geophysics. 1976;41

[7] Sibiryakov BP, Prilous BI, Kopeykin AV. The nature of instability of Blocked Media and Distribution Law of Unstable States. Physical Mesomechanics. 2013;

16:2:141-151. ISSN: 1029-9599

Mechanics. 1941;12:155-164

[8] Biot MA, Willis DJJ. Journal of Applied Mechanics. 1957;24:594-601

[9] Biot MA. General solution of the equations of elasticity and consolidation for a porous material. Journal of Applied

[10] Gassman F. Uber die Elastizitat Poroser Medien: Vier. der Natur. Gesellschaft in Zurich. 1951;96:1-23

[11] Biot MA. Theory of propagation of the elastic waves in a fluid saturated porous solid. 1. Low-frequency range.

propagation of elastic waves in

1990;25(5):600-609

(5):895-921

56

[4] Sibiryakov BP, Prilous BI. The unusual small wave velocities in structural bodies and instability of pore or cracked media by small vibration. WSEAS Transactions on Applied and Theoretical Mechanics. 2007;7:139-144

Fluid simulation is based on Navier-Stokes equations. Efficient simulation codes may rely on the smooth particle hydrodynamic toolbox (SPH), a method that uses kernel density estimation. Many variants of SPH have been proposed to optimize the simulation, like implicit incompressible SPH (IISPH) or predictive-corrective incompressible SPH (PC-ISPH). This chapter recalls the formulation of SPH and focuses on its effective parallel implementation using the Nvidia common unified device architecture (CUDA), while message passing interface (MPI) is another option. The key to effective implementation is a dedicated accelerating structure, and therefore some well-chosen parallel design patterns are detailed. Using a rough model of the ocean, this type of simulation can be used directly to simulate a tsunami resulting from an underwater earthquake.

Keywords: fluid simulation, SPH, CUDA, MPI, Navier-Stokes, tsunami

#### 1. Introduction

Submarine earthquakes may generate tremendous disasters for human, like what occurred during the Tohoku earthquake in 2011. Even if their seismic waves may damage buildings and structures when they occur close to the coast, the tsunami they generally cause are a massive risk for humans. Indeed, the energy produced by a massive undersea quake is transmitted into the water at high speed and results in a high wave when it arrives on the coast.

To avoid human losses, tsunami's simulations can help to inform the governments and society about the risks before and after a submarine earthquake. This chapter presents solutions for implementing such simulations. The main objective is to be able to calculate the propagation of the tsunami wave into the ocean and then to simulate efficiently its effects when the wave reaches the coast. These kinds of simulation can be done in two dimensions considering only the profile of the coast or in three dimensions when all the topography is considered. In both cases, the simulation must handle how water is affected by the earthquake wave.

Viscosity is the measure depicting how a fluid resists deformations. Even water is considered having a non-nil viscosity: so, this parameter must be considered carefully for tsunami simulations. Water simulation relies on Navier-Stokes equations that describe the motion of a viscous fluid. Unfortunately, Navier-Stokes equations cannot be solved directly like it is the case for many differential equations. The only way to obtain a solution at a given time consists of approximating it through simulation. In practice, two family of methods may be used. The first one consists in discretizing the simulation space into small parts and to do the

simulation considering fixed cells in this discrete space (mesh approach). Wellknown methods are the finite element, the finite difference, and the finite volume. A second alternative approach is the smoothed particle hydrodynamics (SPH), introduced in astrophysics in 1977 [1, 2], which is applied in computer graphics [3], oceanography, and many other fields. The latter is particularly interesting for tsunami simulation, since the most important part of the simulation is not in the ocean but rather on the ground. This implies that a part of the fluid will cover the coast. This heterogeneity makes the mesh-free SPH approach more adapted.

Such a solution is quite difficult to use into environment where some large part (like

dt. For a given particle i, the

ui þ ρig (3)

þ g (4)

� � �f xð Þ<sup>i</sup> (5)

∀x∈ R,Wð Þ x ≥0 (6)

W xð Þdx ¼ 1 (7)

� � � (8)

� � � (9)

� � � (10)

Another method to approximate the Navier-Stokes equations is SPH. The Euclidean space is no more discretized. Instead, it considers some moving particles representing the fluid and their interactions. Each particle comes with its specific velocity, pressure, density, and viscosity. Then, the total derivative allows to approach the advection term (those between parentheses on the left part of the

dt � � ¼ �∇pi <sup>þ</sup> <sup>μ</sup>i∇<sup>2</sup>

dt ¼ � <sup>∇</sup>pi ρi

SPH relies on the kernel density estimation [4]. When we only have some samples of a given function, we can estimate its value at a new location using a

> W x � xj � � �

The kernel function W must be a unit positive function. In other words, it must

We denote Wh a kernel function with bounded support 0½ � ; h . This simply means that Whð Þ¼ x 0 for all x < 0 or x≥h. This mathematical tool is used to approximate

where mj and ρ<sup>j</sup> are, respectively, the mass and the density of the jth particle. Useful kernels for liquids are given in [3]. From this simple expression, we can

These formulas allow to calculate the density of any particle, the gradient of the pressure, and the Laplacian of the velocity to approximate a solution of the Navier-

Wh x � xj � � �

∇Wh x � xj � � �

> Wh x � xj � � �

<sup>þ</sup> <sup>μ</sup>i∇<sup>2</sup>ui ρi

ocean) and highly detailed parts must be considered together.

momentum equation) by a single derivative term du

ρi dui

kernel function W and the following estimation:

satisfy the two following properties:

any scalar field Ai for any particle i:

59

Therefore, the acceleration ai of the particle i is given by

ai <sup>¼</sup> dui

fð Þ¼ x

1 <sup>n</sup> <sup>∑</sup> n i¼1

ð R

Ai ¼ ∑ n j¼1 Aj mj ρj

∇Ai ¼ ∑ n j¼1 Aj mj ρj

> Ai ¼ ∑ n j¼1 Aj mj ρj ∇2

Stokes equations. For each particle i, the SPH algorithm follows:

∇2

deduce the estimation of the gradient and the Laplacian of a scalar Ai:

momentum equation becomes

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

2.2 Introduction to SPH

This chapter is organized as follows. Section 2 presents the basics of SPH, detailing the different involved mathematical expressions and steps and previous implementations proposed in the literature. Section 3 presents a parallel implementation of SPH: it recalls the main parallel patterns and how they are used to obtain a reliable and fast simulation. Before the conclusions, Section 4 presents some results for a simple case of tsunami.

#### 2. SPH formulation

SPH is a Lagrangian approach, meaning that particles representing tiny parts of the fluids may move during the simulation. It is based on density estimation applied to moving particles, leading to an approximation of the Navier-Stokes equations. This section recalls these equations and presents the basics of SPH.

#### 2.1 Navier-Stokes equations

Navier-Stokes equations model the dynamics of a fluid. They rely on the Newton second law, stating that the sum of the forces applied on a body equals the product of its mass by its acceleration (∑F ¼ m � a). In practice, it is a system of two equations: the mass continuity equation and the momentum equation. The first one is given by

$$\frac{\partial \rho}{\partial \mathbf{t}} + \nabla(\rho u) = \mathbf{0} \tag{1}$$

where ρ designs the fluid density, ∇ is the gradient operator, and u is the flow velocity. The momentum equation is

$$
\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla \mathbf{p} + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g} \tag{2}
$$

where �<sup>∇</sup> is the divergence operator, <sup>∇</sup><sup>2</sup> the Laplacian operator, p the pressure, <sup>μ</sup> the dynamic viscosity coefficient, and g the gravity term. The right part of the momentum equation represents the sum of the forces that the fluid undergoes, where �∇p is the pressure force, <sup>μ</sup>∇2u the viscosity force, and <sup>ρ</sup><sup>g</sup> the gravitational force. With the fluid velocity function being unknown, it is not possible to compute analytically its divergence. Then, the momentum equation is nonlinear.

Nevertheless, some methods allow to calculate an approximation of these two equations. Most of them regularly discretize the Euclidean space and calculate an approximation by using the finite difference theorem. The advection term (the left part of the momentum equation) is approximated placing particles into the grid and then computing their displacement. In other words, each grid cell contains a given amount of fluid, and the algorithm calculates the exchanges between adjacent cells.

#### Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

simulation considering fixed cells in this discrete space (mesh approach). Wellknown methods are the finite element, the finite difference, and the finite volume. A second alternative approach is the smoothed particle hydrodynamics (SPH), introduced in astrophysics in 1977 [1, 2], which is applied in computer graphics [3], oceanography, and many other fields. The latter is particularly interesting for tsunami simulation, since the most important part of the simulation is not in the ocean but rather on the ground. This implies that a part of the fluid will cover the coast.

This chapter is organized as follows. Section 2 presents the basics of SPH, detailing the different involved mathematical expressions and steps and previous implementations proposed in the literature. Section 3 presents a parallel implementation of SPH: it recalls the main parallel patterns and how they are used to obtain a reliable and fast simulation. Before the conclusions, Section 4 presents some results

SPH is a Lagrangian approach, meaning that particles representing tiny parts of the fluids may move during the simulation. It is based on density estimation applied to moving particles, leading to an approximation of the Navier-Stokes equations.

Navier-Stokes equations model the dynamics of a fluid. They rely on the Newton second law, stating that the sum of the forces applied on a body equals the product of its mass by its acceleration (∑F ¼ m � a). In practice, it is a system of two equations: the mass continuity equation and the momentum equation. The first one

where ρ designs the fluid density, ∇ is the gradient operator, and u is the flow

where �<sup>∇</sup> is the divergence operator, <sup>∇</sup><sup>2</sup> the Laplacian operator, p the pressure, <sup>μ</sup>

Nevertheless, some methods allow to calculate an approximation of these two equations. Most of them regularly discretize the Euclidean space and calculate an approximation by using the finite difference theorem. The advection term (the left part of the momentum equation) is approximated placing particles into the grid and then computing their displacement. In other words, each grid cell contains a given amount of fluid, and the algorithm calculates the exchanges between adjacent cells.

the dynamic viscosity coefficient, and g the gravity term. The right part of the momentum equation represents the sum of the forces that the fluid undergoes, where �∇p is the pressure force, <sup>μ</sup>∇2u the viscosity force, and <sup>ρ</sup><sup>g</sup> the gravitational force. With the fluid velocity function being unknown, it is not possible to compute

analytically its divergence. Then, the momentum equation is nonlinear.

¼ �∇<sup>p</sup> <sup>þ</sup> <sup>μ</sup>∇<sup>2</sup>

þ ∇ð Þ¼ ρu 0 (1)

u þ ρg (2)

This heterogeneity makes the mesh-free SPH approach more adapted.

This section recalls these equations and presents the basics of SPH.

∂ ρ ∂ t

þ u � ∇u 

for a simple case of tsunami.

Seismic Waves - Probing Earth System

2.1 Navier-Stokes equations

velocity. The momentum equation is

ρ ∂u ∂t

is given by

58

2. SPH formulation

Such a solution is quite difficult to use into environment where some large part (like ocean) and highly detailed parts must be considered together.

Another method to approximate the Navier-Stokes equations is SPH. The Euclidean space is no more discretized. Instead, it considers some moving particles representing the fluid and their interactions. Each particle comes with its specific velocity, pressure, density, and viscosity. Then, the total derivative allows to approach the advection term (those between parentheses on the left part of the momentum equation) by a single derivative term du dt. For a given particle i, the momentum equation becomes

$$
\rho\_i \left(\frac{du\_i}{dt}\right) = -\nabla p\_i + \mu\_i \nabla^2 u\_i + \rho\_i \mathbf{g} \tag{3}
$$

Therefore, the acceleration ai of the particle i is given by

$$a\_i = \frac{du\_i}{dt} = -\frac{\nabla p\_i}{\rho\_i} + \frac{\mu\_i \nabla^2 u\_i}{\rho\_i} + \mathbf{g} \tag{4}$$

#### 2.2 Introduction to SPH

SPH relies on the kernel density estimation [4]. When we only have some samples of a given function, we can estimate its value at a new location using a kernel function W and the following estimation:

$$\mathbf{f}(\mathbf{x}) = \frac{1}{n} \sum\_{i=1}^{n} \mathbf{W}(||\mathbf{x} - \mathbf{x}\_{j}||) f(\mathbf{x}\_{i}) \tag{5}$$

The kernel function W must be a unit positive function. In other words, it must satisfy the two following properties:

$$\forall \mathbf{x} \in \mathbb{R}, \ W(\mathbf{x}) \ge \mathbf{0} \tag{6}$$

$$\int\_{\mathbb{R}} \mathcal{W}(\mathbf{x})d\mathbf{x} = \mathbf{1} \tag{7}$$

We denote Wh a kernel function with bounded support 0½ � ; h . This simply means that Whð Þ¼ x 0 for all x < 0 or x≥h. This mathematical tool is used to approximate any scalar field Ai for any particle i:

$$A\_i = \sum\_{j=1}^n A\_j \frac{m\_j}{\rho\_j} W\_h \left( ||\boldsymbol{\omega} - \boldsymbol{\omega}\_j|| \right) \tag{8}$$

where mj and ρ<sup>j</sup> are, respectively, the mass and the density of the jth particle. Useful kernels for liquids are given in [3]. From this simple expression, we can deduce the estimation of the gradient and the Laplacian of a scalar Ai:

$$\nabla A\_i = \sum\_{j=1}^n A\_j \frac{m\_j}{\rho\_j} \nabla \mathcal{W}\_h \left( ||x - x\_j|| \right) \tag{9}$$

$$\nabla^2 A\_i = \sum\_{j=1}^n A\_j \frac{m\_j}{\rho\_j} \nabla^2 \mathcal{W}\_h \left( ||\mathbf{x} - \mathbf{x}\_j|| \right) \tag{10}$$

These formulas allow to calculate the density of any particle, the gradient of the pressure, and the Laplacian of the velocity to approximate a solution of the Navier-Stokes equations. For each particle i, the SPH algorithm follows:

• Compute the density ρi:

$$\rho\_i = \sum\_{j=1}^{n} m\_j \mathcal{W}\_h \left( ||\mathbf{x} - \mathbf{x}\_j|| \right) \tag{11}$$

surrounding the one containing this particle. In dimension 2 this leads to 9 cells

The SPH method described in Section 2.2 has been extended to solve some accuracy problem with incompressible fluids, for instance, predictive-corrective incompressible SPH (PC-ISPH), incompressible SPH (ISPH), and implicit incompressible SPH (IISPH) [5–7]. In Ref. [7], comparisons between these three techniques show that IISPH is faster than PC-ISPH and ISPH, mainly since it allows to use bigger time steps. Hence, this chapter focusses on an implementation of IISPH. This evolved method is also more complex than classical SPH, and then each time step uses more calculations (but they are longer, so it is faster still). More precisely, for each particle it calculates the density ρ<sup>i</sup> and the forces of viscosity, surface tension, and gravity like in SPH method. It adds the calculation of the advection velocity, which is the portion of the velocity independent to the pressure exerted by

(including the cell containing the particle) and 27 in dimension 3.

Δt mi

dii ¼ �Δt

n j

mj uadv

<sup>2</sup> ∑ n j � mj ρ2 j pl j

where l is the iteration number of the corrective loop. Notice that for l ¼ 0,

Then, the IISPH corrective loop continues by computing for each particle the

pi ρ2 i þ pj ρ2 j

!

<sup>ρ</sup><sup>0</sup> � <sup>ρ</sup>adv

aii ¼ ∑ n j

<sup>i</sup> ¼ ρ<sup>i</sup> þ Δt ∑

∑ n j dijpl <sup>j</sup> ¼ Δt

pressure force thanks to the following expression:

f pressure <sup>i</sup> ¼ ∑

where pi is the pressure at a particle i:

, with ω ¼ 0:5.

n j

pi <sup>¼</sup> ð Þ <sup>1</sup> � <sup>ω</sup> pl

�mimj

<sup>i</sup> þ ω aii

f viscosity

<sup>2</sup> ∑ n j mj ρ2 i

<sup>i</sup> <sup>þ</sup> <sup>f</sup> surface

mj dii � dji � �ΔWh xi � xj

<sup>i</sup> � uadv j � � � <sup>Δ</sup>Wh xi � xj

Second, it calculates the following term per particle that will be used many times

ΔWh xi � xj

ΔWh xi � xj

<sup>i</sup> � <sup>∑</sup>pl i � � (25)

The IISPH algorithm continues with the calculation of pressure's forces. It is done through at least two corrective loops to enforce the minimization of the difference between the rest density and the sum of the density of all particles. First,

Wh xi � xj

The IISPH algorithm calculates the advection factor dii and the advection coef-

<sup>i</sup> <sup>þ</sup> <sup>f</sup> gravity i � � (19)

� � (20)

� � (21)

� � (22)

� � (23)

� � (24)

the other particles:

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

ficient aii:

in the next steps:

IISPH uses p<sup>0</sup>

61

uadv <sup>i</sup> ¼ ui þ

this loop calculates the advection density:

ρadv

<sup>i</sup> ¼ ωpi

• Compute the pressure pi :

$$p\_i = k(\rho\_i - \rho\_0) \tag{12}$$

where k is the gas constant and ρ<sup>0</sup> is the rest density.

• Compute fi , the sum of the forces at particle i of pressure, viscosity, surface tension, and gravity:

$$f\_i^{\text{pressure}} = -\sum\_{j=1}^{n} m\_j \frac{p\_i + p\_j}{2\rho\_j} \nabla \mathcal{W}\_h \left( ||\mathbf{x}\_i - \mathbf{x}\_j|| \right) \tag{13}$$

$$f\_i^{\text{viscosity}} = \mu \sum\_{j=1}^{n} m\_j \frac{u\_j - u\_i}{\rho\_j} \nabla^2 \mathcal{W}\_h \left( ||\mathbf{x} - \mathbf{x}\_j|| \right) \tag{14}$$

$$f\_i^{surface} = -\sigma \nabla^2 cs\_i \frac{n\_i}{|n\_i|}\tag{15}$$

$$f\_i^{\text{gravity}} = \rho\_i \mathbf{g} \tag{16}$$

where σ is the tension coefficient relating to the interface between the fluid and the exterior (the air), ni is the normal vector to a particle i, and csi is the color field of the particle i.

• Compute the velocity ui and the new particle position xi using a small integration time step Δt:

$$u\_i = u\_i + \Delta t \frac{f\_i}{\rho\_i} \tag{17}$$

$$\mathbf{x}\_{i} = \mathbf{x}\_{i} + \Delta t \mathbf{u}\_{i} \tag{18}$$

The SPH simulation uses these formulas to compute the positions of the particles for a given time length through an iterative procedure. The particles' interactions are very important: we use a rather small support (small h value) for the kernel function in order to limit the number n of neighboring particles. Then, in any good implementation, one of the key elements is the neighboring handling. Using a parallel processor, this can be achieved with a low complexity, allowing to reach short computation times.

#### 2.3 SPH algorithms

SPH method presented in Section 2.2 is quite immediate to implement [3]. Using a small kernel support, the calculation of the forces that apply to a given particle is quite fast, since only a few numbers of neighbors have to be considered. Nevertheless, the neighborhood needs to be efficiently computed and stored to accelerate the calculations. This needs to be done for each time step. To do that, a regular grid is the faster solution. The size of a grid cell is set as the radius of the kernel support. Then, to find the neighbors of a given particle, it is enough to consider the cells

#### Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

• Compute the density ρi:

Seismic Waves - Probing Earth System

• Compute the pressure pi

tension, and gravity:

f pressure <sup>i</sup> ¼ � ∑

f viscosity <sup>i</sup> ¼ μ ∑

• Compute fi

of the particle i.

integration time step Δt:

short computation times.

2.3 SPH algorithms

60

ρ<sup>i</sup> ¼ ∑ n j¼1

:

where k is the gas constant and ρ<sup>0</sup> is the rest density.

n j¼1 mj

n j¼1 mj

> f gravity

• Compute the velocity ui and the new particle position xi using a small

ui <sup>¼</sup> ui <sup>þ</sup> <sup>Δ</sup><sup>t</sup> fi

The SPH simulation uses these formulas to compute the positions of the particles for a given time length through an iterative procedure. The particles' interactions are very important: we use a rather small support (small h value) for the kernel function in order to limit the number n of neighboring particles. Then, in any good implementation, one of the key elements is the neighboring handling. Using a parallel processor, this can be achieved with a low complexity, allowing to reach

SPH method presented in Section 2.2 is quite immediate to implement [3]. Using a small kernel support, the calculation of the forces that apply to a given particle is quite fast, since only a few numbers of neighbors have to be considered. Nevertheless, the neighborhood needs to be efficiently computed and stored to accelerate the calculations. This needs to be done for each time step. To do that, a regular grid is the faster solution. The size of a grid cell is set as the radius of the kernel support. Then, to find the neighbors of a given particle, it is enough to consider the cells

ρi

f surface <sup>i</sup> ¼ �σ∇<sup>2</sup>

pi þ pj 2ρj

uj � ui ρj

where σ is the tension coefficient relating to the interface between the fluid and the exterior (the air), ni is the normal vector to a particle i, and csi is the color field

∇2

csi ni

mjWh x � xj 

, the sum of the forces at particle i of pressure, viscosity, surface

∇Wh xi � xj 

> Wh x � xj

(11)

(13)

(14)

<sup>∣</sup>ni<sup>∣</sup> (15)

(17)

<sup>i</sup> ¼ ρig (16)

xi ¼ xi þ Δtui (18)

pi ¼ k ρ<sup>i</sup> � ρ<sup>0</sup> ð Þ (12)

surrounding the one containing this particle. In dimension 2 this leads to 9 cells (including the cell containing the particle) and 27 in dimension 3.

The SPH method described in Section 2.2 has been extended to solve some accuracy problem with incompressible fluids, for instance, predictive-corrective incompressible SPH (PC-ISPH), incompressible SPH (ISPH), and implicit incompressible SPH (IISPH) [5–7]. In Ref. [7], comparisons between these three techniques show that IISPH is faster than PC-ISPH and ISPH, mainly since it allows to use bigger time steps. Hence, this chapter focusses on an implementation of IISPH. This evolved method is also more complex than classical SPH, and then each time step uses more calculations (but they are longer, so it is faster still). More precisely, for each particle it calculates the density ρ<sup>i</sup> and the forces of viscosity, surface tension, and gravity like in SPH method. It adds the calculation of the advection velocity, which is the portion of the velocity independent to the pressure exerted by the other particles:

$$u\_i^{adv} = u\_i + \frac{\Delta t}{m\_i} \left( f\_i^{viscosity} + f\_i^{surface} + f\_i^{gravity} \right) \tag{19}$$

The IISPH algorithm calculates the advection factor dii and the advection coefficient aii:

$$\mathbf{d}\_{\rm ii} = -\Delta \mathbf{t}^2 \sum\_{j}^{n} \frac{m\_j}{\rho\_i^2} \,\,\,\mathcal{W}\_h(\mathbf{x}\_i - \mathbf{x}\_j) \tag{20}$$

$$a\_{ii} = \sum\_{j}^{n} m\_{j} \left( d\_{ii} - d\_{ji} \right) \Delta W\_{h} \left( \mathbf{x}\_{i} - \mathbf{x}\_{j} \right) \tag{21}$$

The IISPH algorithm continues with the calculation of pressure's forces. It is done through at least two corrective loops to enforce the minimization of the difference between the rest density and the sum of the density of all particles. First, this loop calculates the advection density:

$$\rho\_i^{adv} = \rho\_i + \Delta t \sum\_j^n m\_j \left( u\_i^{adv} - u\_j^{adv} \right) \cdot \Delta W\_h \left( \varkappa\_i - \varkappa\_j \right) \tag{22}$$

Second, it calculates the following term per particle that will be used many times in the next steps:

$$\sum\_{\mathbf{j}}^{\mathrm{n}} \mathbf{d}\_{\mathbf{j}} \mathbf{p}\_{\mathbf{j}}^{\mathrm{l}} = \Delta \mathbf{t}^{2} \sum\_{\mathbf{j}}^{\mathrm{n}} -\frac{m\_{\mathrm{j}}}{\rho\_{\mathrm{j}}^{2}} \mathbf{p}\_{\mathbf{j}}^{\mathrm{l}} \Delta \mathbf{W}\_{\mathrm{h}} \left(\mathbf{x}\_{\mathrm{i}} - \mathbf{x}\_{\mathrm{j}}\right) \tag{23}$$

where l is the iteration number of the corrective loop. Notice that for l ¼ 0, IISPH uses p<sup>0</sup> <sup>i</sup> ¼ ωpi , with ω ¼ 0:5.

Then, the IISPH corrective loop continues by computing for each particle the pressure force thanks to the following expression:

$$f\_i^{\text{pressure}} = \sum\_{j}^{n} -m\_i m\_j \left(\frac{p\_i}{\rho\_i^2} + \frac{p\_j}{\rho\_j^2}\right) \Delta \mathcal{W}\_h \left(\mathbf{x}\_i - \mathbf{x}\_j\right) \tag{24}$$

where pi is the pressure at a particle i:

$$p\_i = (\mathbf{1} - \alpha)p\_i^l + \frac{\alpha}{a\_{ii}}(\rho\_0 - \rho\_i^{adv} - \sum p\_i^l) \tag{25}$$

This last term is computed using the displacement factors:

$$\sum\_{j} \mathbf{p}\_{i}^{l} = \sum\_{j}^{n} m\_{j} \left( \sum\_{j}^{n} d\_{ij} \mathbf{p}\_{j}^{l} - d\_{jl} \mathbf{p}\_{j}^{l} - \sum\_{k \neq i}^{n} d\_{jk} \mathbf{p}\_{k}^{l} \right) \cdot \mathbf{W}\_{h} \left( \mathbf{x}\_{i} - \mathbf{x}\_{j} \right) \tag{26}$$

All these calculations should be made in parallel to reduce the computation times, using a tuned implementation, for instance, using message passing interface (MPI) for high-performance computing (HPC) or using the Nvidia common unified device architecture (CUDA) on graphics processing unit (GPU) for simpler computers.

#### 3. Parallel SPH implementation

An efficient SPH implementation relies on parallelism at some level. A fully parallel solution may become a very efficient solution, as previous works have shown it. While most of the calculations may be done considering a single particle into a single core, finding the neighboring particles that play a role in the density, the pressure, and the external forces needs collaboration between different cores.

Using the texture mechanism available with GPU, working with the neighbors is quite simple and efficient. Nevertheless, this implies to store all the particles into a regular grid at each time step during the simulation. This part is somewhere the most complicated, and the key step for an efficient implementation.

In many occasions, it is necessary to write the result at a new location, another index. When each possible destination index is used once and only once, we obtain a quite simple parallel pattern called SCATTER. It consists of writing the input data from location i to the destination location map ið Þ, map being a permutation func-

In the same spirit, the GATHER parallel pattern writes at index i data coming from index map ið Þ, using again a permutation function. To differentiate between a SCATTER and a GATHER, you should remember that at first we read contiguous data, while in the second, we write at contiguous location. This is resumed with the

PRAM model is very useful to write efficient algorithms on theory. Nevertheless,

at the end these algorithms run on real computers, with a limited amount of memory and a fixed number of processors. Brent's theorem links the theoretical computation time on PRAM model with the one obtained using only p processors:

cessors. This allows to predict the behavior of (very) simple algorithm on a GPU.

In many cases, some degree of collaboration is needed between processors. This leads to some more complicated parallel patterns. A very common parallel pattern using such a collaboration is the SORT that sorts data according to a given order. It is used in previous SPH implementation for building the neighbors' grid. The SORT pattern is based on the PARTITION pattern that moves values with respect to a given predicate. More precisely, for n values Xi and using the predicate Pi ∈½ � 0; 1 , the PARTITION pattern moves the values Xi for which Pi ¼ 1 at the beginning of the resulting array, the others at the end (see Figure 3 for a simple example).

an algorithm made in <sup>O</sup>ð Þ<sup>1</sup> using <sup>m</sup> processors will run in <sup>O</sup> <sup>m</sup>

Ymap ið Þ ¼ Xi ð Þ SCATTER (28) Yi ¼ Xmap ið Þ ð Þ GATHER (29)

p

using only <sup>p</sup> pro-

tion. Figure 2 illustrates this parallel pattern.

The SCATTER and GATHER patterns move data using a permutation.

following two expressions:

Figure 1.

Figure 2.

Illustration of the MAP parallel pattern.

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

3.1.2 Advanced parallel patterns

63

This section first presents the main parallel patterns (MAP, SORT, SCAN, etc.) and then shows how they can be combined to write a new fast parallel SPH solver.

#### 3.1 Parallel patterns

Writing a parallel algorithm is not as simple as writing a sequential algorithm. This truism is based on the necessary consideration of the collaborations between the different processors of a parallel machine: all the processors must work in concert, and not isolated as in a sequential approach. These collaborative aspects are the main difficulty. How to make sure all these processors expect when it's needed and work to the fullest when no synchronization is required?

Rather than writing a parallel algorithm based on classical sequential patterns, parallel patterns make it possible to write a parallel algorithm directly, abstracting the underlying machine. These patterns rely on very simple parallel architecture, called the parallel random-access memory (PRAM). It assumes a synchronization between an infinite set of processors and an infinite amount of memory [8].

#### 3.1.1 Simple parallel patterns

Simple parallel patterns do not need synchronization. This means that, using a GPU or an HPC, they may be run without any difficulties, even with less processors than needed. The simpler one is the MAP, or transform, that consists in applying a given function f to an input data to obtain the output. The key of this pattern is about the localisation of the data: input and output are generally considered as vectors (or arrays). Then, MAP applied to data at the same index:

$$Y\_i = \mathbf{f}(\mathbf{X}\_i) \quad (\text{MAP}) \tag{27}$$

Figure 1 describes this pattern on small arrays.

#### Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

#### Figure 1.

This last term is computed using the displacement factors:

<sup>j</sup> � djjpl

!

All these calculations should be made in parallel to reduce the computation times, using a tuned implementation, for instance, using message passing interface (MPI) for high-performance computing (HPC) or using the Nvidia common unified device architecture (CUDA) on graphics processing unit (GPU) for simpler

An efficient SPH implementation relies on parallelism at some level. A fully parallel solution may become a very efficient solution, as previous works have shown it. While most of the calculations may be done considering a single particle into a single core, finding the neighboring particles that play a role in the density, the pressure, and the external forces needs collaboration between different cores. Using the texture mechanism available with GPU, working with the neighbors is quite simple and efficient. Nevertheless, this implies to store all the particles into a regular grid at each time step during the simulation. This part is somewhere the

This section first presents the main parallel patterns (MAP, SORT, SCAN, etc.) and then shows how they can be combined to write a new fast parallel SPH solver.

Writing a parallel algorithm is not as simple as writing a sequential algorithm. This truism is based on the necessary consideration of the collaborations between the different processors of a parallel machine: all the processors must work in concert, and not isolated as in a sequential approach. These collaborative aspects are the main difficulty. How to make sure all these processors expect when it's needed

Rather than writing a parallel algorithm based on classical sequential patterns, parallel patterns make it possible to write a parallel algorithm directly, abstracting the underlying machine. These patterns rely on very simple parallel architecture, called the parallel random-access memory (PRAM). It assumes a synchronization between an infinite set of processors and an infinite amount of memory [8].

Simple parallel patterns do not need synchronization. This means that, using a GPU or an HPC, they may be run without any difficulties, even with less processors than needed. The simpler one is the MAP, or transform, that consists in applying a given function f to an input data to obtain the output. The key of this pattern is about the localisation of the data: input and output are generally considered as

Yi ¼ fð Þ Xi ð Þ MAP (27)

most complicated, and the key step for an efficient implementation.

and work to the fullest when no synchronization is required?

vectors (or arrays). Then, MAP applied to data at the same index:

Figure 1 describes this pattern on small arrays.

<sup>j</sup> � ∑ n k6¼i

djkpl k

� Wh xi � xj

� � (26)

mj ∑ n j dijpl

∑pl <sup>i</sup> ¼ ∑ n j

Seismic Waves - Probing Earth System

3. Parallel SPH implementation

computers.

3.1 Parallel patterns

3.1.1 Simple parallel patterns

62

Illustration of the MAP parallel pattern.

#### Figure 2.

The SCATTER and GATHER patterns move data using a permutation.

In many occasions, it is necessary to write the result at a new location, another index. When each possible destination index is used once and only once, we obtain a quite simple parallel pattern called SCATTER. It consists of writing the input data from location i to the destination location map ið Þ, map being a permutation function. Figure 2 illustrates this parallel pattern.

In the same spirit, the GATHER parallel pattern writes at index i data coming from index map ið Þ, using again a permutation function. To differentiate between a SCATTER and a GATHER, you should remember that at first we read contiguous data, while in the second, we write at contiguous location. This is resumed with the following two expressions:

$$Y\_{map(i)} = X\_i \quad \text{(SCATTER)}\tag{28}$$

$$Y\_i = X\_{map(i)} \quad \text{(GATHER)}\tag{29}$$

PRAM model is very useful to write efficient algorithms on theory. Nevertheless, at the end these algorithms run on real computers, with a limited amount of memory and a fixed number of processors. Brent's theorem links the theoretical computation time on PRAM model with the one obtained using only p processors: an algorithm made in <sup>O</sup>ð Þ<sup>1</sup> using <sup>m</sup> processors will run in <sup>O</sup> <sup>m</sup> p using only <sup>p</sup> processors. This allows to predict the behavior of (very) simple algorithm on a GPU.

#### 3.1.2 Advanced parallel patterns

In many cases, some degree of collaboration is needed between processors. This leads to some more complicated parallel patterns. A very common parallel pattern using such a collaboration is the SORT that sorts data according to a given order. It is used in previous SPH implementation for building the neighbors' grid. The SORT pattern is based on the PARTITION pattern that moves values with respect to a given predicate. More precisely, for n values Xi and using the predicate Pi ∈½ � 0; 1 , the PARTITION pattern moves the values Xi for which Pi ¼ 1 at the beginning of the resulting array, the others at the end (see Figure 3 for a simple example).

Figure 3.

Illustration of the PARTITION pattern for nine input values; the values with predicate 1 are put at the beginning of the output, the others at the end.

These complex patterns are built using a fundamental pattern called SCAN. It corresponds to a prefix sum of values, according to the following expression:

$$Y\_i = \bigoplus\_{j=0}^{i} X\_j \quad (\text{INCLUSIVE} - \text{SCAN}) \tag{30}$$

The last programming tool this section covers is the atomic operation notion. A load-modify-write operation cannot be handled in parallel program without caution. Let us consider two processors doing a "plus one" in parallel at the same time. The addition is done by the CPU using registers (local memory to the CPU). Hence the variables to add need to be loaded from the main memory, then added, and then stored into the main memory. If the two processors do the load-modify-write operation at the same time exactly on the same variable, then the result is false. If the processors are not exactly synchronized, the result is certainly false also: to be correct, the two operations must be done sequentially. Atomic operations provide this behavior, performing the read-modify-write operation for one and only one

Obviously, other parallel patterns exist. They are not discussed in this chapter

Previous SPH implementations use the SORT pattern to build the neighbors' grid [7, 9, 10]. The first step consists in calculating the grid index of each particle, using a MAP. Next, the particles are sorted with respect to this index. Then, it is necessary to compute the number of particles per cell and the starting position of each cell. In [9], atomic operations are used for these two operations: the minimum for the first

In Ref. [7], authors follow a similar approach with the particle sort with respect to their cell index but using a MAP to mark the start and the end of each cell with

The main problem is that the sorting algorithm takes a large part of the computation time, near 30% according to [10]. In this chapter, we avoid the full sorting by combining simple parallel patterns and atomic operations. Our grid building algo-

This algorithm uses the Nvidia Thrust API with some freedom to shorten it. First, at line 8 the number of particles per grid cell is set to zero. Next, like with previous methods at line 9, the index of each particle is calculated with a MAP. Using a second MAP at line 10, the particle cell offset is calculated using an atomic addition. More precisely, we use the CUDA int. atomicAdd(int\*cc, int. a) function that adds a to the variable \*cc and returns the old content of \*cc. Since atomic operations are done in sequence, the number of particles per cell is correctly computed. Moreover, each particle receives the old counter value, which is 0 for the first atomic operation execution, 1 for the second, and so on up to m 1 for the last particle added to the

particle into each cell and the addition for the number of particles per cell.

respect to the sorted cell indices, considering their unicity.

rithm is summarized in Figure 5.

Figure 5.

65

Our algorithm to build the neighbors' grid.

since they are not used in our SPH implementation.

processor at a time.

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

3.2 Grid building

The fundamental pattern exists in two versions: inclusive and exclusive ones. The first corresponds to the expression given above, doing a sum-up to the current output position. The exclusive version omits the current position, doing a sum-up to i � 1 and using a nil value for Y<sup>0</sup> (generally, using 0):

$$Y\_i = \bigoplus\_{j=0}^{i-1} X\_i \quad \text{(EXCLUSIVE - SCAN)}\tag{31}$$

Figure 4 shows that these two versions of SCAN are almost the same, except the shift between the resulting arrays: the values obtained with inclusive version correspond to the ones obtained with the exclusive version at the same position plus one.

Another pattern of interest into this chapter is the REDUCE that allows to calculate a single value from an array of values and using any given associative binary function:

$$Y = \bigoplus\_{j=0}^{n-1} X\_i \quad (\text{REDUCE}) \tag{32}$$

For instance, using X ¼ ½ � 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 and the classical integer sum as binary operator, this pattern returns Y ¼ 55, the sum of the 10 first non-nil integers.

These complex patterns have roughly speaking all the same complexity, in <sup>O</sup>ð Þ log <sup>n</sup> on a PRAM machine and O <sup>n</sup> <sup>p</sup> log <sup>n</sup> p using <sup>p</sup> processors only. Nevertheless, since they are built using the SCAN, PARTITION and SORT are in practice more complex and take more time. A fast implementation of the SORT pattern relies on the radix sort algorithm that loops over the number of digits of the maximum key to sort, thus having a practical complexity in Oð Þ 32 log n with 32-bit integers.


Figure 4. Differences between the inclusive and exclusive SCAN patterns.

#### Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

The last programming tool this section covers is the atomic operation notion. A load-modify-write operation cannot be handled in parallel program without caution. Let us consider two processors doing a "plus one" in parallel at the same time. The addition is done by the CPU using registers (local memory to the CPU). Hence the variables to add need to be loaded from the main memory, then added, and then stored into the main memory. If the two processors do the load-modify-write operation at the same time exactly on the same variable, then the result is false. If the processors are not exactly synchronized, the result is certainly false also: to be correct, the two operations must be done sequentially. Atomic operations provide this behavior, performing the read-modify-write operation for one and only one processor at a time.

Obviously, other parallel patterns exist. They are not discussed in this chapter since they are not used in our SPH implementation.

#### 3.2 Grid building

These complex patterns are built using a fundamental pattern called SCAN. It

The fundamental pattern exists in two versions: inclusive and exclusive ones. The first corresponds to the expression given above, doing a sum-up to the current output position. The exclusive version omits the current position, doing a sum-up to

Figure 4 shows that these two versions of SCAN are almost the same, except the shift between the resulting arrays: the values obtained with inclusive version correspond to the ones obtained with the exclusive version at the same position plus one. Another pattern of interest into this chapter is the REDUCE that allows to calculate a single value from an array of values and using any given associative

For instance, using X ¼ ½ � 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 and the classical integer sum as binary operator, this pattern returns Y ¼ 55, the sum of the 10 first non-nil integers. These complex patterns have roughly speaking all the same complexity, in

> <sup>p</sup> log <sup>n</sup> p

since they are built using the SCAN, PARTITION and SORT are in practice more complex and take more time. A fast implementation of the SORT pattern relies on the radix sort algorithm that loops over the number of digits of the maximum key to

sort, thus having a practical complexity in Oð Þ 32 log n with 32-bit integers.

Xj ð Þ INCLUSIVE � SCAN (30)

Xi ð Þ EXCLUSIVE � SCAN (31)

Xi ð Þ REDUCE (32)

using p processors only. Nevertheless,

corresponds to a prefix sum of values, according to the following expression:

Illustration of the PARTITION pattern for nine input values; the values with predicate 1 are put at the

Yi ¼ ⨁ i j¼0

beginning of the output, the others at the end.

Seismic Waves - Probing Earth System

i � 1 and using a nil value for Y<sup>0</sup> (generally, using 0):

Yi ¼ ⨁ i�1 j¼0

> Y ¼ ⨁ n�1 j¼0

binary function:

Figure 4.

64

Figure 3.

<sup>O</sup>ð Þ log <sup>n</sup> on a PRAM machine and O <sup>n</sup>

Differences between the inclusive and exclusive SCAN patterns.

Previous SPH implementations use the SORT pattern to build the neighbors' grid [7, 9, 10]. The first step consists in calculating the grid index of each particle, using a MAP. Next, the particles are sorted with respect to this index. Then, it is necessary to compute the number of particles per cell and the starting position of each cell. In [9], atomic operations are used for these two operations: the minimum for the first particle into each cell and the addition for the number of particles per cell.

In Ref. [7], authors follow a similar approach with the particle sort with respect to their cell index but using a MAP to mark the start and the end of each cell with respect to the sorted cell indices, considering their unicity.

The main problem is that the sorting algorithm takes a large part of the computation time, near 30% according to [10]. In this chapter, we avoid the full sorting by combining simple parallel patterns and atomic operations. Our grid building algorithm is summarized in Figure 5.

This algorithm uses the Nvidia Thrust API with some freedom to shorten it. First, at line 8 the number of particles per grid cell is set to zero. Next, like with previous methods at line 9, the index of each particle is calculated with a MAP. Using a second MAP at line 10, the particle cell offset is calculated using an atomic addition. More precisely, we use the CUDA int. atomicAdd(int\*cc, int. a) function that adds a to the variable \*cc and returns the old content of \*cc. Since atomic operations are done in sequence, the number of particles per cell is correctly computed. Moreover, each particle receives the old counter value, which is 0 for the first atomic operation execution, 1 for the second, and so on up to m 1 for the last particle added to the

Figure 5. Our algorithm to build the neighbors' grid. cell, m being the number of particles added into the cell. These values are used to scatter all the particles to their local position into the grid, at line 13. But, before to do that we need to calculate the global grid offset, corresponding to the position of the first particle of each cell. This is done using an exclusive SCAN at line 11 to compute the global offset, followed by a MAP at line 12 to calculate each particle global offset.

<sup>X</sup>ðÞ¼ <sup>t</sup> Ct � <sup>θ</sup>

<sup>θ</sup>lþ<sup>1</sup> <sup>¼</sup> <sup>θ</sup><sup>l</sup> � <sup>θ</sup><sup>l</sup> � <sup>κ</sup>Ct <sup>þ</sup> <sup>H</sup>

where H and h are the wave height and the water depth, respectively.

More precisely, θ is the solution of the following problem:

X tðÞ¼ <sup>H</sup> κh

coefficient, and θ is given using Newton's method by

tsunami solitary wave.

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

Figure 6.

Figure 7.

67

Tsunami simulation before the solitary wave arrives.

Tsunami wave reaching the building near the beach.

where C is the wave velocity, t is the time in the simulation frame, κ is the decay

<sup>l</sup> <sup>þ</sup> <sup>H</sup>

While this method works on CPU, it is not well-suited for a CUDA implementation of the IISSPH, mainly because the number of iterations of the Newton-Raphson method depends on the input values, and so is not constant per particle. Hence, in this chapter we use a different but simple technique. The wave is produced using a piston wave generator. Here, the piston is a huge virtual object that moves the water to reach the speed of the wave. The length and the speed of the piston movement are calibrated to obtain the good height and speed of the

Figure 6 illustrates such a simple wave simulation, before the tsunami wave arrives. Figure 7 shows the wave arriving at the building at t ¼ 4 s in the simulation frame. In Figure 8, at t ¼ 8 s, the building is completely below the ocean that returns

κ

<sup>h</sup> tan <sup>θ</sup><sup>l</sup>

<sup>h</sup> sech<sup>2</sup> <sup>θ</sup><sup>l</sup> (34)

tanhð Þ κð Þ Ct–X tð Þ (35)

(33)

In practice, this algorithm can be optimized in many ways. First, the device vectors can be allocated only once, and not each time the grid is built. Second, the first two transforms (lines 9 and 10) can be mixed into one. This will limit the memory loading into device registers, known as a major performance limitation with GPU. At last, the last transform (line 12) and the scatter (line 13) can be mixed into a single call again to minimize the memory bandwidth usage. Moreover, the particles' data must be split into multiple arrays for efficiency (one array for position, one for density, one for pressure, etc.) as in [10].

#### 3.3 Main algorithm

The most difficult part of the implementation of the IISPH method is the construction of the neighbors' grid, as for any non-mesh density kernel method. The rest of the calculation is rather simple and relies on two parallel models: the MAP for all the loops on particles and the REDUCE to control the termination of the corrective loop in the calculation of the pressure force.

It is noticeable that the IISPH loop to correct the pressure force runs on the CPU, because there is no available global synchronization on the GPU. Then, the REDUCE is used to return a value from the GPU to the CPU, to decide if more corrections are needed or not. Nevertheless, since this just consists of sending one real value, it is not a big bottleneck.

Moreover, many calculations use data from the neighbors (pressure, density, position, etc.). L1 GPU's memory is used to accelerate these calculations, reducing the computation time around a third in our experiment. Notice also that the IISPH corrective loop amortizes the neighbors' grid building. In our experiments the grid building now represents less than 10 percent of the full computation time.

#### 4. Experiments

The IISPH is a valid solution to simulate a tsunami [11]. Its main advantage regarding a discrete method is that it does not need to refine the mesh near the obstacles, like the coast and the buildings. Moreover, the wave can go everywhere, including interfering with the beach, buildings, infrastructure, etc.

In this chapter, we illustrate the tsunami simulation using IISPH algorithm through a rather simple scenario. It contains a short coast ending with a mountain. We put a building just after the beach. The main difficulty, if either, consists of generating the solitary wave. A tsunami, for instance, is generated by an earthquake at long distance. The produced wave runs at 200 meter per second (720 km/h). We do not need to simulate the propagation of the wave since its epicenter, which is quite difficult with long distance: it needs very long simulation time to see the wave reaching the beach, and obviously it needs a huge amount of memory to handle the sea between the two distant locations. Instead, we simulate the wave into a rather small space. We can predict the time of arrival to the beach, assuming we know the exact distance between the beach and the earthquake location.

In [11], authors solve the solitary wave solution of Boussinesq. They calculate the wave paddle displacement using the equation:

Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

cell, m being the number of particles added into the cell. These values are used to scatter all the particles to their local position into the grid, at line 13. But, before to do that we need to calculate the global grid offset, corresponding to the position of the

In practice, this algorithm can be optimized in many ways. First, the device vectors can be allocated only once, and not each time the grid is built. Second, the first two transforms (lines 9 and 10) can be mixed into one. This will limit the memory loading into device registers, known as a major performance limitation with GPU. At last, the last transform (line 12) and the scatter (line 13) can be mixed into a single call again to minimize the memory bandwidth usage. Moreover, the particles' data must be split into multiple arrays for efficiency (one array for

The most difficult part of the implementation of the IISPH method is the construction of the neighbors' grid, as for any non-mesh density kernel method. The rest of the calculation is rather simple and relies on two parallel models: the MAP for all the loops on particles and the REDUCE to control the termination of the correc-

It is noticeable that the IISPH loop to correct the pressure force runs on the CPU,

Moreover, many calculations use data from the neighbors (pressure, density, position, etc.). L1 GPU's memory is used to accelerate these calculations, reducing the computation time around a third in our experiment. Notice also that the IISPH corrective loop amortizes the neighbors' grid building. In our experiments the grid

The IISPH is a valid solution to simulate a tsunami [11]. Its main advantage regarding a discrete method is that it does not need to refine the mesh near the obstacles, like the coast and the buildings. Moreover, the wave can go everywhere,

In this chapter, we illustrate the tsunami simulation using IISPH algorithm through a rather simple scenario. It contains a short coast ending with a mountain. We put a building just after the beach. The main difficulty, if either, consists of generating the solitary wave. A tsunami, for instance, is generated by an earthquake at long distance. The produced wave runs at 200 meter per second (720 km/h). We do not need to simulate the propagation of the wave since its epicenter, which is quite difficult with long distance: it needs very long simulation time to see the wave reaching the beach, and obviously it needs a huge amount of memory to handle the sea between the two distant locations. Instead, we simulate the wave into a rather small space. We can predict the time of arrival to the beach, assuming we know the

In [11], authors solve the solitary wave solution of Boussinesq. They calculate

because there is no available global synchronization on the GPU. Then, the REDUCE is used to return a value from the GPU to the CPU, to decide if more corrections are needed or not. Nevertheless, since this just consists of sending one

building now represents less than 10 percent of the full computation time.

including interfering with the beach, buildings, infrastructure, etc.

exact distance between the beach and the earthquake location.

the wave paddle displacement using the equation:

first particle of each cell. This is done using an exclusive SCAN at line 11 to compute the global offset, followed by a MAP at line 12 to calculate each particle

position, one for density, one for pressure, etc.) as in [10].

tive loop in the calculation of the pressure force.

real value, it is not a big bottleneck.

global offset.

Seismic Waves - Probing Earth System

3.3 Main algorithm

4. Experiments

66

$$\mathbf{X}(t) = \mathbf{C}t - \frac{\theta}{\kappa} \tag{33}$$

where C is the wave velocity, t is the time in the simulation frame, κ is the decay coefficient, and θ is given using Newton's method by

$$\Theta^{l+1} = \Theta^l - \frac{\Theta^l - \kappa \mathbf{C}t + \frac{H}{\hbar} \tan \left(\Theta^l\right)}{l + \frac{H}{\hbar} \text{sech}^2 \left(\Theta^l\right)} \tag{34}$$

More precisely, θ is the solution of the following problem:

$$\mathbf{X}(\mathbf{t}) = \frac{\mathbf{H}}{\kappa \hbar} \tanh(\kappa(\mathbf{C}t - \mathbf{X}(t))) \tag{35}$$

where H and h are the wave height and the water depth, respectively.

While this method works on CPU, it is not well-suited for a CUDA implementation of the IISSPH, mainly because the number of iterations of the Newton-Raphson method depends on the input values, and so is not constant per particle.

Hence, in this chapter we use a different but simple technique. The wave is produced using a piston wave generator. Here, the piston is a huge virtual object that moves the water to reach the speed of the wave. The length and the speed of the piston movement are calibrated to obtain the good height and speed of the tsunami solitary wave.

Figure 6 illustrates such a simple wave simulation, before the tsunami wave arrives. Figure 7 shows the wave arriving at the building at t ¼ 4 s in the simulation frame. In Figure 8, at t ¼ 8 s, the building is completely below the ocean that returns

Figure 6. Tsunami simulation before the solitary wave arrives.

Figure 7. Tsunami wave reaching the building near the beach.

human living near the coasts. Such phenomena also may produce strong degradation on buildings and structures, in turn inducing human loss as what happened after the Tohoku earthquake in 2011. To avoid these disasters, it is important to be able to validate the robustness of structures and buildings near the dangerous coasts and to inform population after an always unpredictable submarine earthquake. To achieve these goals, it is necessary to produce robust and fast fluid simulator software. To simulate a tsunami wave, a good candidate is the SPH method. Since it does not need the usage of a fix mesh like in discrete techniques, it allows to handle correctly the wave running on the beach and after. Moreover, it correctly handles the contact with buildings and structures, allowing to simulate the forces that they

This chapter recalls the implicit incompressible SPH method, which is one of the fastest among the SPH ones. The parallel implementation for GPU is detailed in depth, with a fast algorithm to build the neighbors' grid, avoiding the classical

At last, this chapter proposes a simple tsunami wave simulation using a piston wave generator, a simple solution for implementing and providing valuable results. It can be used to simulate tsunami generated by submarine earthquake occurring in a pattern of seismic source mechanism when both the location and intensity are

, Emmanuelle Darles<sup>1</sup>

1 XLIM, UMR 7252, CNRS, University of Poitiers, France

2 LMA, UMR 7348, CNRS, University of Poitiers, France

\*Address all correspondence to: lilian.aveneau@univ-poitiers.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

, Pierre-Yves Louis<sup>2</sup>

sorting method which is more time-consuming.

undergo.

Efficient Simulation of Fluids

DOI: http://dx.doi.org/10.5772/intechopen.86619

estimated.

Author details

69

Pierre Thuillier Le Gac<sup>1</sup>

\*

provided the original work is properly cited.

and Lilian Aveneau<sup>1</sup>

Figure 8. The tsunami wave engulfed the building and the coast.

Figure 9. The tsunami wave begins to pull off the coast. With flat coasts, this may take some time.

Figure 10. After a longer time, the tsunami wave has almost completely disappeared.

into its bed after some more time (see Figures 9 and 10). The building is made with fixed particles that nevertheless are considered with the moving particles of the fluid. This allows an interaction between two kinds of particles, and it permits to obtain the pressure and force applied to the building, for instance, to check if it will resist or not.

#### 5. Conclusions

This chapter focusses on the simulation of a tsunami solitary wave. Such a wave is mainly produced by submarine earthquake and may provoke vast disasters for

#### Efficient Simulation of Fluids DOI: http://dx.doi.org/10.5772/intechopen.86619

human living near the coasts. Such phenomena also may produce strong degradation on buildings and structures, in turn inducing human loss as what happened after the Tohoku earthquake in 2011. To avoid these disasters, it is important to be able to validate the robustness of structures and buildings near the dangerous coasts and to inform population after an always unpredictable submarine earthquake.

To achieve these goals, it is necessary to produce robust and fast fluid simulator software. To simulate a tsunami wave, a good candidate is the SPH method. Since it does not need the usage of a fix mesh like in discrete techniques, it allows to handle correctly the wave running on the beach and after. Moreover, it correctly handles the contact with buildings and structures, allowing to simulate the forces that they undergo.

This chapter recalls the implicit incompressible SPH method, which is one of the fastest among the SPH ones. The parallel implementation for GPU is detailed in depth, with a fast algorithm to build the neighbors' grid, avoiding the classical sorting method which is more time-consuming.

At last, this chapter proposes a simple tsunami wave simulation using a piston wave generator, a simple solution for implementing and providing valuable results. It can be used to simulate tsunami generated by submarine earthquake occurring in a pattern of seismic source mechanism when both the location and intensity are estimated.

### Author details

Pierre Thuillier Le Gac<sup>1</sup> , Emmanuelle Darles<sup>1</sup> , Pierre-Yves Louis<sup>2</sup> and Lilian Aveneau<sup>1</sup> \*

1 XLIM, UMR 7252, CNRS, University of Poitiers, France

2 LMA, UMR 7348, CNRS, University of Poitiers, France

\*Address all correspondence to: lilian.aveneau@univ-poitiers.fr

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

into its bed after some more time (see Figures 9 and 10). The building is made with fixed particles that nevertheless are considered with the moving particles of the fluid. This allows an interaction between two kinds of particles, and it permits to obtain the pressure and force applied to the building, for instance, to check if it will resist or not.

The tsunami wave begins to pull off the coast. With flat coasts, this may take some time.

After a longer time, the tsunami wave has almost completely disappeared.

This chapter focusses on the simulation of a tsunami solitary wave. Such a wave is mainly produced by submarine earthquake and may provoke vast disasters for

5. Conclusions

68

Figure 10.

Figure 8.

Figure 9.

The tsunami wave engulfed the building and the coast.

Seismic Waves - Probing Earth System

### References

[1] Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society. 1997;181(3): 375-389. DOI: 10.1093/mnras/181.3.375

[2] Lucy LB. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal. 1977;82: 1013-1024

[3] Müller M, Charypar D, Gross M. Particle-based fluid simulation for interactive applications. In: Proceedings of the 2003 ACM SIGGRAPH/ Eurographics Symposium on Computer Animation. 2003. pp. 154-159

[4] Silverman B. Density Estimation for Statistics and Data Analysis. New York: Routledge; 1998. DOI: 10.1201/ 9781315140919

[5] Solenthaler B, Pajarola R. Predictivecorrective incompressible SPH. In: Hoppe H, editor. ACM SIGGRAPH 2009 papers (SIGGRAPH '09). New York, NY, USA: ACM; 2009. 6 p. DOI: 10.1145/1576246.1531346

[6] Shao S, Lo EYM. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources. 2003;26(6):787-800. DOI: 10.1016/ S0309-1708(03)00030-7

[7] Ihmsen M, Cornelis J, Solenthaler B, Horvath C, Teschner M. Implicit incompressible SPH. IEEE Transactions on Visualization and Computer Graphics. 2014;20(3):426-435. DOI: 10.1109/TVCG.2013.105

[8] Blelloch GE. Vector Models for Data-Parallel Computing. Cambridge, Massachusetts, London, England: MIT Press; 1990

[9] Goswami P, Schlegel P, Solenthaler B, Pajarola R. Interactive SPH simulation and rendering on the GPU. In: Eurographics/ACM SIGGRAPH Symposium on Computer Animation; 2010. pp. 1-10

[10] Bilotta G, Zago V, Hérault A. Design and implementation of particle systems for meshfree methods with high performance. In: High Performance Parallel Computing. IntechOpen; 2018. DOI: 10.5772/intechopen.81755

[11] Sampath R, Montanari N, Akinci N, Prescott S, Smith C. Large-scale solitary wave simulation with implicit incompressible SPH. Journal of Ocean Engineering and Marine Energy. 2016; 2(3):313-329. DOI: 10.1007/s40722-016- 0060-8

**71**

NIED Hi-net

**1. Introduction**

**Chapter 5**

**Abstract**

and S-net Data

Seismic Velocity Structure in and

Derived from Seismic Tomography

Including NIED MOWLAS Hi-net

*Masashi Mochizuki, Toshihiko Kanazawa, Narumi Takahashi,* 

Japanese Islands are composed of four plates, with two oceanic plates subducting beneath the two continental plates. In 2016 the National Research Institute for Earth Science and Disaster Resilience (NIED) Seafloor Observation Network for Earthquakes and Tsunamis along the Japan Trench (S-net) started seismic observation of the offshore Hokkaido to Boso region in the Pacific Ocean, and Dense Oceanfloor Network System for Earthquakes and Tsunamis (DONET) was transferred to NIED. We add the NIED S-net and DONET datasets to NIED highsensitivity seismograph network (Hi-net) and full range seismograph network (F-net) datasets used in the previous study and obtain the three-dimensional seismic velocity structure beneath the Pacific Ocean as well as Japanese Islands. NIED S-net data dramatically improve the resolution beneath the Pacific Ocean at depths of 10–20 km because the seismic stations are located above the earthquakes and on the east side of the Japan Trench. We find a NS high-Vp zone at depths of 20–30 km. The 2018 Eastern Iburi earthquake occurred below the northern part of this high-V zone. The coseismic slip plane of the 2011 Tohokuoki earthquake has low Vp/Vs, but its large slip region has high Vp. The broad

around the Japanese Island Arc

*Makoto Matsubara, Hiroshi Sato, Kenji Uehira,* 

low-Vp/Vs region may play a role in large earthquake occurrence.

both at the plate interfaces and within the plates.

**Keywords:** seismic tomography, failed rift, offshore event, NIED S-net, DONET,

American (NA) plates, and a number of small islands are on the Philippine Sea (PHS) and the Pacific (PAC) plates (**Figure 1**). The PHS and PAC oceanic plates are subducting beneath the EUR and the NA plates. A number of earthquakes occurred

The Japanese Islands are mainly composed of the Eurasian (EUR) and the North

*Kensuke Suzuki and Shin'ichiro Kamiya*

#### **Chapter 5**

References

1013-1024

[1] Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society. 1997;181(3): 375-389. DOI: 10.1093/mnras/181.3.375

Seismic Waves - Probing Earth System

[9] Goswami P, Schlegel P, Solenthaler

simulation and rendering on the GPU. In: Eurographics/ACM SIGGRAPH Symposium on Computer Animation;

[10] Bilotta G, Zago V, Hérault A. Design and implementation of particle systems for meshfree methods with high performance. In: High Performance Parallel Computing. IntechOpen; 2018.

[11] Sampath R, Montanari N, Akinci N, Prescott S, Smith C. Large-scale solitary

incompressible SPH. Journal of Ocean Engineering and Marine Energy. 2016; 2(3):313-329. DOI: 10.1007/s40722-016-

B, Pajarola R. Interactive SPH

DOI: 10.5772/intechopen.81755

wave simulation with implicit

2010. pp. 1-10

0060-8

[2] Lucy LB. A numerical approach to the testing of the fission hypothesis. The

[3] Müller M, Charypar D, Gross M. Particle-based fluid simulation for interactive applications. In: Proceedings

Eurographics Symposium on Computer

[4] Silverman B. Density Estimation for Statistics and Data Analysis. New York:

[5] Solenthaler B, Pajarola R. Predictivecorrective incompressible SPH. In: Hoppe H, editor. ACM SIGGRAPH 2009 papers (SIGGRAPH '09). New York, NY, USA: ACM; 2009. 6 p. DOI:

[6] Shao S, Lo EYM. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources. 2003;26(6):787-800. DOI: 10.1016/

[7] Ihmsen M, Cornelis J, Solenthaler B, Horvath C, Teschner M. Implicit incompressible SPH. IEEE Transactions

[8] Blelloch GE. Vector Models for Data-

on Visualization and Computer Graphics. 2014;20(3):426-435. DOI:

Parallel Computing. Cambridge, Massachusetts, London, England: MIT

Astronomical Journal. 1977;82:

of the 2003 ACM SIGGRAPH/

Animation. 2003. pp. 154-159

Routledge; 1998. DOI: 10.1201/

10.1145/1576246.1531346

S0309-1708(03)00030-7

10.1109/TVCG.2013.105

Press; 1990

70

9781315140919
