2. Motion equation of elastic medium

We consider an isotropic elastic medium with Lame's parameters λ,μ, and a density r. Let us denote x = xjej, ej as the unit vectors of Cartesian coordinate system in the space R<sup>3</sup> ; displacements vector u(x,t) = ujej; stress tensors σij deformation tensor εij. These tensors are connected by Hook's law [1]:

$$
\varepsilon\_{i\rangle} = 0, \mathbf{5} \{ \mu\_{i\cdot j} + \mu\_{j\cdot i} \}, \quad \text{i.j.} \, k = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{1}
$$

$$
\sigma\_{ij} = C\_{ij}^{kl} \varepsilon\_{kl} = C\_{ij}^{kl} u\_{k,l} \tag{2}
$$

ð4Þ

131

ð5Þ

ð6Þ

ð8Þ

ð10Þ

, satisfying the characteristic

Here L is the matrix Lame's operator:

<sup>c</sup><sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi

We note that ν(x,t)=(ν1,ν2,ν3,νt) is a normal vector to F in R<sup>4</sup>

the direction of wave propagation. By virtue of Eq. (7),

consistency conditions for solutions at the wave front:

is the mass force, Δ is the Laplace operator.

µ=r p are the velocities of dilatational and shear waves (c<sup>1</sup> > c2), G(x,t)

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

http://dx.doi.org/10.5772/intechopen.74538

ν<sup>t</sup> ¼ �cj∥ν∥3, j ¼ 1, 2: (7)

½ � u xð Þ ; t Ft ¼ 0 (9)

The system shown in Eq. (4) was fairly well studied by Petrashen [2]. Since the elastic potential of the medium is positive definite, this system is strictly hyperbolic. Such systems can have solutions with discontinuous derivatives. The discontinuity surface <sup>F</sup> in R4 <sup>=</sup> <sup>R</sup><sup>3</sup> � <sup>t</sup>(�<sup>∞</sup> <t< <sup>∞</sup>) coincides with a characteristic surface of the system. It corresponds to a wave front Ft moving

From Eqs. (5) and (7), we get that Ft moves in R<sup>3</sup> at the sound velocity V = c<sup>1</sup> or V = c2.

We introduce a wave vector m = (m1, m2, m3). It is a unit normal vector to Ft in R<sup>3</sup> for fixed t in

Let ν<sup>t</sup> = ν4. The requirement that the displacements be continuous across the wave front, i.e.,

which is associated with the preservation of the continuity of the medium, leads to kinematic

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>µ</sup> � �=r,

in R<sup>3</sup> at the velocity V:

This equation has the roots:

c<sup>1</sup> ¼

q

equation

The elastic constant tensor has the symmetry properties.

$$C\_{ij}^{kl} = C\_{ji}^{kl} = C\_{ij}^{lk} = C\_{kl,j}^{ij}$$

In the case of an isotropic medium, it is equal to

$$C\_{ij}^{kl} = \lambda \delta\_i^j \delta\_l^k + \mu (\delta\_i^k \delta\_j^l + \delta\_j^k \delta\_i^l)\_{,i}$$

and Hook's law has the form

$$\sigma\_{\vec{\eta}} = \lambda \operatorname{div} \boldsymbol{\mu} \delta\_{\vec{\eta}} + \mu (\mu\_{i,j} + \mu\_{j,i})$$

Here δij ¼ δ j <sup>i</sup> is the Kronecker symbol. Everywhere, there are tensor convolutions over of the same name indexes from 1 to 3, ui, <sup>j</sup> ≜ <sup>∂</sup>ui ∂xj .

Motion equations for material continuum

$$\frac{\partial \sigma\_{ij}}{\partial x\_j} + G\_i = \rho \frac{\partial^2 \bar{u}\_i^2}{\partial t^2}, \quad i, j = 1, 2, 3 \tag{3}$$

for elastic medium by using Eqs. (1) and (2) have the form:

General Functions Method in Transport Boundary Value Problems of Elasticity Theory http://dx.doi.org/10.5772/intechopen.74538 131

$$\left(L\_t^\beta \left(\partial\_\mathbf{x}, \partial\_t\right)u\_\beta + G\_\delta = 0\right) \tag{4}$$

Here L is the matrix Lame's operator:

velocities of dilatational and shear waves propagation. This has a large effect on the type of equations and leads to systems of elliptic, hyperbolic, or mixed equations. For transport problems, typical factors are shock effects generated by supersonic loading. At shock fronts, the stresses, displacement rates, and energy density are discontinuous. A convenient research method for such problems is provided by the theory of generalized functions (distributions), which makes it possible to significantly expand the class of processes amenable to study by using singular generalized functions in the simulation of observed phenomena. In this chapter, methods of this theory are used to solve boundary value problems using motion equations of the theory of elasticity in cylindrical domains under the action of transport loads, moving at

We consider an isotropic elastic medium with Lame's parameters λ,μ, and a density r. Let us

ments vector u(x,t) = ujej; stress tensors σij deformation tensor εij. These tensors are connected

σij ¼ λdivuδij þ µ ui, <sup>j</sup> þ uj, <sup>i</sup>

∂xj . <sup>i</sup> is the Kronecker symbol. Everywhere, there are tensor convolutions over of the

, i, j, k <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>: (1)

; displace-

ð2Þ

ð3Þ

denote x = xjej, ej as the unit vectors of Cartesian coordinate system in the space R<sup>3</sup>

εij ¼ 0, 5 ui, <sup>j</sup> þ uj, <sup>i</sup>

supersonic and supersonic speeds.

130 Differential Equations - Theory and Current Research

by Hook's law [1]:

2. Motion equation of elastic medium

The elastic constant tensor has the symmetry properties.

In the case of an isotropic medium, it is equal to

and Hook's law has the form

same name indexes from 1 to 3, ui, <sup>j</sup> ≜ <sup>∂</sup>ui

Motion equations for material continuum

for elastic medium by using Eqs. (1) and (2) have the form:

j

Here δij ¼ δ

$$L\_i^f(\partial\_\mathbf{x}, \partial\_t) = (c\_1^2 - c\_2^2) \frac{\partial}{\partial x\_i} \frac{\partial}{\partial x\_j} + \delta\_i^f \left(c\_2^2 \Delta - \frac{\partial^2}{\partial t^2}\right)$$

c<sup>1</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>µ</sup> � �=r, q <sup>c</sup><sup>2</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffi µ=r p are the velocities of dilatational and shear waves (c<sup>1</sup> > c2), G(x,t) is the mass force, Δ is the Laplace operator.

The system shown in Eq. (4) was fairly well studied by Petrashen [2]. Since the elastic potential of the medium is positive definite, this system is strictly hyperbolic. Such systems can have solutions with discontinuous derivatives. The discontinuity surface <sup>F</sup> in R4 <sup>=</sup> <sup>R</sup><sup>3</sup> � <sup>t</sup>(�<sup>∞</sup> <t< <sup>∞</sup>) coincides with a characteristic surface of the system. It corresponds to a wave front Ft moving in R<sup>3</sup> at the velocity V:

$$\mathcal{V} = -\nu\_t^2 \mathcal{V} \|\nu\|\_3, \qquad \mathcal{V} \|\nu\|\_3 \triangleq \sqrt{\sum\_{k=1}^{39} \nu\_k^2} \tag{5}$$

We note that ν(x,t)=(ν1,ν2,ν3,νt) is a normal vector to F in R<sup>4</sup> , satisfying the characteristic equation

$$\det\left\{ \left( c\_1^2 - c\_2^2 \right) \nu\_i \nu\_j + \delta\_{ij} \left( c\_2^2 \left\| \nu \right\|\_3^2 - \nu\_t^2 \right) \right\} = \left( c\_1^2 \left\| \nu \right\|\_3^2 - \nu\_t^2 \right) \left( c\_2^2 \left\| \nu \right\|\_3^2 - \nu\_t^2 \right)^2 = 0\_\cdot \tag{6}$$

This equation has the roots:

$$\mathbb{L}\nu\_{\ell} = \pm c\_{j} \|\nu\|\_{3^{\nu}} \quad j = 1, 2. \tag{7}$$

From Eqs. (5) and (7), we get that Ft moves in R<sup>3</sup> at the sound velocity V = c<sup>1</sup> or V = c2.

We introduce a wave vector m = (m1, m2, m3). It is a unit normal vector to Ft in R<sup>3</sup> for fixed t in the direction of wave propagation. By virtue of Eq. (7),

$$m\_j = \frac{\nu\_j}{\|\nu\|\_3} = -\nabla \nu\_j / \nu\_t \tag{8}$$

Let ν<sup>t</sup> = ν4. The requirement that the displacements be continuous across the wave front, i.e.,

$$[\mu(\mathbf{x},t)]\_{\mathbb{F}\_t} = \mathbf{0} \tag{9}$$

which is associated with the preservation of the continuity of the medium, leads to kinematic consistency conditions for solutions at the wave front:

$$\left[m\_j \frac{\partial u\_i}{\partial t\_i} + V \frac{\partial u\_i}{\partial x\_i}\right]\_{F\_i} = 0, \quad (i, j) = (1, 2, 3) \tag{10}$$

(the continuity of the tangent derivatives on Ft). Additionally, Eq. (4) implies dynamical consistency conditions for solutions at the wave front, which are equivalent to the momentum conservation law in its neighborhood:

$$\left[\sigma\_{ij}\right]m\_j = -\rho V \left[\frac{\partial u\_i}{\partial t}\right]\_{F\_i}, \quad i, j = 1, 2, 3\tag{11}$$

Since the original system is hyperbolic, Eq. (14) can also have discontinuous solutions. Let F be

moves at one of the sound velocities V = c1,c<sup>2</sup> in the space of (x1,x2,x3). It follows from Eq. (7)

It follows from Eqs. (9) to (11) and Eq. (13) that the kinematic and dynamical consistency conditions for solutions at discontinuities in the mobile coordinate system have the form:

n ¼ f g n1; n2; nz ¼ n<sup>3</sup> is a wave vector, k = 1 for shock dilatational waves, k = 2 for shock shear waves. Here and hereafter, the derivative with respect to xj is denoted by the index j after a

Definition. If c>c2, the solution of the system in Eq. (14) is called classical if it is continuous and twice differentiable everywhere, except for, possibly, wave fronts. The number of fronts is finite at any fixed t and the conditions on the gaps, Eqs. (15) and (16), are satisfied on the wave

At first, we construct the solutions of the transport Lame equation using methods of general-

4. Shock waves as generalized solutions of transport Lame equations:

Consider Eq. (14) and its solutions on the space of generalized vector functions D'<sup>3</sup> (R<sup>3</sup>

Eq. (14) that is twice differentiable, then it is also a generalized solution of Eq. (14). If a vector function u satisfies Eq. (14) in the classical sense almost everywhere, except for some surfaces, on which its derivatives are discontinuous, then, generally speaking, u is not a generalized

Let u(x,z) be a shock wave (x = (x1,x2)), i.e., a classical solution of the Lame transport equations, Eq. (14), that satisfies conditions Eqs. (15) and (16) at the front <sup>F</sup>. Let <sup>b</sup>u xð Þ ; <sup>z</sup> denote the

Proof. Using the rules for differentiating generalized functions with derivatives having jump

Theorem 4.1. The shock wave <sup>b</sup>u xð Þ ; <sup>z</sup> is a generalized solution of the Lame equation in D'<sup>3</sup> (R<sup>3</sup>

discontinuities across some surfaces (see [3]), for the equations of motion in D'<sup>3</sup> (R<sup>3</sup>

(R<sup>3</sup>

½ � u xð Þ ; z <sup>F</sup> ¼ 0 ) nzui, <sup>j</sup> � njui, <sup>z</sup>

0

� �

<sup>σ</sup>ij � �nj ¼ �rckc ui ½ � , <sup>z</sup> <sup>F</sup>; nz ¼ �ck=c, for c <sup>≥</sup> ck; (16)

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

such that it is stationary in this space and

http://dx.doi.org/10.5772/intechopen.74538

133

. Therefore, since c = cj/n<sup>3</sup> and

<sup>F</sup> ¼ 0; (15)

) with

).

), we obtain

) (see [3]). Obviously, if u is a solution of

a discontinuity surface in the space of variables x

that V = cn3, where n = (n1,n2,n3) is the unit normal to F in R<sup>3</sup>


comma in the function notation or by the variable itself.

fronts.

ized functions theory.

solution of Eq. (14).

conditions on wave front

components being generalized functions from D<sup>0</sup>

corresponding regular generalized function.

Definition. A wave is called a shock wave if the jump in the stresses across the wave front is finite: eimj[σij]Ft = 0̸. If mj[σij]Ft = 0, then this is a weak shock wave. If mj[σij]Ft = ∞, then this is a strong shock wave.

Velocity suffers a jump discontinuity across a shock front. At fronts of weak shock waves, the velocities are continuous, but the second derivatives of solutions are not. Strong shock waves (in the sense of the aforementioned definition) do not occur in actual media, since, at large stress jumps, the medium is destroyed and ceases to be elastic. However, strong shock waves in elastic media play an important theoretical role in the construction of solutions, specifically, fundamental solutions of Eq. (4).
