**2. The parameters and motion equations of a two-component Biot's medium**

The equations of motion of a homogeneous isotropic two-component Biot's medium are described by the following system of second-order hyperbolic equations [1–3]:

$$\begin{aligned} (\dot{\boldsymbol{\mu}} + \boldsymbol{\mu}) \text{grad } \text{div} \boldsymbol{u}\_t + \mu \Delta \boldsymbol{u}\_t + \boldsymbol{Q} \operatorname{grad} \operatorname{div} \boldsymbol{u}\_f + \boldsymbol{F}^s(\mathbf{x}, t) &= \rho\_{11} \ddot{\boldsymbol{u}}\_t + \rho\_{12} \ddot{\boldsymbol{u}}\_f \\ \boldsymbol{\mathcal{Q}} \operatorname{grad} \operatorname{div} \boldsymbol{u}\_t + \boldsymbol{R} \operatorname{grad} \operatorname{div} \boldsymbol{u}\_f + \boldsymbol{F}^f(\mathbf{x}, t) &= \rho\_{12} \ddot{\boldsymbol{u}}\_t + \rho\_{22} \ddot{\boldsymbol{u}}\_f \\ (\boldsymbol{\varkappa}, t) \in \boldsymbol{\mathcal{R}}^N \times [0, \infty). \end{aligned} \tag{1}$$

Here *N* is the dimension of the space. At a plane deformation *N* = 2, the total spatial deformation corresponds to *N* = 3, at *N* = 1 the equations describe the dynamics of a porous liquid-saturated rod.

We denote *us* ¼ *usj*ð Þ *x*, *t ej* is a displacement vector of an elastic skeleton, *u <sup>f</sup>* ¼ *ufj*ð Þ *x*, *t ej* is a displacement vector of a liquid, and *ej*ð Þ *j* ¼ 1, … , *N* are basic orts of Lagrangian Cartesian coordinate system (everywhere by repeating indices, there is summation from 1 to *N*).

Constants *ρ*11, *ρ*12, *ρ*<sup>22</sup> have the dimension of mass density, and they are associated with densities of masses of particles, composing a skeleton *ρs*and a fluid *ρ <sup>f</sup>* , by relationships:

$$
\rho\_{11} = (1 - m)\rho\_s - \rho\_{12}, \quad \rho\_{22} = m\rho\_f - \rho\_{12},
$$

where *m* is a porosity of the medium. The constant of attached density *ρ*<sup>12</sup> is related to a dispersion of deviation of micro-velocities of fluid particles in pores from average velocity of fluid flow and depends on pores geometry. Elastic constants *λ*, *μ* are Lama's parameters of an isotropic elastic skeleton, and *Q,R* characterize an interaction of a skeleton with a liquid on the basis of.

### **2.1 Biot's law for stresses**

$$\begin{aligned} \sigma\_{\vec{\imath}\vec{\jmath}} &= \left(\lambda \partial\_k u\_{\cdot k} + Q \,\partial\_k u\_{\vec{f}k}\right) \delta\_{\vec{\imath}\vec{\jmath}} + \mu \left(\partial\_i u\_{\vec{\imath}\vec{\jmath}} + \partial\_j u\_{\vec{\imath}\vec{\imath}}\right) \\ \sigma = -mp &= R \partial\_k u\_{\vec{f}k} + Q \,\partial\_k u\_{\vec{s}k} \end{aligned} \tag{2}$$

*Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot's Medium DOI: http://dx.doi.org/10.5772/intechopen.92064*

Here *σij*ð Þ *x*, *t* are a stress tensor in a skeleton, and *p x*ð Þ , *t* is a pressure in a fluid component. External mass forces acting on a skeleton *<sup>F</sup><sup>s</sup>* <sup>¼</sup> *<sup>F</sup><sup>s</sup> j* ð Þ *x*, *t ej* and on a liquid component *<sup>F</sup> <sup>f</sup>* <sup>¼</sup> *<sup>F</sup> <sup>f</sup> <sup>j</sup>*ð Þ *x*, *t ej*.

Further we use the next notations for partial derivatives: *<sup>∂</sup><sup>k</sup>* <sup>¼</sup> *<sup>∂</sup> <sup>∂</sup>xk* , *<sup>u</sup> <sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>ku <sup>j</sup>*, *<sup>Δ</sup>* <sup>¼</sup> *<sup>∂</sup>k∂<sup>k</sup>* is Laplace operator.

There are three sound speeds in this medium:

the works of Rakhmatullin, Saatov, Filippov, Artykov [6, 7], Erzhanov, Ataliev, Alexeyeva, Shershnev [8, 9], etc. In this regard, it is important to develop effective methods of solution of boundary value problems for such media with the use of

*Mathematical Theorems - Boundary Value Problems and Approximations*

**2. The parameters and motion equations of a two-component Biot's**

The equations of motion of a homogeneous isotropic two-component Biot's medium are described by the following system of second-order hyperbolic

ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∈</sup>*R<sup>N</sup>* � ½ Þ 0, <sup>∞</sup> *:*

Here *N* is the dimension of the space. At a plane deformation *N* = 2, the total spatial deformation corresponds to *N* = 3, at *N* = 1 the equations describe the

We denote *us* ¼ *usj*ð Þ *x*, *t ej* is a displacement vector of an elastic skeleton, *u <sup>f</sup>* ¼ *ufj*ð Þ *x*, *t ej* is a displacement vector of a liquid, and *ej*ð Þ *j* ¼ 1, … , *N* are basic orts of Lagrangian Cartesian coordinate system (everywhere by repeating indices, there is

Constants *ρ*11, *ρ*12, *ρ*<sup>22</sup> have the dimension of mass density, and they are associated with densities of masses of particles, composing a skeleton *ρs*and a fluid *ρ <sup>f</sup>* , by

*ρ*<sup>11</sup> ¼ ð Þ 1 � *m ρ<sup>s</sup>* � *ρ*12, *ρ*<sup>22</sup> ¼ *mρ <sup>f</sup>* � *ρ*12,

where *m* is a porosity of the medium. The constant of attached density *ρ*<sup>12</sup> is related to a dispersion of deviation of micro-velocities of fluid particles in pores from average velocity of fluid flow and depends on pores geometry. Elastic constants *λ*, *μ* are Lama's parameters of an isotropic elastic skeleton, and *Q,R*

*<sup>δ</sup>ij* <sup>þ</sup> *<sup>μ</sup> <sup>∂</sup>iusj* <sup>þ</sup> *<sup>∂</sup> jusi*

characterize an interaction of a skeleton with a liquid on the basis of.

*<sup>σ</sup>* ¼ �*mp* <sup>¼</sup> *<sup>R</sup>∂kufk* <sup>þ</sup> *<sup>Q</sup>∂kusk*

*<sup>σ</sup>ij* <sup>¼</sup> *<sup>λ</sup>∂kusk* <sup>þ</sup> *<sup>Q</sup>∂kufk*

ð Þ¼ *x*, *t ρ*11*u*€*<sup>s</sup>* þ *ρ*12*u*€ *<sup>f</sup>*

(1)

(2)

ð Þ¼ *x*, *t ρ*12*u*€*<sup>s</sup>* þ *ρ*22*u*€ *<sup>f</sup>*

ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup>* grad div*us* <sup>þ</sup> *<sup>μ</sup>*Δ*us* <sup>þ</sup> *<sup>Q</sup>* grad div*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>F</sup><sup>s</sup>*

*<sup>Q</sup>* grad div*us* <sup>þ</sup> *<sup>R</sup>*grad div*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>F</sup> <sup>f</sup>*

dynamics of a porous liquid-saturated rod.

summation from 1 to *N*).

**2.1 Biot's law for stresses**

relationships:

**24**

Periodic on time processes are very widespread in practice. By this cause, here we consider also processes of wave propagation in Biot's medium, posed by the periodic forces of different types. Based on the Fourier transformation of generalized functions, we constructed fundamental solutions of oscillation equations of Biot's medium. It is Green tensor, which describes the process of propagation of harmonic waves at a fixed frequency in the space–time of dimension N = 1, 2, 3, under the action concentrated at the coordinates origin. By using this tensor, we construct generalized solutions of these equations for arbitrary sources of periodic disturbances, which can be described as both regular and singular distributions. They can be used to calculate the stress-strain state of a porous water-saturated

modern mathematical methods.

medium by seismic wave propagation.

**medium**

equations [1–3]:

$$\begin{aligned} c\_1^2 &= \frac{a\_1 + \sqrt{a\_1^2 - 4a\_2a\_3}}{2a\_2}, \\ c\_2^2 &= \frac{a\_1 - \sqrt{a\_1^2 - 4a\_2a\_3}}{2a\_2}, \\ c\_3^2 &= \sqrt{\frac{\rho\_{22}\mu}{a\_2}} \end{aligned} \tag{3}$$

where the next constants were introduced as:

$$\begin{aligned} a\_1 &= \left(\dot{\lambda} + 2\mu\right)\rho\_{22} + R\rho\_{11} - 2Q\rho\_{12}, \\ a\_2 &= \rho\_{11}\rho\_{22} - \left(\rho\_{12}\right)^2, \\ a\_3 &= \left(\dot{\lambda} + 2\mu\right)R - Q^2. \end{aligned}$$

The first two speeds *с*1,*c*2ð Þ *с*<sup>1</sup> >*c*<sup>2</sup> describe the velocity of propagation of two types of *dilatational waves*. The second slower dilatation wave is called *repackaging wave*. A third velocity *c*<sup>3</sup> corresponds to *shear waves* and at *ρ*<sup>12</sup> ¼ 0 coincides with velocity of shear wave propagation in an elastic skeleton (*c*<sup>3</sup> < *c*1).

We introduce also two velocities of propagation of dilatational waves in corresponding elastic body and in an ideal compressible fluid:

$$c\_s = \sqrt{\frac{\lambda + 2\mu}{\rho\_{11}}}, \qquad c\_f = \sqrt{\frac{R}{\rho\_{22}}}$$
