**10. Calculation of the stress state of Biot's medium**

Using Biot's law (2), we can define the generalized stresses in skeleton and a pressure in a liquid:

$$\begin{aligned} \sigma\_{\vec{\eta}} &= \left(\lambda \partial\_l U\_l^k \ast F\_{ks} + Q \partial\_l U\_l^{k+N} \ast F\_{k\vec{\eta}}\right) \delta\_{\vec{\eta}} + \\ &+ \mu \left(\partial\_l U\_j^k \ast F\_{ks} + \partial\_j U\_i^{k+N} \ast F\_{k\vec{\eta}}\right) \\ \sigma = -mp &= R \partial\_l U\_l^{k+N} \ast F\_{k\vec{\eta}} + Q \partial\_l U\_l^k \ast F\_{k\vec{\eta}} \end{aligned} \tag{30}$$

These formulas also can be written in integral form by using the same rules. But we can apply here the next lemma, which was proved in [11].

**Lemma.** *Fourier transformations of the divergences of Green tensor have the next form:*

$$\begin{aligned} \text{by } k &= 1, \ldots, N\\ F\left[\partial\_j U^k\_j\right] &= D\_1 i\xi\_k \left(b \,\_f f \text{q}\_1(\xi, o) - b \,\_f f \text{q}\_2(\xi, o)\right) \\ F\left[\partial\_j U^k\_{j+N}\right] &= D\_1 i\xi\_k \left(d\_3 f \text{q}\_1(\xi, o) - d\_3 f \text{q}\_2(\xi, o)\right) \\ j &= 1, \ldots, N. \end{aligned}$$

$$\begin{aligned} \text{by } k &= N+1, \ldots, 2N \end{aligned}$$

$$F\left[\partial\_j U^k\_j\right] = i\xi\_{k-N} D\_1 \big(d\_3 f u(\xi, o) - d \text{q} f u(\xi, o)\big) F\left[\partial\_j U^k\_{j+N}\right] = i\xi\_{k-N} D\_1 (b f u(\xi, o) - b \text{q} f u(\xi, o)) \big| \qquad j = 1, \ldots, N - 1, \quad \text{for } k = 1, \ldots, N - 1$$

From this lemma, we can prove easily the next theorem.

**Theorem 4.** *Divergences of elastic and liquid displacement of Green tensor have the next form:*

for *k* ¼ 1, … , *N ∂ jU<sup>k</sup> <sup>j</sup>* ¼ �*D*<sup>1</sup> *<sup>b</sup> <sup>f</sup>* <sup>1</sup>*∂k***Φ**<sup>01</sup> � *<sup>b</sup> <sup>f</sup>* <sup>2</sup>*∂k***Φ**<sup>02</sup> � � *∂ jU<sup>k</sup> <sup>j</sup>*þ*<sup>N</sup>* ¼ �*D*1ð Þ *<sup>d</sup>*31*∂k***Φ**<sup>01</sup> � *<sup>d</sup>*32*∂k***Φ**<sup>02</sup> *j* ¼ 1, … , *N:*

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

 $for \ k = N + 1, \ldots, 2N$ 
$$\partial\_j U\_j^k = -D\_1 (d\_{31} \partial\_{k-N} \Phi\_{01} - d\_{32} \partial\_{k-N} \Phi\_{02})$$

$$\partial\_j U\_{j+N}^k = -D\_1 (b\_{i1} \partial\_{k-N} \Phi\_{01} - b\_{i2} \partial\_{k-N} \Phi\_{02})$$
 $j = 1, \ldots, N$ .

Substituting these formulas in (30), we define the stresses in the skeleton and the pressure in the liquid of Biot's medium.

If we paste Φ02ð Þ *x*, *ω* instead of **Φ**02ð Þ *x*, *t* in formulas of this theorem, then formula (30) expresses complex amplitudes of stress tensor and pressure by periodic oscillations. It is used to determine stresses and pressure by solving the periodic problems (4).

## **11. Conclusion**

Here the internal integral is taken over sphere with center in the point *x*, and its

If components of acting forces *F x*ð Þ , *t* are double differentiable vector function,

Substituting the formulas of Theorem 4 into (29), we obtain displacements and stresses of skeleton and liquid in Biot's medium in spaces of dimension *N* = 1, 2, 3. Calculation of these convolutions by using these formulas essentially depends on the form of acting forces and gives possibility to construct regular presentation of generalized solution for wide class of acting forces, which are the classic solution of

Using Biot's law (2), we can define the generalized stresses in skeleton and a

*<sup>l</sup>* <sup>∗</sup> *Fks* <sup>þ</sup> *<sup>Q</sup>∂lU<sup>k</sup>*þ*<sup>N</sup>*

*<sup>j</sup>* <sup>∗</sup> *Fks* <sup>þ</sup> *<sup>∂</sup> jU<sup>k</sup>*þ*<sup>N</sup>*

� �

These formulas also can be written in integral form by using the same rules. But

<sup>¼</sup> *<sup>D</sup>*1*iξ<sup>k</sup> <sup>b</sup> <sup>f</sup>* <sup>1</sup>*f*01ð Þ� *<sup>ξ</sup>*, *<sup>ω</sup> <sup>b</sup> <sup>f</sup>* <sup>2</sup>*f*02ð Þ *<sup>ξ</sup>*,*<sup>ω</sup>* � �

¼ *D*1*iξk*ð Þ *d*31*f*01ð Þ� *ξ*, *ω d*32*f*02ð Þ *ξ*,*ω*

*j*þ*N* h i

� �

**Lemma.** *Fourier transformations of the divergences of Green tensor have the next*

*by k* ¼ *N* þ 1, … , 2*N*

**Theorem 4.** *Divergences of elastic and liquid displacement of Green tensor have the*

*<sup>j</sup>* ¼ �*D*<sup>1</sup> *<sup>b</sup> <sup>f</sup>* <sup>1</sup>*∂k***Φ**<sup>01</sup> � *<sup>b</sup> <sup>f</sup>* <sup>2</sup>*∂k***Φ**<sup>02</sup>

*<sup>j</sup>*þ*<sup>N</sup>* ¼ �*D*1ð Þ *<sup>d</sup>*31*∂k***Φ**<sup>01</sup> � *<sup>d</sup>*32*∂k***Φ**<sup>02</sup>

� �*δij*<sup>þ</sup>

*<sup>l</sup>* ∗ *Fkf*

*<sup>i</sup>* ∗ *Fkf*

*<sup>l</sup>* ∗ *Fks*

*<sup>l</sup>* <sup>∗</sup> *Fkf* <sup>þ</sup> *<sup>Q</sup>∂lU<sup>k</sup>*

<sup>ð</sup>**Ф**2*<sup>m</sup>* <sup>∗</sup> *F x*ð Þ , *<sup>t</sup>* Þ ¼ **<sup>Ф</sup>**2*m*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>∗</sup> *<sup>∂</sup>*<sup>2</sup>

*F ∂x <sup>j</sup>∂xk*

(30)

¼ *iξk*�*ND*1ð*bs*1*f*01ð Þ� *ξ*,*ω bs*2*f*02ð Þ *ξ*,*ω*Þ *j* ¼

it is convenient to use the property of differentiation of convolution [10]:

*∂*2 *∂x <sup>j</sup>∂xk*

*Mathematical Theorems - Boundary Value Problems and Approximations*

**10. Calculation of the stress state of Biot's medium**

*<sup>σ</sup>ij* <sup>¼</sup> *<sup>λ</sup>∂lU<sup>k</sup>*

<sup>þ</sup>*<sup>μ</sup> <sup>∂</sup>iU<sup>k</sup>*

we can apply here the next lemma, which was proved in [11].

*F ∂ jU<sup>k</sup> j* h i

*F ∂ jU<sup>k</sup>*

�

*j*þ*N* h i

<sup>¼</sup> *<sup>i</sup>ξk*�*ND*<sup>1</sup> *<sup>d</sup>*31*f*01ð Þ� *<sup>ξ</sup>*,*<sup>ω</sup> <sup>d</sup>*32*f*02ð Þ *<sup>ξ</sup>*,*ω*<sup>Þ</sup> *<sup>F</sup> <sup>∂</sup> jU<sup>k</sup>*

*∂ jU<sup>k</sup>*

*∂ jU<sup>k</sup>*

From this lemma, we can prove easily the next theorem.

*j* ¼ 1, … , *N:*

*<sup>σ</sup>* ¼ �*mp* <sup>¼</sup> *<sup>R</sup>∂lU<sup>k</sup>*þ*<sup>N</sup>*

radius is equal to *cmτ*.

Biot's equation.

pressure in a liquid:

by *k* ¼ 1, … , *N*

*form:*

*F ∂ jU<sup>k</sup> j* h i

*next form:*

**34**

for *k* ¼ 1, … , *N*

*j* ¼ 1, … , *N:*

*∂*2 **Ф**2*<sup>m</sup> ∂x <sup>j</sup>∂xk*

∗ *F x*ð Þ¼ , *t*

The obtained solutions give possibility to study the dynamics of porous waterand gas-saturated media and rods under actions of disturbance sources of different forms and can be used for solutions of boundary value problems in porous media by using boundary element method.

These solutions can be used for describing wave processes by explosions and earthquakes. In these cases mass forces are described by using singular generalized function, such as multipoles, simple and double layers, and others.

## **Acknowledgements**

This work was financially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant AP05132272).
