5. Fundamental Green's tensors and generalized solutions of transport Lame equations

The matrix of fundamental solutions U x b ð Þ ; z satisfies Eq. (14) with a delta function in the mass force:

$$L\_i^j \left(\frac{\partial}{\partial \mathbf{x'}}\right) \hat{U}\_j^k + \delta(\mathbf{x'}) \delta\_i^k = 0, \qquad i, j = 1, 2, 3 \tag{21}$$

This matrix is called Green's tensor for the transport Lame equations if it satisfies the decay conditions at infinity

$$
\hat{U}\_i^k \to 0, \quad \partial\_j \hat{U}\_i^k \to 0 \quad , \quad \mathbf{x} \stackrel{\sim}{\to} \approx \text{,} \quad i, j, k = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{22}
$$

For a fixed k, its components describe the displacements of the elastic medium under a concentrated force moving at the velocity c along the axis Z = X<sup>3</sup> and acting in the Xk direction.

Green's tensor can be obtained by taking the Fourier transform of Eq. (17) and solving the corresponding system of linear algebraic equations for the Fourier transforms U(ξ1,ξ2,ξ3). It is reduced to the form (see [4]).

$$O\_4^j = \frac{M\_2^2 \delta\_t^j}{\left(\|\xi\|^2 - M\_2^2 \xi\_3^2\right)} + \frac{\xi\_3 \xi\_j}{\xi\_3^2} \left(\frac{1}{\|\xi\|^2 - M\_2^2 \xi\_3^2} - \frac{1}{\|\xi\|^2 - M\_1^2 \xi\_3^2},\right) \tag{23}$$

It can be seen that U x b ð Þ ; z has no classical inverse Fourier transform since it has non-integrable singularities in its denominators. This is associated with the fact that the matrix of fundamental solutions is defined, generally speaking, up to solutions of the homogeneous system of equations. The functions

$$\tilde{f}\_{k m} = \xi^{-m} \left( \|\xi\|^2 - M\_m^2 \xi\_3^2 \right)^{-1}, \quad m = 0, 1, 2.$$

are of crucial importance in the construction of the original Green's tensor. It is easy to see that f ¯ <sup>0</sup><sup>m</sup> is the Fourier transform of the fundamental solution to the equation

$$\frac{\partial^2 \bar{f}\_{0k}}{\partial x\_1^2} + \frac{\partial^2 \bar{f}\_{0k}}{\partial x\_2^2} + (1 - M\_k^2) \frac{\partial^2 \bar{f}\_{0k}}{\partial z^2} + \delta(x)\delta(z) = 0 \tag{24}$$

This equation is similar to the elliptic Laplace equation at subsonic speeds if Mk < 1 and to the wave equation at supersonic speeds if Mk > 1. At the sound speed (Mk ¼ 1), the variable z disappears from the equation and the equation becomes parabolic, since the space dimension is higher by one, which determines the type of Eq. (14), as noted earlier, since the solutions contain waves of two types. Green's tensor for the Lame transport equation was constructed by Alekseyeva [4] by applying fundamental solutions of the Laplace and wave equations and regularization functions f ¯ km, which depends on the speed of transport load. Green's tensor has the regular form:

$$\mathcal{U}\_i^j(\mathbf{x}, z) = c\_2^{-2} \delta\_{\vec{\vartheta}}^j f\_{02}(\|\mathbf{x}\|, z) + c^{-2} \left( f\_{21^i \vec{\vartheta}}(\|\mathbf{x}\|, z) - f\_{22^i \vec{\vartheta}}(\|\mathbf{x}\|, z) \right), \tag{25}$$

If the following convolution exists,

of the type of single layers g ¼ gj

function [3].

properties

determined by its value.

<sup>b</sup>ui <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> i

ð Þ x � y; z � τ gj

ð Þ y; τ ejδDð Þ y; τ , then

ð Þ x � y; z � τ gj

If mass forces are regular, then Eq. (28) has an integral presentation:

Uj i

ð

D�

ð

�

Uj i

D

uið Þ¼ x; z

<sup>b</sup>uið Þ¼ <sup>x</sup>; <sup>z</sup>

loads moving along the strip in an elastic medium.

∗Gjð Þ x; z =rc

it is easy to prove that it is the generalized solution of the transport Lame equations, Eq. (14).

If mass forces are concentrated on surface D and described by singular generalized functions

Moreover, by the Du Bois-Reymond lemma [3], these solutions are classical. For other types of singular mass forces, to calculate Eq. (28), we use the definition of convolution of a generalized

It is easy to see from Eqs. (23) to (25) that the solution is represented as a composition of fundamental solutions distributed over the support of the function f(x,z); their intensities are

In Alexeyeva and Kayshibayeva's paper [5], there are some numerical examples of calculation of the dynamic of elastic medium at subsonic, transonic, and supersonic speed of transport

In the subsonic case from Eq. (25), we obtain the components of Green's tensor in the form:

6. Subsonic Green's tensor, fundamental stress tensors, and their

<sup>2</sup> (27)

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

ð Þ y; τ dy1dy2dτ ¼ uið Þ x; z (28)

(29)

137

ð Þ y; τ dD yð Þ¼ ; τ uið Þ x; z

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

where the type of basic function depends on velocity c.

In subsonic case (Mk < 1):

$$\begin{aligned} 4\pi f\_{o\bar{j}}(r,z) &= \frac{1}{\sqrt{z^2 + m\_{\bar{j}}^2 r^2}}, \quad 4\pi f\_{1\bar{j}} = \text{sgn} \quad |z| \ln \left(\frac{|z| + \sqrt{z^2 + m\_{\bar{j}}^2 r^2}}{m\_{\bar{j}} r}\right), \\\\ 4\pi f\_{2\bar{j}} &= |z| \ln \left(\frac{|z| + V\_{\bar{j}}}{m\_{\bar{j}} r}\right) - V\_{\bar{j}} + m\_{\bar{j}} ||\mathbf{x}|| \end{aligned}$$

In sonic case (Mk = 1):

$$f\_{ck}(|\mathbf{x}|,z) = -0.5\ \delta(z)|\mathbf{x}| \quad f\_{1k} = 0.5\ \theta(z)|\mathbf{x}| \quad f\_{2k} = 0.5\ \mathbf{z}\ \theta(z)|\mathbf{x}|.$$

In supersonic case (Mk > 1):

$$f\_{\neg j}(r,z) = \frac{\theta(z-m\_{\uparrow}r)}{2\pi\sqrt{z^2-m\_{\uparrow}^2r^2}}, \quad f\_{1\circ j} = \frac{\theta(z-m\_{\uparrow}r)}{2\pi} \cdot \ln\left(\frac{z+V\_j^-}{m\_{\uparrow}r}\right), \quad f\_{2\circ j} = \frac{\theta(z-m\_{\uparrow}r)}{2\pi} \left(z\ln\left(\frac{z+V\_j^-}{m\_{\uparrow}r}\right) \quad -V\_j^-\right), \quad z \to -\infty$$

Here and hereafter, we use the following notation: θð Þz is the Heaviside step function,

$$m\_k = \sqrt{1 - M\_{k'}^2} \quad V\_k = \sqrt{z^2 + m\_k^2 r^2}, \quad V\_k^- = \sqrt{z^2 - m\_k^2 r^2}, \quad r = \sqrt{\mathbf{x}\_1^2 + \mathbf{x}\_2^2} = \|\mathbf{x}\|\boldsymbol{\rho}$$

The dilatational and shear components of U x b ð Þ ; z are easy to write out

$$\begin{aligned} \mathcal{U}\_i^j(\mathbf{x}, z) &= \mathcal{U}\_{i1}^j(\mathbf{x}, z) + \mathcal{U}\_{i2}^j(\mathbf{x}, z) \\ \mathcal{U}\_{i1}^j &= \mathbf{c}^{-2} f\_{21 \cdot \hat{\imath}}(\|\mathbf{x}\|, z), \quad \mathcal{U}\_{i2}^j(\mathbf{x}, z) = \mathbf{c}\_2^{-2} \delta\_{\mathbf{y}}^j f\_{02}(\|\mathbf{x}\|, z) - f\_{22 \cdot \hat{\imath}}(\|\mathbf{x}\|, z) \end{aligned} \tag{26}$$

In the supersonic case, the support of the functions is the cone z>mk∥x∥. This determines a radiation condition as physical considerations imply that there are no displacements of the elastic medium outside this cone since the perturbations have a finite propagation velocity, which cannot be higher than the corresponding sound velocity for a particular type of deformation. At the fronts of shock waves (z = mk∥x∥), Green's tensor grows to infinity.

If the following convolution exists,

disappears from the equation and the equation becomes parabolic, since the space dimension is higher by one, which determines the type of Eq. (14), as noted earlier, since the solutions contain waves of two types. Green's tensor for the Lame transport equation was constructed by Alekseyeva [4] by applying fundamental solutions of the Laplace and wave equations and

km, which depends on the speed of transport load. Green's tensor has

j j z þ

, f <sup>2</sup><sup>j</sup> <sup>¼</sup> <sup>θ</sup> <sup>z</sup> � mjr � �

, r ¼

f <sup>02</sup>ð Þ� k kx ; z f <sup>22</sup>, ijð Þ k kx ; z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> k r2

b ð Þ ; z are easy to write out

0 @ q

mjr

�<sup>2</sup> <sup>f</sup> <sup>21</sup>, ijð Þ� k k<sup>x</sup> ; <sup>z</sup> <sup>f</sup> <sup>22</sup>, ijð Þ k k<sup>x</sup> ; <sup>z</sup> � �, (25)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> j r2

<sup>2</sup><sup>π</sup> <sup>z</sup> ln <sup>z</sup> <sup>þ</sup> <sup>V</sup>�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2

q

j mjr � �

� �

¼ k kx ,

� V� j

(26)

,

1 A,

regularization functions f

In subsonic case (Mk < 1):

In sonic case (Mk = 1):

<sup>f</sup> ojð Þ¼ <sup>r</sup>; <sup>z</sup> <sup>θ</sup> <sup>z</sup> � mjr � � 2π

mk ¼

In supersonic case (Mk > 1):

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> j r2

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>M</sup><sup>2</sup> k

q

Uj i

Uj

Uj i ð Þ¼ x; z c

136 Differential Equations - Theory and Current Research

4πf ojð Þ¼ r; z

the regular form:

¯

�2 <sup>2</sup> δ j i

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> j r2

mjr � �

where the type of basic function depends on velocity c.

<sup>4</sup>π<sup>f</sup> <sup>2</sup><sup>j</sup> <sup>¼</sup> j j <sup>z</sup> ln j j <sup>z</sup> <sup>þ</sup> Vj

<sup>q</sup> , f <sup>1</sup><sup>j</sup> <sup>¼</sup> <sup>θ</sup> <sup>z</sup> � mjr � �

, Vk ¼

The dilatational and shear components of U x

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>U</sup><sup>j</sup>

f <sup>02</sup>ð Þþ k kx ; z c

<sup>q</sup> , <sup>4</sup>π<sup>f</sup> <sup>1</sup><sup>j</sup> <sup>¼</sup> sgn j j <sup>z</sup> ln

� Vj þ mjk kx ,

f okð Þ¼� j j x ; z 0:5 δð Þz j j x , f <sup>1</sup><sup>k</sup> ¼ 0:5 θð Þz j j x , f <sup>2</sup><sup>k</sup> ¼ 0:5 zθð Þz j j x :

j mjr � �

q

<sup>2</sup> δ j i

<sup>2</sup><sup>π</sup> ln <sup>z</sup> <sup>þ</sup> <sup>V</sup>�

Here and hereafter, we use the following notation: θð Þz is the Heaviside step function,

, V� <sup>k</sup> ¼

<sup>i</sup>2ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>c</sup>�<sup>2</sup>

In the supersonic case, the support of the functions is the cone z>mk∥x∥. This determines a radiation condition as physical considerations imply that there are no displacements of the elastic medium outside this cone since the perturbations have a finite propagation velocity, which cannot be higher than the corresponding sound velocity for a particular type of defor-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>m</sup><sup>2</sup> k r2

<sup>i</sup>2ð Þ x; z

mation. At the fronts of shock waves (z = mk∥x∥), Green's tensor grows to infinity.

q

<sup>i</sup>1ð Þþ <sup>x</sup>; <sup>z</sup> <sup>U</sup><sup>j</sup>

<sup>i</sup><sup>1</sup> <sup>¼</sup> <sup>c</sup>�<sup>2</sup><sup>f</sup> <sup>21</sup>, ijð Þ k k<sup>x</sup> ; <sup>z</sup> , U<sup>j</sup>

$$
\widehat{\boldsymbol{\mu}}\_i = \widehat{\mathbf{U}}\_i^{\boldsymbol{\zeta}} \ast \mathbf{G}\_{\boldsymbol{\zeta}}(\mathbf{x}, \boldsymbol{z}) / \rho \boldsymbol{\varepsilon}^2 \tag{27}
$$

it is easy to prove that it is the generalized solution of the transport Lame equations, Eq. (14). If mass forces are regular, then Eq. (28) has an integral presentation:

$$u\_i(\mathbf{x}, z) = \int\_{D^-} \mathcal{U}\_i^j(\mathbf{x} - \mathbf{y}, z - \tau) \mathbf{g}\_j(\mathbf{y}, \tau) d\mathbf{y}\_1 d\mathbf{y}\_2 d\tau = u\_i(\mathbf{x}, z) \tag{28}$$

If mass forces are concentrated on surface D and described by singular generalized functions of the type of single layers g ¼ gj ð Þ y; τ ejδDð Þ y; τ , then

$$\widehat{u}\_i(\mathbf{x}, z) = \int\_D \left( \mathcal{U}\_i^j(\mathbf{x} - \mathbf{y}, z - \tau) \mathbf{g}\_j(\mathbf{y}, \tau) dD(\mathbf{y}, \tau) = u\_i(\mathbf{x}, z) \tag{29}$$

Moreover, by the Du Bois-Reymond lemma [3], these solutions are classical. For other types of singular mass forces, to calculate Eq. (28), we use the definition of convolution of a generalized function [3].

It is easy to see from Eqs. (23) to (25) that the solution is represented as a composition of fundamental solutions distributed over the support of the function f(x,z); their intensities are determined by its value.

In Alexeyeva and Kayshibayeva's paper [5], there are some numerical examples of calculation of the dynamic of elastic medium at subsonic, transonic, and supersonic speed of transport loads moving along the strip in an elastic medium.
