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

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

$$\begin{aligned} \dot{U}\_1^1 &= \frac{1}{4\pi c^2} \left( \frac{1}{V\_2} + \frac{z^2 x\_1^2 W\_{12}}{r^4 M\_2^2} - \frac{x\_2^2 V\_{12}}{r^4 M\_2^2} \right), \\ \dot{U}\_2^2 &= \frac{1}{4\pi c^2} \left( \frac{1}{V\_2} + \frac{z^2 x\_2^2 W\_{12}}{r^4 M\_2^2} - \frac{x\_1^2 V\_{12}}{r^4 M\_2^2} \right), \\ \dot{U}\_3^3 &= \dot{U}\_2^1 = \frac{(x\_1 x\_2 \quad \left(z^2 W\_{12} + \; V\_{12}\right), \; \dot{U}\_3^3 = \frac{1}{4\pi c^2} \left(\frac{1}{V\_1} - \frac{m\_2^2}{V\_2}\right), \\ \dot{U}\_1^3 &= \dot{U}\_3^1 = -\frac{x\_1 z}{4\pi c^2 r^2} W\_{12}, \quad \dot{U}\_2^3 = \dot{U}\_3^2 = -\frac{x\_2 z W\_{12}}{4\pi c^2 r^2}, \\ \dot{V}\_2 &= V\_1 - V\_2, \quad V\_i = \sqrt{z^2 + m\_1^2 r^2}, \quad m\_i^2 = 1 - M\_i^2, \quad W\_{12} = V\_1^{-1} - V\_2^{-1}, \; r = \sqrt{x\_1^2 + x\_2^2 + x\_3^2} \end{aligned}$$

They are regular functions. Since by x 0 ! 0 [6]:

$$V\_{12} \sim \frac{r^2(m\_1^2 - m\_2^2)}{2\left|z\right|}, \quad W\_{12} \sim \frac{r^2(m\_2^2 - m\_1^2)}{2\left|z\right|}, \quad \frac{z^2 x\_1^2}{r^4} W\_{12} - \frac{x\_2^2}{r^4} V\_{12} \sim \frac{\left(m\_2^2 - m\_1^2\right)}{2\left|z\right|}\tag{30}$$

7. Statement of subsonic transport boundary value problems. Uniqueness

Let D� be an elastic medium bounded by a cylindrical surface D with generator parallel to the axis X3; let S� be the cross-section of the cylindrical domain; let S be its boundary, and let n be the unit outward normal of D. Obviously, n = n(x) and n<sup>3</sup> = 0. We assume that G is an integrable

> 2 pi

A vector function u(x,z) satisfying the aforementioned conditions is referred to as a classical

W � 0:5rc

2

ð

Dz,�<sup>∞</sup>

0:5rc 2

<sup>∥</sup>p xð Þ ; <sup>z</sup> <sup>∥</sup> <sup>≤</sup> O zj j�1�ε<sup>2</sup> � � for <sup>∣</sup>z<sup>∣</sup> ! <sup>∞</sup>, x∈S: (38)

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

� = {(x,z): x ∈D�,a<z<b}. The two useful energetic equalities have

<sup>2</sup> u, k k<sup>z</sup> <sup>2</sup> � ð Þ <sup>G</sup>; <sup>u</sup> � � dx1dx2dz<sup>þ</sup>

ui, <sup>z</sup> � <sup>σ</sup>i3ð Þ <sup>x</sup>; <sup>b</sup> � � ð Þ <sup>x</sup>;<sup>b</sup> uið Þ <sup>x</sup>; <sup>b</sup> �

ð Þ P; u, <sup>z</sup> dD xð Þþ ; z

dx1dx2dz <sup>¼</sup> <sup>0</sup> �

P; ui, ð Þ<sup>z</sup> dx1dx2dz þ

<sup>∥</sup>u, <sup>z</sup>∥<sup>2</sup> � <sup>W</sup> � ð Þ <sup>G</sup>; <sup>u</sup> � �dx1dx2dz

ð

D�

ab ¼ ð Þ x; z : x ∈ D� f g ; a < z < b :

� �

ð

G; u, ð Þ<sup>z</sup> dx1dx2dz

ð Þ G; u, <sup>z</sup> dV xð Þ ; z (39)

D� z,�<sup>∞</sup>

∥ ! ∞, x<sup>0</sup> ∈ D� þ D: (35)

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

ð Þ x; z , xð Þ ; z ∈ D (36)

ð37Þ

139

<sup>∥</sup>�ð Þ <sup>3</sup>þ<sup>ε</sup> � � for∥x<sup>0</sup>

There is the subsonic transport load P(x,z) moving along the boundary D (c<c2):

σijð Þ x; z njð Þ¼ x P xð Þ¼ ; z rc

Theorem 7.1. Classic solution of transport BVP satisfying to the equalities:

ð

D� ab

� � � � � � �

<sup>z</sup> dx1dx<sup>2</sup> ¼

ð

D�

D

ð Þ P; u dD xð Þ� ; z

� � ð Þ <sup>x</sup>;<sup>a</sup> uið Þ� <sup>x</sup>; <sup>a</sup> <sup>r</sup><sup>c</sup>

ð Þ P xð Þ ; z ; u xð Þ ; z dD xð Þ¼ ; z

2 <sup>∥</sup>u, <sup>z</sup>∥<sup>2</sup> � �dV xð Þ¼ <sup>ð</sup>

Dab ¼ f g ð Þ x; z : x∈ D; a ≤ z ≤ b , D�

ui, <sup>z</sup> � σi<sup>3</sup>

<sup>2</sup> u, k k<sup>z</sup> <sup>2</sup> � <sup>σ</sup>i3ui, <sup>z</sup> � � �<sup>∞</sup>

W þ 0; 5rc

� �

of solution

vector function and ∃ε > 0 such that

We assume that ∃ε<sup>i</sup> > 0:

solution of the BVP. Let Cab

been proved by Alexeyeva [6].

ð

Dab

rc 2

þ ð

ð

S�

S�

W þ 0; 5rc

ð

D

ð

S�

∥G xð Þ ; z ∥ ≤ O ∥x<sup>0</sup>

these components are bounded for ð Þ x; z 6¼ ð Þ 0; 0; 0 . At the point ð Þ¼ x; z ð Þ 0; 0; 0 , they have a weak singularity of order R�<sup>1</sup> , R <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>p</sup> . It has a similar asymptotic at infinity. Accordingly, R�<sup>2</sup> is the order of the tensor derivatives asymptotic and the behavior of at ∞.

Tensor Ub generates next fundamental stress tensors if we use Hook's law (Eq. (2)):

$$\begin{aligned} \Sigma^i\_{\mathbb{jk}}(\mathbf{x}, z) &= \lambda \, \mathcal{U}^i\_{l^\cdot l^\cdot} \delta\_{\mathbb{jk}} + \mu \left( \mathcal{U}^i\_{l^\cdot k} + \mathcal{U}^i\_{k^\cdot j} \right), \quad \Gamma^i\_{\mathbb{j}}(\mathbf{x}, z, n) = \Sigma^i\_{\mathbb{jk}}(\mathbf{x}, z) n\_k \\\\ \widehat{T}^i\_{\mathbb{j}}(\mathbf{x}, z, n) &= -\left(\rho c^2\right)^{-1} \Gamma^i\_i(\mathbf{x}, z, n) \end{aligned} \tag{31}$$

Then the elastic constant tensor is presented in the form

$$
\widehat{T}\_i^{\dagger}(\mathbf{x}, z, \mathfrak{n}) = \widehat{\mathsf{C}}\_{km}^{\vec{\mathsf{I}}} \widehat{\mathsf{U}}\_{i \, \cdot \, m}^{\mathbf{k}} \mathfrak{n}\_{l \prime} \quad \widehat{\mathsf{C}}\_{km}^{\vec{\mathsf{I}}} = \mathsf{C}\_{km}^{\vec{\mathsf{I}}} / \{\rho c^2\} \tag{32}
$$

Tensor Γ<sup>i</sup> j ð Þ x; z; n describes the stresses at the plate with normal n in a point x 0 = (x,z). Tensor Tb have some remarkable properties.

Theorem 6.1. Fundamental stress tensor T is the generalized solution of the transport Lame equation b with singular mass forces of the multipole type:

$$
\rho c^2 L\_i^j(\partial\_{\mathbf{x'}}) \widehat{T}\_j^k + K\_k^i(\partial\_{\mathbf{x'}}, n) \delta \left(\mathbf{x'}\right) = \mathbf{0} \tag{33}
$$

where

$$K\_i^l(\mathfrak{d}\_{\mathfrak{x'}}, n) = \lambda n\_i \mathfrak{d}\_l + \mu m\_j \left(\delta\_i^l \mathfrak{d}\_j + \delta\_j^l \mathfrak{d}\_i\right).$$

For any closed Lyapunov's surface D, bounding a domain D� ⊂ R<sup>3</sup>

$$\delta\_i^j H\_D^-(\mathbf{x}, z) = V.P.\int\_D \left( T\_i^j(\mathbf{x} - \mathbf{y}, \tau - z, n(\mathbf{y}, \tau)) + \mathcal{U}\_{i^\*z}^j(\mathbf{x} - \mathbf{y}, \tau - z) n\_z(\mathbf{y}, \tau) \right) dS(\mathbf{y}, \tau) \tag{34}$$

where H� <sup>D</sup>ð Þ x; z is the characteristic function of D�, which is equal to 0.5 at D; n yð Þ ; τ is a unit normal vector to D. The integrals are regular for xð Þ ; z ∉D and are taken in value principle sense for ð Þ x; z ∈ D.

These formulas have been proved by Alexeyeva [6]. The formula in Eq. (35) can be referred to as a dynamic analog of the well-known Gauss formula for a double-layer potential of the fundamental solution of Laplace's equation ([3]: 403). It plays a fundamental role in the solution of transport boundary value problems (BVP).
