11. Fundamental supersonic antiderivative stress tensor Hb and its properties

We introduce antiderivative stress tensor

W ci <sup>j</sup> <sup>¼</sup> <sup>X</sup> 2

144 Differential Equations - Theory and Current Research

Fj ∗

2πW<sup>1</sup>

2πW<sup>2</sup>

Tensor W ci

it into the terms:

The tensors Wis

ing tensor Wis

shifts

Uj i ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>U</sup><sup>j</sup>

<sup>1</sup> <sup>¼</sup> <sup>z</sup> <sup>r</sup><sup>4</sup> <sup>x</sup><sup>2</sup> <sup>1</sup> � <sup>x</sup><sup>2</sup> 2

<sup>2</sup> ¼ � <sup>z</sup>

<sup>r</sup><sup>4</sup> <sup>x</sup><sup>2</sup> <sup>1</sup> � <sup>x</sup><sup>2</sup> 2

2πW<sup>3</sup>

2πW<sup>2</sup>

it continues on fronts Kj. Wj

W<sup>i</sup> j

<sup>j</sup> , Wid

i

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>W</sup>is

k¼1 W ci jk <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>i</sup> j

<sup>3</sup> <sup>¼</sup> <sup>θ</sup>1ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>1</sup>

<sup>j</sup> has the same support as <sup>U</sup><sup>b</sup> <sup>i</sup>

<sup>z</sup> θð Þz . After calculation, we define its components as:

� �ðθ1V<sup>1</sup> � <sup>θ</sup>2V2Þ þ <sup>0</sup>, <sup>5</sup>m<sup>2</sup>

m1r

i

<sup>j</sup> ð Þþ <sup>x</sup>; <sup>z</sup> <sup>W</sup>id

2πc<sup>2</sup>Wis

2πc<sup>2</sup>Wis

Uj i

ð Þ¼ <sup>y</sup>; <sup>x</sup>; <sup>z</sup> <sup>U</sup><sup>i</sup>

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

have the following symmetry properties around the Z-axis:

j

But for the components with indices (i,j) = (1,3), (3,1), (2,3), (3,2)

i

ð Þ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> , W<sup>j</sup>

� �ðθ1V<sup>1</sup> � <sup>θ</sup>2V2Þ þ <sup>0</sup>, <sup>5</sup>m<sup>2</sup>

<sup>þ</sup> <sup>m</sup><sup>2</sup>

<sup>1</sup> <sup>¼</sup> zx1x2r�<sup>4</sup>ð Þ <sup>θ</sup>1V<sup>1</sup> � <sup>θ</sup>2V<sup>2</sup> , <sup>2</sup>πW<sup>3</sup>

<sup>2</sup>θ2ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

<sup>∗</sup>δð Þ <sup>x</sup><sup>1</sup> <sup>δ</sup>ð Þ <sup>x</sup><sup>2</sup> <sup>θ</sup>ð Þ¼ <sup>z</sup> <sup>U</sup><sup>b</sup> <sup>i</sup>

They are also fundamental solutions of Eq. (14) for the mass forces of the corresponding

<sup>1</sup>θ1ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>1</sup> m1r

<sup>m</sup>2<sup>r</sup> , <sup>2</sup>πW<sup>3</sup>

<sup>j</sup> but as at the cone Kj

singularity on Z with respect to ∥x∥ by x = 0. To single out these singularities, we decompose

2

k¼1

and have a logarithmic singularity with respect to ∥x∥ on the Z-axis. Unlike the generat-

i

ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>W</sup><sup>j</sup>

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

<sup>j</sup>1ð Þ¼� <sup>x</sup> <sup>δ</sup>i3δj<sup>3</sup> <sup>þ</sup> <sup>0</sup>; <sup>5</sup>m<sup>2</sup>

ð Þ <sup>x</sup> � <sup>y</sup>; <sup>z</sup> , W<sup>j</sup>

i

<sup>j</sup>2ð Þ¼ð <sup>x</sup> <sup>δ</sup>i3δj<sup>3</sup> <sup>þ</sup> <sup>δ</sup>ij <sup>0</sup>; <sup>5</sup>m<sup>2</sup>

<sup>1</sup> <sup>θ</sup>1ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>1</sup> m1r

j ∗ z

θð Þz , W ci j , <sup>z</sup> <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>i</sup>

<sup>þ</sup> <sup>M</sup><sup>2</sup>

<sup>2</sup> ¼ �x2r

<sup>þ</sup> <sup>M</sup><sup>2</sup>

<sup>1</sup> ¼ �x1r�<sup>2</sup>ð Þ <sup>θ</sup>1V<sup>1</sup> � <sup>θ</sup>2V<sup>2</sup>

ð Þ x; z has weak singularity by x' = 0 and weak logarithmic

jkð Þþ <sup>x</sup> <sup>W</sup>id

2

<sup>j</sup> ð Þ x; z ,

<sup>θ</sup>kð Þ <sup>z</sup> � mkr <sup>W</sup>is

<sup>j</sup> of diagonal form are independent of z inside the sonic cones Kl(l = 1, 2)

<sup>j</sup> has bounded jumps on the Kl. One can readily see that the tensor

ð Þ¼ x; y; z W

i

<sup>1</sup>ð Þ 1 � δi<sup>3</sup> δij � �lnm1r,

<sup>1</sup>ð Þ� <sup>1</sup> � <sup>δ</sup>i<sup>3</sup> <sup>M</sup><sup>2</sup>

� �lnm2r

cj i ð Þ x � y; z

ð Þ¼ <sup>y</sup>; <sup>x</sup>; <sup>z</sup> <sup>W</sup><sup>i</sup>

j

ð Þ x; y; z , i, j ¼ 1, 2 (52)

<sup>2</sup> � <sup>0</sup>; <sup>5</sup>m<sup>2</sup> 2 � �θ2ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

�2

<sup>2</sup> � <sup>0</sup>; <sup>5</sup>m<sup>2</sup> 2 � �θ2ln <sup>z</sup> <sup>þ</sup> <sup>V</sup><sup>2</sup>

ð Þ θ1V<sup>1</sup> � θ2V<sup>2</sup>

<sup>j</sup> (48)

m2r

m2r

(49)

ð50Þ

(51)

$$\begin{aligned} \hat{\Sigma}\_{i3}^{j} &= \hat{\Sigma}\_{j}^{i} \* \boldsymbol{\theta}(\mathbf{z}) \boldsymbol{\delta}(\mathbf{x}) \\ \hat{H}\_{j}^{i} &= \hat{T}\_{j}^{i} \* \boldsymbol{\theta}(\mathbf{z}) \boldsymbol{\delta}(\mathbf{x}) = \hat{T}\_{j}^{i} \* \boldsymbol{\theta}(\mathbf{z}), \hat{H}\_{j}^{i}{}\_{z} = \hat{T}\_{j}^{i} \end{aligned} \tag{54}$$

This tensor can be obtained in a different way, by analogy with T, using Hooke's law, except that the Green tensor should be replaced with its antiderivative W. By using the presentation of the basic functions of Green's tensor construction (Eq. (25)) in the supersonic case, it can be presented in the following form:

$$\hat{H}\_{i}^{j}(x,z,n) = \left(2M\_{i}^{2} - M\_{2}^{2}\right)n\_{j}f\_{1i,i} - M\_{2}^{2}\left(\delta\_{ij}\frac{\partial f\_{1i}}{\partial n} + n\_{i}f\_{1i,j}\right) - 2\frac{\partial}{\partial n}\left(f\_{1i,i,j} - f\_{3i,i,j}\right),$$

$$2\pi f\_{1k}f\_{i}\left(\left\|x\right\|, z\right) = \frac{\partial\_{k}}{V\_{k}^{-1}}\left(\delta\_{ik} - \frac{z}{\left\|x\right\|}r\_{,i}\right),$$

$$2\pi f\_{3k,ij}\left(\left\|x\right\|, z\right) = \left(\delta\_{i3}\delta\_{jk} + 0, 5m\_{k}^{2}\delta\_{ij}\epsilon\_{\left\|x\right\|}\right)\theta\_{k}\ln\frac{z + \left|V\_{k}^{-1}\right|}{m\_{k}\left\|x\right\|}$$

$$-\frac{V\_{k}^{-2}\theta\_{k}}{\left\|x\right\|}\left(\delta\_{ik}r\_{,j} + \delta\_{j3}r\_{,i} + \frac{z}{\left\|x\right\|}\left(r\_{,i}r\_{,j} - 0, 5\delta\_{j}\epsilon\_{\left\|x\right\|}\right)\right)$$

Obviously, for z < τ, all the introduced shifted tensors are zero. It has the following symmetry properties around the Z-axis:

$$H\_i^j(\mathbf{x}, y, z, m) = -H\_i^j(y, \mathbf{x}, z, m) = -H\_i^j(\mathbf{x}, y, z, -m)$$

except for (i,j) = (1,3), (2,3), (3,1):

$$H\_i^3(\mathbf{x}, y, z) = H\_i^3(y, \mathbf{x}, z), \quad H\_i^3(\mathbf{x}, y, z) = H\_i^3(y, \mathbf{x}, z), \quad i = 1, 2.1$$

Components H<sup>i</sup> j ð Þ <sup>x</sup>; <sup>z</sup> have weak singularities on the fronts of the type <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> j∥x∥<sup>2</sup> � ��1=<sup>2</sup> , but more stronger singularity of the type of ∥x∥�<sup>1</sup> on the axis Z. If we put Eq. (51) in Hook's law, then we can again single out two terms in H<sup>i</sup> j ð Þ x; z :

$$H\_j^i(\mathbf{x}, z) = H\_j^{is}(\mathbf{x}, z) + H\_j^{id}(\mathbf{x}, z) = \sum\_{k=1}^2 \Theta\_k(z - m\_k r) \left( H\_{jk}^{is}(\mathbf{x}) + H\_{jk}^{id}(\mathbf{x}, z) \right) \tag{55}$$

Since the tensors His jkð Þx independent of z inside the sonic cones Kl (l = 1,2), we conventionally say that they are stationary. Accordingly, the tensors Hid <sup>j</sup> ð Þ x; z are said to be dynamic, because they depend essentially on z, although they are regular functions. The aforementioned symmetry properties hold for both stationary and dynamic terms in the tensors.

(see full proof). Then together with conditions given in Eq. (59) of decay of the solutions at

The energy density E is a positive definite quadratic form of ui,j by construction. Therefore, by virtue of the decay of the solution at infinity, the relation only holds if ui,j = 0 for all i and j. Hence, we obtain u = 0; i.e., the solutions coincide. The proof of the theorem is complete.

To solve the problem, we also use the method of generalized functions. We introduce here the

Also using the properties of differentiation of regular generalized functions with gaps at D, and taking into account the boundary conditions and the conditions on the fronts, we obtain the transport Lame equations (Eq. (14)) on the space of distributions with singular mass forces:

By using the properties of convolutions with the Green tensor and the boundary conditions,

By analog with the subsonic case, if we use fundamental stress tensor, then the right-hand side of Eq. (61) may be represented in the form of a surface integral over the boundary of the

> ð Þ� <sup>y</sup>; <sup>τ</sup> <sup>T</sup><sup>j</sup> i

This formula is similar to the Somigliana formula in the static theory of elasticity ([1]: 146), but it is impossible to use this formula to determine the solution of the boundary value problem in the case of supersonic loads, because the second term contains strong non-integrable singularities of the tensor T on the shock wave fronts of fundamental solutions; therefore, the integrals are divergent. To construct a regular integral representation of the formula, we must regularize

<sup>∗</sup> <sup>λ</sup>umnmδij <sup>þ</sup> <sup>μ</sup> uinj <sup>þ</sup> ujni

<sup>b</sup>ujð Þ¼ <sup>x</sup>; <sup>z</sup> ujð Þ <sup>x</sup>; <sup>z</sup> <sup>H</sup>�

δSð Þx θð Þþ z λuknkδij þ μ uinj þ ujni

k, i

þ:

σi3ui, <sup>z</sup>ð Þ x; z dx1dx<sup>2</sup> ! 0 by z ! ∞

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

<sup>3</sup> <sup>R</sup><sup>3</sup> � � and its generalized solution

� � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þ<sup>z</sup> � �, <sup>i</sup> (60)

ð Þ x; y; z � τ; n yð Þ ; τ ujð Þ y; τ � �dD yð Þ ; <sup>τ</sup> (62)

� � � � <sup>δ</sup>Sð Þ<sup>x</sup> <sup>θ</sup>ð Þ<sup>z</sup> (61)

<sup>S</sup> ð Þx θð Þz (59)

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

147

infinity and the zero conditions for z = 0, ð

S�

13. Statement of supersonic BVP in D<sup>0</sup>

regular generalized function with support on D�

<sup>b</sup>gj <sup>¼</sup> pj

<sup>2</sup>buk <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

ð

Uj i

D<sup>þ</sup>

rc

<sup>S</sup> ð Þx θð Þ¼ z

it. For this, we use the tensor H.

uiH�

we obtain the generalized solution of BVP in the form:

<sup>k</sup>∗PjδSð Þ<sup>x</sup> <sup>θ</sup>ð Þþ <sup>z</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

domain. In our notation, on the boundary, it acquires the form

ð Þ x; y; z � τ pj

E xð Þ ; z dx1dx<sup>2</sup> ¼

ð

S�

Theorem 12.1 holds for both exterior and interior boundary value problems.

For this type of tensors, the next theorem was proved (see [7]).

Theorem 11.1. The fundamental stress tensor H satisfies the relation

$$\begin{aligned} \delta\_i^j H\_S^- (\mathbf{x}) \theta(z) &= \int\_0^z d\tau \int\_S \left. H\_i^j(y-\mathbf{x}, \tau, n(y)) \right| dS(y) + \\ &+ \int\_S \left( \left( \rho c^2 \right)^{-1} \tilde{\Sigma}\_{i3}^{\dot{j}}(\mathbf{x}-y, z) - \mathcal{U}\_{i^\*z}^{\dot{j}}(y-\mathbf{x}, z) \right) dy\_1 dy\_2 \end{aligned}$$

For x∉D all integrals are regular; for x ∈ D the first integral is singular, calculated in value principle sense.

This theorem enables us to obtain solvable singular boundary integral equations for a supersonic transport boundary value problem.
