7. Statement of subsonic transport boundary value problems. Uniqueness of solution

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 vector function and ∃ε > 0 such that

$$\|\|G(\mathbf{x}, z)\|\| \le O\left(\|\mathbf{x'}\|\|^{-(3+\varepsilon)}\right) \text{ for} \|\mathbf{x'}\| \to \infty, \quad \mathbf{x'} \in D^- + D. \tag{35}$$

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

$$
\sigma\_{\vec{\eta}}(\mathbf{x}, z) n\_{\vec{\jmath}}(\mathbf{x}) = P(\mathbf{x}, z) = \rho c^2 p\_i(\mathbf{x}, z), \quad (\mathbf{x}, z) \in D \tag{36}
$$

We assume that ∃ε<sup>i</sup> > 0:

ð30Þ

(31)

They are regular functions. Since by x

138 Differential Equations - Theory and Current Research

weak singularity of order R�<sup>1</sup>

Σi

Tbi j

have some remarkable properties.

with singular mass forces of the multipole type:

Tensor Γ<sup>i</sup> j

where

δ j i H�

where H�

<sup>D</sup>ð Þ¼ x; z V:P:

jkð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>λ</sup>U<sup>i</sup>

l

ð Þ¼� <sup>x</sup>; <sup>z</sup>; <sup>n</sup> <sup>r</sup>c<sup>2</sup> � ��<sup>1</sup>

Then the elastic constant tensor is presented in the form

Tbj i 0

, R <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

, <sup>l</sup> <sup>δ</sup>jk <sup>þ</sup> <sup>μ</sup> <sup>U</sup><sup>i</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>n</sup> <sup>C</sup>~jl

rc 2 Lj <sup>i</sup> <sup>∂</sup><sup>x</sup> ð Þ0 <sup>T</sup>b<sup>k</sup>

Kl

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

Tj i

ð

D

solution of transport boundary value problems (BVP).

Γj i ð Þ x; z; n

ingly, 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)):

j , <sup>k</sup> <sup>þ</sup> <sup>U</sup><sup>i</sup> k, j � �

kmU<sup>b</sup> <sup>k</sup>

ð Þ x; z; n describes the stresses at the plate with normal n in a point x

<sup>i</sup> , mnl, <sup>C</sup>~jl

Theorem 6.1. Fundamental stress tensor T is the generalized solution of the transport Lame equation b

<sup>k</sup> ∂<sup>x</sup> ð Þ <sup>0</sup> ; n δ x

<sup>j</sup> <sup>þ</sup> <sup>K</sup><sup>i</sup>

<sup>i</sup> <sup>∂</sup><sup>x</sup> ð Þ¼ <sup>0</sup> ; <sup>n</sup> <sup>λ</sup>ni∂<sup>l</sup> <sup>þ</sup> <sup>μ</sup>т<sup>j</sup> <sup>δ</sup><sup>l</sup>

<sup>ð</sup><sup>x</sup> � <sup>y</sup>; <sup>τ</sup> � <sup>z</sup>; n yð Þ ; <sup>τ</sup> Þ þ <sup>U</sup><sup>j</sup>

! 0 [6]:

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

, Γ<sup>i</sup> j

km <sup>¼</sup> Cjl

<sup>0</sup> � �

i <sup>∂</sup><sup>j</sup> <sup>þ</sup> <sup>δ</sup><sup>l</sup> j ∂i � �

i

� �

<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

<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. Accord-

ð Þ¼ <sup>x</sup>; <sup>z</sup>; <sup>n</sup> <sup>Σ</sup><sup>i</sup>

km= rc

:

, <sup>z</sup>ð Þ x � y; τ � z nzð Þ y; τ

jkð Þ x; z nk

<sup>2</sup> � � (32)

= (x,z). Tensor Tb

dS yð Þ ; τ (34)

0

¼ 0 (33)

$$\left\|\left|u^{D}(x,z)\right|\right\| \leq O(\left|z\right|^{-\epsilon\_{1}}) \quad \text{ for } \left|z\right| \to \ast, \ x \in \mathbb{S},\tag{37}$$

$$\|\|p(\mathbf{x}, z)\|\| \le O\left(\left|z\right|^{-1-\varepsilon 2}\right) \text{ for } |z| \to \infty, \mathbf{x} \in \mathbf{S}.\tag{38}$$

A vector function u(x,z) satisfying the aforementioned conditions is referred to as a classical solution of the BVP. Let Cab � = {(x,z): x ∈D�,a<z<b}. The two useful energetic equalities have been proved by Alexeyeva [6].

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

ð Dab ð Þ P; u dD xð Þ� ; z ð D� ab W � 0:5rc <sup>2</sup> u, k k<sup>z</sup> <sup>2</sup> � ð Þ <sup>G</sup>; <sup>u</sup> � � dx1dx2dz<sup>þ</sup> þ ð S� rc 2 ui, <sup>z</sup> � σi<sup>3</sup> � � ð Þ <sup>x</sup>;<sup>a</sup> uið Þ� <sup>x</sup>; <sup>a</sup> <sup>r</sup><sup>c</sup> 2 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> � � � dx1dx2dz <sup>¼</sup> <sup>0</sup> � � � ð S� W þ 0; 5rc <sup>2</sup> u, k k<sup>z</sup> <sup>2</sup> � <sup>σ</sup>i3ui, <sup>z</sup> � � �<sup>∞</sup> <sup>z</sup> dx1dx<sup>2</sup> ¼ ð Dz,�<sup>∞</sup> P; ui, ð Þ<sup>z</sup> dx1dx2dz þ ð D� z,�<sup>∞</sup> G; u, ð Þ<sup>z</sup> dx1dx2dz � � � � � � � ð D ð Þ P xð Þ ; z ; u xð Þ ; z dD xð Þ¼ ; z ð D� 0:5rc 2 <sup>∥</sup>u, <sup>z</sup>∥<sup>2</sup> � <sup>W</sup> � ð Þ <sup>G</sup>; <sup>u</sup> � �dx1dx2dz ð S� W þ 0; 5rc 2 <sup>∥</sup>u, <sup>z</sup>∥<sup>2</sup> � �dV xð Þ¼ <sup>ð</sup> D ð Þ P; u, <sup>z</sup> dD xð Þþ ; z ð D� ð Þ G; u, <sup>z</sup> dV xð Þ ; z (39) Dab ¼ f g ð Þ x; z : x∈ D; a ≤ z ≤ b , D� ab ¼ ð Þ x; z : x ∈ D� f g ; a < z < b :

The following assertion is its corollary.

Theorem 7.2. The solution of the subsonic transport boundary value problem is unique.

Proof. Since the problem is linear, it suffices to prove the uniqueness of the zero solution. Let u (x,z) satisfy the zero boundary conditions P(x,z) = 0 on D and be a solution of the homogeneous Lame equations ((Eq. (14)) by G(x,z) = 0.

Then for ∀z

$$\int\_{S^{-}} (W + \rho c^2 \|\mu\_{,z}\|^2)dV(\mathbf{x}) = 0$$

By using the properties of the differentiation of regular generalized functions with jumps on D,

ð Þ <sup>∂</sup>x; <sup>∂</sup>z; <sup>b</sup>ujð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>G</sup>bi<sup>þ</sup>

<sup>D</sup>ð Þ x; z , δDð Þ¼ x; z δSð Þx 1ð Þz , 1ð Þ� z 1, is a simple layer on D. Since n<sup>3</sup> = 0 on D, it

follows from the properties of the Green tensor that an analog of the Somigliana formula holds

<sup>l</sup> þ μ njul þ nluj � � � �δD<sup>∗</sup>U<sup>b</sup> <sup>l</sup>

> i , <sup>m</sup><sup>∗</sup> C~kl

� �dD yð Þ ; <sup>τ</sup> ,

If we write out this convolution in integral form with regard to the notation introduced here and Eqs. (1) and (2), then we obtain a formula, whose form coincides with the Somigliana

i

This formula permits one to determine displacements in the medium on the basis of known boundary values of displacements and stresses. But the integrals are regular only for xð Þ ; z ∉D

The following assertion provides a solution for the aforementioned boundary value problems.

Theorem 9.1. If the solution u(x;z) of subsonic transport BVP satisfies the Holder condition on D;

� �

i

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

ðx; y; z; τ; nðy; τÞÞujðy; τÞ

ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>τ</sup>; <sup>n</sup> <sup>T</sup><sup>j</sup>

i

, x ∈S, y∈S,

ð Þ x � y; z � τ; n :

, <sup>j</sup> <sup>þ</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> i ∗GjH� D,

jmuknlδDð Þ x; z (43)

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

j

<sup>δ</sup>Dð Þþ <sup>x</sup>; <sup>z</sup> <sup>U</sup><sup>b</sup> <sup>j</sup>

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

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

9. Singular boundary integral equations of subsonic transport BVP

<sup>k</sup>ujðx, zÞ � ujðy, tÞk <sup>≤</sup> <sup>C</sup>kðx, <sup>z</sup>Þ�ðy,tÞk<sup>β</sup>

ð42Þ

141

(44)

rc 2 Lj i

∗Pjδ<sup>D</sup> þ λuknkδ

which we write in a form more suitable for transformation as:

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

ð Þ x; y; z; τ pj

formula for problems of elastostatics ([1]: 605):

ð

Uj i

D

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

i

we obtain the equation for <sup>b</sup>u xð Þ ; <sup>z</sup> :

in the space of generalized functions:

<sup>2</sup>bui <sup>¼</sup> <sup>U</sup><sup>b</sup> <sup>j</sup> i

rc

uiH�

<sup>D</sup>ð Þ¼ x; z

where we introduce the shift tensors:

i, j ¼ 1, 2, 3

Uj i

and do not exist forð Þ x; z ∈ D.

namely,

Gb ¼ GH�

It follows from the formula (Eq. (40)) of Theorem 7.1. The integrand is a positive quadratic form in ui,j, since the elastic potential satisfies the relation W ≥ 0 ([1]: 589, 591); moreover, W = 0 only for displacements of the medium treated as an absolutely rigid body. Therefore, Eq. (40) is true only if ui,j = 0 for all i, j. This, together with the decay of solutions at infinity and the arbitrary choice of z, implies that u = 0.

The proof of the theorem is complete. It is valid both for the internal and external boundary value problem. The asymptotic requirements on G and the boundary functions may be weakened.

### 8. General functions method: statement of subsonic transport BVP in D0 <sup>3</sup> <sup>R</sup><sup>3</sup> � �

Our aim is to construct the solution of BVP by using boundary integral equations (BIE) for u(x,z). The construction of an analog of Green's formula for solutions of elliptic equations ([3]: 366), which permits one to determine the values of the desired function inside the domain on the basis of the boundary values of the function and its normal derivative, is the key point in the construction of BIE of boundary value problems. An analog of this formula for equations of the static theory of elasticity is referred to as the Somigliana formula [1]. It determines the function u(x,z) in the domain D�, if the boundary values of displacements uD(x,z) and stresses p(x,z) are given. We construct a dynamic analog of that formula in the case of transport solutions. To this end, we use the method of generalized functions (GFM).

We introduce the regular generalized solution of BVP

$$
\widehat{\boldsymbol{\mu}}(\mathbf{x}, \mathbf{z}) = \boldsymbol{\mu}(\mathbf{x}, \mathbf{z}) H\_D^-(\mathbf{x}) = \boldsymbol{\mu}(\mathbf{x}, \mathbf{z}) H\_S^-(\mathbf{x}) \mathbf{1}(\mathbf{z}).\tag{41}
$$

which defines it as a regular vector function on all space R<sup>3</sup> . Here H� <sup>D</sup>ð Þ x; z is the characteristic function of the set D: 1(z) � 1, H� <sup>S</sup> ð Þx is the characteristic function of S�, which is equal to 0.5 at S: ∂jHS �ð Þ¼� x njð Þx δSð Þx , where nj(x)δS(x) is a simple layer at S.

By using the properties of the differentiation of regular generalized functions with jumps on D, we obtain the equation for <sup>b</sup>u xð Þ ; <sup>z</sup> :

$$
\rho c^2 L\_i^j(\mathbf{\hat{o}}\_x, \mathbf{\hat{o}}\_z) \hat{u}\_j(x, z) = \hat{\mathbf{G}}\_i + \\
$$

$$
+(\rho c^2 n\_3 \hat{u}\_{i,z} - P\_i)\delta\_D + (n\_3 \hat{u}\_i \delta\_D)\_{;z} - \left(C\_{ij}^{kl} \hat{u}\_k n\_l \delta\_D\right)\_{;j} \tag{42}
$$

Gb ¼ GH� <sup>D</sup>ð Þ x; z , δDð Þ¼ x; z δSð Þx 1ð Þz , 1ð Þ� z 1, is a simple layer on D. Since n<sup>3</sup> = 0 on D, it follows from the properties of the Green tensor that an analog of the Somigliana formula holds in the space of generalized functions:

$$\rho \boldsymbol{\sigma}^2 \widehat{\boldsymbol{u}}\_i = \widehat{\boldsymbol{\mathcal{U}}}\_i^j \boldsymbol{\ast} \boldsymbol{P}\_j \boldsymbol{\delta}\_D + \left( \left( \lambda \boldsymbol{u}\_k \boldsymbol{n}\_k \boldsymbol{\delta}\_l^j + \mu \left( \boldsymbol{n}\_j \boldsymbol{u}\_l + \boldsymbol{n}\_l \boldsymbol{u}\_j \right) \right) \boldsymbol{\delta}\_D \,^\* \widehat{\boldsymbol{\mathcal{U}}}\_i^l \right)\_{\cdot j} + \widehat{\boldsymbol{\mathcal{U}}}\_i^j \boldsymbol{\ast} \boldsymbol{G}\_j \boldsymbol{H}\_{D'}^- $$

which we write in a form more suitable for transformation as:

$$
\widehat{\boldsymbol{\mu}}\_{i} = \widehat{\boldsymbol{\mathcal{U}}}\_{i}^{j} \ast \boldsymbol{p}\_{j} \delta\_{\boldsymbol{D}}(\mathbf{x}, \boldsymbol{z}) + \widehat{\boldsymbol{\mathcal{U}}}\_{i^{\ast} \cdot \boldsymbol{m}}^{j} \prescript{\widetilde{\mathbf{C}}}{}{\boldsymbol{\mathcal{U}}}\_{jm} \boldsymbol{\mu}\_{k} \boldsymbol{n}\_{l} \delta\_{\boldsymbol{D}}(\mathbf{x}, \boldsymbol{z}) \tag{43}
$$

If we write out this convolution in integral form with regard to the notation introduced here and Eqs. (1) and (2), then we obtain a formula, whose form coincides with the Somigliana formula for problems of elastostatics ([1]: 605):

$$\begin{aligned} u\_i H\_D^-(\mathbf{x}, z) &= \int\_D \left( \mathcal{U}\_i^j(\mathbf{x}, y, z, \tau) \, p\_j(y, \tau) - T\_i^j(\mathbf{x}, y, z, \tau, n(y, \tau)) u\_j(y, \tau) \right) dD(y, \tau), \\ \mathbf{i}, j &= 1, 2, 3 \end{aligned} \tag{44}$$

where we introduce the shift tensors:

The following assertion is its corollary.

140 Differential Equations - Theory and Current Research

Lame equations ((Eq. (14)) by G(x,z) = 0.

arbitrary choice of z, implies that u = 0.

Then for ∀z

weakened.

D0 <sup>3</sup> <sup>R</sup><sup>3</sup> � �

S: ∂jHS

Theorem 7.2. The solution of the subsonic transport boundary value problem is unique.

W þ rc 2

ð

S�

Proof. Since the problem is linear, it suffices to prove the uniqueness of the zero solution. Let u (x,z) satisfy the zero boundary conditions P(x,z) = 0 on D and be a solution of the homogeneous

It follows from the formula (Eq. (40)) of Theorem 7.1. The integrand is a positive quadratic form in ui,j, since the elastic potential satisfies the relation W ≥ 0 ([1]: 589, 591); moreover, W = 0 only for displacements of the medium treated as an absolutely rigid body. Therefore, Eq. (40) is true only if ui,j = 0 for all i, j. This, together with the decay of solutions at infinity and the

The proof of the theorem is complete. It is valid both for the internal and external boundary value problem. The asymptotic requirements on G and the boundary functions may be

Our aim is to construct the solution of BVP by using boundary integral equations (BIE) for u(x,z). The construction of an analog of Green's formula for solutions of elliptic equations ([3]: 366), which permits one to determine the values of the desired function inside the domain on the basis of the boundary values of the function and its normal derivative, is the key point in the construction of BIE of boundary value problems. An analog of this formula for equations of the static theory of elasticity is referred to as the Somigliana formula [1]. It determines the function u(x,z) in the domain D�, if the boundary values of displacements uD(x,z) and stresses p(x,z) are given. We construct a dynamic analog of that formula in the case of transport

<sup>D</sup>ð Þ¼ x u xð Þ ; z H�

<sup>S</sup> ð Þx 1ð Þz , (41)

<sup>D</sup>ð Þ x; z is the characteristic

. Here H�

<sup>S</sup> ð Þx is the characteristic function of S�, which is equal to 0.5 at

8. General functions method: statement of subsonic transport BVP in

solutions. To this end, we use the method of generalized functions (GFM).

<sup>b</sup>u xð Þ¼ ; <sup>z</sup> u xð Þ ; <sup>z</sup> <sup>H</sup>�

�ð Þ¼� x njð Þx δSð Þx , where nj(x)δS(x) is a simple layer at S.

We introduce the regular generalized solution of BVP

function of the set D: 1(z) � 1, H�

which defines it as a regular vector function on all space R<sup>3</sup>

<sup>∥</sup>u, <sup>z</sup>∥<sup>2</sup> � �dV xð Þ¼ <sup>0</sup> (40)

$$\mathcal{U}\_i^j(\mathbf{x}, y, z, \tau) = \mathcal{U}\_i^j(\mathbf{x} - y, z - \tau), \quad \mathcal{T}\_i^j(\mathbf{x}, y, z, \tau, n) = \mathcal{T}\_i^j(\mathbf{x} - y, z - \tau, n).$$

This formula permits one to determine displacements in the medium on the basis of known boundary values of displacements and stresses. But the integrals are regular only for xð Þ ; z ∉D and do not exist forð Þ x; z ∈ D.
