Open access peer-reviewed chapter

# General Functions Method in Transport Boundary Value Problems of Elasticity Theory

Written By

Lyudmila Alexeyeva

Submitted: November 6th, 2017 Reviewed: January 26th, 2018 Published: May 23rd, 2018

DOI: 10.5772/intechopen.74538

From the Edited Volume

## Differential Equations

Edited by Terry E. Moschandreou

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## Abstract

The Lame system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, and supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Transport boundary value problem for an elastic medium bounded by a cylindrical surface of arbitrary cross section and subjected to transport loads is considered in the subsonic and supersonic case with regard to shock waves. To solve problems, the generalized functions method is developed. In the space of generalized functions, generalized solutions are constructed and their regular integral presentations are obtained. Singular boundary equations solving the boundary value problems are presented.

### Keywords

• elastic medium
• subsonic
• transonic
• supersonic speed
• shock waves
• boundary value problem
• generalized functions method
• generalized solutions
• singular boundary equations

## 1. Introduction

A widespread source of wave generation in continuous media is transport loading, i.e., moving loads whose form does not change over time. The velocity of a transport load has a large effect on the type of differential equations describing the dynamics of the medium. The equations depend parametrically on the Mach numbers, i.e., on the ratio of the speed of motion to the propagation speeds of perturbations in the medium (sound speeds). It is well known [1] that, in an isotropic elastic medium, there are two sound speeds (c1,c2), which determine the velocities of dilatational and shear waves propagation. This has a large effect on the type of equations and leads to systems of elliptic, hyperbolic, or mixed equations. For transport problems, typical factors are shock effects generated by supersonic loading. At shock fronts, the stresses, displacement rates, and energy density are discontinuous. A convenient research method for such problems is provided by the theory of generalized functions (distributions), which makes it possible to significantly expand the class of processes amenable to study by using singular generalized functions in the simulation of observed phenomena. In this chapter, methods of this theory are used to solve boundary value problems using motion equations of the theory of elasticity in cylindrical domains under the action of transport loads, moving at supersonic and supersonic speeds.

## 2. Motion equation of elastic medium

We consider an isotropic elastic medium with Lame’s parameters λ,μ, and a density ρ. Let us denote x = xjej, ej as the unit vectors of Cartesian coordinate system in the space R3; displacements vector u(x,t) = ujej; stress tensors σij deformation tensor εij. These tensors are connected by Hook’s law [1]:

εij=0,5ui,j+uj,i,i,j,k=1,2,3.E1
E2

The elastic constant tensor has the symmetry properties.

In the case of an isotropic medium, it is equal to

and Hook’s law has the form

σij=λdivuδij+µui,j+uj,i

Here δij=δij is the Kronecker symbol. Everywhere, there are tensor convolutions over of the same name indexes from 1 to 3, ui,juixj.

Motion equations for material continuum

E3

for elastic medium by using Eqs. (1) and (2) have the form:

E4

Here L is the matrix Lame’s operator:

c1=λ+2µ/ρ,c2=µ/ρ are the velocities of dilatational and shear waves (c1 > c2), G(x,t) is the mass force, ∆ is the Laplace operator.

The system shown in Eq. (4) was fairly well studied by Petrashen [2]. Since the elastic potential of the medium is positive definite, this system is strictly hyperbolic. Such systems can have solutions with discontinuous derivatives. The discontinuity surface F in R4 = R3 × t(−∞ < t < ∞) coincides with a characteristic surface of the system. It corresponds to a wave front Ft moving in R3 at the velocity V:

E5

We note that ν(x,t) = (ν123t) is a normal vector to F in R4, satisfying the characteristic equation

E6

This equation has the roots:

νt=±cjν3,j=1,2.E7

From Eqs. (5) and (7), we get that Ft moves in R3 at the sound velocity V = c1 or V = c2.

We introduce a wave vector m = (m1, m2, m3). It is a unit normal vector to Ft in R3 for fixed t in the direction of wave propagation. By virtue of Eq. (7),

E8

Let νt = ν4. The requirement that the displacements be continuous across the wave front, i.e.,

uxtFt=0E9

which is associated with the preservation of the continuity of the medium, leads to kinematic consistency conditions for solutions at the wave front:

E10

(the continuity of the tangent derivatives on Ft). Additionally, Eq. (4) implies dynamical consistency conditions for solutions at the wave front, which are equivalent to the momentum conservation law in its neighborhood:

E11

Definition. A wave is called a shock wave if the jump in the stresses across the wave front is finite: eimj[σij]Ft = 0̸. If mj[σij]Ft = 0, then this is a weak shock wave. If mj[σij]Ft = ∞, then this is a strong shock wave.

Velocity suffers a jump discontinuity across a shock front. At fronts of weak shock waves, the velocities are continuous, but the second derivatives of solutions are not. Strong shock waves (in the sense of the aforementioned definition) do not occur in actual media, since, at large stress jumps, the medium is destroyed and ceases to be elastic. However, strong shock waves in elastic media play an important theoretical role in the construction of solutions, specifically, fundamental solutions of Eq. (4).

## 3. Lame transport equations and Mach numbers

Suppose that the force affecting the medium moves at a constant velocity c along the X3 axis (for convenience, in its negative direction) and, in a moving coordinate system x=x1x2z=x3+ct it does not depend on t:

Gxz=Gjx1x2x3+ctejE12

Transport solutions are solutions of Eq. (4) with the same structure:

u=ux1x2x3+ct=uxzE13

The speed of transport loads is called subsonic if c < c2,transonic if c2 < c < c1, and supersonic if c > c1. A speed is called the first or second sound speed if c = cj, j = 1, 2, respectively.

In the new variables, the equations of motion are brought to the form

E14

Here gj=ρc21Gj;Mj=c/cj are Mach numbers: (M1 < M2).

As Mj < 1(j = 1, 2) the load is subsonic and the system of equations is elliptic. If the load is supersonic, i.e., Mj > l, j = 1, 2, then the system becomes hyperbolic. In the case of transonic speeds, i.e., M1 < 1 and M2 > 1, the equations are hyperbolic-elliptic. In the case of sound speeds, the equations are parabolic-elliptic if M2 = 1 and parabolic-hyperbolic if M1 = 1. We will show this later when considering fundamental solutions of Eq. (14).

Since the original system is hyperbolic, Eq. (14) can also have discontinuous solutions. Let F be a discontinuity surface in the space of variables x such that it is stationary in this space and moves at one of the sound velocities V = c1,c2 in the space of (x1,x2,x3). It follows from Eq. (7) that V = cn3, where n = (n1,n2,n3) is the unit normal to F in R3. Therefore, since c = cj/n3 and |n3| ≤ 1, such surfaces can arise only at supersonic speeds: c ≥ cj.

It follows from Eqs. (9) to (11) and Eq. (13) that the kinematic and dynamical consistency conditions for solutions at discontinuities in the mobile coordinate system have the form:

uxzF=0nzui,jnjui,zF=0;E15
σijnj=ρckcui,zF;nz=ck/c,forcck;E16

n=n1n2nz=n3 is a wave vector, k = 1 for shock dilatational waves, k = 2 for shock shear waves. Here and hereafter, the derivative with respect to xj is denoted by the index j after a comma in the function notation or by the variable itself.

Definition. If c > c2, the solution of the system in Eq. (14) is called classical if it is continuous and twice differentiable everywhere, except for, possibly, wave fronts. The number of fronts is finite at any fixed t and the conditions on the gaps, Eqs. (15) and (16), are satisfied on the wave fronts.

At first, we construct the solutions of the transport Lame equation using methods of generalized functions theory.

## 4. Shock waves as generalized solutions of transport Lame equations: conditions on wave front

Consider Eq. (14) and its solutions on the space of generalized vector functions D3 (R3) with components being generalized functions from D(R3) (see [3]). Obviously, if u is a solution of Eq. (14) that is twice differentiable, then it is also a generalized solution of Eq. (14). If a vector function u satisfies Eq. (14) in the classical sense almost everywhere, except for some surfaces, on which its derivatives are discontinuous, then, generally speaking, u is not a generalized solution of Eq. (14).

Let u(x,z) be a shock wave (x = (x1,x2)), i.e., a classical solution of the Lame transport equations, Eq. (14), that satisfies conditions Eqs. (15) and (16) at the front F. Let ûxz denote the corresponding regular generalized function.

The shock wave ûxz is a generalized solution of the Lame equation in D3 (R3).

Proof. Using the rules for differentiating generalized functions with derivatives having jump discontinuities across some surfaces (see [3]), for the equations of motion in D3 (R3), we obtain

σ̂ijxjρc22ûixz2+Gi=σijhjρc2hzuizFδF++xjλukhkδij+μ(uihj+ujhiFδFzuiFhzδF,E17

Here, the right-hand side involves singular generalized functions, namely, single layers δF (x,z) and double layers on F. By virtue of conditions Eqs. (15) and (16), the densities of these layers are equal to zero, so the right-hand side of Eq. (17) vanishes; i.e., the shock wave satisfies the same equations, Eq. (14), but in the generalized sense.

As a result, we obtain a simple formal method for deriving conditions at jumps in solutions and their derivatives across the shock fronts in hyperbolic equations. Namely, these equations are written in the space of generalized functions and the densities of the singular functions corresponding to single, double, etc., layers are set to zero.

Define as follows the kinetic energy density

K=0.5ρu,t2=0.5ρc2u,z2E18

and elastic potential

W=0.5σijui,j=0.5σijεijE19

Consider the following functions: the energy density E = K + W of elastic deformations and the Lagrangian Λ = K − W.

If G is continuous, then the Lagrangian Λ is continuous at the shock waves fronts.

(ΛF=0) and the jump in the energy density satisfies the relation

hzEF=σijhjui,zF,E20

First formula is equivalent to the equality:

EFck=ckchjkσijui,zFck,k=1,2

where ck is the sound velocity corresponding to front F, hjk is the components of the wave vector to F.

The last formula may be easy to get if we write the equation for E in D3R3 in the form

Ê,z=E,z+EFhzδF=σijui,z,j+ρGu,z+σijui,zFδFhj+ρGuFhzδFEFhz=σijui,zFhj

as uF=0. For the gaps of these functions, the theorem has been proved on the basis of classic methods (see [4, 5]). For full proof of this theorem, see [6].

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

The matrix of fundamental solutions Ûxz satisfies Eq. (14) with a delta function in the mass force:

E21

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

E22

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 = X3 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¯(ξ123). It is reduced to the form (see [4]).

E23

It can be seen that Ûxz 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

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

E24

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=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:

Uijxz=c22δijf02xz+c2f21,ijxzf22,ij(xz),E25

where the type of basic function depends on velocity c.

In subsonic case (Mk < 1):

4πfojrz=1z2+mj2r2,4πf1j=sgnzlnz+z2+mj2r2mjr,4πf2j=zlnz+VjmjrVj+mjx,

In sonic case (Mk = 1):

fokxz=0.5δzx,f1k=0.5θzx,f2k=0.5zθzx.

In supersonic case (Mk > 1):

fojrz=θzmjr2πz2mj2r2,f1j=θzmjr2πlnz+Vjmjr,f2j=θzmjr2πzlnz+VjmjrVj,

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

mk=1Mk2,Vk=z2+mk2r2,Vk=z2mk2r2,r=x12+x22=x,

The dilatational and shear components of Ûxz are easy to write out

Uijxz=Ui1jxz+Ui2jxzUi1j=c2f21,ijxz,Ui2jxz=c22δijf02xzf22,ijxzE26

In the supersonic case, the support of the functions is the cone z > mkx∥. 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 = mkx∥), Green’s tensor grows to infinity.

If the following convolution exists,

ûi=ÛijGjxz/ρc2E27

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:

uixz=DUijxyzτgjyτdy1dy2=uixzE28

If mass forces are concentrated on surface D and described by singular generalized functions of the type of single layers g=gjyτejδDyτ, then

ûixz=D(UijxyzτgjyτdDyτ=uixzE29

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.

## 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:

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

E30

these components are bounded for xz000. At the point xz=000, they have a weak singularity of order R1,R=z2+r2. It has a similar asymptotic at infinity. Accordingly, R−2 is the order of the tensor derivatives asymptotic and the behavior of at .

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

Σjkixz=λUli,lδjk+μUji,k+Uki,j,Γjixzn=ΣjkixznkT̂jixzn=ρc21ΓijxznE31

Then the elastic constant tensor is presented in the form

T̂ijxzn=C˜kmjlÛik,mnl,C˜kmjl=Ckmjl/ρc2E32

Tensor Γjixzn describes the stresses at the plate with normal n in a point x = (x,z). Tensor T̂ have some remarkable properties.

Fundamental stress tensor T̂ is the generalized solution of the transport Lame equation with singular mass forces of the multipole type:

ρc2Lijx'T̂jk+Kkix'nδx'=0E33

where

Kilx'n=λnil+μтjδilj+δjli.

For any closed Lyapunov’s surface D, bounding a domain DR3

δijHDxz=V.P.DTij(xyτznyτ)+Uij,z(xyτz)nz(yτ)dSyτE34

where HDxz is the characteristic function of D, which is equal to 0.5 at D; nyτ is a unit normal vector to D. The integrals are regular for xzD and are taken in value principle sense for xzD.

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).

## 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 n3 = 0. We assume that G is an integrable vector function and ∃ε > 0 such that

GxzOx3+εforx,xD+D.E35

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

σijxznjx=Pxz=ρc2pixz,xzDE36

We assume that ∃εi > 0:

E37
pxzOz1ε2forz,xS.E38

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

Classic solution of transport BVP satisfying to the equalities:

DabPudDxzDabW0.5ρc2u,z2Gudx1dx2dz+
+Sρc2ui,zσi3xauixaρc2ui,zσi3xbxbui(xb)dx1dx2dz=0
SW+05ρc2u,z2σi3ui,zz±dx1dx2=Dz,±Pui,zdx1dx2dz+Dz,±Gu,zdx1dx2dz
DPxzuxzdDxz=D0.5ρc2u,z2WGudx1dx2dz
SW+05ρc2u,z2dVx=DPu,zdDxz+DGu,zdVxzE39
Dab=xz:xDazb,Dab=xz:xDa<z<b.

The following assertion is its corollary.

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

SW+ρc2u,z2dVx=0E40

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 D3'R3

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

ûxz=uxzHDx=uxzHSx1z,E41

which defines it as a regular vector function on all space R3. Here HDxz is the characteristic function of the set D: 1(z) ≡ 1, HSx is the characteristic function of S, which is equal to 0.5 at S: jHSx=njxδSx, 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 ûxz:

ρc2Lijxzûjxz=Ĝi+
E42

Ĝ=GHDxz,δDxz=δSx1z,1z1, is a simple layer on D. Since n3 = 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:

ρc2ûi=ÛijPjδD+λuknkδlj+μnjul+nlujδDÛil,j+ÛijGjHD,

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

ûi=ÛijpjδDxz+Ûij,mC˜jmkluknlδDxzE43

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):

uiHDxz=DUijxyzτpj(yτ)Tij(xyzτn(yτ))uj(yτ)dDyτ,i,j=1,2,3E44

where we introduce the shift tensors:

Uijxyzτ=Uijxyzτ,Tijxyzτn=Tijxyzτn.

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 xzD and do not exist for xzD.

## 9. Singular boundary integral equations of subsonic transport BVP

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

If the solution u(x;z) of subsonic transport BVP satisfies the Holder condition on D; namely,

uj(x,z)uj(y,t)C(x,z)(y,t)β,xS,yS,

then u(x;z) satisfies the singular boundary integral equation

0,5uixz=ĝjÛijxz+DUijxyzτpjyτdDyτV.P.DTijxyzτnyτujyτdDyτi,j=1,2,3E45

Proof. Let consider Eq. (45) for xzD. Let (x,z) ∈ D, x → (x,z). Then, using Theorem 6.1, we have

limxzxzuixz=uixz=ĝjÛijxz+limxzxzDUijxyzτpjyτdDyτlimxzxzDTijxyzτnyτujyτuj(xz)dDyτ++ujxzlimxzxzDTijxyzτnyτdDyτ==ĝjÛijxz+DUijxyzτpjyτdDyτV.P.DTijxyzτnyτujyτuj(xz)dDyτ++ujxzlimxzxzδijDUij,zxyzτ)nzydD(yτ)==ĝjÛijxz+DUijxyzτpjyτdDyτV.P.DTijxyzτnyτujyτuj(xz)dDyτ++ujxzδij=ĝjÛijxz+DUijxyzτpjyτdDyτV.P.DTijxyzτnyτujyτdDyτ0,5uixz+uixz.

In the last relation, we have used the obvious properties: integrals with Uij exist by virtue of the Holder property of u on D and weak singularity Uij at D. Then if the surface integral exists, its value coincides with the principal value; the principal value of the integral containing the difference of integrated functions is equal to the difference of the principal values of integrals corresponding to each of these functions if they exist.

By transposing the last two terms to the left-hand side of the relation, we obtain the formula of the theorem for the boundary points. The proof of the theorem is complete.

This theorem gives us resolving system of integral equations for defining unknown values of boundary displacements.

Note also that the subsonic analog of the Somigliana formula was obtained for generalized functions. But since they are regular, from the Dubois-Reymond lemma ([3]: 97), the solution is classical. However, if the acting loads are described by singular generalized functions, which often takes place in physical problems, then one should use a representation of a generalized solution in the convolution form (Eq. (43)) with the evaluation of convolutions by the definition (see [3]: 133).

## 10. Supersonic green’s tensor and its antiderivative with respect to z

From Eq. (25), we get the regular representation of Ûij in the supersonic case which has the form

2πU11=θ2V2+z2x12r4M22θ1V1θ2V2x22r4M22θ1V1θ2V2,2πU22=θ2V2+z2x22r4M22θ1V1θ2V2x12r4M22θ1V1θ2V2,E46
2πU12=x1x2r4z2θ1V1θ2V2+θ1V1θ2V2,2πU33=θ1V1+θ2m22V2,2πU13=x1zr2θ1V1θ2V2,2πU23=x2zr2θ1V1θ2V2

Here θj=θzmjx,Vj=z2mj2x2,mj=Mj21. It satisfies the radiation conditions:

suppzUxzz>0,Uik0,Ui'jk0 byx.E47

One can readily see that its components are zero outside the sonic cones:

Kl+=xz:z>mlx,l=1,2.

On the surfaces of the cones, the components U13 have singularities of the type (z2 − m2jr2)−1/2.

For solution of supersonic problems, we introduce the tensor Ŵjixz, which is the antiderivative of Ûji with respect to z:

Ŵji=k=12Ŵjki=Ûjiδx1δx2θz=Ûjizθz,Ŵji,z=ÛjiE48

They are also fundamental solutions of Eq. (14) for the mass forces of the corresponding Fjzθz. After calculation, we define its components as:

2πW11=zr4x12x22θ1V1θ2V2+0,5m12θ1lnz+V1m1r+M2205m22θ2lnz+V2m2r2πW22=zr4x12x22θ1V1θ2V2+0,5m12θ1lnz+V1m1r+M2205m22θ2lnz+V2m2rE49
2πW33=θ1lnz+V1m1r+m22θ2lnz+V2m2r,2πW23=x2r2θ1V1θ2V22πW12=zx1x2r4θ1V1θ2V2,2πW13=x1r2θ1V1θ2V2

Tensor Ŵji has the same support as Ûji but as at the cone Kj

E50

it continues on fronts Kj. Wjixz has weak singularity by x’ = 0 and weak logarithmic singularity on Z with respect to ∥x∥ by x = 0. To single out these singularities, we decompose it into the terms:

Wjixz=Wjisxz+Wjidxz=k=12θkzmkrWjkisx+Wjidxz,2πc2Wj1isx=δi3δj3+05m121δi3δijlnm1r,2πc2Wj2isx=(δi3δj3+δij05m121δi3M22lnm2rE51

The tensors Wjis of diagonal form are independent of z inside the sonic cones Kl(l = 1, 2) and have a logarithmic singularity with respect to ∥x∥ on the Z-axis. Unlike the generating tensor Wjis, Wjid has bounded jumps on the Kl. One can readily see that the tensor shifts

Uijxyz=Ûijxyz,Wijxyz=Ŵijxyz

have the following symmetry properties around the Z-axis:

Uijxyz=Uijyxz=Ujixyz,Wijxyz=Wijyxz=Wjixyz,i,j=1,2E52

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

Uijxyz=Uijyxz,Wijxyz=WijyxzE53

## 11. Fundamental supersonic antiderivative stress tensor Ĥ and its properties

We introduce antiderivative stress tensor

Σ˜i3j=Σ̂jiθzδxĤji=T̂jiθzδx=T̂jizθz,Ĥji,z=T̂jiE54

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:

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

Hijxyzm=Hijyxzm=Hijxyzm

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

Hi3xyz=Hi3yxz,Hi3xyz=Hi3yxz,i=1,2.

Components Hjixz have weak singularities on the fronts of the type z2m2jx21/2, but more stronger singularity of the type of ∥x−1 on the axis Z. If we put Eq. (51) in Hook’s law, then we can again single out two terms in Hjixz:

Hjixz=Hjisxz+Hjidxz=k=12θkzmkrHjkisx+Hjkid(xz)E55

Since the tensors Hjkisx independent of z inside the sonic cones Kl (l = 1,2), we conventionally say that they are stationary. Accordingly, the tensors Hjidxz 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.

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

The fundamental stress tensor H satisfies the relation

δijHSxθz=0zSHijyxτnydSy++Sρc21Σ˜i3j(xyz)Uij,z(yxz)dy1dy2

For xD 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.

## 12. Statement of supersonic transport BVP: uniqueness of solutions

We suppose here that supersonic transport loads, moving at supersonic velocity c > c1, are known on the boundary D:

P=σijniej=ρc2pjxzejθz,x=x1x2S,i,j=1,2,3E56

Functions pj(x,z) are integrable on D+. We assume here G = 0 and

uxz=0,ui,zxz=0,z0,xSE57

For ∥(x,z)∥ → ∞

uj0,ε>0:ju<Oxz1+ε,j=1,2,zE58

The jump conditions, Eqs. (15) and (16) are satisfied on the shock wave fronts.

The solution of the supersonic transport boundary value problem is unique.

Proof. Suppose that there exist two solutions. Since the problem is linear, it follows that their difference u(x,z) satisfies the zero boundary conditions, i.e., P(x;z) = 0, and is a solution of the homogeneous equations of motion (G = 0). We note, that Lemma 8.1 is also true in the supersonic case for shock waves as there is Theorem 3.2 for the gaps of energy on their fronts (see full proof). Then together with conditions given in Eq. (59) of decay of the solutions at infinity and the zero conditions for z = 0,

SExzdx1dx2=Sσi3ui,zxzdx1dx20byz

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.

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

## 13. Statement of supersonic BVP in D3'R3 and its generalized solution

To solve the problem, we also use the method of generalized functions. We introduce here the regular generalized function with support on D+:

ûjxz=ujxzHSxθzE59

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:

ĝj=pjδSxθz+λuknkδij+μuinj+ujniδSxθz,iE60

By using the properties of convolutions with the Green tensor and the boundary conditions, we obtain the generalized solution of BVP in the form:

ρc2ûk=ÛkjPjδSxθz+Ûkj,iλumnmδij+μuinj+ujniδSxθzE61

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

uiHSxθz=D+Uijxyzτpj(yτ)Tij(xyzτn(yτ))uj(yτ)dDyτE62

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 it. For this, we use the tensor H.

## 14. Dynamic analog of the Somigliana formula in supersonic case

For regularization of Eq. (61), we put W,z instead of U in the second term and use the property of differentiation of convolution:

ρc2ûk=ÛkjPjδSxθz+Ŵkj,izλumnmδij+μuinj+ujniδSxθz==ÛkjPjδSxθz+Wkj,iλum,znmδij+μui,znj+uj,zniδSxθz++Ŵkj,iλumnmδij+μuinj+ujniδSxδz==ÛkjPjδSxθz+Ŵkj,mCjmilui,znlxδSxθz+CjmilŴkj,mxuix.0nlxδSxE63

From here on, we use Eq. (57) we get the formula which can be written in integral form.

The generalized solution of supersonic transport BVP can be presented in the form:

ûk=ÛkjpjδSxθz+C˜jmilŴkj,mui,znlxδSxθzE64

which for xS has the next integral presentation

uiHSxθz=k=12SθzmkrdSy0zmkrUijxyzτpj(yτ)Hijd(xyzτny)uj,z(yτ)SHijsxyznyujyzmkrdSyE65
r=xy

Proof. Formula (65) follows from Eq. (64) in virtue of Eqs. (61) and (32). Its integral form is

uiHSxθz=0zUijxyzτpj(yτ)Hij(xyzτny)uj,z(yτ)

If we use Eq. (55) for Hjixz as the support of Hjisxz,Hjidxz, we get

uiHSxθz=k=12SθzmkrdSy0zmkrUijxyzτpj(yτ)(Hikjd(xyzτny+Hikjs(xyny))uj,z(yτ)E66

Note that

0zmkrHikjsxynyuj,τyτ=Hikjsxyny)(ujyzmkrujy0==Hikjsxyny)ujyzmkr

In virtue of this equity, we get from Eq. (66) the last formula of the theorem.

All integrals exist; indeed, the integrands are integrable everywhere, including the fronts of fundamental solutions, because the kernels of the integrands have weak singularities on the fronts of the form z2m2jx21/2 in virtue of the properties of kernels U and H. The proof is completed.

This formula is a dynamic analog of Somigliana formula for supersonic loads. It defines the displacement in elastic medium by using boundary values of stresses and velocity of displacements of boundary surface.

This formula also preserves its form for xzD with regard to the definition of HSxθz on D.

## 15. Singular boundary integral equations of supersonic transport BVP

Тheorem 15.1. If the classical solution of BVP satisfies the Holder’s conditions at D+, i.e., C > 0, β > 0 that

ujxzuj(yz)<Cxyβ,x,yS.

then it satisfies the singular boundary integral equation at D+

0,5uixz=k=12Szkx'θzmkrdSy0zmkrUijxyzτpj(yτ)
Hijd(xyzτny)uj,z(yτ)
V.P.SzkxHijsxyznyujyzmkrdSy,r=xy

Proof. The desired assertion follows from Theorem 14.1 and Theorem 11.1 for tensor H by analogy of the proof of Theorem 12.1 about singular boundary integral equations in the subsonic case. Full proofs of these theorems can be found in [7].

This theorem gives us a resolving system of integral equations for definition of unknown values of boundary displacements in the supersonic case.

Moreover, the Somigliana formula for displacements was obtained for generalized functions. But since they are regular, from the Dubois-Reymond lemma ([3]: 97), this solution is classical. However, if the acting loads are described by singular generalized functions, which often takes place in physical problems, then one should use a representation of a generalized solution in the convolution form (Eq. (65)) with the evaluation of convolutions by the definition (see [3]: 133).

## 16. Conclusion

The constructed singular boundary integral equations in the supersonic case are not classical equations because the solution inside a domain is determined by the boundary values of stresses and displacement rates rather than displacements themselves, unlike the Somigliana formula. In addition, the domain of integration over a boundary surface substantially depends on z, which is specific for hyperbolic equations. This complicates finding solutions of such problems by the successive approximation method. However, for the numerical discretization of singular boundary integral equations, the method of boundary elements makes it possible to use standard methods of computational mathematics for a computer implementation of the solution of such problems. The aforementioned boundary value problems model the dynamics of underground structures like transport tunnels and extended excavations subjected to the dynamic influence of moving vehicles and seismic loads. They permit one to study the dynamics of a rock mass in a neighborhood of underground structures depending on its physical-mechanical properties, the velocity of moving transport, specific features of the transport load, and the geometric properties of structures in technical computations of displacements and the stress-strain state of the mass away from the tunnel.

## References

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3. 3. Vladimirov VS. Obobshchennye funktsii v matematicheskoi fizike (Generalized Functions in Mathematical Physics), Moscow; 1978
4. 4. Alekseyeva LA. Fundamental solutions in an elastic space in the case of moving loads. Journal of Applied Mathematics and Mechanics. 1991;55(5):714-723
5. 5. Alexeyeva LA, Kayshibayeva GK. Transport solutions of the Lame equations and shock elastic waves. Computational Mathematics and Mathematical Physics. 2016;56(7):1343-1354
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7. 7. Alexeyeva LA. Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads. Differential Equations. 2017;53(3):327-342

Written By

Lyudmila Alexeyeva

Submitted: November 6th, 2017 Reviewed: January 26th, 2018 Published: May 23rd, 2018