c Pleiades Publishing, Ltd., 2017. ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 3, pp. 317–332. c L.A. Alexeyeva, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 3, pp. 327–342. Original Russian Text
PARTIAL DIFFERENTIAL EQUATIONS
Singular Boundary Integral Equations of Boundary Value Problems of the Elasticity Theory under Supersonic Transport Loads L. A. Alexeyeva Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic Kazakhstan, Almaty, 050010 Kazakhstan e-mail:
[email protected] Received January 22, 2016
Abstract—We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem. DOI: 10.1134/S0012266117030041
The most common wave generation sources in continuous media are transport sources related to moving loads whose shape is preserved in the course of time. The study of such processes by mathematical modeling methods is reduced to the solution of boundary value problems for systems of differential equations parametrically dependent on the moving load velocity. The motion velocity substantially affects the type of differential equations depending on the Mach numbers, that is, the ratios of the load velocity to the perturbation propagation velocities in the medium (the sonic velocities). There can be several sonic velocities depending on the type of the medium deformation. For small (subsonic) velocities, the equations belong to the class of elliptic equations, for which the theory of boundary value problems is most developed. For supersonic velocities of motion of transport loads, the equations become hyperbolic, shock waves of various types appear in the media, and the strains and medium displacement rates experience jumps on the fronts of these waves. These processes have been little studied yet; therefore, their mathematical modeling based on the construction of transport solutions of the corresponding boundary value problems for systems of hyperbolic equations and the analysis of their properties is quite topical. In [1], transport boundary value problems were considered for an isotropic elastic medium bounded by a cylindrical surface on which a load is moving at a constant subsonic velocity, the shape of the load being preserved in the course of time (a transport load). This class of problems models the dynamics of underground structures like transport tunnels as well as ground road transport and can be reduced to the solution of elliptic boundary value problems for the system of Lam´e equations in a moving coordinate system fixed to the transport load. In the present paper, we consider a similar problem in the supersonic case, where the load velocity exceeds the longitudinal and transverse wave propagation velocities. We suggest a method for the derivation of conditions on the stress, velocity, and energy jumps on the shock wave fronts based on the theory of generalized functions. We pose the corresponding transport boundary value problem and prove the uniqueness of its solution with regard to shock waves. We use the method of generalized functions to construct its solution. We pose the corresponding boundary value problem in the space of generalized functions, construct its generalized solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem. 317
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ALEXEYEVA
´ EQUATIONS. SHOCK WAVES 1. LAME Consider an elastic medium described by the Lam´e parameters λ and μ and the density . We denote the displacement vector by u = uj (x, t)ej (j = 1, 2, 3); further, x = xj ej , where the xj are the Lagrangian coordinates of medium points and the ej are the unit vectors of the Cartesian coordinate system. (Throughout the following, repeated indices imply tensor convolution.) The components σij and εij of the stress and strain tensors are related by Hooke’s law σij = λ div uδij + 2μεij , 1 i, j, k = 1, 2, 3. εij = (ui ,j +uj ,i ), 2
(1) (2)
∂ui ∂ , ∂j = , and δij = δij is the Kronecker delta. ∂xj ∂xj By virtue of relations (1) and (2), the equations of motion of a continuous medium, which read [2, p. 64] ∂ 2 ui i, j = 1, 2, 3, (3) ∂j σij + Gi = 2 , ∂t can be reduced to the form (4) Lji (∂x , ∂t )uj (x, t) + Fi (x, t) = 0, Here and in the following, ui,j =
where Lji is the Lam´e differential operator Lji (∂x , ∂t ) = (c21 − c22 )∂i ∂j + δij (c22 Δ3 − ∂t ∂t ), c1 = (λ + 2μ)/ and c2 = μ/ are the propagation velocities of bulk compression–expansion waves and shear waves, respectively, in the elastic medium, G = Fi ei is the bulk force density, and ΔN is the Laplace operator in RN . System (4) was comprehensively studied by Petrashen’ [3]. It is strictly hyperbolic, because the elastic potential of the medium is positive definite. The determinant of its characteristic matrix Lji (ν, νt ) = (c21 − c22 )νi νj + δij (c22 ν23 − νt2 ),
ν23 = νj νj ,
has six real roots (counting multiplicities) νt = ±cj ν3 ,
j = 1, 2.
(5)
Δ
Here (ν, νt ) = {ν1 , ν2 , ν3 , ν4 } (v4 = νt ) is the normal vector to the characteristic surface F in R4 satisfying the equation (6) Det{L(ν, νt )} = 0, which defines movable surfaces Ft , that is, wave fronts in R3 , on which the derivatives of the displacements u(x, t) experience jumps. The front Ft moves at the velocity V = −νt /ν3 .
(7)
It follows from relations (5)–(7) that the wave front moves in space at one of the sonic velocities (c1 or c2 ); i.e., in the elastic medium there are shock waves of two types related to specific features of its deformation. Bulk compression–expansion waves (longitudinal waves) propagate at the velocity c1 , and slower shear strain waves (transverse waves) propagate at the velocity c2 [2, p. 55]. Let us introduce the wave vector m = (m1 , m2 , m3 ), that is, the unit normal to the front Ft in R3 at a given time t directed in the wave propagation direction. By virtue of relation (7), we have mj =
νj νj V =− , ν3 νt
j = 1, 2, 3,
V = c1 , c2 .
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The requirement of continuity of the displacement across the wave front, which is related to the preservation of medium continuity, (9) [u]Ft = 0, leads to kinematic conditions of compatibility of solutions on the moving fronts, [mj ui ,t +V ui ,j ]Ft = 0,
i, j = 1, 2, 3
(10)
(the condition of continuity of the tangential derivatives ui on Ft ), where [f ]Ft is the jump of a function on the surface Ft . In addition, from Eq. (4) we obtain dynamic conditions on the jump of stresses on the front, which are equivalent to the momentum conservation law in its neighborhood [3], i = 1, 2, 3; V = c1 , c2 . (11) [σij mj + V ui ,t ]Ft = 0, Definition 1. A wave is called a shock wave if the stress jump of the wave front is finite, i.e., if ei mj [σij ]Ft = 0. If ei mj [σij ]Ft = 0 on the wave front and the medium before the wave front is not perturbed, then the wave is called a weak shock wave. If mj [σij ]Ft = ∞, then the wave is called a strong shock wave. In the case of shock waves, there is a velocity jump on the wave front. Strong shock waves (in the sense of the above definition) do not occur in real media, because, for large stress jumps, the medium undergoes fracture and becomes nonelastic. However, strong shock waves in elastic media play an important theoretical role in the construction of solutions of various boundary value problems, in particular, the fundamental solutions of the Lam´e equations. ´ EQUATIONS. MACH NUMBERS 2. TRANSPORT SOLUTIONS OF LAME Let a force acting in a medium move at a constant velocity c along the coordinate axis X3 (here, for convenience of calculations, in the opposite direction) and be independent of t in the moving coordinate system: (12) Gj = Gj (x1 , x2 , x3 + ct). Such loads are naturally called transport loads. Accordingly, we seek solutions of the Lam´e equations (4) of the same structure, (13) uj = uj (x1 , x2 , x3 + ct), which are called transport solutions. Let us introduce the moving coordinate system x = (x1 , x2 , x3 ) = (x1 , x2 , x3 + ct) = (x1 , x2 , z). In the new variables, the equations of motion (4) acquire the form 2 ∂2 2 2 2 2 ∂ δij uj + Fi = 0. (c1 − c2 ) + c2 Δ3 − c ∂xi ∂xj ∂x 23
(14)
Since the original system is hyperbolic, it follows that Eqs. (14) can have solutions with discontinuous derivatives as well. Let F be a discontinuity surface in a moving reference system (where it is fixed), which moves at one of the sonic velocities V = c1 , c2 in the original space of the variables (x1 , x2 , x3 ). It follows from relation (7) that V = ch3 , where h = (h1 , h2 , h3 ) is the unit normal to F in R3 . Since c = cj /|h3 |,
|h3 | ≤ 1,
we find that such surfaces can arise only at supersonic velocities c ≥ cj . By virtue of the choice of the motion direction (opposite to the Z-axis), we have h3 = −ck /c, DIFFERENTIAL EQUATIONS
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Definition 2. A velocity c is said to be subsonic if c < c2 , transonic if c2 < c < c1 , and supersonic if c > c1 . If c = cj , j = 1, 2, then the velocity is called the first sonic velocity or second sonic velocity, respectively. We divide Eq. (14) by c2 , ∂2 ∂2 ∂ −2 −2 −2 uj ≡ (M1 − M2 ) + M2 Δ − 2 δij uj + gi = 0. Λij ∂x ∂xi ∂xj ∂x 3
(16)
Here g = G/(c2 ) and Mj = c/cj are the Mach numbers. The properties of this system were studied in [4, 5]. If Mj < 1 (j = 1, 2), then the load is subsonic, and system (16) has the elliptic type. For supersonic velocities Mj > 1 (j = 1, 2), the equations become hyperbolic. If M1 < 1 and M2 > 1, then the load is transonic, and the type of equations is hyperbolic-elliptic. For the sonic velocities, the equations are parabolic-elliptic if M2 = 1 and parabolic-hyperbolic if M1 = 1 [4]. As follows from relations (9)–(11) with regard to (13), the kinematic and dynamic compatibility conditions for the solutions on the discontinuities in the space of the variables x acquire the form [uj ]F = 0, whence we obtain hz [ui ,j ]F = hj [ui ,z ]F , hj [σij ]F = c2 hz [ui ,z ]F = −ck c[ui ,z ]F .
(17) (18)
Definition 3. For c > c1 , a solution of Eq. (16) is said to be classical if it is continuous and is twice differentiable everywhere outside the wave fronts, that is, moving two-dimensional surfaces in R3 whose number is finite at any time on any closed set in R3 and on which the jump conditions (17) and (18) are satisfied. ´ EQUATIONS 3. SHOCK WAVES AS GENERALIZED SOLUTIONS OF THE LAME Consider Eq. (16) and its solutions on the space D3 (R3 ) = {fˆ = (fˆ1 , . . . , fˆ3 )} of generalized vector functions whose components are generalized functions in D (R3 ) [6]. Let u(x ) ˆ(x, z) we denote the corresponding regular generalized function be a classical solution in R3 . By u u ˆ(x, z) = u ˆ(x ). By the rules of differentiation of regular generalized functions [6], we obtain the equations of motion on D3 (R3 ) in the form 2 ˆi ∂σ ˆij 2∂ u − c + Gi ∂xj ∂z 2 ∂ui 2 = σij hj − c hz δF + {[λuk hk δij + μ(ui hj + uj hi )]F δF },j − {hz [ui ]F δF },z , ∂z F
(19)
where δF is a simple layer on F whose density is defined by the displacement jump on the wave front. By virtue of conditions (17) and (18) on the shock wave fronts, the right-hand side of Eq. (19) is zero; i.e., the generalized function u ˆ satisfies the same equations (16) but in the generalized sense. Therefore, shock waves are generalized solutions of the Lam´e equations. Hence we obtain a simple formal method for the derivation of conditions for the jumps of solutions and their derivatives on the shock wave fronts: one should differentiate them by the rules of differentiation of generalized functions and equate the densities of independent layers with zero. 4. ENERGY CONSERVATION LAW We represent Hooke’s law in the form σij = Cijkl uk ,l ,
Cijkl = λδij δkl + μ(δkj δil + δil δkj ), DIFFERENTIAL EQUATIONS
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and introduce the following notation for the energy characteristics of an elastic medium: E = K+W is the energy density, L = K − W is the Lagrange function, K = 0.5u,t 2 = 0.5c2 u,z 2 is the kinetic energy density, and W = 0.5σij εij = σij ui ,j = Cijkl uk ,l ui ,j is the elastic strain energy density, that is, the elastic potential, which is a positive definite function of the strains [2, p. 106]. Since the stresses and velocities experience jumps on the shock wave fronts, let us compute the jumps in these energy functions. Theorem 1. If G(x, z) is a continuous vector function, then, in the differentiability domain of the solutions, one has 2L = c2 (ui ,z ui ),z + Gi ui − (σij ui ),j ,
E,z = (σij ui ,z ),j + (G, u,z ),
(20)
and the jumps in the energy E and the Lagrange function L on the shock wave fronts are given by [E]Fk = −
chkj [σij ui ,z ]Fk , ck
[L]Fk = 0,
k = 1, 2,
(21)
where the hkj are the components of the unit normal to the wave front and ck is the corresponding sonic velocity; i.e., the Lagrange function is continuous on the shock wave fronts. Proof. For the first formula in the domain of differentiability of the solutions, we take the convolution of the equations of motion with ui ,z , σij ,j ui ,z − c2 ui ,zz ui ,z − Gi ui ,z = 0. Since σij ,j ui ,z = (σij ui ,z ),j − σij ui ,jz , we have the relation
K,z = 0.5 c2 (ui ,z ui ,z ),z ,
W,z = σij ui ,jz ,
(σij ui ,z ),j − E,z + (G, u,z ) = 0.
Hence we obtain the first relation in (20). To prove the second formula, we take the convolution of the equations of motion with ui , σij ,j ui − c2 ui ,zz ui − Gi ui = 0, whence we obtain (σij ui ),j − σij ui ,j − c2 (ui ,z ui ),z + c2 (ui ,z ui ,z ) − Gi ui = 2L + (σij ui ),j − c2 (ui ,z ui ),z − Gi ui = 0; i.e., we have the second equation in (20). Formulas (21) for the jumps in E and L follow from (20) if we write them in D3 (R3 ). The equation for E, that is, E,z + [E]hz δF = (σij ui ,z ),j + [σij ui ,z ]hj δF + (G, u,z ), preserves the form (20) if [E]hz = [σij ui ,z ]hj δF . This, together with relation (15), implies the first relation in (21). By virtue of the conservation of momentum on the shock wave fronts (18), the equation for L in D3 (R3 ), that is, 2L = Gi ui + c2 (ui ,z ui ),z − (σij ui ),j + c2 [ui ,z ui ]hz δF − [σij ui ]hj δF preserves its form. Consequently, the function L is continuous on the shock wave fronts. DIFFERENTIAL EQUATIONS
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Let us prove conditions for the jump in the energy by the classical method with the use of conditions (17) and (18) on the fronts and the following easy-to-prove relation for the jump of the product of discontinuous functions: [ab] = a+ [b] + [a]b− , where the signs + and − indicate the values of the variables before and behind the wave front, respectively. Then c k c c k 2 k σij hj ui ,z = σij ui ,j + hj ui ,z + ui ,z c ui ,z + σij hj 2[E] + 2 ck ck ck − c k c k c k − [σij ] = [σij ] ui ,j + hj ui ,z + σij ui ,j + hj ui ,z + ui ,j + hj ui ,z ck ck ck − c c c 2 k − 2 k 2 k + [ui ,z ] c ui ,z + σij hj + ui ,z c ui ,z + σij hj + [ui ,z ] c ui ,z + σij hj ck ck ck − − c c = ui ,j + hkj ui ,z [σij ] + [ui ,z ] c2 ui ,z + σij hkj ck ck c − k c k = u− h [σij ] + c2 [ui ,z ] + u− σ h [ui ,z ] i ,z i ,j [σij ] + ck j ck ij j c − k c k − σij [hj ui ,z ] = u− h σ [ui ,j ] = u− i ,j [σij ] + i ,j [σij ] + ck ck z ij − − − − − = u− i ,j [σij ] − σij [ui ,j ] = −ui ,j σij + σij ui ,j = 0. Here the index k is fixed and corresponds to the longitudinal or transverse wave front. The proof of the theorem is complete. We see that this is a much more complicated procedure, especially if the jump on the front is unknown. In a similar way, one can prove the continuity of the function L. We use Theorem 1 and obtain the energy conservation law with regard to shock waves. To this end, consider the domain D − = {(x, z) : x ∈ S − ⊂ R2 , −∞ ≤ z ≤ ∞} bounded by a cylindrical surface D = {(x, z) : x ∈ S − ⊂ R2 }, where S is the boundary of S − in the class of Lyapunov − = {(x, z) : x ∈ S ⊂ R2 , a ≤ z ≤ b}. surfaces and Dab We denote the stresses on D by P = pi ei = σij (x, z)nj (x)ei ; obviously, the unit normal n to the surface D is independent of z, n = (n1 (x), n2 (x), 0). Theorem 2. If (G, u,z ) ∈ L1 (D + D− ) and (P, u,z ) ∈ L1 (D), then (E(x, b) − E(x, a)) dx1 dx2 = (G, u,z ) dx1 dx2 dz + (P, u,z ) dD(x, z) S−
− Dab
Dab
(σi3 ui ,z (x, b) − σi3 ui ,z (x, a)) dx1 dx2
+ S−
for arbitrary a and b, a < b, and (E(x, z) − E(x, 0)) dx1 dx2 S−
=
(G, u,z ) dx1 dx2 dz +
− D0z
(P, u,z ) dD(x, z) +
D0z
σi3 ui ,z (x, z) dx1 dx2 . S−
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− Proof. We integrate Eq. (20) over Dab = {(x, z) : x ∈ D− , 0 < a ≤ z ≤ b} with the use of the Gauss divergence theorem in the domains between fronts, where the solutions are differentiable, take into account the conditions (21) for the energy jump on the fronts and the relation n = (0, 0, ∓1) in the cross-sections S − , and obtain the relations (x,b) E,z dx1 dx2 dz = E(x, z)|(x,a) dx1 dx2 + [E]hlz dFl (x, z) − Cab
l
S−
=
nj σij ui ,z |(x,a) dx1 dx2 (x,b)
σij ui ,z nj dD + S−
Dab
+
Fl
l
[σij ui ,z ]hlj
dFl +
(G, u,z ) dx1 dx2 dz. − Dab
Fl
This implies the first formula in the theorem. The second formula follows from the first one if we set a = 0 and b = z. The proof of the theorem is complete. Theorem 3. If (G, u) ∈ L1 (D + D− ) and (P, u) ∈ L1 (D), then 2 L dx1 dx2 dz = c (ui ,z ui (x, b) − ui ,z (x, a)ui (x, a)) dx1 dx2 2 − Cab
S−
+
(σiz ui (x, b) − σiz ui (x, a)) dx1 dx2 +
(P, u) dD(x, z) + S−
Dab
L dx1 dx2 dz = −c
2
2 − D+
ui ,z (x, 0)ui (x, 0) dx1 dx2
S−
σij ui nj dD(x, z) −
+
(G, u) dx1 dx2 dz,
− Dab
σiz ui (x, a) dx1 dx2 + S−
D+
(G, u) dx1 dx2 dz. − D+
Proof. We integrate the second equation in (20) over a bounded cylinder with the use of the Gauss divergence theorem and conditions (17) on the shock wave fronts and obtain the desired relation, (x,b) L dx1 dx2 dz = c2 (ui ,z (x, z)ui (x, z))|(x,a) dx1 dx2 2 − Cab
S−
+
σiz ui |(x,a) dx1 dx2 + (x,b)
σij ui nj dD(x, z) + S−
Dab
− Dab
+
[σij hlj − c2 hlz ui ,z ]ui dFl
Fl
(ui ,z ui (x, b) − ui ,z (x, a)ui (x, a)) dx1 dx2
S−
(σiz ui (x, b) − σiz ui (x, a)) dx1 dx2 +
(P, u) dD(x, z) + Dab
l
(G, u) dx1 dx2 dz = c2
+
S−
(G, u) dx1 dx2 dz. − Dab
The second relation in the theorem follows from the first one with a = 0 and b = +∞. The proof of the theorem is complete. DIFFERENTIAL EQUATIONS
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5. STATEMENT OF BOUNDARY VALUE PROBLEMS. UNIQUENESS OF SOLUTIONS Assume that an elastic isotropic medium fills a domain D − ⊂ R3 bounded by a smooth cylindrical Lyapunov surface D whose generators are parallel to the Z-axis. The set S − ⊂ R2 is a perpendicular cross-section of D − , S is its boundary, D − = S − × Z, D = S × Z, n(x) = (n1 (x), n2 (x), 0) is the unit outward normal to D, D+ = S × Z+ , and Z+ = {z ∈ R1 : z ≥ 0}. Transport boundary value problem. Assume that transport loads moving at a constant velocity c are defined on the boundary D, P (x, z), x = (x1 , x2 ), z = x3 + ct, x ∈ S, P = σij ni ej = pj (x, z)ej θ(z),
j = 1, 2, 3,
(22)
where θ(z) is the Heaviside function. We assume that the functions P (x, z) are integrable on D+ , pj (x, z) ∈ L1 (D+ ). The relations ui (x, 0) = 0,
σi3 (x, 0) = 0,
x ∈ S,
hold for z = 0, the convergence uj → 0 takes place as (x, z) → ∞, and ∂j u < O((x, z)1+ε ),
j = 1, 2, z,
(23)
for some ε > 0. The jump conditions (17) and (18) are satisfied on the shock wave fronts. Theorem 4. The solution of the 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. Then Theorem 3, together with conditions (23) of decay of the solutions at infinity and the zero conditions for z = 0, implies that E(x, z) dx1 dx2 = σi3 ui ,z (x, z) dx1 dx2 −→ 0. z→∞
S−
S−
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 4 holds for both the exterior and interior boundary value problem. Now let us construct a solution of this boundary value problem. 6. STATEMENT OF THE PROBLEMS IN THE SPACE OF GENERALIZED FUNCTIONS, AND THEIR GENERALIZED SOLUTION To solve the problem, we use the method of generalized functions. To this end, we pass to the statement of the boundary value problem in S3 (R3 ) = {fˆ = (fˆ1 , . . . , fˆ3 ), fˆk ∈ S (R3 ), k = 1, 2, 3}, that is, the space S (R3 ) of tempered distributions defined on the Schwartz space S(R3 ) [6]. We introduce the regular generalized function − (x) = u(x, z)HS− (x)θ(z). u ˆ(x, z) = u(x, z)HD − (x) is the Here HS− (x) is the characteristic function of the set S − , which is equal to 0.5 on S; HD characteristic function of the half-cylinder D − = {(x, z) : z > 0, x ∈ S − }, S − ∈ R2 is an open set bounded by the curve S, and D = {(x, z) : z > 0, x ∈ S}, ⎧ for (x, z) ∈ D− , ⎨1 − − HD (x) = HS (x)θ(z) = 1/2 for (x, z) ∈ D, ⎩ 0 for (x, z) ∈ / D− + D.
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We use the properties of differentiation of regular generalized functions with a jump on S, take into account the boundary conditions and the conditions on the fronts, and obtain the following equation for u ˆ: ∂ j ˆ + (c2 nz ui , z − pi )δD (x, z) + (nz ui δD (x, z)),z u ˆj (x ) = −G Li ∂x ∂ui − (λuk nk δij + μ(nj ui + ni uj ))δD (x, z),j + σiz (x, 0) − c (24) δ(z)HS− (x). ∂z z=0 ˆ = GH − (x, z)θ(z), and δD (x, z) = δS (x)θ(z) is a simple layer on Here G S D+ = {(x, z) : x ∈ S, z ≥ 0}. Since n3 = 0 on D+ , we use the properties of the convolution with the Green tensor and the boundary conditions (22) and obtain a representation of the solution of the boundary value problem in the space of generalized functions in the form ˆik ∗ G ˆk + U ˆik ∗ pk δD + λuk nk δD ∗ U ˆil ,l + μ(nl uj + nj ul )δD ∗ U ˆij ,l . ˆi = U c2 u
(25)
If we introduce the fundamental stress tensor Tˆij (x, z, n) related to stress tensors ˆli ,l δjk + μ(U ˆji ,k +U ˆki ,j ), Σijk (x, z) = λU
ˆ ji (x, z, n) = Σj (x, z)nk Γ ik
ˆ i by the formula generated by U j ˆ i (x, z, n), Tˆij (x, z, n) = Γ j then the right-hand side of formula (25) can 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
z ˆik ∗ G ˆ k + θ(z) ˆi (x, z) = U c u 2
(Uij (x, y, z, τ )pi (y, τ ) + Tij (x, y, z, τ, n(y))uj (y, τ ) dS(y),
dτ 0
S
where we have introduced the shifted tensors ˆji (x − y, z − τ ), Uji (x, y, z, τ ) = U
Tji (x, y, z, τ, n) = Tˆji (x − y, z − τ, n).
That formula is similar to the Somigliana formula in the static theory of elasticity [2, p. 146], which permits one to determine displacements in the medium on the basis of known boundary values of the displacements and stresses. However, 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 nonintegrable singularities of the tensor Tij on the shock wave fronts; therefore, the integrals are divergent. To construct a regular integral representation of the formula, let us first describe the properties of the fundamental tensors Uji and Tji . ´ TRANSPORT EQUATION 7. GREEN TENSOR OF THE LAME AND ITS ANTIDERIVATIVE WITH RESPECT TO z ˆki }3×3 is a solution of the equation The tensor U = {U
∂ j ˆjk + δ(x )δik = 0, i, j = 1, 2, 3, U Λi ∂x and satisfies the radiation condition at infinity, ˆ k = 0, U i ˆ Uik → 0, DIFFERENTIAL EQUATIONS
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It describes the displacements of the elastic medium under the action of a force with components [k] Gi = c2 δ(x)δ(z)δi , which is concentrated at the point x = 0, acts in the direction of the coordinate axis Xk , and moves at the velocity c along the X3 -axis. For an arbitrary regular force with components Gk = c2 gk , the corresponding solution has the form of the convolution k ˆ ˆik (x − y, z − τ )gk (y, τ ) dy1 dy2 dτ, i, k = 1, 2, 3, u ˆi = Ui ∗ gˆk = U R3
if it exists. ˆik for various velocities c was obtained with the use of the Fourier The representation of U transform of generalized functions and studied in [4, 5]. In the supersonic case (for c > c1 ), it has the form 2 2 2 M z θ θ x θ x2 2 1 2 2 1 1 ˆ = − 2 4 (θ1 V1− − θ2 V2− ), 2π U 1 − + − − − 4 x x V2 V V2 1 2 2 2 2 ˆ22 = M2 θ2 + z x2 θ1 − θ2 − x1 (θ1 V1− − θ2 V2− ), 2π U x4 V1− V2− x4 V− 2 2 θ2 θ1 ˆ 3 = x1 z ˆ 3 = θ1 + m2 θ2 , 2π U − , 2π U 3 1 x2 V1− V2− V1− V2− x2 z θ2 θ1 3 ˆ12 , U ˆ32 = U ˆ23 , U ˆ31 = U ˆ13 , ˆ21 = U ˆ − , U 2π U2 = − x2 V1− V2− x1 x2 θ2 θ1 2 2 − − ˆ − z + (θ1 V1 − θ2 V2 ) . 2π U1 = x4 V1− V2− Here we have used the notation θj = θ(z − mj x), Vj− = z 2 − m2j x2 , and mj = Mj2 − 1. ˆij , It is convenient to single out the bulk and shear components of the tensor U j j ˆi1 ˆi2 ˆij = U +U , U
which describe the bulk and shear strains separately. One can readily write them by using Eq. (19) and by collecting terms with like subscripts in θj . ˆ j are zero outside the sonic cones One can readily see that the components of the tensors U il Kl+ = {(x, z) : z > ml x},
l = 1, 2.
The medium is at rest outside the zone K1+ . The deformation is purely bulk in the zone between the cones, and the bulk deformation is supplemented with the shear deformation in the zone K2+ . On the surfaces of the cones Kj = {(x, z) : z = mj x}, that is, on the wave fronts ˆ i are singular and have singularities of the type of of the fundamental solutions, the components U j 2 2 2 −1/2 . In addition, the following symmetry properties hold: (z − mj x ) ˆij (x, z) = U ˆ i (−x, z). ˆ i (x, z) = U U j j ˆ ji , that is, the antiderivative of Uji with Next, to regularize the integrals, we need the tensor W respect to z, ˆji ∗ δ(x1 )δ(x2 )θ(z) = U ˆji ∗ θ(z). ˆ ji = U W z
It has the form ˆ ij = 1 (θ1 V − − θ2 V − ) z (r,i r,j −0.5ε[i]3 δij ) − δi3 r,j −δj3 r,i 2π W 1 2 x r − z + V1 z + V2− (0.5m21 ε[i]3 δij + δi3 δj3 ) − θ2 ln ((0.5m22 ε[i]3 − M22 )δij + δi3 δj3 ), + θ1 ln m1 x m2 x DIFFERENTIAL EQUATIONS
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where r,j = xj /x, j = 1, 2, r,3 = 0, and εij = (1 − δij ). The notation ε[i]j stands for a fixed value of the index i (we do not take the convolution with respect to i in the product). Here it is also convenient to single out the bulk and shear parts, ˆ ij (x, z) = W
2
ˆj = W ik
k=1
2
z θ(z − mk x)
k=1
ˆ j (x, τ ) dτ. U ik
mk x
ˆij , but since mj x = z on Kj+ , we have ˆ ij has the same support as U Obviously, the tensor W Vj− (x, z) = 0, whence it follows that z + Vj = 0; ln mj x ˆ ij has no strong consequently, it is continuous on the fronts Kj . Therefore, for z = 0 the tensor W singularity on the Z-axis; however, it has a weak logarithmic singularity with respect to x. ˆ ij into the terms To single out these singularities, we decompose W ˆ ijs + W ˆ ijd = ˆ ij (x, z) = W W
2
ˆ jd (x, z), ˆ js (x) + W θk W ik ik
k=1
where
ˆ i1js (x) = −(δi3 δj3 + 0.5m2 δij ε[i]3 ) ln(m1 x), 2π W 1 ˆ i2js (x) = −(δi3 δj3 + δij (0.5m22 ε[i]3 − M22 )) ln(m2 x). 2π W
ˆ js of diagonal form are independent of z inside the sonic cones K + (l = 1, 2) and The tensors W l il ˆ ij , have a logarithmic singularity with respect to x on the Z-axis. Unlike the generating tensor W ˆ jd have bounded jumps on the fronts Kl . ˆ js and W the tensors W ik ik ˆij (x−y, z−τ ) and Wij (x, y, z−τ ) = One can readily see that the shifted tensors Uij (x, y, z−τ ) = U ˆ ij (x − y, z − τ ) have the following symmetry properties around the Z-axis : W Uij (x, y, z − τ ) = Uij (y, x, z − τ ),
Wij (x, y, z − τ ) = Wij (y, x, z − τ )
except for the components with indices (i, j) = (1, 1, 3), (1, 2, 3), (1, 3, 1), for which we have Ui3 (x, y, z − τ ) = Ui3 (y, x, z − τ ),
Wi3 (x, y, z − τ ) = Wi3 (y, x, z − τ ),
i = 1, 2.
ˆ ij also has a physical meaning. It describes the displacement of an elastic medium The tensor W [k] under the action of a force components Gi = c2 δ(x)θ(z)δi concentrated on the X3 -semiaxis, acting in the direction of the coordinate axis Xk , and moving along X3 at the velocity c. ˆ 8. FUNDAMENTAL STRESS TENSORS Tˆ AND H Consider the stress tensors generated by the Green tensor, ˆ k ,m δij + μ(U ˆ k ,j + U ˆ k ,i ), ˆ k (x, z) = λU Σ ij m i j j k k ˆ ˆ ˆ ij (x, z, n). ˆ Ti (x, z, n) = Γ Γi (x, z, n) = Σij (x, z)nj ,
(26)
ˆ ki (x, z, n) describes stresses on an area element with normal n at a point (x, z). The tensor Γ Theorem 5. The tensor Tˆij is a generalized solution of the Lam´e transport equation with a mass force of the multipole type, Λji (∂x )Tˆjk + Kki (∂x , n)δ(x ) = 0, where Kil (∂x , n) = λni ∂l + μmj (δil ∂j + δjl ∂i ). DIFFERENTIAL EQUATIONS
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ALEXEYEVA
ˆ i = K l (∂x , n)U ˆ i (x, z). Then Proof. It follows from relations (26) that Tˆij = Γ j j l ˆ j = Λji (∂x )K l (∂x , n)U ˆj Λji (∂x )Tˆjk = Λji (∂x )Γ k k l j l l ˆ = K (∂x , n)Λi (∂x )U (x, z) = −K l (∂x , n)δl δ(x ) = −K i (∂x , n)δ(x ). k
j
k
i
k
The proof of the theorem is complete. Since the introduced tensor satisfies the equations of motion, we find that it describes the displacements of an elastic medium under the action of a running lumped mass force of a multipole type. However, unlike the subsonic case [5], its antiderivative with respect to z rather than the tensor itself plays a key role in the solution of boundary value problems; in this antiderivative, it is again convenient to single out the bulk and shear components, ˆ ji = H
2
ˆ j = Tˆji ∗ δ(x1 , x2 )θ(z) = Tˆji ∗ θ(z), H ik
k=1
z
∂f12 1 ˆj Hi (x, z, n) = (2M12 − M22 )nj f11 ,i − M22 δij + ni f12 ,j μ ∂n where
θk 2πf1k ,i (x, z) = − Vk
−2
∂ (f31 ,ij −f32 ,ij ), ∂n
z r,i , δi3 − x
z + Vk− 2πf3k ,ij (x, z) = (δi3 δj3 + 0.5m2k δij ε[i]3 )θk ln mk x − z Vk θ k δi3 r,j + δj3 r,i + (r,i r,j − 0.5δij ε[i]3 ) . − x x That tensor can be obtained in a different way, by analogy with T , with the use of Hooke’s law, except that the Green tensor should be replaced with its antiderivative W . ˆ ij , Here we again single out two terms in H ˆ ijs + H ˆ ijd = ˆ ij (x, z, n) = H H
2
ˆ js (x) + H ˆ jd (x, z). θk (z − mk x)H ik ik
(27)
k=1
ˆ js are independent of z inside the sonic cones K + (l = 1, 2), we convenˆ js and H Since the tensors W l il ik ˆ jd are said to be dynamic, ˆ ijd and H tionally say that they are stationary; accordingly, the tensors W ik because they depend essentially on z, but they are regular. ˆ ijd , ˆ ij has weak singularities on the fronts of the type of (z 2 − m2j x2 )−1/2 . Unlike H The tensor H ˆ ijs has a stronger singularity of the type of x−1 on the Z-axis. It has the following the tensor H symmetry properties around the Z-axis in terms of the shifted tensors: Hij (x, y, z − τ, m) = −Hij (y, x, z − τ, m) = −Hij (x, y, z − τ, −m) except for (i, j) = (1, 3), (2, 3), (3, 1); Hi3 (x, y, z − τ ) = Hi3 (y, x, z − τ ),
Hi3 (x, y, z − τ ) = Hi3 (y, x, z − τ ),
i = 1, 2.
Obviously, for z < τ all the introduced shifted tensors are zero. The above-represented symmetry properties hold for both stationary and dynamic terms in the tensors. ˆ ij satisfies the relation Lemma 1. The tensor H − Hki (y, x, z, n(y)) dS(y, τ ) + (Σki3 (x, y, z) − c2 Uik (x, y, z)) dV (y) = c2 δki HD (x), Sz (x)
z
Sz− (x)
/ D, then the integrals are regular ; if x ∈ D, then the integrals over where dV (y) = dy1 dy2 . If x ∈ Sz (x) are singular with a singularity at y = x and are understood in the sense of the principal value. DIFFERENTIAL EQUATIONS
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Proof. We take the convolution of the equation k ˆ k ,zz = δk δ(x, z) ,j − U (c2 )−1 Sˆij i i
ˆ with δ(x)θ(z) and obtain the following equation for W ˆ : for U ˆ ik ,zz = δik δ(x)θ(z). ˆ kij ,j − W (c2 )−1 Σ Next, we take the convolution of this equation with HS− (x)δ(z) and use the property of differentiation of a convolution, ˆ k ,z ∗ H − (x) = δk H − (x)θ(z). ˆ k ∗ nj δS (x) + (c2 )−1 Σ ˆ k ,z ∗ H − (x) − U (c2 )−1 Σ S S S ij i3 i i x
x
/ D can be written in the integral form Since S is bounded, it follows that this equation for x ∈ given in the lemma, where all integrals exist. The integration domains on the integrals are the supports of the integrands, which depend on z. ± : r < ε, Let (x, z) ∈ D. We denote half-neighborhoods of the point x by Γ± ε (x) = {y ∈ S r = x − y}. Likewise, we obtain the relations Hki (y, x, z, m(y, τ )) dS(y, τ ) + (Σki3 (x, y, z) − c2 Uik (x, y, z)) dV (y) = c2 δki , S − +Γ+ (x)
Sε +Γ+ ε (x)
Hik (y, x, z, m(y, τ )) dS(y, τ )
(Σki3 (x, y, z) − c2 Uik (x, y, z)) dV (y)
+
Sε +Γ− ε (x)
z
S − +Γ− ε (x)
= 0. z
Here m(y, τ ) is the unit outward normal to the integration surface, which coincides with n on Sε and is directed along the radius of the ε-neighborhood on Γ+ ε . We add these two relations and pass to the limit as ε → 0. The inner integrals over Sε tend to integrals in the sense of the principal value, which exist by virtue of the symmetry properties (24) and (25) and the asymptotic properties as r → 0 for z > 0. − The limit of the sum of inner integrals over Γ+ ε and Γε tends to zero because m(y, τ ) takes the + − same values at points Γε and Γε opposite with respect to x. The integrals over Sε− (x) and Sε+ (x) tend to zero by virtue of the weak singularity of the integrands. If we divide the limit relation by 2 − on D, then we obtain the desired assertion. The proof of the lemma and use the definition of HD is complete. If the inequality z ≥ maxy∈S (m1 x − y) holds for given (x, z), then the integration domains become constant, Sz− (x) = S − , Sz (x) = S. Lemma 2. If z > m1 and ε > 0, then y−x k dS(y, τ ) = c2 δik . Hi y, x, z, lim ε→0 r
(28)
r=ε
Proof. Since the integration domain is independent of z for such values of z and the integrands, together with their derivatives, are continuous inside the ε-neighborhood, it follows that one can perform the differentiation with respect to z in the integrand in formula (28) written for the abovementioned set and obtain y−x k dS(y, τ ) + (Σki3 (y, x, z) − c2 Uik (x, y, z)),z dV (y) = c2 δik . Hi y, x, z, r r=ε
r c1 , the solution of the boundary value problem satisfies the relations 2
c
− ui (x, z)HD (x)
=
2
z−m kr
θ(z − mk r) dS(y)
k=1 S j (x, y, z − Hik
j (Uik (x, y, z − τ )pj (y, τ ) 0
− τ, n(y))∂τ uj (y, τ )) dτ
(30)
for (x, z) ∈ / D and 2
0.5c ui (x, z)θ(z) =
2
z−m kr
θ(z − mk r) dS(y)
k=1 S
j Uik (x, y, z − τ )pj (y, τ ) dτ 0 z−m kr
θ(z − mk r) dS(y)
− v.p. S
j Hik (x, y, z − τ, n(y))∂τ uj (y, τ ) dτ
(31)
0
for (x, z) ∈ D+ , where the last integral is singular and is treated in the sense of the principal value over the contour S or its part. Proof. The integral representation of the generalized solution (25) has the form (30). 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 (z 2 − m2j x2 )−1/2 . Let us show that the formula also preserves its form for (x∗ , z ∗ ) ∈ D with regard − (x, z) on D. to the definition of HD For a domain with a deleted neighborhood of a boundary point, from the properties (29) we obtain 0=
2 k=1
z−m kr
j j (Uik (x∗ , y, z ∗ − τ )pj (y, t) − Hik (x∗ , y, z ∗ − τ, n(y))uj ,z (y, τ )) dτ.
θ(z − mk r) dS(y)
Sε +Γ− ε
0
In this domain, consider the integral ∗
∗
Iε (x , z ) =
2 k=1
Γ− ε
∗
z ∗ −mk r
j Hik (x∗ , y, z ∗ − τ, n(y))uj ,z (y, τ ) dτ.
θ(z − mk r) dS(y) 0
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Let ε < z ∗ /m1 . Since the normal to Γ− ε is n = m = (x − y)/ε, it can be represented in the form 2
Iε (x∗ , z ∗ ) =
k=1
z ∗ −mk ε
θ(z ∗ − mk ε) dS(y)
Γ− ε
0
2
=
k=1 2
=
k=1
j (Hik (x∗ , y, z ∗ − τ, m(y)) dτ uj (y, τ )
j Hik (x∗ , y, mk ε, m(y))uj (y, z ∗ − mk ε) dS(y)
Γ− ε
j Hik (x∗ , y, mk ε, m(y))(uj (y, z ∗ − mk ε) − uj (x∗ , z ∗ )) dS(y)
Γ− ε ∗
∗
+ uj (x , z )
2 k=1
j Hik (x∗ , y, mk ε, m(y)) dS(y).
Γ− ε
We pass in the last relation to the limit as ε → 0. The last term has a weak singularity with respect to r, which can be removed, because uj (y, z ∗ − mk r) →ε→0 uj (x∗ , z ∗ ). Since Γ− ε → 0, it follows that the second term tends to zero as well. The integral in the second term tends to the integral over a half-circle. Consequently, by Lemma 2, Iε (x∗ , z ∗ ) → 0.5uj (x∗ , z ∗ ). ε→0
We transpose this term to the left-hand side of the equation and use the definition of Hs− (x) on S; then we obtain the second formula of the theorem. The proof of the theorem is complete. The kernels of the integrands have weak singularities on the fronts, and the integrals in formula (30) admits the change of the integration order. Therefore, by virtue of the symmetry properties of the kernels, the formula can be represented in the form 2
z
2
c ui (x, z) =
k=1 0
j j (Uik (y, x, z − τ )pj (y, τ ) − Hik (y, x, z − τ, n(y))∂z uj (y, τ )) dS(y)
dτ Sτk (x)
for z > 0 and x ∈ S − . 10. SINGULAR BOUNDARY INTEGRAL EQUATIONS AT SUPERSONIC VELOCITIES For (x, z) ∈ D, formula (31) is a system of boundary integral equations for the solution of the boundary value problem. After the computation of the unknown boundary values of the displacement rate with the use of this system, we find the solution inside the medium from the first formula in Theorem 6. To compute the singular integrals, it is convenient to transform formula (31) by expanding kernels in accordance with the strain generated by them. Theorem 7. On the domain boundary, the solution of the boundary value problem satisfies the singular boundary integral equations of the form ˆj ∗ U ˆij 0.5c2 ui (x, z)θ(z) = G 2 z j jd + θ(z) dτ (Uik (y, x, z − τ )pj (y, τ ) − Hik (y, x, z − τ, n(y))∂z uj (y, τ )) dS(y) k=1 0
− θ(z) v.p.
Sτk (x) js Hik (y, x, n(y))uj (y, z − mk r) dS(y)
Szk (x)
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ALEXEYEVA
for (x, z) ∈ D+ , Sτk (x ) = {(y, τ ) : mk r < z − τ }, and Szk (x) = {(y) : mk r < z}, where the last integral is singular and is treated in the sense of the principal value over the contour S or its part. Proof. The desired assertion follows from the decomposition of the tensor H into dynamic and stationary components in accordance with the type of strains (27). We substitute relation (27) into the formula of the theorem and integrate with respect to τ in the term containing u,z ; then we obtain the formula in the theorem. By virtue of the weak singularities of the kernels of integrands ˆ jd , the first integral is regular, and the second integral is understood in the sense of containing H ik the principal value but along the contour of the cross-section rather than over the surface of the cylinder. Note that the solution (29) is represented in the space of generalized functions. It admits the integral representation (30) expressed via regular generalized functions. Since two regular functions on the left and right are equal, it follows that the solution is classical by virtue of the Du BoisReymond lemma [6, p. 17] and is unique by Theorem 4. CONCLUSION The constructed singular boundary integral equations are equations of a nonclassical type, 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 [2, p. 146]. 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 a problem. The above-considered 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, it is unnecessary to solve singular integral equations, and one can use the formula 2
z
2
c ui (x, z) =
k=1 0
Uij (x, y, z − τ )pj (y, τ ) dS(y)
dτ Sτk (x)
by virtue of the higher asymptotics of the singular kernel with respect to r of the order of 1/r 2 . Accordingly, the computation error for the stresses and strains is of the order of 1/r 3 . REFERENCES 1. Alexeyeva, L.A., Singular boundary integral equations of boundary value problems for elastic dynamics in the case of subsonic running loads, Differ. Equations, 2010, vol. 46, no. 4, pp. 515–522. 2. Novatskii, V., Teoriya uprugosti (Theory of Elasticity), Moscow, 1975. 3. Petrashen’, G.I., Foundations of mathematical theory of the propagation of elastic waves, in Voprosy dinamicheskoi teorii rasprostraneniya seismicheskikh voln (Problems of Dynamical Theory of Propagation of Seismic Waves), Leningrad, 1978, no. 18. 4. Alekseyeva, L.A., Fundamental solutions in an elastic space in the case of moving loads, J. Appl. Math. Mech., 1991, vol. 55, no. 5, pp. 714–723. 5. Alekseeva, L.A., Generalized solutions of Lam´e equations in the case of running loads, Mat. Zh., 2009, vol. 9, no. 1, pp. 16–25. 6. Vladimirov, V.S., Obobshchennye funktsii v matematicheskoi fizike (Generalized Functions in Mathematical Physics), Moscow, 1978.
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