MOS Subject Classification: 31 A 10, 35 J 55, 73 B 40, 73 C 35, 73 M 25

THE INTERFACE CRACK PROBLEM FOR ANISOTROPIC BODIES

David Natroshvili & Shota Zazashvili

Department of Mathematics, Georgian Technical University Kostava str. 77, Tbilisi 380075, Republic of Georgia

Abstract The two-dimensional interface crack problem is investigated for anisotropic bodies in the Comninou formulation. It is established that, as in the isotropic case, properly incorporating contact zones at the crack tips avoids contradictions connected with the oscillating asymptotic behaviour of physical and mechanical characteristics leading to the overlapping of material. Applying the special integral representation formulae for the displacement field the problem in question is reduced to the scalar singular integral equation with the index equal to −1. The analysis of this equation is given. The comparison with the results of previous authors shows that the integral equations corresponding to the interface crack problems in the anisotropic and isotropic cases are actually the same from the point of view of the theoretical and numerical analysis.

1 Introduction

This paper is concerned with two-dimensional mathematical problems of solid-solid interaction. We consider in IR^{2} two elastic half-planes x_{2} > 0 and x_{2} < 0 filled up by different anisotropic materials which are coupled via the x_{1} axis. In addition, this solid structure contains an interface crack (cut) of finite length. The conventional formulation of the problem, which assumes that the two faces of the crack are traction free and outside of the crack the rigid contact conditions (continuity of the displacements and stresses) are given, leads to contradictions. This is connected with the oscillating asymptotic behaviour of solutions near the tips of the crack which exibits overlapping of material (for details see [7], [12], [3], [4], [5], and references therein).

M.Comninou in [3], [4] considered an alternative version of the above mixed interface problem in isotropic case. The basic idea of the new approach is that the crack faces are always in contact at the very tips of the crack. Mathematically it means that the crack (cut) interval is devided into three subintervals: two of them are contact intervals, one-sided small vicinities of the tips of the crack, and the third one (the middle interval) corresponds to the open traction free part of the crack. On these contact intervals special type transmission conditions are given which describe that the normal components of the displacement and the stress vectors are continuous on the contact interval, while the tangent components of the stress vector vanish on the same contact interval. The Comninou formulation of the interface crack problem is free of the above contradictions. Moreover, we note that the interface crack problem in Comninou formulation is uniquely solvable (see [15]). This approach has been considered and studied by a lot of authors. An exhaustive information concerning theoretical and numerical results for the isotropic case can be found in [3], [4], [5], [6], [8], [9], [10], [15].

As to the general anisotropic case, to the authors’ best knowlege, the above problem has not been treated systematically in the scientific literature. In the present paper we apply special potential type intregral representation formulae for the displacement vector (obtained in [1], [2]) and investigate the above interface crack problem in the Comninou formulation. We reduce the problem to the scalar singular integral equation of index −1 and give the corresponding analysis. The equation in anisotropic case is quite similar to the equation obtained by Gautesen and Dundurs in [9], [10] for the isotropic case. Therefore, applying the same approach as in [8], [9], and [10] one can solve explicitly the singular integral equation corresponding to the anisotropic case.

2 Formulation of the Problem

Denote by S^{(0)} the upper half-plane (x_{2} > 0) and by S^{(1)} the lower half-plane (x_{2} < 0). We assume that the domains S^{(j)} are filled up by anisotropic materials with elastic constants A_{11}^{(j)}, A_{12}^{(j)}, A_{13}^{(j)}, A_{22}^{(j)}, A_{23}^{(j)}, and A_{33}^{(j)}, j = 0, 1. The common boundary (x_{1}-axis) of the above two half-planes will be referred to as the contact line l. In what follows we will use the superscript (j) with the physical characteristics corresponding to the domain S^{(j)}. Sometimes we will omit the superscript (j) when this causes no confusion.

The system of equations of elastostatics in the anisotropic case, sans body forces, reads as ([11])

A_{11} \frac{∂^{2}u_{1}}{∂x_{1}^{2}} + 2A_{13} \frac{∂^{2}u_{1}}{∂x_{1}∂x_{2}} + A_{33} \frac{∂^{2}u_{1}}{∂x_{2}^{2}} + A_{13} \frac{∂^{2}u_{2}}{∂x_{1}^{2}} + (A_{12} + A_{33})\frac{∂^{2}u_{2}}{∂x_{1}∂x_{2}} + A_{23} \frac{∂^{2}u_{2}}{∂x_{2}^{2}} = 0,

A_{13} \frac{∂^{2}u_{1}}{∂x_{1}^{2}} + (A_{12} + A_{33})\frac{∂^{2}u_{1}}{∂x_{1}∂x_{2}} + A_{23} \frac{∂^{2}u_{1}}{∂x_{2}^{2}} + A_{33} \frac{∂^{2}u_{2}}{∂x_{1}^{2}} + 2A_{23} \frac{∂^{2}u_{2}}{∂x_{1}∂x_{2}} + A_{22} \frac{∂^{2}u_{2}}{∂x_{2}^{2}} = 0, (2.1)

where u = (u_{1}, u_{2})^{⊤} is the displacement vector, and x = (x_{1}, x_{2}) ∈ S^{(j)} ⊂ IR^{2}. Here and in what follows ⊤ denotes transposition.

The stress components σ_{x_{1}}, σ_{x_{2}}, τ_{x_{1}x_{2}}, and the strain components ε_{x_{1}}, ε_{x_{2}}, ε_{x_{1}x_{2}} are related by Hook’s law

σ_{x_{1}} = A_{11}ε_{x_{1}} + A_{12}ε_{x_{2}} + A_{13}ε_{x_{1}x_{2}},
σ_{x_{2}} = A_{12}ε_{x_{1}} + A_{22}ε_{x_{2}} + A_{23}ε_{x_{1}x_{2}},
τ_{x_{1}x_{2}} = A_{13}ε_{x_{1}} + A_{23}ε_{x_{2}} + A_{33}ε_{x_{1}x_{2}},  (2.2)

where

ε_{x_{1}} = \frac{∂u_{1}}{∂x_{1}} , ε_{x_{2}} = \frac{∂u_{2}}{∂x_{2}} , ε_{x_{1}x_{2}} = \frac{∂u_{1}}{∂x_{2}} + \frac{∂u_{2}}{∂x_{1}} . (2.3)

The positive definiteness of the potential energy implies that ([11])

\begin{matrix}A_{11} > 0, A_{22} > 0, A_{33} > 0, \\ A_{11}A_{22} − A_{12}^{2} > 0, A_{11}A_{33} − A_{13}^{2} > 0, \\ A_{22}A_{33} − A_{23}^{2} > 0, \end{matrix} Δ = det [ \begin{matrix}A_{11} & A_{12} & A_{13} \\ A_{12} & A_{22} & A_{23} \\ A_{13} & A_{23} & A_{33} \end{matrix} ] > 0. (2.4)

In the isotropic case

A_{11} = A_{22} = λ + 2μ, A_{12} = λ, A_{13} = A_{23} = 0, A_{33} = μ, Δ = 4μ^{2}(λ + μ),

where λ and μ are the Lamé constants.

As it has been indicated in the introduction, we follow the mathematical model of M.Comninou developed in ([3], [4]) for the interface crack problem and assume that we have an interface crack (cut) along the segment [−L, L] on the contact line. According to the Comninou approach outside of the cut (|x_{1}| > L) we have the perfect bond (the rigid contact conditions) between the two materials. Moreover, the crack interval (−L, L) is devided into the three subintervals (−L, −a), (−a, b), and (b, L), where the different boundary and contact conditions are given describing that the middle (open) part (−a, b) of the crack is free of tractions, while the two remaining intervals represent small one-sided vicinites of the tips of the crack and are the contact zones with special contact conditions (see below the formulation of problem (C)). We observe that in this model the constants a and b are unknowns and they must be determined in the course of solution.

In addition, we provide that the following conditions

σ_{x_{2}}^{∞} = T, τ_{x_{1}x_{2}}^{∞} = S, (2.5)

where T and S are given real constant numbers, are fulfilled at infinity.

Problem (C). Find regular solutions u^{(j)} to the system (2.1) in S^{(j)} (j = 0, 1) satisfying the following interface and boundary conditions on the contact line l

i) [u^{(0)}]^{+} = [u^{(1)}]^{−}, [σ_{x_{2}}^{(0)}]^{+} = [σ_{x_{2}}^{(1)}]^{−}, [τ_{x_{1}x_{2}}^{(0)}]^{+} = [τ_{x_{1}x_{2}}^{(1)}]^{−},  (2.6)
x_{2} = 0, |x_{1}| > L,
ii) [u_{2}^{(0)}]^{+} = [u_{2}^{(1)}]^{−}, [σ_{x_{2}}^{(0)}]^{+} = [σ_{x_{2}}^{(1)}]^{−}, [τ_{x_{1}x_{2}}^{(0)}]^{+} = [τ_{x_{1}x_{2}}^{(1)}]^{−} = 0,  (2.7)
x_{2} = 0, x_{1} ∈ (−L, −a) ∪ (b, L),
iii) [σ_{x_{2}}^{(0)}]^{+} = [σ_{x_{2}}^{(1)}]^{−} = 0, [τ_{x_{1}x_{2}}^{(0)}]^{+} = [τ_{x_{1}x_{2}}^{(1)}]^{−} = 0, x_{2} = 0, x_{1} ∈ (−a, b),  (2.8)

where a and b are unknown real numbers (0 < a < L, 0 < b < L), and where the symbols [·]^{+} and [·]^{−} denote limits on l from S^{(0)} and S^{(1)}, respectively.

By a regular solution to the system (2.1) is understood a two-dimensional vector u^{(j)} = (u_{1}^{(j)}, u_{2}^{(j)})^{⊤} such that:

a) u^{(j)} ∈ C(\overline{S^{(j)}}) ∩ C^{2}(S^{(j)}),

b) the corresponding stress components σ_{x_{1}}^{(j)}, σ_{x_{2}}^{(j)}, and τ_{x_{1}x_{2}}^{(j)} (see (2.2)) are continuously extendible on the whole x_{1} axis except the points {−L; −a; b; L} in the vicinity of which they have integrable singularities, c) for sufficiently large |x| = (x_{1}^{2} + x_{2}^{2})^{1/2}

u^{(j)}(x) − u_{∞}^{(j)}(x) = O(1),  (2.9)
\frac{∂}{∂x_{k}} [u^{(j)}(x) − u_{∞}^{(j)}(x)] = O(|x|^{−2}), k = 1, 2,  (2.10)

where

u_{∞}^{(j)}(x) = \frac{1}{A_{22}^{(j)}A_{33}^{(j)} − (A_{23}^{(j)})^{2}} [ \begin{matrix}A_{22}^{(j)}S − A_{23}^{(j)}T \\ A_{33}^{(j)}T − A_{23}^{(j)}S \end{matrix} ] x_{2}. (2.11)

3 Preliminary material

In this section we collect some auxiliary material connected with the potential method in the two-dimensional elasticity theory of anisotropic bodies.

3.1. First let us note that the conditions (2.2) and (2.4) imply

ε_{x_{1}} = a_{11}σ_{x_{1}} + a_{12}σ_{x_{2}} + a_{13}τ_{x_{1}x_{2}},
ε_{x_{2}} = a_{12}σ_{x_{1}} + a_{22}σ_{x_{2}} + a_{23}τ_{x_{1}x_{2}},
ε_{x_{1}x_{2}} = a_{13}σ_{x_{1}} + a_{23}σ_{x_{2}} + a_{33}τ_{x_{1}x_{2}},  (3.1)

where a_{11}, a_{12}, ⋯, a_{33} are the entries of the matrix [a_{kj}]_{3×3} inverse to [A_{kj}]_{3×3} and satisfy the conditions similar to (2.4).

In the isotropic case we have

a_{11} = a_{22} = \frac{λ + 2μ}{4μ(λ + μ)}, a_{12} = −\frac{λ}{4μ(λ + μ)}, a_{13} = a_{23} = 0, a_{33} = \frac{1}{μ}.

We recall that the so-called characteristic equation of the system (2.1) (see [11])

a_{11}α^{4} − 2a_{13}α^{3} + (2a_{12} + a_{33})α^{2} − 2a_{23}α + a_{22} = 0 (3.2)

possesses only the complex roots

α_{k} = a_{k} + ib_{k}, ᾱ_{k} = a_{k} − ib_{k}, (b_{k} > 0), k = 1, 2. (3.3)

In the isotropic case α_{1} = α_{2} = i.

3.2. The fundamental matrix of solutions to the system (2.1) has been constructed by M.Basheleishvili in [1]

Γ(z, t) = Im ∑_{k=1}^{2} [ \begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix} ] ln σ_{k}, (3.4)

where

A_{k} = −\frac{2}{Δa_{11}} (A_{22}α_{k}^{2} + 2A_{23}α_{k} + A_{33})d_{k},

B_{k} = \frac{2}{Δa_{11}} (A_{23}α_{k}^{2} + (A_{12} + A_{33})α_{k} + A_{13})d_{k},

C_{k} = −\frac{2}{Δa_{11}} (A_{33}α_{k}^{2} + 2A_{13}α_{k} + A_{11})d_{k},

d_{1}^{−1} = (α_{1} − ᾱ_{1})(α_{1} − α_{2})(α_{1} − ᾱ_{2}), d_{2}^{−1} = (α_{2} − α_{1})(α_{2} − ᾱ_{1})(α_{2} − ᾱ_{2}); (3.5)

here z = x_{1} + ix_{2} and t = y_{1} + iy_{2} are arbitrary points in the plane, and

σ_{k} = z_{k} − t_{k}, z_{k} = x_{1} + α_{k} x_{2}, t_{k} = y_{1} + α_{k} y_{2}.

In the sequel we will use the notation v(z) for the function v(x_{1}, x_{2}) of the variables x_{1} and x_{2}.

The coefficients A_{k}, B_{k}, and C_{k} satisfy the equations

A_{k}C_{k} − B_{k}^{2} = 0, k = 1, 2. (3.6)

Next, we set (see [1])

C = 2i ∑_{k=1}^{2} d_{k}, A = 2i ∑_{k=1}^{2} α_{k}d_{k}, B = 2i ∑_{k=1}^{2} α_{k}^{2}d_{k}. (3.7)

It is easy to show that

C > 0, B > 0, BC − A^{2} > 0. (3.8)

From (3.7) it follows that

d_{k} = −\frac{i}{2}(Cs_{k} − Ar_{k}), α_{k}d_{k} = −\frac{i}{2}(Ap_{k} − Cq_{k}),

α_{k}^{2}d_{k} = −\frac{i}{2}(Bp_{k} − Aq_{k}), Ap_{k} − Cq_{k} = As_{k} − Br_{k},  (3.9)

where

r_{k} = \frac{(−1)^{k}}{α_{1} − α_{2}} , s_{k} = \frac{(−1)^{k}α_{1}α_{2}}{α_{k}(α_{1} − α_{2})}, p_{k} = −α_{k}r_{k}, q_{k} = −α_{k}s_{k}. (3.10)

These equations yield

[ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] = \frac{2id_{k}}{BC − A^{2}} [ \begin{matrix}α_{k}^{2} & −α_{k} \\ −α_{k} & 1 \end{matrix} ] [ \begin{matrix}C & A \\ A & B \end{matrix} ] ,

∑_{k=1}^{2} [ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] =: E.  (3.11)

In what follows we shall also use the abridged notation

P_{(k)} := [ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] . (3.12)

The stress vector T (∂_{z}, n)u(z), acting on an arc element with the unit normal vector n = (n_{1}, n_{2}), and the corresponding stress operator T (∂_{z}, n) are calculated by the formulae

[T (∂_{z}, n)u(z)]_{1} = σ_{x_{1}}n_{1}(z) + τ_{x_{1}x_{2}}n_{2}(z), [T (∂_{z}, n)u(z)]_{2} = τ_{x_{1}x_{2}}n_{1}(z) + σ_{x_{2}}n_{2}(z), T (∂_{z}, n) = [T_{kj}(∂_{z}, n)]_{2×2},

T_{11}(∂_{z}, n) = n_{1}(z) (A_{11} \frac{∂}{∂x_{1}} + A_{13} \frac{∂}{∂x_{2}} ) + n_{2}(z) (A_{13} \frac{∂}{∂x_{1}} + A_{33} \frac{∂}{∂x_{2}} ) ,

T_{12}(∂_{z}, n) = n_{1}(z) (A_{13} \frac{∂}{∂x_{1}} + A_{12} \frac{∂}{∂x_{2}} ) + n_{2}(z) (A_{33} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ) ,

T_{21}(∂_{z}, n) = n_{1}(z) (A_{13} \frac{∂}{∂x_{1}} + A_{33} \frac{∂}{∂x_{2}} ) + n_{2}(z) (A_{12} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ) ,

T_{22}(∂_{z}, n) = n_{1}(z) (A_{33} \frac{∂}{∂x_{1}} + A_{23} \frac{∂}{∂x_{2}} ) + n_{2}(z) (A_{23} \frac{∂}{∂x_{1}} + A_{22} \frac{∂}{∂x_{2}} ) .  (3.13)

Taking into account (3.6) we get

[T (∂_{t}, n)Γ(z, t)]^{⊤} = Im ∑_{k=1}^{2} [ \begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix} ] \frac{∂ ln σ_{k}}{∂s(t)} , (3.14)

where

M_{k} = −\frac{2}{a_{11}} (a_{11}α_{k}^{2} − a_{13}α_{k} + a_{12})d_{k}, R_{k} = −\frac{2}{a_{11}α_{k}} (a_{12}α_{k}^{2} − a_{23}α_{k} + a_{22})d_{k},

N_{k} = −α_{k}M_{k}, L_{k} = −α_{k}R_{k}, \frac{∂}{∂s(t)} = n_{1}(t)\frac{∂}{∂y_{2}} − n_{2}(t)\frac{∂}{∂y_{1}} .  (3.15)

We have also the following relationship between the above coefficients

[ \begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix} ] = [ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] + 2ωd_{k} [ \begin{matrix}α_{k} & α_{k}^{2} \\ −1 & −α_{k} \end{matrix} ] , (3.16)

where

ω = b_{1}b_{2} − a_{1}a_{2} + \frac{a_{12}}{a_{11}} . (3.17)

The equations (3.9) imply

[ \begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix} ] = [ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] [ \begin{matrix}1 − iωA & −iωB \\ iωC & 1 + iωA \end{matrix} ] . (3.18)

Further, since α_{k} represents the root of the equation (3.2), with the help of the Vieta’s theorem we see that

A_{k} = \frac{a_{11}}{2d_{k}} M_{k}^{2}, B_{k} = \frac{a_{11}}{2d_{k}} M_{k}R_{k}, C_{k} = \frac{a_{11}}{2d_{k}} R_{k}^{2}. (3.19)

Whence due to (3.11) and (3.16)

[ \begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix} ] = [ \begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix} ] \{\frac{ia_{11}}{BC − A^{2}} [ \begin{matrix}C & A \\ A & B \end{matrix} ] + ωa_{11} [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ]\} . (3.20)

Therefore, from (3.18) and (3.20) we derive

∑_{k=1}^{2} [ \begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix} ] = [ \begin{matrix}1 − iωA & −iωB \\ iωC & 1 + iωA \end{matrix} ] ,  (3.21)
∑_{k=1}^{2} [ \begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix} ] = \frac{im}{BC − A^{2}} [ \begin{matrix}C & A \\ A & B \end{matrix} ] ,  (3.22)

where

m = a_{11}(1 − ω^{2}(BC − A^{2})). (3.23) We note that the constants ω and m introduced above are positive [1]

ω > 0, m > 0. (3.24) Furthermore, let us introduce the so-called pseudo-stress operator which will be essen-

tially amployed in the next sections

N(∂_{z}, n)u(z) = T (∂_{z}, n)u(z) + κ_{N} [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] \frac{∂u(z)}{∂s(z)}, (3.25)

where

κ_{N} = ω(BC − A^{2})m^{−1} > 0. (3.26) By the direct calculations we get

[N(∂_{t}, n)Γ(z, t)]^{⊤} = Im ∑_{k=1}^{2} [ \begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix} ] \frac{∂ ln σ_{k}}{∂s(t)} . (3.27)

Here the coefficients E_{k}, F_{k}, G_{k}, and H_{k}, due to (3.20), can be written as

[ \begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix} ] = \frac{a_{11}}{m} [ \begin{matrix}N_{k} & M_{k} \\ L_{k} & R_{k} \end{matrix} ] [ \begin{matrix}1 + iωA & −iωC \\ iωB & 1 − iωA \end{matrix} ] ,

whence

E_{(k)} := [ \begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix} ] = −\frac{i}{m} [ \begin{matrix}A_{k} & B_{k} \\ B_{k} & C_{k} \end{matrix} ] [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] . (3.28)

Notice that

E_{(1)} + E_{(2)} = E, (3.29)

where E is the 2 × 2 identity matrix.

We note that the columns of the matrices (3.14) and (3.27), considered as vector functions of the variable z (i.e., of the variables x_{1} and x_{2}), solve the system (2.1) for z ≠ t. Moreover, by equations (3.21) we can show that the entries of the matrix (3.14) have singularities of type |z − t|^{−1} and on arbitrary C^{1+α}-smooth curve (with α > 0) they represent singular kernels in variables z and t. Making use of (3.29) it can also be shown that the entries of the matrix (3.27) on a C^{1+α}-smooth curve have integrable singularities of type |z − t|^{−1+α}.

For our purposes we need also the so-called hypersingular kernels constructed by the fundamental solution (3.4). The direct calculations lead to the formulae

T (∂_{z}, n)[T (∂_{t}, n)Γ(z, t)]^{⊤} = −a_{11}^{−1} Re ∑_{k=1}^{2} P_{(k)} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] \frac{∂^{2} ln σ_{k}}{∂s(z)∂s(t)},  (3.30)

T (∂_{z}, n)[N(∂_{t}, n)Γ(z, t)]^{⊤} = Im ∑_{k=1}^{2} [ \begin{matrix}E′_{k} & F′_{k} \\ G′_{k} & H′_{k} \end{matrix} ] \frac{∂^{2} ln σ_{k}}{∂s(z)∂s(t)},  (3.31)

where

E′_{(k)} := [ \begin{matrix}E′_{k} & F′_{k} \\ G′_{k} & H′_{k} \end{matrix} ] = −\frac{i}{m} [ \begin{matrix}N_{k} & L_{k} \\ M_{k} & R_{k} \end{matrix} ] [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] , k = 1, 2. (3.32)

These matrices can be also represented as follows

E′_{(k)} = [ \begin{matrix}p_{k} & q_{k} \\ r_{k} & s_{k} \end{matrix} ] \{κ_{N} [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] − \frac{i}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ]\} , k = 1, 2. (3.33)

The formulae (3.32) and (3.33) yield

E′_{k} = −α_{k}G′_{k}, F′_{k} = −α_{k}H′_{k},

[ \begin{matrix}E_{k} & F_{k} \\ G_{k} & H_{k} \end{matrix} ] = \{ωa_{11} [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] + \frac{ia_{11}}{BC − A^{2}} [ \begin{matrix}C & A \\ A & B \end{matrix} ]\} [ \begin{matrix}E′_{k} & F′_{k} \\ G′_{k} & H′_{k} \end{matrix} ] ,

∑_{k=1}^{2} [ \begin{matrix}E′_{k} & F′_{k} \\ G′_{k} & H′_{k} \end{matrix} ] = κ_{N} [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] − \frac{i}{m} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] , k = 1, 2. (3.34)

We remark that the coefficients A, B, C, ω, m, and κ_{N} defined by (3.7), (3.17), (3.23), and (3.26), respectively, in the isotropic case read as

A = 0, B = C = \frac{1}{2}, ω = 2\frac{κ − 1}{κ + 1}, m = \frac{κ}{2μ(κ + 1)}, κ_{N} = \frac{μ(κ − 1)}{κ}, (3.35)

where κ = (λ + 3μ)(λ + μ)^{−1}.

3.3. Let l_{0} be a simple, closed, nonselfintersecting, C^{1+α}-smooth curve which surrounds a bounded domain Ω^{+}. Further, let Ω^{−} = IR^{2}\\overline{Ω^{+}}, \overline{Ω^{+}} = Ω^{+} ∪ l_{0}, l_{0} = ∂Ω^{±}. Throughout this paper by n(z) (z ∈ l_{0}) we denote the outward unit normal vector with respect to Ω^{+}.

We introduce the following generalized potentials

V (g)(z) := \frac{1}{π} ∫_{l_{0}} Γ(z, t) g(t) ds,  (3.36)

W (g)(z) := \frac{1}{π} ∫_{l_{0}} [T (∂_{t}, n)Γ(z, t)]^{⊤} g(t) ds,  (3.37)

W̃ (g)(z) := \frac{1}{π} ∫_{l_{0}} [N(∂_{t}, n)Γ(z, t)]^{⊤} g(t) ds,  (3.38)

where g = (g_{1}, g_{2})^{⊤} is a real-valued density; V, W and W̃ will be referred to as a single-layer potential, a double-layer potential of the first kind and a double-layer potential of the second kind, respectively. Clearly, the potentials (3.36)-(3.38) are solutions of the system (2.1) in Ω^{+} and Ω^{−} for arbitrary integrable density g. Properties of the above potentials in the space of Hölder-continuous vector functions have been investigated in [1]. THEOREM 3.1 [1] Let l_{0} ∈ C^{2+α}, 0 < α < 1, and 0 < β < α. Then

i) the single-layer potential V (g) is continuous in the whole plane if g ∈ C(l_{0}) and V (g) ∈ C^{k+1+β}(\overline{Ω^{±}}) if g ∈ C^{k+β}(l_{0}), with k = 0, 1, 2;

ii) the double-layer potentials W (g) and W̃(g) belong to the spaces C^{k+β}(\overline{Ω^{±}}) if g ∈ C^{k+β}(l_{0}) with k = 0, 1, 2; moreover, for arbitrary g ∈ C^{β}(l_{0}) and t_{0} ∈ l_{0} there hold the following jump formulae

[W (g)(t_{0})]^{±} = ±g(t_{0}) + \frac{1}{π} ∫_{l_{0}} [T (∂_{t}, n(t))Γ(t_{0}, t)]^{⊤}g(t)ds,  (3.39)

[W̃(g)(t_{0})]^{±} = ±g(t_{0}) + \frac{1}{π} ∫_{l_{0}} [N(∂_{t}, n(t))Γ(t_{0}, t)]^{⊤}g(t)ds;  (3.40)

iii) for arbitrary g ∈ C^{β}(l_{0}) and t_{0} ∈ l_{0} there hold the jump relations

[T (∂_{t_{0}}, n)V (g)(t_{0})]^{±} = ∓g(t_{0}) + \frac{1}{π} ∫_{l_{0}} [T (∂_{t_{0}}, n(t_{0}))Γ(t_{0}, t)] g(t)ds,  (3.41)

[N(∂_{t_{0}}, n)V (g)(t_{0})]^{±} = ∓g(t_{0}) + \frac{1}{π} ∫_{l_{0}} [N(∂_{t_{0}}, n(t_{0}))Γ(t_{0}, t)] g(t)ds;  (3.42)

iv) for arbitrary g ∈ C^{1+β}(l_{0}) the vectors T (∂_{z}, n)W (g)(z) and T (∂_{z}, n)W̃(g)(z) are continuously extendible on l_{0} from Ω^{±} and for arbitrary t_{0} ∈ l_{0} there hold

[T (∂_{t_{0}}, n)W (g)(t_{0})]^{±} = \frac{1}{πa_{11}} Re ∑_{k=1}^{2} P_{(k)} [ \begin{matrix}B & −A \\ −A & C \end{matrix} ] ∫_{l_{0}} \frac{∂ ln σ_{k}}{∂s(t_{0})} \frac{∂g(t)}{∂s(t)}ds,  (3.43)

[T (∂_{t_{0}}, n)W̃(g)(t_{0})]^{±} = ∓κ_{N} [ \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ] \frac{∂g(t_{0})}{∂s(t_{0})} − \frac{1}{π} Im ∑_{k=1}^{2} E′_{(k)} ∫_{l_{0}} \frac{∂ ln σ_{k}}{∂s(t_{0})} \frac{∂g(t)}{∂s(t)}ds. (3.44)

We note that the integral operators in (3.39) and (3.41) have singular kernels, while the integral operators in (3.40) and (3.42) have weakly singular kernels (cf. [13], [14], [1]).

3.4. Finally, we recall the Green formulae in anisotropic elasticity (see, e.g., [1]). Let u ∈ C^{1}(\overline{Ω^{+}}) ∩ C^{2}(Ω^{+}) represents a solution to the system (2.1) in Ω^{+}. Then the following equations hold

∫_{Ω^{+}} T (u, u)dσ = ∫_{l_{0}} [u]^{+}[T (∂_{t}, n)u]^{+}ds,  (3.45)
∫_{Ω^{+}} N(u, u)dσ = ∫_{l_{0}} [u]^{+}[N(∂_{t}, n)u]^{+}ds,  (3.46)

where T (u, u) and N(u, u) are positive definite quadratic forms and read as

T (u, u) = A_{11}ε_{x_{1}}^{2} + 2A_{12}ε_{x_{1}}ε_{x_{2}} + A_{22}ε_{x_{2}}^{2} + 2A_{13}ε_{x_{1}}ε_{x_{1}x_{2}} +
+2A_{23}ε_{x_{2}}ε_{x_{1}x_{2}} + A_{33}ε_{x_{1}x_{2}}^{2},  (3.47)
N(u, u) = A_{11}ε_{x_{1}}^{2} + 2(A_{12} + κ_{N})ε_{x_{1}}ε_{x_{2}} + A_{22}ε_{x_{2}}^{2} +
+2A_{13}ε_{x_{1}}ε_{x_{1}x_{2}} + 2A_{23}ε_{x_{2}}ε_{x_{1}x_{2}} + (A_{33} − \frac{1}{2}κ_{N})ε_{x_{1}x_{2}}^{2} + \frac{1}{2}κ_{N}ω_{x_{1}x_{2}}^{2},  (3.48)

where ε_{x_{1}}, ε_{x_{2}}, and ε_{x_{1}x_{2}} are given by (2.3), and

ω_{x_{1}x_{2}} = \frac{∂u_{2}}{∂x_{1}} − \frac{∂u_{1}}{∂x_{2}} .

It can be easily seen that the equatoin T (u, u) = 0 implies u(z) = (e_{1} − e_{3}x_{2}, e_{2} + e_{3}x_{1}), while the equation N(u, u) = 0 yields u(z) = (e_{1}, e_{2}), where e_{1}, e_{2}, and e_{3} are arbitrary real constants.

In the case of the domain Ω^{−} the formulae (3.45) and (3.46) (with the sign ”–” in the left-hand side) remain also valid if, in addition, the vector u meets the conditions

u(z) = O(1), \frac{∂u(z)}{∂x_{k}} = O(|z|^{−2}), k = 1, 2, (3.49)

at infinity.

3.5. For the interface crack problem formulated above the following uniqueness theorem holds.

THEOREM 3.2 Let a pair (u^{(0)}, u^{(1)}) of vector functions u^{(0)} = (u_{1}^{(0)}, u_{2}^{(0)})^{⊤} and u^{(1)} = (u_{1}^{(1)}, u_{2}^{(1)})^{⊤} be a regular solution to the homogeneous version of Problem (C) (i.e., T = 0 and S = 0 ).

Then u^{(0)} = (e_{1}, e_{2})^{⊤} in S^{(0)} and u^{(1)} = (e_{1}, e_{2})^{⊤} in S^{(1)} where e_{1} and e_{2} are arbitrary real constants.

Proof. We start with the remark that u^{(0)} and u^{(1)} satisfy conditions (3.49) at infinity, since in the case under consideration u_{∞}^{(j)}(x) = 0, j = 0, 1 (see (2.9) and (2.10)). We proceed as follows. Denote by K_{R} the circle centered at the origin and radius R, and let l_{R}^{+} and l_{R}^{−} be the corresponding upper and lower semicircles. We put S_{R}^{(0)} = S^{(0)} ∩ K_{R} and S_{R}^{(1)} = S^{(1)} ∩ K_{R}. Further, due to formula (3.45) and conditions (2.6)-(2.8), we have

∫_{S_{R}^{(0)}} T^{(0)}(u^{(0)}, u^{(0)})dσ + ∫_{S_{R}^{(1)}} T^{(1)}(u^{(1)}, u^{(1)})dσ =

= ∫_{l_{R}^{+}} [u^{(0)}][T^{(0)}(∂_{t}, n)u^{(0)}]ds + ∫_{l_{R}^{−}} [u^{(1)}][T^{(1)}(∂_{t}, n)u^{(1)}]ds,  (3.50)

where n is the exterior unit normal vector to l_{R} = l_{R}^{+} ∪ l_{R}^{−}. Taking into account the behaviour of the vectors u^{(j)} at infinity, the positive definiteness of the forms T^{(0)}(u^{(0)}, u^{(0)}) and T^{(1)}(u^{(1)}, u^{(1)}), and passing to the limit in (3.50) as R → +∞, we conclude that the right-hand side integrals in (3.50) vanish and, therefore,

T^{(j)}(u^{(j)}, u^{(j)}) = 0, j = 0, 1.

Whence

u^{(j)}(z) = (e_{1}^{(j)} − e_{3}^{(j)}x_{2}, e_{2}^{(j)} + e_{3}^{(j)}x_{1}), j = 0, 1,

with arbitrary constants e_{k}^{(j)}, k = 1, 2, 3, and j = 0, 1. The behaviour of the displacement fields at infinity implies that e_{3}^{(j)} = 0, j = 0, 1, while from the contact conditions it follows that e_{k}^{(0)} = e_{k}^{(1)}, k = 1, 2.

REMARK 3.3 We note that, if, in addition, the displacement fields in S^{(0)} and S^{(1)} vanish at invinity, then the homogeneous Problem (C) possesses only the trivial solution.

4 The Basic Interface Problem

In this section we present the explicit solution of the basic interface problem for the piecewise homogeneous anisotropic elastic plane \overline{S^{(0)}} ∪ \overline{S^{(1)}} introduced above (see Section 2). This problem can be formulated as follows [1]: Find regular solutions u^{(j)} to the system (2.1) in the domains S^{(j)} (j = 0, 1) satisfying on the interface x_{2} = 0 (i.e., on the line l) the transmission conditions

[u^{(0)}]^{+} − [u^{(1)}]^{−} = f(x_{1}),
[τ_{x_{1}x_{2}}^{(0)}]^{+} − [τ_{x_{1}x_{2}}^{(1)}]^{−} = φ_{1}(x_{1}),
[σ_{x_{2}}^{(0)}]^{+} − [σ_{x_{2}}^{(1)}]^{−} = φ_{2}(x_{1}), −∞ < x_{1} < +∞,  (4.1)

where f = (f_{1}, f_{2})^{⊤} ∈ C^{1+α}(l) and φ = (φ_{1}, φ_{2})^{⊤} ∈ C^{α}(l) are given vector functions with the following asymptotics at infinity

f(x_{1}) = p_{0} + \frac{q_{0}}{|x_{1}|^{ε}} , φ(x_{1}) = \frac{r_{0}}{|x_{1}|^{1+η}}; (4.2)

here p_{0}, q_{0}, r_{0}, ε > 0, and η > 0 are real constants.

In addition, we assume that the resultant vector vanishes, i.e.,

∫_{−∞}^{+∞}φ(x_{1}) dx_{1} = 0, (4.3)

and that the stress components vanish at infinity as well.

The solution of the above basic interface problem is then representable in the form (for details see [1], [2])

u^{(j)}(z) = \frac{κ_{N}^{(1)} − κ_{N}^{(0)}}{Δ_{0}} [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] K + \frac{1}{π} Im ∑_{k=1}^{2} E_{(k)}^{(j)} \{(X^{(j)} + iY^{(j)}) ∫_{−∞}^{+∞}\frac{f(t)dt}{t − z_{kj}} +

+ (X̃^{(j)} + iỸ^{(j)}) ∫_{−∞}^{+∞}φ(t) ln(t − z_{kj})dt} , j = 0, 1,  (4.4)

where K is an arbitrary real constant, E_{(k)}^{(j)} is given by (3.28), and z_{kj} = x_{1}+α_{k}^{(j)}x_{2}; moreover,

Δ_{0} = 2κ_{N}^{(0)}κ_{N}^{(1)} + \frac{B^{(0)}C^{(0)} − (A^{(0)})^{2}}{m^{(0)}a_{11}^{(0)}} + \frac{B^{(1)}C^{(1)} − (A^{(1)})^{2}}{m^{(1)}a_{11}^{(1)}} +

+\frac{B^{(1)}C^{(0)} + B^{(0)}C^{(1)} − 2A^{(0)}A^{(1)}}{m^{(0)}m^{(1)}} > 0;

X^{(0)} + iY^{(0)} = \frac{1}{Δ_{0}} \{(\frac{B^{(1)}C^{(1)} − (A^{(1)})^{2}}{m^{(1)}a_{11}^{(1)}} + κ_{N}^{(0)}κ_{N}^{(1)}) E+

+\frac{1}{m^{(0)}m^{(1)}} [ \begin{matrix}B^{(1)}C^{(0)} − A^{(0)}A^{(1)} & A^{(0)}C^{(1)} − A^{(1)}C^{(0)} \\ A^{(0)}B^{(1)} − A^{(1)}B^{(0)} & B^{(0)}C^{(1)} − A^{(0)}A^{(1)} \end{matrix} ] −

−i (\frac{κ_{N}^{(1)}}{m^{(0)}} [ \begin{matrix}A^{(0)} & −C^{(0)} \\ B^{(0)} & −A^{(0)} \end{matrix} ] + \frac{κ_{N}^{(0)}}{m^{(1)}} [ \begin{matrix}A^{(1)} & −C^{(1)} \\ B^{(1)} & −A^{(1)} \end{matrix} ])\} ,

X̃^{(0)} + iỸ^{(0)} = \frac{1}{Δ_{0}} \{(κ_{N}^{(0)} − κ_{N}^{(1)}) [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] +

+i (\frac{1}{m^{(1)}} [ \begin{matrix}C^{(1)} & A^{(1)} \\ A^{(1)} & B^{(1)} \end{matrix} ] + \frac{1}{m^{(0)}} [ \begin{matrix}C^{(0)} & A^{(0)} \\ A^{(0)} & B^{(0)} \end{matrix} ])\};  (4.5)

the constant matrices X^{(1)} + iY^{(1)} and X̃^{(1)} + iỸ^{(1)} are obtained from the above formulae for X^{(0)} + iY^{(0)} and X̃^{(0)} + iỸ^{(0)} by the interchange of the superscripts (0) and (1). Note that

X^{(0)} + X^{(1)} = E, Y^{(0)} = Y^{(1)}, X̃^{(0)} + X̃^{(1)} = 0, Ỹ^{(0)} = Ỹ^{(1)}. (4.6)

Making use of the relationships between the coefficients established in Subsection 3.2, from (4.4) we derive the following representation formulae for the stress components σ_{x_{2}}^{(j)} and τ_{x_{1}x_{2}}^{(j)}

[ \begin{matrix}τ_{x_{1}x_{2}}^{(j)} \\ σ_{x_{2}}^{(j)} \end{matrix} ] = −\frac{1}{π} Im ∑_{k=1}^{2} E_{(k)}^{′(j)} \{(X^{(j)} + iY^{(j)}) ∫_{−∞}^{+∞}\frac{f′(t)dt}{t − z_{kj}} − (X̃^{(j)} + iỸ^{(j)}) ∫_{−∞}^{+∞}\frac{φ(t)dt}{t − z_{kj}} \} , (4.7)

where j = 0, 1, and E_{(k)}^{′(j)} is given by (3.32).

5 Reduction of the Mixed Interface Problem to the Integral Equation

In this section we shall essentially use the representation formulae (4.4) and (4.7) together with the equations (3.28), (3.29), (3.32), (3.33), (3.34), (4.5), and (4.6), to investigate the mixed interface problem (2.6)-(2.8).

Taking into account the interface conditions (2.6)-(2.8) we easily conclude that [σ_{x_{2}}^{(0)}]^{+} − [σ_{x_{2}}^{(1)}]^{−} = 0 and [τ_{x_{1}x_{2}}^{(0)}]^{+} − [τ_{x_{1}x_{2}}^{(1)}]^{−} = 0 on the whole contact line l. Therefore, due to formula (4.4), we look for the displacement fields in the domains S^{(j)} in the form

u^{(j)}(z) = u_{∞}^{(j)}(z) + \frac{1}{π} Im ∑_{k=1}^{2} E_{(k)}^{(j)}(X^{(j)} + iY^{(j)}) [ \begin{matrix}∫_{−L}^{L} \frac{u_{1}^{(0)}(t)−u_{1}^{(1)}(t)}{t−z_{kj}}dt \\ ∫_{−a}^{b} \frac{u_{2}^{(0)}(t)−u_{2}^{(1)}(t)}{t−z_{kj}}dt \end{matrix} ] , (5.1)

where z ∈ S^{(j)}, j = 0, 1.

In (5.1) the first summand u_{∞}^{(j)}(z), given by (2.11), represents the solution of (2.1) which satisfies conditions (2.5) at infinity. Here and throughout this section we use the notations u^{(0)}(t) := [u^{(0)}(t, 0)]^{+} and u^{(1)}(t) := [u^{(1)}(t, 0)]^{−} for −∞ < t < +∞.

Clearly, the difference u_{1}^{(0)}(t) − u_{1}^{(1)}(t) is unknown in the interval (−L, L), while the difference u_{2}^{(0)}(t) − u_{2}^{(1)}(t) is unknown in the interval (−a, b).

It is evident that the conditions

u_{1}^{(0)}(±L) − u_{1}^{(1)}(±L) = 0,  (5.2)
u_{2}^{(0)}(−a) − u_{2}^{(1)}(−a) = 0, u_{2}^{(0)}(b) − u_{2}^{(1)}(b) = 0,  (5.3)

are sufficient for the above displacement vectors u^{(j)} to be continuously extendible on the whole contact line l (see [14]).

Further, let

u_{1}^{(0)}(x_{1}) − u_{1}^{(1)}(x_{1}) = − ∫_{−L}^{x_{1}} B_{1}(t)dt,  (5.4)

u_{2}^{(0)}(x_{1}) − u_{2}^{(1)}(x_{1}) = − ∫_{−a}^{x_{1}}B_{2}(t)dt.  (5.5)

These equations yield

B(x_{1}) = −\frac{d}{dx_{1}} (u^{(0)}(x_{1}) − u^{(1)}(x_{1})), (5.6)

where B(x_{1}) = (B_{1}(x_{1}), B_{2}(x_{1}))^{⊤} is the so-called dislocation vector.

We require that

∫_{−L}^{L} B_{1}(t)dt = 0,  (5.7)

∫_{−a}^{b} B_{2}(t)dt = 0,  (5.8)

which guarantee the conditions (5.2) and (5.3).

Moreover, we provide (cf. [3], [4])

B_{2}(−a) = B_{2}(b) = 0, (5.9)

which shows that the normal displacements u_{2}^{(0)} and u_{2}^{(1)} have a ”smooth contact” at the points −a and b of the contact line l. From (5.1) it follows that the stress components can be represented in terms of the dislocation vector

[ \begin{matrix}τ_{x_{1}x_{2}}^{(j)} \\ σ_{x_{2}}^{(j)} \end{matrix} ] = [ \begin{matrix}S \\ T \end{matrix} ] + \frac{1}{π} Im ∑_{k=1}^{2} E_{(k)}^{′(j)}(X^{(j)} + iY^{(j)}) [ \begin{matrix}∫_{−L}^{L} & \frac{B_{1}(t)dt}{t−z_{kj}} \\ ∫_{−a}^{b} & \frac{B_{2}(t)dt}{t−z_{kj}} \end{matrix} ] . (5.10)

The limits of the displacement vectors (5.1) on the contact line are expressed by the formulae

u^{(j)}(x_{1}) = \frac{1}{π} Y^{(j)} [ \begin{matrix}∫_{−L}^{L} \frac{u_{1}^{(0)}(t)−u_{1}^{(1)}(t)}{t−x_{1}}dt \\ ∫_{−a}^{b} \frac{u_{2}^{(0)}(t)−u_{2}^{(1)}(t)}{t−x_{1}}dt \end{matrix} ] , |x_{1}| > L, j = 0, 1.

Since Y^{(0)} = Y^{(1)}, due to (4.6), we have u^{(0)}(x_{1}) = u^{(1)}(x_{1}) for |x_{1}| > L, and, therefore, we can check that the first equation in (2.6) is automatically satisfied.

Next, applying the formula (5.10) we arrive at the equation on l

[ \begin{matrix}τ_{x_{1}x_{2}}^{(0)} \\ σ_{x_{2}}^{(0)} \end{matrix} ]^{+} − [ \begin{matrix}τ_{x_{1}x_{2}}^{(1)} \\ σ_{x_{2}}^{(1)} \end{matrix} ]^{−} = \frac{1}{π} Im ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)})−

−E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) [ \begin{matrix}∫_{−L}^{L} & \frac{B_{1}(t)dt}{t−x_{1}} \\ ∫_{−a}^{b} & \frac{B_{2}(t)dt}{t−x_{1}} \end{matrix} ] , |x_{1}| > L.

With the help of the equality

Im ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) − E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) = 0, (5.11)

which follows from the fomulae (3.32), (3.33), (3.34), (4.5), and (4.6), we can easily show that the stress components (5.10) automatically satisfy the second and the third equations in (2.6).

In that case when x_{1} ∈ (−L, −a) ∪ (b, L) we have

u^{(j)}(x_{1}) = (−1)^{j}X^{(j)} [ \begin{matrix}u_{1}^{(0)}(x_{1}) − u_{1}^{(1)}(x_{1}) \\ 0 \end{matrix} ] + \frac{Y^{(j)}}{π} [ \begin{matrix}∫_{−L}^{L} \frac{u_{1}^{(0)}(t)−u_{1}^{(1)}(t)}{t−x_{1}}dt \\ ∫_{−a}^{b} \frac{u_{2}^{(0)}(t)−u_{2}^{(1)}(t)}{t−x_{1}}dt \end{matrix} ] .

Whence it follows that

u^{(0)}(x_{1}) − u^{(1)}(x_{1}) = (X^{(0)} + X^{(1)}) [ \begin{matrix}u_{1}^{(0)}(x_{1}) − u_{1}^{(1)}(x_{1}) \\ 0 \end{matrix} ] = [ \begin{matrix}u_{1}^{(0)}(x_{1}) − u_{1}^{(1)}(x_{1}) \\ 0 \end{matrix} ] .

Therefore, u_{2}^{(0)}(x_{1}) = u_{2}^{(1)}(x_{1}) for x_{1} ∈ (−L, −a) ∪ (b, L), and the first equation in (2.7) is also fulfilled.

Further, we derive the following relation

[ \begin{matrix}τ_{x_{1}x_{2}}^{(0)} \\ σ_{x_{2}}^{(0)} \end{matrix} ]^{+} − [ \begin{matrix}τ_{x_{1}x_{2}}^{(1)} \\ σ_{x_{2}}^{(1)} \end{matrix} ]^{−} =

= Re ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) + E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) [ \begin{matrix}B_{1}(x_{1}) \\ 0 \end{matrix} ] +

+\frac{1}{π} Im ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) − E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) [ \begin{matrix}∫_{−L}^{L} & \frac{B_{1}(t)dt}{t−x_{1}} \\ ∫_{−a}^{b} & \frac{B_{2}(t)dt}{t−x_{1}} \end{matrix} ] ,

x_{1} ∈ (−L, −a) ∩ (b, L).

Now the equations (5.11) and the equality

Re ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) + E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) = 0 (5.12)

imply that the second condition in (2.7) and the equation [τ_{x_{1}x_{2}}^{(0)}]^{+} = [τ_{x_{1}x_{2}}^{(1)}]^{−} are also fulfilled. Finally, from (5.10) we deduce for x_{1} ∈ (−a, b)

[ \begin{matrix}τ_{x_{1}x_{2}}^{(0)} \\ σ_{x_{2}}^{(0)} \end{matrix} ]^{+} − [ \begin{matrix}τ_{x_{1}x_{2}}^{(1)} \\ σ_{x_{2}}^{(1)} \end{matrix} ]^{−} =

=Re ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) + E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) [ \begin{matrix}B_{1}(x_{1}) \\ B_{2}(x_{1}) \end{matrix} ] +

+\frac{1}{π} Im ∑_{k=1}^{2} (E_{(k)}^{′(0)}(X^{(0)} + iY^{(0)}) − E_{(k)}^{′(1)}(X^{(1)} + iY^{(1)})) [ \begin{matrix}∫_{−L}^{L} & \frac{B_{1}(t)dt}{t−x_{1}} \\ ∫_{−a}^{b} & \frac{B_{2}(t)dt}{t−x_{1}} \end{matrix} ] .

Whence one can easily show that σ_{x_{2}}^{(0)} = σ_{x_{2}}^{(1)} and τ_{x_{1}x_{2}}^{(0)} = τ_{x_{1}x_{2}}^{(1)} for x_{1} ∈ (−a, b). Therefore, the conditions (2.6)-(2.8) will be completely fulfilled if

[σ_{x_{2}}^{(0)}]^{+} = 0, x_{1} ∈ (−a, b),  (5.13)
[τ_{x_{1}x_{2}}^{(0)}]^{+} = 0, x_{1} ∈ (−L, L).  (5.14)

These boundary conditions together with the formula (5.10) lead to the system of integral equations

QB_{1}(x_{1}) + \frac{R_{21}}{π} ∫_{−L}^{L} \frac{B_{1}(t)dt}{t − x_{1}} + \frac{R_{22}}{π} ∫_{−a}^{b} \frac{B_{2}(t)dt}{t − x_{1}} = T, x_{1} ∈ (−a, b),

Q(H(x_{1} − b) − H(x_{1} + a))B_{2}(x_{1}) + \frac{R_{11}}{π} ∫_{−L}^{L} \frac{B_{1}(t)dt}{t − x_{1}} +

+\frac{R_{12}}{π} ∫_{−a}^{b} \frac{B_{2}(t)dt}{t − x_{1}} = S, x_{1} ∈ (−L, L),  (5.15)

where H stands for the Heviside step function, and

Q = \frac{1}{Δ_{0}} (κ_{N}^{(1)} \frac{B^{(0)}C^{(0)} − (A^{(0)})^{2}}{m^{(0)}a_{11}^{(0)}} − κ_{N}^{(0)} \frac{B^{(1)}C^{(1)} − (A^{(1)})^{2}}{m^{(1)}a_{11}^{(1)}} ) ,

[ \begin{matrix}R_{11} & R_{12} \\ R_{21} & R_{22} \end{matrix} ] = \frac{1}{Δ_{0}} (\frac{B^{(1)}C^{(1)} − (A^{(1)})^{2}}{m^{(0)}m^{(1)}a_{11}^{(1)}} [ \begin{matrix}B^{(0)} & −A^{(0)} \\ −A^{(0)} & C^{(0)} \end{matrix} ] +

+\frac{B^{(0)}C^{(0)} − (A^{(0)})^{2}}{m^{(0)}m^{(1)}a_{11}^{(0)}} [ \begin{matrix}B^{(1)} & −A^{(1)} \\ −A^{(1)} & C^{(1)} \end{matrix} ]) .

Remark that

R_{11} > 0, R_{22} > 0, R_{12} = R_{21}. (5.16)

Thus, we have to find the unknown functions B_{1}(x_{1}) and B_{2}(x_{1}) satisfying the system of integral equations (5.15) and conditions (5.7)-(5.9).

It is easy to check that in the isotropic case (cf. [3], [4])

R_{12} = R_{21} = 0, R_{11} = R_{22} = \frac{2μ_{1}(1 + α_{∗})}{(κ_{1} + 1)(1 − β_{∗}^{2})} = \frac{2μ_{0}(1 − α_{∗})}{(κ_{0} + 1)(1 − β_{∗}^{2})} = C_{∗},

Q = β_{∗}C_{∗}, β_{∗} = \frac{μ_{0}(κ_{1} − 1) − μ_{1}(κ_{0} − 1)}{μ_{0}(κ_{1} + 1) + μ_{1}(κ_{0} + 1)}, α_{∗} = \frac{μ_{0}(κ_{1} + 1) − μ_{1}(κ_{0} + 1)}{μ_{0}(κ_{1} + 1) + μ_{1}(κ_{0} + 1)}.

Therefore, in that case, the system (5.15) reads as

β_{∗}B_{1}(x_{1}) + \frac{1}{π} ∫_{−a}^{b} \frac{B_{2}(t)dt}{t − x_{1}} = \frac{T}{C_{∗}} , x_{1} ∈ (−a, b),

β_{∗}(H(x_{1} − b) − H(x_{1} + a))B_{2}(x_{1}) + \frac{1}{π} ∫_{−L}^{L} \frac{B_{1}(t)dt}{t − x_{1}} = \frac{S}{C_{∗}} , x_{1} ∈ (−L, L).  (5.17)

This system coincides with the equations obtained in [6] for the mixed interface problem in question in the isotropic case (i.e., when the domains S^{(0)} and S^{(1)} in the problem (2.6)- (2.8) are isotropic half-planes with distinct Lamé constants). The same system has been investigated in [10] where the explicit solution is also constructed (see also [8], [9]).

We follow to [10] and make in (5.15) the change of independent variables

x_{1} = L\frac{s − γ}{1 − sγ} , t = L\frac{r − γ}{1 − rγ} , γ = \frac{L(a − b)}{L^{2} − ab + \sqrt{(L^{2} − a^{2})(L^{2} − b^{2})}}. (5.18)

Next we introduce the new unknown functions A_{1} and A_{2} rather than B_{1} and B_{2}

R_{11}B_{1}(x_{1}) = (1 − sγ)^{2}A_{1}(s), R_{22}B_{2}(x_{1}) = (1 − sγ)^{2}A_{2}(s). (5.19)

After these manipulations the equations (5.15) are recast into the system

\frac{Q}{R_{11}} A_{1}(s) + \frac{R_{21}}{πR_{11}} ∫_{−1}^{1} \frac{(1 − rγ)A_{1}(r)}{(1 − sγ)(r − s)}dr + \frac{1}{π} ∫_{−c}^{c} \frac{(1 − rγ)A_{2}(r)}{(1 − sγ)(r − s)}dr =

= \frac{T}{(1 − sγ)^{2}} , |s| < c,

−\frac{Q}{R_{22}} H(c^{2} − s^{2})A_{2}(s) + \frac{1}{π} ∫_{−1}^{1} \frac{(1 − rγ)A_{1}(r)}{(1 − sγ)(r − s)}dr +

+\frac{R_{12}}{πR_{22}} ∫_{−c}^{c} \frac{(1 − rγ)A_{2}(r)}{(1 − sγ)(r − s)}dr = \frac{S}{(1 − sγ)^{2}} , |s| < 1,  (5.20)

where

c = \frac{a\sqrt{L^{2} − b^{2}} + b\sqrt{L^{2} − a^{2}}}{L(\sqrt{L^{2} − b^{2}} + \sqrt{L^{2} − a^{2}})} < 1.

The conditions (5.7)-(5.9) now read as

∫_{−1}^{1} A_{1}(r)dr = 0,  (5.21)

∫_{−c}^{c} A_{2}(r)dr = 0,  (5.22)
A_{2}(−c) = A_{2}(c) = 0.  (5.23)

Applying the equality

∫_{−a}^{a} \frac{(1 − rγ)A(r)}{(1 − sγ)(r − s)}dr = ∫_{−a}^{a} \frac{A(r)dr}{r − s} − \frac{γ}{1 − sγ} ∫_{−a}^{a} A(r)dr,

and bearing in mind that A_{1} and A_{2} are subjected to the conditions (5.21)-(5.23), from the system (5.20) we infer

\frac{Q}{R_{11}} A_{1}(s) + \frac{R_{21}}{πR_{11}} ∫_{−1}^{1} \frac{A_{1}(r)dr}{r − s} + \frac{1}{π} ∫_{−c}^{c} \frac{A_{2}(r)dr}{r − s} = \frac{T}{(1 − sγ)^{2}} , |s| < c,

−\frac{Q}{R_{22}} H(c^{2} − s^{2}) A_{2}(s) + \frac{1}{π} ∫_{−1}^{1} \frac{A_{1}(r)dr}{r − s} + \frac{R_{12}}{πR_{22}} ∫_{−c}^{c} \frac{A_{2}(r)dr}{r − s} = \frac{S}{(1 − sγ)^{2}} , |s| < 1. (5.24)

Clearly, the unknown functions A_{1} and A_{2} have to meet the conditions (5.21)-(5.23). Now we show that this problem can be reduced to a single integral equation (cf. [10]). To this end let us rewrite the second equation in (5.24) as follows

\frac{1}{π} ∫_{−1}^{1} \frac{A_{1}(r)dr}{r − s} = \frac{Q}{R_{22}} H(c^{2} − s^{2}) A_{2}(s) − \frac{R_{12}}{πR_{22}} ∫_{−c}^{c} \frac{A_{2}(r)dr}{r − s} + \frac{S}{(1 − sγ)^{2}} , |s| < 1. (5.25)

Next, we shall apply the general theory of singular integral equations on arcs developed in [14] (Chapter 5). According to this theory, if we look for a solution A_{1} to the equation (5.25) in the class of functions which are unbounded at the both ends ±1 and satisfy the condition (5.21), we can invert the left-hand side operator in (5.25). As a result we obtain

A_{1}(s) = −\frac{X(s)}{π} ∫_{−1}^{1} (\frac{Q}{R_{22}} H(c^{2} − r^{2})A_{2}(r) − \frac{R_{12}}{πR_{22}} ∫_{−c}^{c} \frac{A_{2}(t)dt}{t − r}+

+\frac{S}{(1 − rγ)^{2}} ) \frac{dr}{X(r)(r − s)}, |s| < 1,  (5.26)

where X(s) = (1 − s^{2})^{−1/2}. On account the additional conditions (5.22) and (5.23) we can simplify (5.26)

A_{1}(s) = −\frac{R_{12}}{R_{22}} H(c^{2} − s^{2})A_{2}(s) − \frac{QX(s)}{πR_{22}} ∫_{−c}^{c} \frac{A_{2}(r)dr}{X(r)(r − s)} + SX(s)\frac{d}{dγ} (\frac{\sqrt{1 − γ^{2}}}{1 − sγ} ) . (5.27)

Substitution of this expression for A_{1} into the first equation of the system (5.24) implies

\frac{1}{π} ∫_{−c}^{c} \frac{A_{2}(r)}{r − s}(\sqrt{1 − s^{2}} − ν^{2}\sqrt{1 − r^{2}})dr = \frac{d}{dγ} (\frac{γT_{∗}\sqrt{1 − s^{2}} − δS\sqrt{1 − γ^{2}}}{1 − sγ} ) , (5.28)

where |s| < c and

ν^{2} = \frac{Q^{2}}{R_{11}R_{22} − R_{12}^{2}} > 0, T_{∗} = \frac{R_{22}(R_{11}T − R_{12}S)}{R_{11}R_{22} − R_{12}^{2}}, δ = \frac{QR_{22}}{R_{11}R_{22} − R_{12}^{2}} .

Thus, we have obtained the single integral equation (5.28) together with the conditions (5.22)-(5.23) to find the unknown A_{2} in the class of functions bounded at the both ends ±c.

Let us single out the so-called characteristic (dominant singular) part of the equation (5.28)

\frac{1}{π} ∫_{−c}^{c} \frac{A_{2}(r) dr}{r − s} + \frac{1}{π} ∫_{−c}^{c} K(s, r)A_{2}(r) dr = Φ(s), −c < s < c, (5.29)

where

K(s, r) = \frac{ν^{2}}{1 − ν^{2}} \frac{r + s}{\sqrt{1 − s^{2}}(\sqrt{1 − s^{2}} + \sqrt{1 − r^{2}})},

Φ(s) = \frac{1}{1 − ν^{2}} \frac{1}{\sqrt{1 − s^{2}}} \frac{d}{dγ} (\frac{γT_{∗}\sqrt{1 − s^{2}} − δS\sqrt{1 − γ^{2}}}{1 − sγ} ) .

Obviously, K(s, r) ∈ C^{1/2}([−c, c] × [−c, c]) and Φ ∈ C^{1/2}([−c, c]), i.e., the kernel K and the right-hand side function Φ are Hölder-continuous functions with the exponent 1/2.

We look for solutions of the equation (5.29) in the class of functions which are bounded at the both ends −c and c. Due to [14], we denote this class by h(−c, c). We emphasize that, in addition, the solution has to satisfy the contitions (5.22) and (5.23). It is evident that (see [14]), if equation (5.29) possesses a solution, then this solution automatically satisfies the conditions (5.23) since the function Φ is Hölder-continuous. Thus, we have to look for a solution to the equation (5.29) in the class h(−c, c) with the only additional condition (5.22).

It can be easily seen that the index of the equation (5.29) in the class h(−c, c) is equal to −1. Furthermore, applying the uniqueness Theorem 3.2 and Remark 3.3 we can show that the homogeneous version of equation (5.29) has no nontrivial solutions in the class h(−c, c). Therefore, the homogeneous equation

\frac{1}{π} ∫_{−c}^{c} \frac{σ(r)}{r − s}(\sqrt{1 − s^{2}} − ν^{−2}\sqrt{1 − r^{2}})dr = 0, (5.30)

which is adjoint to equation (5.29), has only one linearly independent solution in the class of functions unbounded at the both ends −c and c. We denote this nontrivial solution by σ_{0}(s). Due to the Noether’s theorems the condition

∫_{−c}^{c} Φ(r) σ_{0}(r) dr = 0 (5.31)

is then necessary and sufficient for the equation (5.29) (i.e., (5.28)) to be solvable (cf. [14]). The unknown parameters c and γ (i.e., a and b) are to be defined by the conditions (5.22) and (5.31).

A direct comparison shows that the equations (5.28), (5.22), and (5.23) are quite similar to the equations obtained by Gautesen and Dundurs (see equations (2.18), (2.13), and (2.19) in [10]). In fact, the above integral equation (5.28) and the equation (2.18) in [10] coincide term by term within the constant coefficients involved in these equations. Therefore, the analysis given in [10] can also be applied word for word to our case to construct the explicit solution of the equation (5.28) and to define the unknown parameters a and b.

The above results show that the theoretical and numerical analysis of the singular integral equations, corresponding to the Comninou formulation of the interface crack problem, are quite similar in the isotropic and in the general anisotropic cases.

ACKNOWLEDGEMENT This research was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number 436 GEO 17/2/97.

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