The problem is reduced to

The problem is reduced to solving the equilibrium equations in each of the regions:where r and θ are the polar coordinates, \'s are the displacements along the z axis. The perfect contact conditions are imposed: ( are the tangential stresses), along with the conditions along the crack line: where τ(r) is an unknown function. The solution to the problem is sought for in the form of Mellin integrals: where the transforms for the displacements and the stresses are determined by the following formulae: Since regularity conditions are imposed for r → 0 and r → ∞, the elastase inhibitor path L is parallel to the imaginary axis in the strip From conditions (3) and (4) we obtain the equalities where The functions и are regular and have no zeroes in the half-plane to the left of the path L, and the same is true for in the right half-plane [16]. Let us substitute Eq. (6) into the left-hand sides of Eq. (7) and into conditions (2) modified by Mellin transform. After excluding the quantities (p) and (p), we arrive to the Wiener–Hopf equation: An imaginary axis can be taken here as the path L, while the function F(p) has the form The elastic properties of the composite are reproduced in these formulae through a single bielastic constant m: where represents the relative stiffness of the inclusion (0 ≤ μ < ∞). This quantity satisfies the inequality m  ≤ 1 for all combinations of shear moduli of the materials. If the inclusion material is harder than the matrix material, then 0 < m < 1; otherwise (for a soft inclusion), the bielastic constant lies in the interval The value m = 0 corresponds to a homogeneous medium and the values govern an absolutely hard inclusion and a wedge-shaped notch.
Solution of the Wiener–Hopf equation Let us arrange function (10) in the form Factorization of the function X(p) is carried out in an elementary way [16]: where Γ(x) is the gamma function. Along the imaginary axis and with p=it, function (14) is continuous, has no zeros and poles, its index is equal to zero, and at t  → ∞ it tends exponentially to unity if . Therefore, in accordance with the results obtained in Refs. [9,16], the equalities hold true. Since the function Φ(p) is even, the analytical functions in the regions Ω+ and Ω− can be written in the form Using formulae (13)–(15), rearranging the terms in Eq. (9), and applying the Liouville theorem [16], we obtain: where
Estimating the terms in equality (16) atp → ∞, we obtain that the analytic function . Then, from Eq. (16) we find Since with p → ∞, we obtain the asymptotic . From this, by the Abel-type theorem [16], we can conclude that the asymptotic of the stresses at has the form
Stress intensity factor Let us define the stress intensity factor (SIF) in the crack tip by the formula Then, using the asymptotic formula (20), we obtain the following formula: For the purpose of constructing Green\'s function, we shall assume that self-balanced concentrated forces T0 are applied to the edges of the crack at a distance r0 from its tip, i.e., where δ(r) is Dirac\'s delta function, and ɛ < r0 < ∞. Then, calculating the function by formula (8), combining the integration path with the imaginary axis in expression (19) and using the theorem of residues in the region , according to formula (21), we obtain: where is the SIF in the tip of a crack located in an unbounded homogeneous medium; the prime denotes the derivative with respect to the variable p, and denotes the positive zeroes of function (12). The roots of the equation located in the strip , have been analyzed in detail in Ref. [17]. It was found that depending on the composite parameters α, β and m, Eq. (23) can have one root p1 < 0.5 or p1 > 0.5 in this strip, as well as two roots: or Fig. 2 shows the variation ranges of the angular parameters for which the first root of Eq. (23) is greater or less than 0.5 for a relatively hard (0 < m < 1) or soft () medium 1.