The asymptotic form for the SIF provided that

The asymptotic form for the SIF provided that ɛ → 0 follows from (22): Thus, the behavior of the SIF at small distances from the crack tip to the inclusion vertex is governed by the value of the first root of Eq. (23). The following three variants are possible if ɛ → 0:
Similar behavior of the SIF for a mode I crack approaching the material interface was observed in Refs. [1,5,9]. In case of a symmetric structure, the characteristic equation takes the form and has a single root in the interval (0, 1). This root exceeds 0.5 for m > 0, and lies in the range 0 < p1 < 0.5 for m < 0. Fig. 3 shows the dependence of the normalized SIF value on the relative distance ɛ/r0 for the hard and the soft inclusion. The above data prove that the SIF is lower for a crack located in a soft material (Fig. 3a) than the SIF in a homogeneous medium, and decreases with decreasing distance from the crack tip to the interface. If the crack is located in a hard material (Fig. 3b), the SIF has an inverse dependence on the relative distance. A similar change in the SIF in a plane problem with a bimaterial interface was obtained in Ref. [18] based on a numerical procedure. In the case of an asymmetric structure (β ≠ 0), the behavior of the SIF for small ε is rather more complex. If the vertex angle of the inclusion α ∈ (π, 2π), the dependence of the SIF on ε/r0 will be similar to the symmetric case, that is, crack growth will always be stable for a hard inclusion, and unstable for a soft one. If a crack approaches a hard inclusion from a soft medium at an angle 0 < α < π, then, if the inequality is satisfied, characteristic equation (23) will necessarily have the first root p1 < 0.5 and, consequently, KIII → ∞ if ɛ → 0. If this SAR 405 inequality is violated, KIII → 0 if ɛ → 0, and the crack will be stable. If the crack approaches a soft inclusion from a hard medium, the situation will be reversed: the crack experiences stable growth with the composite parameters satisfying inequality (25), while for the parameters that violate inequality (25), KIII → ∞ if ɛ → 0, and the growth of the crack is unstable. These conclusions are illustrated for the hard and the soft inclusion in Fig. 4 (curves 2 and 3 in Fig. 4a, curve 3 in Fig. 4b).
Stress singularity in the angular point of the inclusion Based on Eq. (7) for the stresses along the crack line, we have: Using formulae (13), (15), (17) and (18), we obtain the inequality Taking into account this representation, stresses (26) take the form Based on Eq. (11), we can arrange the characteristic equation determining the poles of the integrand in the following form: Since by substituting m with –m the function is transformed into , it suffices to consider, for example, only the roots of the equation
Besides, the following equality takes place: It follows then that it is necessary to analyze the roots of Eq. (28), for example, only for the soft inclusion (m < 0) with 0 < α ≤ π. Then the root values for the hard inclusion (m > 0) are obtained by substituting α with 2π – α. The analysis shows that Eq. (28) has a single real root in the interval (0.5, 1.0) with 0 < α < π for the case of the soft inclusion, and with π < α < 2π for the case of the hard inclusion. Next, we combine the integration path in expression (27) with the imaginary axis, apply the theorem of residues, taking into account that the poles of the integrand in the region are determined by the negative zeroes of the function f(p). Then we obtain the following expression:
It is obvious that the asymptotic form of the stresses for r → 0 is determined by the first term in series (29), that is, the first positive zero of the function f(p). In the case of a composite with an asymmetric structure (β ≠ 0), the stress singularity in the angular point of the inclusion is weak and holds for any value α ≠ π and for any relative hardness μ ≠ 1.