When the parameter is close

When the parameter μ is close to zero, we will have:
From here it follows that in the case of a low relative hardness of the composite, the SIF limit values are expressed by formulae (29) for a homogeneous wedge at . Notice that asymptotes (31) and (32) can also be obtained by solving the limiting problems for (9) occurring at μ → ∞ and μ → 0. Let us introduce normalized SIFs reflecting the influence that the inhomogeneity of the composite structure has on the SIF. Fig. 2 shows the curves of the normalized SIFs in the tip of a crack loaded with concentrated forces, at α1=π/2, α2=3π/2 and r0=4, versus the relative hardness parameter of the wedge. With this choice of angles, Problem 1 corresponds to the case of the interaction between two orthogonal cracks, Problem 2 to that between a crack and a hard inclusion and Problem 3 to that of a crack and a detached hard inclusion. The data presented indicates that the inhomogeneity of the composite can cause the SIF to both increase and decrease in comparison with the homogeneous medium. These effects are the strongest at low and high values of relative hardness in Problem 3.
Stress fields near the wedge vertex Let us examine the stress fields in the vertex of the wedge at r → 0. Based on formulae (7) and conditions (3), the tangential stresses on the line θ=0 have the form
Using Eqs. (15) and (16), the irak pathway for the integrand can be written as follows:
As a result of applying the residue theorem to the poles determined by the negative roots of the equations (see (20) and (25)) we obtain:
The coefficients of this series are found from the formulae where
It follows from (35) that the singular stress fields near the wedge vertex are generated by the roots of Eqs. (34), located in the strip . Similar to the calculations presented in [10], we can demonstrate that all of the numbers in this strip are real and single. Functions (20) and (25) have the following properties:
It follows from equalities (36) and (38) that in Problems 1 and 3, it is sufficient to consider, for example, the roots of Eqs. (34) at α1 < α2 for . Due to property (37) the roots in Problem 2 are identical to the roots in Problem 1 if the parameter m is substituted by −m. As we are not going to analyze the roots of Eqs. (34) in detail in this paper, let us note only some of the properties of the roots located in the (0, 1) interval. Depending on the parameters of the composite structure, Eqs. (34) may have either no roots, or one, or two roots in this interval. These roots can generate both weak and strong singularities in the wedge vertex. If there are two roots in the interval, only one of the two situations is possible: either or . In other words, such root distributions in the interval cause either two weak stress field singularities at r → 0, or a strong and a weak one. The existence of two strong singularities in the stress asymptote is impossible. There are no singular terms in Problems 1 and 2 in stress representations (35) for example, at any for the apex angles of the wedge that do not exceed π/2. In Problem 3, a similar situation occurs for the angles in the same interval at m ≤ 0. For geometrically symmetric composite wedges in Problems 1 and 2, Eqs. (34) have only one root in the (0, 1) interval, if α > π/2. This root does not depend on the bielastic constant and is identical to the root for the homogeneous wedge . In Problem 3, here are three possibilities for this case. Firstly, for the angles , there are no roots smaller than unity. Secondly, for the angles satisfying the inequality , there is one root in the interval in question. Finally, for the angles , the stresses in the angular point of the wedge contain two singular terms generated by the roots
Fig. 3a shows the plots of the first two roots of Eq. (34) in Problem 1 versus the α1angle for different values of the relative hardness, when the composite medium occupies the entire plane, and contains two interfacial cracks. At 0 < μ < 1 (the bielastic constant m > 0), Eq. (34) has two roots in the (0, 1) interval: if 0 < α1 < π/2, and one root if π/2 < α1 ≤ π. In the case when μ > 1 (m < 0) the roots exceed 0.5 in the entire variation range of the α1 angle. This means that the stress asymptote in tip of the crack coinciding with the angular point of the interface may contain one or two singular terms. The singularity exponent is different from the conventional value of 0.5. The conventional result is retained for two collinear cracks .