The results of the calculations performed

The results of the calculations performed for two cases of boundary conditions for the temperature are presented as bifurcation curves in Fig. 5. For the heat-conducting walls (curve 1) this AZD 0530 curve has a pronounced maximum =7.7° at r=3.3. These values are close to those obtained in the calculations of the bifurcation curve for a circular cylinder with heat-conducting walls [13,14], but still differ from them. The bifurcation curve for the heat-insulated walls corresponds to the results obtained in Refs. [15,16]. The calculations in [15] were carried out by the Petrov–Galerkin method, which used up to seventy basis functions. Chebyshev polynomials served as these functions. The individual points of the bifurcation curves for three Prandtl numbers obtained in [16] are consistent with the results of Ref. [15] for the case of the air. Our calculations, performed on a relatively coarse grid, yielded good agreement with the results presented in Ref. [15], which indicates the reliability of our findings.
Conclusion
Introduction One of the first works in this direction was the paper by Erdogan and Gupta [1], which considered an interface crack in a composite wedge, obtained a singular integral equation and used a numerical method for solving it. Ref. [2] constructed an exact solution for the case of multiple collinear cracks located on the symmetry axis of an isotropic wedge. Shahani [3] analyzed a crack emanating from the vertex of an anisotropic wedge. Based on the dislocation method, Faal et al. [4] derived a system of singular integral equations for several arbitrarily oriented cracks in an isotropic wedge and developed a numerical procedure for solving this system. Wu et al. [5] studied by the method of complex potentials the equilibrium of a wedge consisting of two transversely isotropic piezoelectric materials with an interfacial finite-length crack. Ref. [6] established that the solution for the antiplane problem with an anisotropic composite wedge can be obtained from the solution for the respective problem with isotropic materials, by converting the polar coordinates. For this reason, the solutions obtained for the isotropic components of the composite are of particular importance. An effective method for constructing exact solutions for this type of problems was demonstrated in the works of Beom and Jang [7,8] for an interfacial crack and a crack lying outside the interface. In the case of wedge faces loaded by the concentrated forces, the authors reduced the problem to a scalar Wiener–Hopf equation. However, the method of infinite products was used in solving this equation for the factorization of functions, which somewhat reduces the efficiency of the approach. The present study investigated the equilibrium of a piecewise-homogeneous wedge with a semi-infinite interfacial mode III crack with a self-balanced load applied at the sides. Three variants of the boundary conditions at the sides of the wedge are discussed:
Each of the variants reduces this problem, by using the integral Mellin transform, to the Riemann problem [9]. Its exact solution in quadratures has been constructed based on the procedure developed in [10]. Function factorization was performed using the Cauchy-type integrals. The simplest formulae for the stress intensity factor (SIF) at the crack tip have been obtained in the case of a geometrically symmetric structure of the composite. We have examined limiting situations where the relative hardness of the composite tends to zero or to infinity and studied the stress singularity of the stresses in the vertex of the wedge. It turns out that, unlike the case of a homogeneous material, the stress asymptotics near this singular point may contain two singular terms for certain parameter values of the composite. The previous studies on this subject have not mentioned this fact [11,12].
Setting the problem and reducing Prokaryotic to Riemann\'s problem Let us examine a semi-infinite mode III interfacial crack between two inhomogenous wedges with angles α1 and α2 (Fig. 1). Let us assume the materials of the wedges in the areas Ω1 and Ω2 to be isotropic, homogeneous and having shear moduli µ1 and µ2. The contact of the materials outside the crack is assumed to be ideal. Without loss of generality, we are going to suppose that the crack tip is located at a unit length\'s distance from the vertex of the composite wedge. A self-balanced load g(r) is applied to the sides of the crack (r is the polar radius).