A similar law of the shear module

A similar law of the shear module variation in the antiplane problem for a system of perfectly bonded wedges was used to construct singular solutions in Ref. [12]. Additionally, this article suggested an approximate method for determining the order of singularity in wedge-like areas, based on a piecewise-constant approximation of the shear module of FGM. However, the roots of the characteristic equation were not analyzed in Refs. [11,12]. The present work studies the stress state in the vertex of a composite wedge consisting of two homogeneous materials under the conditions of an antiplane problem. Instead of the traditional straight interface we examine a wedge-like FGM-filled area. We modeled the interface in this manner coming from the physical assumption that there is a diffusion of materials during their process connection [11,13,14]. This leads to the elastic modulus varying continuously; in cetp inhibitors with Refs. [11,12] the modulus was assumed to depend quadratically on the polar angle in the transition area. This functional relationship for the shear module allows to obtain a characteristic equation of the problem in an explicit form and to analyze the equation roots causing the singularities depending on the composition parameters.
Problem statement We are going to analyze the stress state of a composite wedge consisting of three wedge-shaped areas (k=1, 2, 3). The materials of two areas (Fig. 1): (r and θ are the polar coordinates) are considered homogeneous and isotropic with shear moduli μ1 and μ2, respectively. The geometrical parameters defining the boundaries of these areas must satisfy the following inequalities:
The third (intermediate) area consists of an FGM that is modeled by an inhomogeneous material without accounting for its microstructure. The shear modulus of the functionally graded interphase μ3 is assumed to depend only on the polar angle. The functional dependence μ3(θ) is such that the elastic modulus of the composite is continuous on the θ=±β boundaries, while its derivatives with respect to the angle θ have discontinuities on these boundaries. The composite wedge is in equilibrium under antiplane deformation when subjected to self-equilibrating concentrated forces of magnitude Т0 applied to the wedge sides at a distance r0 from its vertex (see Fig. 1). The contact of materials at the interface is assumed to be perfect. From a mathematical standpoint, the problem is reduced to solving harmonic equations of equilibrium in each of the areas (k=1, 2):
The shear stresses are found from the displacements using the formulae
In the area of the intermediate FGM with a shear modulus varying in the transverse direction the equilibrium equation takes the form
Let us examine a special type of a functional relationship between the interphase shear modulus and the polar angle:
A similar type of material inhomogeneity was discussed in Ref. [14] in a Cartesian coordinate system. Then, searching for displacements in the area Ω3 in the form we shall obtain from Eq. (3) that the function w3(r, θ) also satisfies Eq. (1), i.e. is harmonic. Using the formulae (3), we arrive at the following expressions for the stresses:
The constants a and b included in the expression (4) are found from the continuity conditions of the shear modulus at the interfaces and have the form
For these constants the shear modulus μ3(θ) is a monotonic function for which the following formula holds on the interphase symmetry axis:
The solutions of Eqs. (1) and (3) must satisfy the boundary conditions for the perfect interphase contact: and the boundary conditions at the wedge sides: where δ(r) is the Dirac delta function.
The solution in Mellin transforms Subjecting Eqs. (1) to an integral Mellin transform, for the displacement transforms we obtain the equations
The general solutions of these equations have the form
Let us subject the functions (9) to the Mellin-transformed boundary conditions (7) and (8); then, using the formulae (2), (4)–(6) we will arrive at a system of six linear algebraic equations for the quantities and (k=1, 2, 3). After performing an inverse transform