539.3 Within the framework of the approach proposed in the first part of the work, the two-dimensional boundary-value problem for an isotropic body with noncanonical elastic inclusion is reduced to a finite system of linear algebraic equations. It is shown that the solution of this problem for elastic inclusions with small radius of curvature at the tip and/or cusps describes the intensity and concentration of stresses in the composition. For some special examples, we reveal the influence of elastic properties of the components of the composition and configuration of the inclusions on its local stress-strain state. It is also established that, unlike the method of perturbation of the shape of the boundary, this method is applicable to the determination of the concentration and intensity of stresses in the vicinity of the tips of elastic inclusions with small radii of curvature, including the inclusions whose elastic properties are close to the elastic properties of the matrix.In the first part of the present work [1], we suggested a method for the determination of the stress-strain state of isotropic bodies with curvilinear foreign elastic inclusions in the two-dimensional case. Here, we present some examples of application of this method. For inclusions of noncanonical shapes, the problem was reduced to the solution of a system of linear algebraic equations. It was shown that the investigated approach can be used for the determination of the concentration and intensity of stresses near the tips of inclusions with small radii of curvature. The influence of the configuration and elastic properties of inclusions on the local stress-strain state of the composition is analyzed for several special cases.
Reduction of the Two-Dimensional Problem to a Finite System of Linear Algebraic EquationsLet us now return to the statement of the problem from [ 1 ]. We restrict ourselves to the case of a smooth contour L and present basic relationships of the suggested approach for the evaluation of a single complex stress potential ___l (Z). In what follows, the functions of z are denoted by underlined symbols.Let % (~) and ~l (~) be the limiting values on the contour ~/ (in a sense of transformation (8) The functions tp2(• ) and ~2 (0)
Within the framework of the approach proposed by L.P. Mazurak, L. T. Berezhnyts'kyi, and P. S. Kachur ["Method for determination of elastic equilibrium of isotropic bodies with curvilinear inclusions. Part 1. Mathematical foundations," Fiz.- Khim. Mekh. Mater., 33, No. 6, 21-31 (1997)], we construct a new method for the determination of elastic equilibrium of cylindrical bodies with noncanonical curvilinear foreign elastic inclusions under conditions of longitudinal shear. Unlike the method of perturbation of the form of a boundary, this method imposes no restrictions on the form of inclusions. The method is based on a procedure of determination of contour integrals of the Cauchy type by using the Faber polynomials.
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