Application of a Modified Boundary-Element Method to Problems of the Elasticity Theory in Piecewise Homogeneous Anisotropic Media

Abstract

UDC 539.3 We study the problem of determination of the stress-strain state of a thin elastic orthotropic plate with orthotropic inclusions of any shape under loads applied to its edges. To solve the problem, we propose a modified direct boundary-element method capable of the simultaneous determination of all necessary quantities on the contact surfaces of each inclusion with the matrix by using a two-dimensional approximation of the components of the vector of displacements and stress tensor in every element of the discretization. The governing input relations are obtained for all components of the vector of displacements and stress tensor on the boundary and at inner points of the body under consideration. Recent years are marked by the extensive application of materials and structural elements with anisotropic properties. The use of these materials makes it possible to get the required operating characteristics of strength and stiffness. For their efficient and safe application, we need the exact and reliable numerical methods aimed at the determination of their stress-strain state. The customary numerical methods based on the discretization of the domain itself (finite-element method) or its boundary (boundary-element method) exhibit high efficiency in the case where the corresponding fundamental solutions of the problem under study are available. The presence of anisotropy increases the number of elastic constants in Hooke’s law and, hence, complicates the procedure of construction of the fundamental solutions. However, the fundamental solutions of the two-dimensional problems of elasticity theory are, as a rule, available [5, 6]. The analysis of the stress-strain state for these class of problems was carried out, e.g., in [4, 7, 8]. If we use the ordinary boundary-element method, then the solution of boundary integral equations enables one to find only a part of the components of the stress tensor, namely, the components appearing in the contact conditions. To find the other components of the stress tensor, it is necessary to perform the limit transition both in the integral representations of the vector of displacements and in their differentiated representations. The two-dimensional approximation of displacements on the boundaries of inclusions [1] removes the necessity of differentiation of the representations and makes it possible to determine all components of the vector of displacements and stress tensor on the boundaries of the analyzed bodies already directly from the boundary integral equations. In what follows, we apply this approach to the construction of constitutive relations for the plane problem of elasticity for orthotropic bodies. Consider a thin elastic orthotropic plate with orthotropic inclusions of any shape. The plate suffers the action of muss forces Xi and distributed forces pi applied to its edges. Thus, it can be regarded as an elastic orthotropic matrix with elastic orthotropic inclusions.