Effective defining relations of inelastic composites

In this paper we consider the first special boundary value problem in the mechanics of inhomogeneous deformable solids, when the defining relations connecting the stress tensor with the strain tensor are a nonlinear operator from the strain tensor. The type of the defining operator in an inhomogeneous body depends on at which point the stresses are determined. At the boundary of the body, at each boundary point, the displacements are defined as a convolution of an arbitrary constant symmetric tensor of rank 2 with the coordinates of this point. In our study, it is assumed that the deformations, arising in the body from such a boundary action are small. As a consequence, the average value of the strain tensor in the body coincides with the constant tensor defined at the boundary, independently of type of the defining relations. The displacement of a point inside the body is represented as a sum of two terms. The first term is the convolution of the boundary tensor with the point coordinates, and the second term is an unknown vector function (structural function) that depends on the coordinates of the point and the boundary tensor. This function is zero at the boundary of the body. A nonlinear operator differential equation is obtained for the structural function in the general case. To solve this equation, the method of successive approximations is applied and approximate expressions for the structural functions and, through them, the strains and stresses at each point of the body are found. Stresses are then averaged over the body volume and compared with average strains, i.e., the type of effective defining relations expressing average stresses through average strains is determined. The case of an inhomogeneous in thickness, infinite in plan, plate is considered in detail.