Modeling of structures in the mechanics of composites

Abstract

One possible general statement of a quasi-static problem in the mechanics of composites is considered. It is assumed that a composite is characterized not only by the heterogeneity of a regular structure, but also by the presence of imperfections, impurities, cracks, and the roughness of surfaces, which are partly taken into account by introducing appropriate couple stresses. Two statements, “in displacements” and “in stresses,” are considered together with the statement of the same problems in the case where the constitutive relations are linear integral operators. The boundary-value problem remains nonlinear due to the nonlinearity of a scattering function which enters into the heat equation. The theory of effective moduli for a nonpolar medium is discussed in more detail. The equilibrium equations for a homogeneous medium with reduced characteristics and the equation of heat inflow, introduced in nonlinear (in an explicit form) and linear variants, are examined. For a simple laminated composite, all effective mechanical and thermophysical characteristics are found in an explicit form. The effective material functions for a transversely isotropic medium are constructed on the basis of a unique dimensionless relaxation kernel with the use of several Il'ushin kernels. Based on the known solution of the boundary-value problem for the reduced medium, the stress and strain concentration tensors, at any point of a simple laminated composite, are also constructed in an explicit form. In this case, the changes in the structure are taken into account.