Local mechanical properties of inhomogeneous materials

Full text: Materials having discrete inhomogeneities, e.g. composites, are of increasing interest for the modern science and technology. The peculiarity of such materials is stochastic microstructure, that requires to use the probabilistic approaches for modeling of their properties. There are two main problems in the mechanics of composites. The first one deals with the determination of the macroscopic law of Hook: The effective elastic modules, connecting the tensors of average stresses and strains, are required to be calculated in this case. Another one is the evaluation of local fields of stresses (strains), i.e. the problem of the calculation of tensor operators of their concentration. These operators connect the average (external) stresses (strains) in the material with their internal (local) values. These problems are mutually connected, because in any known approximations the solution of stochastic differential equations (the equilibrium equations for elastic mediums with a microstructure) are required in the both case. The determination of local fields of stresses, which exceed the breaking point of the matrix or inclusions, is required for modeling of local damage of composites. Thus, the dependence of local fields of stresses (strains) on a distance between inclusions should be considered as the important property of composites. In this work only matrix composites are discussed. Assuming, that the composite is macroscopically homogeneous, the nonlinear dependence of average distance between inclusions on their bulk concentration is established. The various kinds of approximations are obtained for the evaluation of local fields of stresses and strains, which are similar to approximations of Voigt, Reuss and other authors for the calculation of effective elastic properties. Application of these results allows to establish the dependence between operators of the concentration of stresses (strains) and the average distance between inclusions. In conclusion, our simulation has shown, that: 1. average distance between inclusions is inverse proportional to a cube root of their bulk concentration; 2. the operators of the concentration of stresses have essentially nonlinear dependence on the average distances between inclusions; 3. for composites, consisting of isotropic components, the average distances, where the perturbations of stresses caused by the inclusion, should disappear, are established. (author)