Effective Elastic Coefficients of an Inhomogeneous Solids

Аннотация

The first special boundary value problem (SBVP) of the theory of elasticity for an inhomogeneous body is considered. The effective elasticity coefficients are found from the solution of the SBVP. They form a fourth-rank tensor, namely the tensor of effective elasticity moduli that makes it possible to express the volume average stresses via the mean deformations. It is shown that the solution of the first SBVP and hence the effective coefficients of elasticity are expressed in terms of the integrals of the Green tensor. The integrals of the Green tensor with respect to one of the variables are called the structural functions. The auxiliary equations, the solutions of which are determined by the functional dependence of the elastic characteristics on the coordinates, have been obtained for these structural functions. It is shown that in the case when the elastic moduli are periodic functions of one, two, or three coordinates, then the structural functions, far from the body boundary, are also periodic functions of the same coordinates. The structural functions are transformed to be equal to zero on the all body border approaching the boundary. In other words, in an inhomogeneous body with a periodic structure, it is possible to distinguish the boundary layer, which separates the regions of periodic values of structural functions from non-periodic ones. The thickness of this layer is of the order of the characteristic size of the periodicity cell. The effective tensors are found due to the structural functions. It is proved that the tensor of effective elastic moduli satisfies all the conditions of symmetry and positive definiteness. The case of an infinite plate with non-uniform thickness is considered in detail.