On the Influence of Initial Stresses on the Velocity of Elastic Waves in Composites

Abstract

The paper is devoted to the problem of propagation of elastic waves in composites with initial stresses. We suppose initial stresses are well within the elastic regime. We deal with the long-wave case and use the asymptotic homogenization technique based on the two-scale asymptotic approach. The main problem lies in solving the local (cell) problem, i.e., boundary value problem on a periodically repeating fragment of a composite. In general, the local problem cannot be solved explicitly. In our work, it is obtained for any initial stresses formulas, which is convenient for solving by standard codes. An analytical solution is obtained for small initial stresses. Asymptotic expansions used a small parameter characterizing the smallness of the initial stresses. In the zero approximation, composites without initial stresses are considered; the first approximation takes into account their influence on waves propagation. Two particular cases are considered in detail: laminated media and frame (honeycomb cell) composites. The analyzed frame composite can be used for the modeling of porous media. We select these two cases for the following reasons. First, the laminated and porous material are widely used in practice. Second, for these materials, the homogenized coefficients may be computed in the explicit form for an arbitrary value of the initial stresses. The dependence of the velocity of elastic waves on the initial stresses in laminated and homogeneous bodies differs. The initial tension increases the velocity of elastic waves in both cases, but the quantitative effect of the increase can vary greatly. For frame composites modeling porous bodies, the initial tension can increase or decrease the velocity of elastic waves (the initial tension decreases the velocity of elastic waves in the porous body with an inverted honeycomb periodicity cell). The decrease of the velocity of elastic waves is impossible in homogeneous media. The problem under consideration is related, in particular, to the core sample analysis in the geophysics. This question is discussed in the paper. We also analyzed some features of applications of asymptotic homogenization procedure for the dynamical problem of stressed composite materials, i.e., the nonadditivity of homogenization of sum of operators.