A UNIFIED INVARIANT DESCRIPTION OF MICROMECHANICALLY-BASED EFFECTIVE ELASTIC PROPERTIES FOR 2-DIMENSIONAL DAMAGED SOLIDS

This paper is devoted to an unified invariant description of the effec tive elastic compliance, say S, of a two-dimensional (2D) damaged soli d with an isotropic matrix. A macro- and micromechanically combined ap proach is proposed, which consists of three major steps. First, the in ternal variables, i.e., the damage tensors are correlated with the mic roscopical geometry and distribution of the damage. It is shown that t he damage due to microcracks and holes affect S through a symmetric se cond-order tensor, d, and an irreducible fourth-order tensor, D. Secon d, the general formulation of the effective compliance S as an isotrop ic tensor function of d and D is phenomenologically given by making us e of several results in the theory of censor function representation. The linearization and second-rank nonlinearization from the general fo rmulation are derived, where one and six new parameters (named damage constants), respectively, are consistently introduced. It is shown tha t the linearization provides the most general invariant formulation fo r micromechanically-based non-interacting or dilute estimations of the effective compliance, and nonlinear models are accounted for the inte racting effect of damage. Third, we relate these damage constants to s hape and morphology factors of the damage via micromechanical analyses or computer numerical experiments for problems with simple and regula r damage. The above-mentioned approach allows invariant consistent mod els of effective elastic compliance which are not only capable of desc ribing complex and irregular damage, but also indicate in detail the r ole of the real damage. Thus, these invariant models may further const itute a solid basis of predicating damage evolution. As an application of the proposed approach, the most general second-rank interacting mo del for damage due to microcracks is established, which is available f or any distribution of microcracks.