Effective Elastic Compliances and Engineering Constants for Damaged Isotropic Solids

In continuum damage mechanics, damaged solids have been represented by the effective elastic stiffness into which local damages are smoothly smeared. In parallel to the effective elastic stiffness representation, damaged solids may be represented in terms of effective elastic compliance. It then becomes easier to derive the effective engineering constants (i.e., effective elastic moduli and Poisson's ratios) for damaged solids, all in closed forms, from the effective elastic compliance rather than from the effective elastic stiffness. Thus, in this paper, by using a continuum modeling approach previously developed by the author based on both the principle of strain energy equivalence and the equivalent elliptical micro-crack representation of a local damage, the effective elastic compliance and effective engineering constants are derived in closed forms for both damaged two- and three-dimensional isotropic solids. They are derived in terms of the undamaged (virgin) elastic properties and a scalar damage variable.