Solids with cracks and non-spherical pores : proper parameters of defect density and effective elastic properties

Abstract

The work is motivated by inadequacy of the conventional defect density parameters, such as porosity (relative volume of pores) or the usual crack density in situations that are frequently encountered in applications: of non-spherical pores of diverse shapes, fluid-filled cracks/pores, pores in an anisotropic matrix. We call a defect density parameter proper if it correctly reflects the individual defect contributions into the effective elastic properties. Only in terms of such parameters can these properties be uniquely expressed. Their identification is non-trivial even in the framework of the non-interaction approximation; defect interactions further complicate the problem. We show that the proper parameters are identified by the structure of the elastic potential. Besides being necessary, the proper parameters yield the following benefits: (1) anisotropy due to non-randomly oriented defects is established; (2) expressions for the effective moduli cover, in a unified way, all mixtures of defects of diverse shapes and arbitrary orientational distributions; (3) they provide guidance for the proper interpretation of experimental data on elasticity of porous materials. For certain types of defects (field of pores of complex, but identical shapes, for example), the general results in terms of tensorial parameters reduce, for each particular orientational distribution, to expressions in terms of the conventional parameters. However, in other situations (non-spherical pores of diverse shapes, fluid-filled cracks/pores, pores in an anisotropic matrix) such a reduction cannot, generally, be done, even for a particular orientational distribution.