Explicit bounds on elastic moduli of solids containing isotropic mixture of cracks and voids

A variational formulation has been recently proposed by the authors (Xu X.F. and Stefanou, G. (2011). Int J Multiscale Comput Engng 9(3) pp. 347363) to evaluate effective elastic moduli of randomly cracked solids. Using this formulation, explicit expressions have been obtained for the upper bounds of the elastic moduli in the case of penny-shaped and slit-like random cracks with parallel and random orientations subject to a spherical exclusion constraint. In this paper, it is shown that the bounds derived previously are directly applicable to solids containing isotropic mixture of cracks with no exclusion constraint, which is a more general case for a cracked solid. The bounds are further extended to solids containing spheroidal voids that are characterized with finite value of aspect ratios.