Stress-based estimates and bounds of effective elastic properties: The case of cracked media with unilateral effects

For direct insight into the stress concentration rule, stress-based boundary conditions are most appropriate. In this framework, the dilute and Mori–Tanaka estimates are shown to be identical for frictionless closed cracks. This extends a result already known for of opened cracks. The link with strain-based homogenization is also provided. In order to take into account the effects of the spatial distribution of cracks, a stress-based lower bound of the effective compliance dual of the strain-based upper bound of the stiffness proposed in [9] is derived. Illustrations for parallel closed cracks are presented.