Overall moduli and constitutive relations of bodies containing multiple bridged microcracks

The overall moduli and non-linear constitutive relations of solids containing randomly oriented and parallel multiple bridged microcracks are studied in this paper. For both linear and non-linear bridging laws, two non-dimensional mixed parameters which play an important role in determining the overall moduli and constitutive properties are identified. It is shown that the overall moduli of a body containing bridged cracks can be derived from those of the body containing unbridged cracks of the same configuration with the crack density parameter being discounted by these two non-dimensional parameters. For non-linear bridging, attention is paid to the power-law bridging which represents the existing bridging laws in the literature. Particular emphasis is placed on the widely used square-root type bridging law. In this case, closed-form expressions for the overall non-linear constitutive relations are obtained. For a general power-law bridging, it is found that the overall constitutive behaviour of the cracked body is governed by Ramberg–Osgood type relations under strong bridging.