Tangential stiffness of elastic materials with systems of growing or closing cracks

Although much has been learned about the elastic properties of solids with cracks, virtually all the work has been confined to the case when the cracks are stationary, that is, neither grow nor shorten during loading. In that case, the elastic moduli obtained are the secant moduli. The paper deals with the practically much more important but more difficult case of tangential moduli for incremental deformations of the material during which the cracks grow while remaining critical, or shorten. Several families of cracks of either uniform or random orientation, characterized by the crack density tensor, are considered. To simplify the solution, the condition of crack criticality, i.e. the equality of the energy release rate to the energy dissipation rate based on the fracture energy of the material, is imposed only globally for all the cracks in each family, rather than individually for each crack. Sayers and Kachanov's approximation of the elastic potential as a tensor polynomial that is quadratic in the macroscopic stress tensor and linear in the crack density tensor, with coefficients that are general nonlinear functions of the first invariant of the crack density tensor, is used. The values of these coefficients can be obtained by one of the well-known schemes for elastic moduli of composite materials, among which the differential scheme is found to give more realistic results for post-peak strain softening of the material than the self-consistent scheme. For a prescribed strain tensor increment, a system of N+6 linear equations for the increments of the stress tensor and of the crack size for each of N crack families is derived. Iterations of each loading step are needed to determine whether the cracks in each family grow, shorten, or remain stationary. The computational results are qualitatively in good agreement with the stress-strain curves observed in the testing of concrete.