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 l oading. In that case, the elastic moduli obtained are the secant modul i. The paper deals with the practically much more important but more d ifficult case of tangential moduli for incremental deformations of the material during which the cracks grow while remaining critical, or sh orten. Several families of cracks of either uniform or random orientat ion, characterized by the crack density tensor, are considered. To sim plify the solution, the condition of crack criticality, i.e. the equal ity 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. S ayers and Kachanov's approximation of the elastic potential as a tenso r polynomial that is quadratic in the macroscopic stress tensor and li near in the crack density tensor, with coefficients that are general n onlinear functions of the first invariant of the crack density tensor, is used. The values of these coefficients can be obtained by one of t he 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-consiste nt scheme. For a prescribed strain tensor increment, a system of N + 6 linear equations for the increments of the stress tensor and of the c rack 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 qu alitatively in good agreement with the stress-strain curves observed i n the testing of concrete. (C) 1997 Elsevier Science Ltd.