On coupling anisotropic damage and internal friction in modeling brittle materials

An internal friction-damage coupled model is derived from a micromechanically based description of the response of brittle materials. The material is modeled as an elastic isotropic matrix containing a statistically uniform distribution of crack-like defects. Under the simplifying assumption of non-interacting and self-similar propagating flat cracks having the same shape and random locations and orientations, displacement discontinuities due to microcrack opening and sliding are treated as inelastic contributions to the mean strain. The model is based on two tensor-valued internal variables, representing damage and frictional contact tractions and governing the microscopic mechanisms responsible for the global inelastic response. The use of a tensor-valued variable for damage, in particular, makes the model to be capable of describing the load-induced anisotropy of damage in brittle materials. In the framework of thermodynamics with internal variables, overall frictional sliding and crack growth criteria with associated flow rules are introduced to complement the model that should be formulated in incremental form, because of nonlinearity and stress-path-dependence of the constitutive response to arbitrary stress applied to a brittle material at an arbitrary current state. As an example, the model response to proportional loading has been analysed. In this case no crack changes its status and explicit solutions are possible. On the basis of these results, biaxial and triaxial failure envelopes, together with some characteristic stress-strain curves, have been obtained and used for both identification and validation of the model. It is straightforward that, because of the assumption of statistically uniformly distributed damage as an ensemble of non-interacting microcracks, the model is inherently incapable of predicting the onset of macroscopic failure, occurring as damage localization into bands and coalescence of microdefects into a macrocrack. Practical application of the model is then restricted to the hardening phase of the mechanical response of brittle materials. Furthermore, due to the self-similar crack growth, the characteristic feature of dilatancy cannot be captured. On the other hand, in this model effects of anisotropic damage and frictional sliding for prescribed (even complex) loading are combined in a relatively simple formulation which takes the physics of the problem into account.