An equilibrium bifurcation approach to the compressive failure of microcracked elastic solids

In the framework of micromechanical approaches to modeling brittle materials subjected to compression, damage is postulated both to have a uniform distribution and to emanate from inherent microcracks embedded in an elastic matrix. Several models have been developed for crack growth based on the solution for a single sliding crack in an infinite sheet. When uniaxial compression is considered, such models give proper information about either the dilatancy or the damage evolution process and determine a stable microcrack growth. It follows that a limit value of compression cannot be obtained to be assumed as the compressive strength of the material. On the other hand, taking into account the microcrack interaction leads to a limiting compressive stress for microcrack stable propagation, which cannot be easily interpreted from a physical point of view. As shown by one of the authors in a previous paper, where a discrete model for fracture evolution is proposed, even though the damage evolution results in the initiation of macrocracks parallel to the loading directions, equilibrated configurations can be always found. It follows that compression failure can be interpreted as a more complex phenomenon than the simple consequence of damage evolution. In this paper damage evolution is considered to decrease the elastic solid stiffness until a critical condition of equilibrium bifurcation is reached. The related value of the stress is defined as the compressive strength. The problem of instability of a rectangular elastic orthotropic solid is analysed, where a perturbation technique is used on the non linear differential equations with the related boundary conditions, referred to the initial state before any deformation occurs. Analytical periodic solutions for the perturbed displacement field are assumed, from which the critical stress can be obtained related to the current damaged configuration. Basing on the micromechanical damage constitutive model developed by the first author, incremental elastic moduli are calculated for the brittle orthotropic solid to be used in the instability analysis. The damage evolution and the current configuration are obtained by means of an equivalent crack model; either crack propagation or monolateral contact forces are not taken into account for the perturbed configuration. With reference to concrete-like materials, the proposed approach can properly be used to simulate the failure process. Interesting results are driven by the made assumptions. First of all, realistic values of the compressive strength are related to extensive damage patterns. Furthermore, the calculated compressive strength turns out not to depend on the specimen shape ratio and to asympthotically approach the tangential elastic modulus. The theoretical compressive strength seems to be a material property even though it has been obtained by means of an instability analysis. A more detailed analysis of the effect of both imperfections and the hypothesis of uniformly distributed damage on the compressive failure behaviour is considered.