Modelling of quasi-brittle behaviour: a discrete approach coupling anisotropic damage growth and frictional sliding

The paper provides development of the model of anisotropic damage by microcracking proposed by Bargellini et al. 2006. This model is based on a discrete approach, which introduces a finite set of microcrack densities associated with fixed directions. This approach avoids inconveniences encountered when using a single second order tensor damage variable D (non uniqueness of the free energy) and strain decomposition into positive and negative parts (spurious dissipation at crack closure). Frictional sliding on closed microcracks is introduced as an additional dissipative mechanism; it is represented by a second order sliding variable in each damage direction. Corresponding sliding criteria and non-associated sliding evolution laws, formulated in the strain space for the model coherence, permit to account for hysteretic phenomena. Unilateral effect is taken into account; Young's and shear moduli are correctly restored at microcrack closure. The crucial requirements of continuity of the energy and of stress–strain response are ensured through relevant conditions on parameters and sliding variables values at opening-closure. The discrete approach, associated with some hypotheses concerning damage evolution, permits to couple damage and dissipative sliding. The pertinence of the proposed theory is illustrated by simulating first elastic properties at constant damage, then by considering a specific loading path involving both damage and friction evolutions.