A directional microcrack damage theory for brittle solids based on continuous hyperplasticity

We present a novel microcrack-damage theory for brittle solids under compression. Instead of using internal variables like zeroth, second or fourth rank damage tensors, the state of material damage is represented by an internal function that encapsulates the information of direction, density and size of microcracks. Just like other internal variables, the evolution of this state function must obey the second law of thermodynamics for arbitrary loading paths. This is done by casting the model in the framework of continuous hyperplasticity and enforcing a non-negative dissipation rate functional. The proposed framework offers predictions on the continuous evolution of microcrack density and the induced material anisotropy along with the macroscopic stress-strain curves. The use of continuous damage function grants the model significantly enhanced resolution in characterizing the direction-dependent response of cracked solids compared to classical models that are based on damage tensors. Two scenarios are considered in developing the theory, one assumes frictionless cracks and the other incorporates friction between crack surfaces. The results highlight that inelasticity, pressure dependence, and loading-unloading hysteresis exhibited by brittle solids are natural consequences of frictional microcracks. The proposed theory offers a generic and versatile framework to upscale micromechanical processes operating at individual crack scale to explain the macroscopic behavior of cracked solids.