A dynamic description of the smoothing gradient damage model for quasi-brittle failure

Quasi-static simulations are of limited interest because cracks, if they are not severely con-strained, propagate dynamically. When natural disasters such as earthquakes or explosions happen, structures made of quasi-brittle or brittle materials can suffer from failures activated by, for instance, loading at a high rate. Dynamic fractures, especially dynamic crack branching, are often observed during those events. We present in this paper, for the first time, a dynamic description of the smoothing gradient-enhanced damage model towards the simulation of quasi-brittle failure localization under time-dependent loading conditions. We introduce two efficient rate-dependent damage laws and various equivalent strain formulations to analyze the complicated stress states and inertia effects of dynamic regime, enhancing the capability of the adopted approach in modeling dynamic fracture and branching. The study is carried out using low-order finite elements, and the merits of the developed approach are examined through our numerical experiments, including mixed-mode fracture and dynamic crack branching simulations.