Quasi-static simulations are of limited interest because cracks, if they are not severely constrained, 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.

准静态模拟的兴趣有限，因为如果没有严格约束，裂纹会动态传播。当发生地震或爆炸等自然灾害时，由准脆性或脆性材料制成的结构可能会因高速率加载等而引发故障。在这些事件期间经常观察到动态断裂，特别是动态裂纹分支。我们在本文中首次提出了平滑梯度增强损伤模型的动态描述，用于模拟瞬态载荷条件下的准脆性失效局部化。我们引入了两种有效的速率相关损伤定律和各种等效应变公式来分析动态状态的复杂应力状态和惯性效应，增强了所采用方法模拟动态断裂和分支的能力。该研究是使用低阶有限元进行的，并通过我们的数值实验（包括混合模式断裂和动态裂纹分支模拟）检验了所开发方法的优点。