This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively $ 1 $-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

中文翻译：

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粘弹性材料动态相场断裂的离散近似

该贡献涉及对具有动态效应的粘弹性材料中的相场断裂模型的分析。损伤的演化以两种不同的方式处理：作为具有二次耗散势的粘性演化和作为具有正 $1$-均质耗散势的速率无关定律。这两个进化定律都编码了一个非平滑约束，以确保损坏的单向性，从而使材料无法愈合。在这两种设置中都引入了合适的解决方案概念。解的存在性是使用空间和时间上的离散近似方案获得的。基于能量泛函的凸性特性和位移的规律性，由于它们的粘性演化，还获得了内部变量关于时间的改进规律性结果：