A rate-independent damage model which features two damage variables coupled through a penalty term in the stored energy is considered. Since the energy functional is nonconvex, solutions may be discontinuous in time. This calls for suitable notions of (weak) solutions which allow for jumps. We resort to a vanishing-viscosity approach based on an $L^2(\mathrm{\Omega })-$arclength parametrization, where parametrized solutions arise as a limit of the (reparametrized) graphs of the viscous solutions in the extended state space. This enables us to prove that vanishing-viscosity solutions exist and belong to the class of parametrized solutions. We show that the latter can be characterized in various different ways. These alternative formulations highlight the influence of the viscous effects at the jump points, while at the continuity points, the evolution displays a rate-independent behavior. As it turns out, the behavior of the system at each jump point is described by an ordinary differential equation in Banach space during which the physical time is constant.

中文翻译：

---

速率无关的两场梯度损伤模型的消失粘度解

考虑了一个与速率无关的损伤模型，该模型具有通过存储能量中的惩罚项耦合的两个损伤变量。由于能量泛函是非凸的，解在时间上可能是不连续的。这需要适当的（弱）解决方案概念，以允许跳转。我们采用基于消失粘度的方法${升}^2(\mathrm{\Omega })-$弧长参数化，其中参数化解作为扩展状态空间中粘性解的（重新参数化）图的限制出现。这使我们能够证明粘度消失解存在并且属于参数化解的类别。我们表明后者可以以各种不同的方式表征。这些替代公式突出了跳跃点处粘性效应的影响，而在连续点处，演化显示出与速率无关的行为。事实证明，系统在每个跳跃点的行为由巴拿赫空间中的常微分方程描述，在该方程期间物理时间是常数。