Rate-independent systems in solid mechanics and their coupling with other dissipative systems

在第一个资助期内，该项目的目标是推导出一个物理和数学上合理的一般框架，通过分析/数学和建模/物理之间的密切合作，对固体中与速率无关的损伤演化进行建模。在第二个资助期内，这一框架将扩展到多物理问题-再次通过数学和力学之间的密切合作。更具体地说，在第一个资助期内制定的不依赖于费率的原型损坏模型将与其他模型相结合，这些模型也是不依赖于费率的（RI）或依赖于费率的（RD）。从一般的数学观点来看，所采用的本构框架是基于能量泛函和耗散泛函的引入。然而，与第一个资助期形成鲜明对比的是，由于不同系统的耦合，所涉及的耗散泛函现在可能是超线性的（RD）和状态依赖的。虽然从韧性损伤已知的损伤/断裂和塑性的耦合被选为RI-RI型的原型模型，氢辅助开裂损伤耦合扩散建模分析RI-RD-系统。预计从力学和数学之间的跨学科合作，特别是，物理实验的比较，数值方案的预测具有数学保证的收敛性能将得到新的见解的辩论，以解决方案的概念（例如充满活力的解决方案或BV解决方案）和离散化策略是最合适的从物理的角度来看。拟议的项目完全符合DFG优先计划的主题。最重要的是，它代表了数学和工程专家的共同努力，以进一步推进与速率无关或速率相关系统耦合的速率无关系统的建模。同样重要的是，变分方法在这两个领域中发挥着至关重要的作用。更准确地说，能量最小化是压倒一切的原则，就建模而言，它开辟了应用变分分析的演化问题的可能性。该项目符合所有三个主要研究方向。虽然研究方向（A）“维度的耦合”（例如，损伤传播可以解释为体能量和表面能量之间的竞争）和研究方向（C）“结构和演化的耦合”（例如，在第一个供资期内，研究方向（B）是“过程的耦合”，例如，在不同的时间尺度上发挥作用。

During the first funding period, the goal of this project was the derivation of a physically and mathematically sound general framework for the modeling of rate-independent damage evolution in solids by a strong collaboration between analysis/mathematics and modeling/physics. Within the second funding period, this framework is to be extended to multi-physics problems - again by a strong collaboration between mathematics and mechanics. More specifically, the rate-independent prototype damage model elaborated within the first funding period will be coupled with additional models which are either also rate-independent (RI) or rate-dependent (RD). From a general mathematical point of view, the employed constitutive framework is based on the introduction of an energy functional and a dissipation functional. However, and in sharp contrast to the first funding period, the involved dissipation functionals might now be super-linear (RD) and state-dependent due the coupling of different systems. While the coupling of damage/fracture and plasticity as known from ductile damage is chosen as a prototype model of type RI-RI, hydrogen-assisted cracking modeled by damage coupled to diffusion is analyzed for RI-RD-systems. It is expected that the results obtained from the interdisciplinary cooperation between Mechanics and Mathematics and, in particular, the comparison of physical experiments to predictions of numerical schemes having mathematically guaranteed convergence properties will give new insights into the debate as to which solution concept (e.g. energetic solutions or BV solutions) and which discretization strategy is most appropriate from the physical point of view. The proposed project perfectly matches the topics of the DFG priority program. Most important, it represents a concerted effort by experts in both mathematics and engineering in order to further advance the modeling of rate-independent systems coupled to rate-independent or rate-dependent systems. Equally important, variational methods play a crucial role within both fields. To be more precise, energy minimization is the overriding principle as far as the modeling is concerned and it opens up the possibility of applying variational analysis to evolution problems. The project fits into all three major research directions. While research direction (A) "Coupling of dimensions" (e.g., damage propagation can be interpreted as a competition between bulk and surface energies) and research direction (C) "Coupling of structure and evolution" (e.g., global versus local energy minimization) were addressed in the first funding period, emphasis is now on research direction (B) "Coupling of processes" acting, for instance, on different time scales.