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

在第一个资助期，该项目的目标是通过分析/数学和建模/物理之间的强有力合作，为固体中与速率无关的损伤演变建立一个物理和数学上合理的总体框架。在第二个资助期内，这一框架将扩展到多物理问题--同样是通过数学和力学之间的强有力的合作。更具体地说，在第一个资助期内阐述的与费率无关的原型损害模型将与其他模型相结合，这些模型要么也是费率无关的(RI)，要么是费率依赖的(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)和状态相关的。以韧性损伤的损伤/断裂与塑性的耦合作为RI-RI型的原型模型，对RI-RD体系的氢辅助断裂进行了损伤与扩散耦合模型的分析。预计力学和数学之间的跨学科合作所取得的结果，特别是物理实验与具有数学保证的收敛性质的数值方案的预测的比较，将使人们对从物理角度看哪种解的概念(例如能量解或BV解)以及哪种离散化策略最合适的争论有新的见解。拟议的项目与DFG优先方案的主题完美匹配。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依赖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都扮演着至关重要的角色。更准确地说，就建模而言，能量最小化是压倒一切的原则，它打开了将变分分析应用于进化问题的可能性。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，损伤传播可解释为体能和表面能之间的竞争)和研究方向(C)“结构和演化的耦合”(例如，全局能量最小化与局部能量最小化)在第一个资助期中得到了解决，但现在的重点是研究方向(B)“过程的耦合”，例如，在不同的时间尺度上起作用。

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.