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

点击翻译按钮获取中文摘要

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.