Novel Approaches for the Multidimensional Convexification of Inelastic Variational Models for Fracture

损伤建模和仿真具有重要的工程意义。在宏观尺度上，损伤表现为应力和应变软化效应，以及裂纹的形成和扩展方面的断裂。对于它的建模，通常使用经典的连续介质损伤模型，其中微观损伤被内部变量唯象地捕捉到。然而，当达到一定程度的微观损伤时，这些模型会遇到相关增量变分公式的凸性损失，这从根本上限制了它们的实用性，特别是在它们的数值评估方面。事实证明，基于凸化的松弛方法在克服这一问题方面非常有效。松弛(凸化)模型保证了与网格无关的解，而且，它们经常描述均匀的微结构，从而允许对损伤现象进行细观力学解释。最近，已有研究表明，即使是应变软化，也就是材料的应力随应变增加而减小的行为，即使在有限应变下也能被这种类型的模型所捕捉，这使得它们也适用于软材料。虽然最近在有效的数值凸化方案方面取得了重大进展，但对三维复杂工程结构的计算到今天还不可行。因此，本研究项目的主要目标之一是开发离线和在线(机器)学习策略，以便能够使用松弛损伤模型进行相关的计算模拟。此外，基于偏微分方程组或多凸化的新的凸化方法，而不是逼近一阶凸包络，是三维松弛模型的有希望的替代方案，这些模型将被开发以提高效率。对于更复杂的力学问题，包括宏观裂纹扩展意义上的脆性和延性断裂，这些方法预期的速度将是必要的，其中仅学习策略可能会变得更昂贵。因此，本研究项目的最终主要目标是扩展增量变分公式，以包括脆性和延性断裂相关问题的塑性效应和损伤。

Damage modelling and simulation is of fundamental engineering interest. At the macroscale, damage manifests itself through stress- and strain-softening effects as well as fracture in terms of the formation and propagation of cracks. For its modelling, classical continuum damage models are usually applied, where the microscopic damage is phenomenologically captured by internal variables. However, when reaching certain degrees of microscopic damage, these models encounter a loss of convexity of the associated incremental variational formulation, which limits their usefulness fundamentally, in particular with respect to their numerical evaluation. Relaxation approaches based on convexification have proven very powerful in overcoming this problem. Relaxed (convexified) models guarantee mesh-independent solutions and, moreover, they often describe homogenized microstructures, thus allowing a micro-mechanical interpretation of the damage phenomena. Recently, it has been shown that even strain-softening, i.e., material behavior showing decreasing stresses with increasing strains, can be captured by such type of models, even at finite strains, which makes them also appropriate for soft materials. Although significant steps forward with respect to efficient numerical convexification schemes have recently been made, computations for complex engineering structures in three dimensions are infeasible as of today. Hence, one of the main goals in this research project is to exploit offline- and online (machine) learning strategies to enable relevant computational simulations using relaxed damage models. Moreover, novel convexification approaches based on PDE formulations or polyconvexification rather than approximating the rank-one convex envelope constitute promising alternatives for relaxed models in three dimensions, which are to be developed with improved efficiency. The speed-up to be expected from these approaches will be necessary for more complex mechanical problems including brittle and ductile fracture in the sense of macroscopic crack propagation, where learning strategies alone may become more expensive. Therefore, the final major aim of this research project is to extend the incremental variational formulations to capture plastic effects combined with damage for brittle and ductile fracture-related problems.