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

断裂非弹性变分模型多维凸化的新方法

• 批准号: 441154176
• 负责人: Professor Dr.-Ing. Daniel Balzani
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441154176?language=en
• 关键词: Novel; Approaches; Multidimensional; Convexification; Inelastic

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.

损伤建模与仿真是一项基础工程研究。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。对于其建模，通常采用经典的连续损伤模型，其中微观损伤由内部变量在现象学上捕获。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，特别是在数值评估方面。基于凸化的松弛方法已被证明在克服这个问题方面非常有效。松弛（凸化）模型保证了网格无关的解决方案，而且，它们通常描述均匀的微观结构，从而允许对损伤现象进行微观力学解释。最近，有研究表明，即使在有限应变下，这种类型的模型也可以捕获均匀应变软化，即材料表现出随应变增加而减小的应力行为，这使得它们也适用于软材料。尽管最近在有效的数值凸化方案方面取得了重大进展，但到目前为止，复杂工程结构的三维计算是不可行的。因此，本研究项目的主要目标之一是利用离线和在线（机器）学习策略来实现使用松弛损伤模型的相关计算模拟。此外，基于PDE公式或多凸化的新颖凸化方法，而不是近似1阶凸包络，为三维松弛模型提供了有希望的替代方法，这些方法有待于提高效率。对于更复杂的机械问题，包括宏观裂纹扩展意义上的脆性和延性断裂，这些方法所期望的加速将是必要的，在这些问题中，单独学习策略可能会变得更加昂贵。因此，本研究项目的最终主要目标是扩展增量变分公式，以捕获脆性和延性断裂相关问题的塑性效应和损伤。