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

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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.