Adaptive convexification of microsphere-based incremental damage for stress and strain softening at finite strains

Abstract

The brief history of relaxation in continuum mechanics ranges from early application of non-convex plasticity and phase transition formulations to small and large strain continuum damage mechanics. However, relaxed continuum damage mechanics formulations are still limited in the following sense that their material response lack to model strain softening and the convexification of the non-convex incremental stress potential is computationally costly. This paper presents a reduced model for relaxed continuum damage mechanics at finite strains which includes strain softening by a fiber-specific damage in the microsphere approach. Computational efficiency is achieved by novel adaptive algorithms for the fast convexification of the one-dimensional fiber material model. The algorithms are benchmarked against state-of-the-art methods, and the choice of quadrature schemes for the microsphere approach is discussed. This contribution is finalized by a mesh independence test.

Relaxed Incremental Formulations for Damage at Finite Strains Including Strain Softening

“…This optimization problem can be efficiently solved by the discrete convexification scheme reported in [16] with linear complexity in the number of discretization points. The multi-dimensional setting is more sophisticated, since the chosen (semi)convex notion drastically influences the complexity of the construction of the associated hull.…”

“…This issue can be circumvented by methods which utilize the fact that rank-one convexity corresponds to convexity along rank-one lines. Across rank-one lines, performant one-dimensional relaxation methods can be used as presented in [16]. This idea originates from [17] and was extended in [18].…”

“…The schemes of [10,13] kept the convex hull fixed after loss of convexity because the evolution of the internal variable in the steps afterwards is simply unknown. In [19], the concept of reconvexification was introduced due to the possibility to construct the convex hull in each incremental step by the methods of [16]. Within the publication, the convex hull is constructed in each incremental step and the strongly damaged phase, associated with the deformation gradient F + , is allowed to evolve.…”

“…However, the emulation of the microstructures requires assumptions and the obtained relaxed energy is a convex envelope which neglects compatibility. Due to the discrete convexification scheme of [16], the construction of a one-dimensional convex envelope is significantly accelerated. This can be exploited in the one-and multidimensional relaxation setting.…”