Variationally consistent microstructure evolution and microsphere-based reconvexification for damage with application to arterial tissues

Modeling the softening of arterial tissue, as it can be observed in terms of strain softening preceding atherosclerotic plaque rupture or in terms of stress-softening during balloon angioplasty, places special demands on the material formulations employed. In the context of finite-element discretized boundary value problems, the softening of the material is typically associated with a non-convex strain energy, which leads to an ill-posed behavior, i.e., mesh-dependency. In this work, we discuss the applicability of a material model for strain-softening that is based on the repeated construction of the convex envelope of an initially non-convex strain energy. The model possesses an evolving property that is exploited for the description of material softening. We propose an accurate and efficient convexification method, which is based on the reformulation of the convexity requirement in terms of a new optimization problem. Furthermore, we adopt the formulation to model softening in arterial tissue by combining a microsphere approach with a new orientation distribution function representing the dispersion of collagen fibers and show its capability to represent experimental data. Finally, we provide the derivation of a variationally sound approach to damage microstructure evolution and analyze some unique aspects significantly limiting the applicability to strain-softening. • An accurate, fast and robust one-dimensional convexification approach is proposed. • A simple enhancement of the reconvexification approach is introduced, which significantly reduces load-step sensitivity. • A new probability distribution function for the modeling of fiber dispersion in soft collagenous tissues is suggested. • The resulting microsphere model enables an accurate representation of experimental data of aortic tissue. • A variationally consistent microstructure evolution based on the Wasserstein-1 distance is derived for an incremental variational damage formulation.