Narrow Results

This paper revolves around a newly introduced weak solvability concept for rate-independent systems, alternative to the notions of Energetic ($\mathrm{\text{E}}$) and Balanced Viscosity ($\mathrm{\text{BV}}$) solutions. Visco-Energetic ($\mathrm{\text{VE}}$) solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for $\mathrm{\text{E}}$ solutions, but perturbed by a “viscous” correction term, as in the case of $\mathrm{\text{BV}}$ solutions. However, for VE solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion turns out to describe a kind of evolution in between Energetic and BV evolution. In this paper we aim to investigate the application of $\mathrm{\text{VE}}$ solutions to nonsmooth rate-independent processes in solid mechanics such as damage and plasticity at finite strains. We also address the limit passage, in the $\mathrm{\text{VE}}$ formulation, from an adhesive contact to a brittle delamination system. The analysis of these applications reveals the wide applicability of this solution concept, in particular to processes for which $\mathrm{\text{BV}}$ solutions are not available, and confirms its intermediate character between the $\mathrm{\text{E}}$ and $\mathrm{\text{BV}}$ notions.

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.