An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

TL;DR

This paper investigates an abstract inf-sup problem related to limit analysis in plasticity, establishing conditions for no gap between inf-sup and sup-inf, and applying results to continuum mechanics problems.

Contribution

It introduces a generalized inf-sup condition for convex cones, extends the Babuska-Brezzi condition, and develops a regularization method with a computable majorant for numerical analysis.

Findings

Established conditions for inf-sup and sup-inf equivalence.

Developed a regularization method for solving the inf-sup problem.

Applied the abstract results to limit load problems in plasticity.

Abstract

This work 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 Babuska-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.