Variational Inequalities in Critical-State Problems

TL;DR

This paper explores the use of variational inequalities in modeling critical-state problems such as sandpile growth, lake formation, superconductor magnetization, and elastoplasticity, highlighting new dual formulations.

Contribution

It introduces new dual variational formulations for sandpiles and superconductors, clarifying the common structure of these models.

Findings

Derived new dual variational formulations for sandpiles.

Clarified the origin of similarity among different critical-state models.

Outlined main steps in deriving these models.

Abstract

Similar evolutionary variational inequalities appear as convenient formulations for continuous quasistationary models for sandpile growth, formation of a network of lakes and rivers, magnetization of type-II superconductors, and elastoplastic deformations. We outline the main steps of such models derivation and try to clarify the origin of this similarity. New dual variational formulations, analogous to mixed variational inequalities in plasticity, are derived for sandpiles and superconductors.