Existence and construction of bipotentials for graphs of multivalued laws

TL;DR

This paper advances bipotential theory in convex analysis by establishing conditions for the existence and methods for constructing bipotentials for nonsmooth mechanics laws, enhancing modeling of dissipative materials.

Contribution

It provides the first comprehensive study on the existence and construction of bipotentials for nonsmooth constitutive laws in mechanics.

Findings

Established existence criteria for bipotentials in nonsmooth laws

Developed methods to construct bipotentials for specific material models

Enhanced the mathematical framework for modeling dissipative materials

Abstract

This is a first paper in convex analysis dedicated to the bipotential theory, based on an extension of Fenchel's inequality. Introduced by the second author, bipotentials lead to a succesful new treatment of the constitutive laws of some dissipative materials: frictional contact, non-associated Drucker-Prager model, or Lemaitre plastic ductile damage law. We solve here the problems of existence and construction of a bipotential for a nonsmooth mechanics constitutive law.