Symplectic bipotentials

TL;DR

This paper extends a symplectic variational principle to non-associated plasticity and frictional contact by introducing symplectic bipotentials, enabling new variational formulations for complex dissipative systems.

Contribution

It introduces the concept of symplectic bipotentials and generalizes the variational principle to non-associated dissipative laws in dynamical systems.

Findings

Successfully extended the symplectic formalism to non-associated plasticity.

Derived a method to construct symplectic bipotentials from bipotentials.

Applied the framework to Coulomb friction and unilateral contact laws.

Abstract

In a previous paper, we proposed a symplectic version of Brezis-Ekeland-Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non associated plasticity. For this aim, we generalize the concept of bipotential to dynamical systems. The keystone idea is to define a symplectic bipotential. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis-Ekeland-Nayroles principle to non associated dissipative laws. We apply it to the non associated plasticity and to the unilateral contact law with Coulomb's dry friction.