A class of non--associated materials: n--monotone materials

TL;DR

This paper explores n-monotone materials, a class between standard and bipotential models, using Fitzpatrick's functions to characterize and classify materials based on monotonicity properties.

Contribution

It introduces n-monotone materials governed by maximal n-monotone operators and demonstrates how Fitzpatrick's functions can be used to model and classify these materials.

Findings

Fitzpatrick's functions provide a constructive modeling approach.

n-monotonicity is a key criterion for material classification.

The model generalizes standard and bipotential material frameworks.

Abstract

Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. G\'ery de Saxc\'e's Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class of n-monotone materials governed by maximal n-monotone operators and modeled by Fitzpatrick's functions. Revisiting the model of elastic materials initiated by Robert Hooke, and insisting on the linearity, coaxiality and monotonicity properties of the constitutive law, we illustrate that Fitzpatrick's representation of n-monotone operators coming from convex analysis provides a constructive method to discover the best bipotential modeling a n-monotone material. Giving up the symmetry of the linear constitutive laws, we find out that n-monotonicity is a relevant criterion for the materials characterization and classification.