A Class of Non-associated Materials: n-Monotone Materials—Hooke’s Law of Elasticity Revisited

Abstract

Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. Géry de Saxcé's Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class of n-monotone materials governed by maximal nmonotone 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.2000 Mathematics Subject Classification. Primary: 74D10 and 47H05 and Secondary: 47H04.

Construction of a bipotential representing a linear non-associated constitutive law

“…Coaxial laws are non-associated constitutive laws [8,10,11,[13][14][15]. They respect all the principles originally enacted by Robert Hooke to model the behaviour of materials called 'elastic': linearity, monotonicity and coaxiality (neither isotropy, nor existence of a potential are required).…”

“…In this linear case, the monotonicity condition reduces [9][10][11] to the positive definiteness of the symmetric part S of A.…”

“…Under this condition, we say that the constitutive law y = Ax is n-monotone [9] and also that the material is n-monotone [11]. The 2-monotonicity coincides with the classical monotonicity.…”