Existence and construction of bipotentials for multivalued constitutive laws

Bipotentials are non smooth mechanics tools, used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Given by a graph M representing a multivalued constitutive law, we state a simple necessary and sufficient condition for the existence of a bipotential b for which M is the set of (x, y) such that b(x, y) = 〈x, y〉. If this condition is fulfilled, we give a method to construct such a bipotential, generalizing a theorem due to Rockafellar, which states that a multivalued constitutive law admits a superpotential if and only if it’s graph is cyclically monotone. 1 Bipotentials and multivalued constitutive laws The bipotential theory, based on an extension of Fenchel’s inequality, leads to a succesful new writing of the constitutive laws of some dissipative materials: frictional contact [5], non-associated Drucker-Prager model [1], or Lemaitre plastic ductile damage law [2]. In all these papers, bipotentials for certain multivalued constitutive laws are constructed. Nevertheless, it was not previously known how general is the bipotential approach, namely: what are the conditions to be satisfied by a constitutive law in order that it can be expressed with the help of a bipotential. 2 Notations and definitions X and Y are topological, locally convex, real vector spaces of dual variables x ∈ X and y ∈ Y , with the duality product (x, y) 7→ 〈x, y〉. We shall suppose that X,Y have topologies compatible with the duality product, that is: any continuous linear functionals on X (resp. Y ) has the form x 7→ 〈x, y〉, y ∈ Y (resp. y 7→ 〈x, y〉, x ∈ X). ”Simion Stoilow” Institute of Mathematics of the Romanian Academy, PO BOX 1-764,014700 Bucharest, Romania, e-mail: [email protected] Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, Cité Scientique, F-59655 Villeneuve d’Ascq cedex, France, e-mail: [email protected]