
Journal of Convex Analysis 15 (2008), No. 1, 087104 Copyright Heldermann Verlag 2008 Existence and Construction of Bipotentials for Graphs of Multivalued Laws Marius Buliga Inst. of Mathematics, Romanian Academy, P.O. Box 1764, 014700 Bucharest, Romania Marius.Buliga@imar.ro Géry de Saxcé Lab. de Mécanique, UMR CNRS 8107, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d'Ascq, France gery.desaxce@univlille1.fr Claude Vallée Lab. de Mécanique des Solides, UMR 6610  UFR SFASP2MI, Bvd. M. et P. Curie, Téléport 2  BP 30179, 86962 FuturoscopeChasseneuil, France vallee@lms.univpoitiers.fr [Abstractpdf] Based on an extension of Fenchel inequality, 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).\par Let $X$, $Y$ be dual locally convex spaces, with duality product $\langle \cdot , \cdot \rangle: X \times Y \rightarrow \mathbb{R}$. Given the graph $\displaystyle M \subset X\times Y$ of a multivalued law $\displaystyle T \colon X\rightarrow 2^{Y}$, 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 $b(x, y) = \langle x, y\rangle$.\par If this condition is fulfilled, we use convex lagrangian covers in order 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 its graph is cyclically monotone. [ Fulltextpdf (158 KB)] for subscribers only. 