Abstract

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). Let XX, YY be dual locally convex spaces, with duality product ⟨⋅,⋅⟩:X×Y→R\langle \cdot, \cdot \rangle: X \times Y \rightarrow \mathbb{R}. Given the graph M⊂X×Y\displaystyle M \subset X\times Y of a multivalued law T ⁣:X→2Y\displaystyle T \colon X\rightarrow 2^{Y}, we state a simple necessary and sufficient condition for the existence of a bipotential bb for which MM is the set of (x,y)(x, y) such that b(x,y)=⟨x,y⟩b(x, y) = \langle x, y\rangle. 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

Author information

Contact details are reproduced from the original publication and may be historical.

Marius Buliga

Inst. of Mathematics, Romanian Academy, P.O. Box 1-764, 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@univ-lille1.fr

Claude Vallée

Lab. de Mécanique des Solides, UMR 6610 - UFR SFA-SP2MI, Bvd. M. et P. Curie, Téléport 2 - BP 30179, 86962 Futuroscope-Chasseneuil, France

vallee@lms.univ-poitiers.fr

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Suggested citation

M. Buliga, G. de Saxcé, C. Vallée. “Existence and Construction of Bipotentials for Graphs of Multivalued Laws.” Journal of Convex Analysis 15 (2008), No. 1, 87–104. https://doi.org/10.68381/jca15007

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press.