Convex analysis and thermodynamics

Nelly Point; Silvano Erlicher

doi:10.3934/krm.2013.6.945

Article Contents

2013, Volume 6, Issue 4: 945-954. Doi: 10.3934/krm.2013.6.945

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Convex analysis and thermodynamics

Nelly Point^1, and
Silvano Erlicher^2,

1.
Université Paris-Est/Laboratoire Navier, Ecole des Ponts ParisTech, 6 et 8 av. Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée, Cedex 2
2.
Direction Technique et Scientifique, EGIS Industries, 4, rue Dolores Ibarruri, 93188 Montreuil Cedex

Received: July 31, 2013

Revised: August 31, 2013

Published: November 2013

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Abstract

Convex analysis is very useful to prove that a material model fulfills the second law of thermodynamics. Dissipation must remains non-negative and an elegant way to ensure this property is to construct an appropriate pseudo-potential of dissipation. In such a case, the corresponding material is said to be a Standard Generalized Material and the flow rules fulfill a normality rule (i.e. the dissipative thermodynamic forces are assumed to belong to an admissible domain and the flow of the corresponding state variables is orthogonal to the boundary of this domain). The sum of the pseudo-potential with its Legendre-Fenchel conjugate fulfills the Fenchel's inequality and as the actual value of the dual pair forces-flows minimizes this inequality, this can be used as a convergence criterium for numerical applications. Actually, some very commonly used and effective models do not fit into that family of Standard Generalized Materials. A procedure is here proposed which permits to retrieve the normality assumption and to construct a pair of dual pseudo-potentials also for these non-standard material models. This procedure was first presented by the authors for non-associated plasticity. Now it is extended to a large range of mechanical problems.

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Mathematics Subject Classification: Primary: 74C05, 74A15, 74A20.

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