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December  2013, 6(4): 945-954. doi: 10.3934/krm.2013.6.945

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, France
2. 

Direction Technique et Scientifique, EGIS Industries, 4, rue Dolores Ibarruri, 93188 Montreuil Cedex

Received  August 2013 Revised  September 2013 Published  November 2013

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.
Keywords: pseudo-potentials, dissipation., thermodynamics, Convex analysis, non-associated flow rules, Legendre-Fenchel conjugate.
Mathematics Subject Classification: Primary: 74C05, 74A15, 74A2.
Citation: Nelly Point, Silvano Erlicher. Convex analysis and thermodynamics. Kinetic & Related Models, 2013, 6 (4) : 945-954. doi: 10.3934/krm.2013.6.945
References:
[1]

J. J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, Série II, 271 (1970), 608-611 (in French). Google Scholar

[2]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Journal de Mécanique, 14 (1975), 39-63 (in French).  Google Scholar

[3]

S. Erlicher and N. Point, On the Associativity of the Drucker -Prager Model, VIII International Conference on Computational Plasticity - Fundamentals and Applications, Barcelona, Spain, 2005. Google Scholar

[4]

S. Erlicher and N. Point, Endochronic theory, non-linear kinematic hardening rule and generalized plasticity: A new interpretation based on generalized normality assumption, International Journal of Solids and Structures, 43 (2006), 4175-4200. doi: 10.1016/j.ijsolstr.2005.03.022.  Google Scholar

[5]

N. Point and S. Erlicher, Pseudo-potentials and loading surfaces for an endochronic plasticity theory with isotropic damage, Journal of Engineering Mechanics ASCE, 134 (2008), 832-842. Google Scholar

[6]

N. Point and S. Erlicher, Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact, Discret and Continuous Dynamical Systems - Series S, 6 (2013), 567-590.  Google Scholar

[7]

G. de Saxcé, A generalization of Fenchel's inequality and its applications to the constitutive laws, Comptes Rendus de Académie des Sciences, Série II, 314 (1992), 125-129. Google Scholar

[8]

J. F. Babadjian, G. Francfort and M. G. Mora, Quasistatic evolution in non-associative plasticity: The cap model, SIAM J. Math. Anal, 44 (2012), 245-292. doi: 10.1137/110823511.  Google Scholar

[9]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[10]

H. Ziegler and C. Wehrli, The derivation of constitutive equations from the free energy and the dissipation function, Advances in Applied Mechanics, 25 (1987), 183-237. doi: 10.1016/S0065-2156(08)70278-3.  Google Scholar

[11]

M. Jirásek and Z. P. Bazant, Inelastic Analysis of Structures, Wyley, Chichester, 2002. Google Scholar

[12]

M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002.  Google Scholar

[13]

J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990. Google Scholar

show all references

References:
[1]

J. J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, Série II, 271 (1970), 608-611 (in French). Google Scholar

[2]

B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Journal de Mécanique, 14 (1975), 39-63 (in French).  Google Scholar

[3]

S. Erlicher and N. Point, On the Associativity of the Drucker -Prager Model, VIII International Conference on Computational Plasticity - Fundamentals and Applications, Barcelona, Spain, 2005. Google Scholar

[4]

S. Erlicher and N. Point, Endochronic theory, non-linear kinematic hardening rule and generalized plasticity: A new interpretation based on generalized normality assumption, International Journal of Solids and Structures, 43 (2006), 4175-4200. doi: 10.1016/j.ijsolstr.2005.03.022.  Google Scholar

[5]

N. Point and S. Erlicher, Pseudo-potentials and loading surfaces for an endochronic plasticity theory with isotropic damage, Journal of Engineering Mechanics ASCE, 134 (2008), 832-842. Google Scholar

[6]

N. Point and S. Erlicher, Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact, Discret and Continuous Dynamical Systems - Series S, 6 (2013), 567-590.  Google Scholar

[7]

G. de Saxcé, A generalization of Fenchel's inequality and its applications to the constitutive laws, Comptes Rendus de Académie des Sciences, Série II, 314 (1992), 125-129. Google Scholar

[8]

J. F. Babadjian, G. Francfort and M. G. Mora, Quasistatic evolution in non-associative plasticity: The cap model, SIAM J. Math. Anal, 44 (2012), 245-292. doi: 10.1137/110823511.  Google Scholar

[9]

R. T. Rockafellar, Convex Analysis, Reprint of the 1970 original. Princeton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[10]

H. Ziegler and C. Wehrli, The derivation of constitutive equations from the free energy and the dissipation function, Advances in Applied Mechanics, 25 (1987), 183-237. doi: 10.1016/S0065-2156(08)70278-3.  Google Scholar

[11]

M. Jirásek and Z. P. Bazant, Inelastic Analysis of Structures, Wyley, Chichester, 2002. Google Scholar

[12]

M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002.  Google Scholar

[13]

J. Lemaitre and J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990. Google Scholar

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