December  2013, 6(6): 1641-1649. doi: 10.3934/dcdss.2013.6.1641

Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws

1. 

Université de Poitiers, Institut P', UPR CNRS 3346, Boulevard Marie et Pierre Curie, Téléport 2, BP 30179, 86962 Futuroscope Cedex, France, France

2. 

21, rue du Hameau du Cherpe, 86280 Saint-Benoît, France

3. 

Université de Lomé, Département de Physique, Laboratoire sur l'Énergie Solaire, BP 1515, Lomé, Togo

4. 

Université Ibn Zohr, Faculté des Sciences, Département de Physique, Laboratoire d'Electronique, de Traitement de Signal et de Modélisation Physique, Cité Dakhla, BP 8106, 80000 Agadir, Morocco

Received  June 2012 Revised  September 2012 Published  April 2013

We analyze the relation between Géry de Saxcé's bipotentials representing non-associated constitutive laws and Fitzpatrick's functions representing maximal monotone multifunctions. Revisiting the model of elastic materials initiated by Robert Hooke, we illustrate that Fitzpatrick's representation of monotone operators coming from convex analysis provides a constructive method to discover the best bipotential for modelling an Implicit Standard Material.
Citation: Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641
References:
[1]

S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich and X. Wang, Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative,, Nonlinear Analysis, 66 (2007), 1198. doi: 10.1016/j.na.2006.01.013. Google Scholar

[2]

M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, Journal of Convex Analysis, 15 (2008), 87. Google Scholar

[3]

S. Fitzpatrick, Representing monotone operators by convex functions,, in, 20 (1988), 59. Google Scholar

[4]

J. J. Moreau, "Fonctionnelles Convexes,", Istituto poligrafico e Zecca dello stato S. p. A., (2003). Google Scholar

[5]

G. de Saxcé and Z. Q. Feng, New inequation and functional for contact with friction: the implicit standard material approach,, International Journal Mechanics of Structures and Machines, 19 (1991), 301. doi: 10.1080/08905459108905146. Google Scholar

[6]

G. de Saxcé and L. Bousshine, Implicit standard materials,, in, 432 (2002). Google Scholar

[7]

C. Vallée, C. Lerintiu, D. Fortuné, K. Atchonouglo and M. Ban, Representing a non associated constitutive law by a bipotential issued from a Fitzpatrick sequence,, Archives of Mechanics (Arch. Mech. Stos.), 61 (2009), 325. Google Scholar

show all references

References:
[1]

S. Bartz, H. H. Bauschke, J. M. Borwein, S. Reich and X. Wang, Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative,, Nonlinear Analysis, 66 (2007), 1198. doi: 10.1016/j.na.2006.01.013. Google Scholar

[2]

M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, Journal of Convex Analysis, 15 (2008), 87. Google Scholar

[3]

S. Fitzpatrick, Representing monotone operators by convex functions,, in, 20 (1988), 59. Google Scholar

[4]

J. J. Moreau, "Fonctionnelles Convexes,", Istituto poligrafico e Zecca dello stato S. p. A., (2003). Google Scholar

[5]

G. de Saxcé and Z. Q. Feng, New inequation and functional for contact with friction: the implicit standard material approach,, International Journal Mechanics of Structures and Machines, 19 (1991), 301. doi: 10.1080/08905459108905146. Google Scholar

[6]

G. de Saxcé and L. Bousshine, Implicit standard materials,, in, 432 (2002). Google Scholar

[7]

C. Vallée, C. Lerintiu, D. Fortuné, K. Atchonouglo and M. Ban, Representing a non associated constitutive law by a bipotential issued from a Fitzpatrick sequence,, Archives of Mechanics (Arch. Mech. Stos.), 61 (2009), 325. Google Scholar

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