In many practical situations, uncertainties affect the mechanical behavior that is given by a family of graphs instead of a single graph. In this paper, we show how the bipotential method is able to capture such blurred constitutive laws, using bipotential convex covers.
Halphen, B., and Nguyen Quoc, S.Sur les matériaux standard généralisés. Journal de Mécanique, 14, 39-63 (1975).
2.
de Saxcé, G., and Feng, ZQNew inequation and functional for contact with friction: the implicit standard material approach. Mechanics of Structures and Machines, 19, 301-325 (1991).
3.
de Saxcé, G., and Bousshine, L.On the extension of limit analysis theorems to the non associated flow rules in soils and to the contact with Coulomb’s friction. Proceedings of the 11th Polish Conference on Computer Methods in Mechanics at Kielce, Vol. 2, pp. 815-822 (1993).
4.
de Saxcé, G.The bipotential method, a new variational and numerical treatment of the dissipative laws of materials. Proceedings of the 10th International Conference on Mathematical and Computer Modelling and Scientific Computing at (Boston, MA, 1995).
5.
de Saxcé, G.Une généralisation de l’inégalité de Fenchel et ses applications aux lois constitutives. Comptes-Rendus de l’Académie des Sciences de Paris Série II, 314, 125-129 (1992).
6.
Bodovillé, G., and de Saxcé , G.Plasticity with non linear kinematic hardening: modelling and shakedown analysis by the bipotential approach. European Journal of Mechanics A/Solids, 20, 99-112 (2001).
7.
Hjiaj, M., Bodovillé, G., and de Saxcé, G.Matériaux viscoplastiques et loi de normalité implicites. Comptes-Rendus de l’Académie des Sciences de Paris Série II Mécanique Physique Astronomie , 328, 519-524 (2000).
8.
Bodovillé, G.On damage and implicit standard materials . Comptes-Rendus de l’Académie des Sciences de Paris Série II Mécanique Physique Astronomie, 327, 715-720 (1999).
9.
de Saxcé, G., and Bousshine, L.Implicit standard materials, in Inelastic behaviour of structures under variable repeated loads, pp. 59-76, ed., D. Weichert and G. Maier , CISM Courses and Lectures432, Springer, Wien, 2002 .
10.
Vallée, C., Lerintiu, C., Fortuné , D., Ban , M., and de Saxcé, G.Hill’s bipotential, in ed. M. Mihailescu-Suliciu , New Trends in Continuum Mechanics, Theta Series in Advanced Mathematics, pp. 339-351, Theta Foundation, Bucarest, 2005.
11.
Bousshine, L., Chaaba, A., and de Saxcé , G.Plastic limit load of plane frames with frictional contact supports. International Journal of Mechanical Science, 44, 2189-2216 (2002 ).
12.
Feng, ZQ, Hjiaj, M., de Saxcé, G., and Mróz, Z.Effect of frictional anisotropy on the quasistatic motion of a deformable solid sliding on a planar surface. Computational Mechanics , 37, 349-36 (2006).
13.
Fortin, J. , Hjiaj, M., and de Saxcé , G.An improved discrete element method based on a variational formulation of the frictional contact law. Computational Geotechnics, 29, 609-640 ( 2002).
14.
Hjiaj, M. , Feng, ZQ, de Saxcé , G., and Mróz, Z.Three dimensional finite element computations for frictional contact problems with on-associated sliding rule. International Journal of Numerical Methods in Engineering, 60, 2045-2076 (2004).
15.
de Saxcé, G., and Feng, ZQThe bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms . Mathematical and Computer Modelling, 28, 225-245 ( 1998).
16.
Laborde, P., and Renard, Y.Fixed points strategies for elastostatic frictional contact problems. Mathematical Methods in Applied Science , 31, 415-441 (2008).
17.
Buliga, M. , de Saxcé, G., and Vallée , V.Existence and construction of bipotentials for graphs of multivalued laws. Journal of Convex Analysis, 15, 87-104 (2008).
18.
Buliga, M. , de Saxcé, G., and Vallée , V.Bipotentials for non monotone multivalued operators: fundamental results and applications. Acta Applicandae Mathematicae, DOI 10.1007/s10440-009-9488-3 ( 2009).
19.
Buliga, M. , de Saxcé, G., and Vallée , V.Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb’s dry friction law. Journal of Convex Analysis, 17 (2010).
20.
Babuska, I., and Rheinboldt , WCA posteriori estimates for the finite element method. International Journal of Numerical Methods Engineering, 12, 1597-1615 (1978 ).
21.
Babuska, I., and Rheinboldt , WCAdaptive approaches and reliability estimators in finite element analysis. Computational Methods in Applied Mechanical Engineering, 17/18, 519-540 ( 1979).
22.
Kelly, DW , Gago, J., Zienkiewicz, OC, and Babuska, I.A posteriori error analysis and adaptative processes in finite element method. Part 1: Error analysis. International Journal of Numerical Methods in Engineering, 19, 1593-1619 (1983).
23.
Gago, J., Kelly, DW, Zienkiewicz, OC, and Babuska, I.A posteriori error analysis and adaptative processes in finite element method. Part 2: Adaptative mesh refinement. International Journal of Numerical Methods in Engineering, 19, 1921-1956 (1983 ).
24.
Oden, JT, Demkowicz, L., Rachowicz, W., and Westermann, TAToward a universal h-p adaptative finite element strategy, Part 2: A posteriori error estimation. Computational Methods in Applied Mechanical Engineering, 77, 113-180 (1989).
25.
Zienkiewicz, OC, and Zhu, JZA simple error estimator and adaptative procedure for practical engineering analysis. International Journal of Numerical Methods in Engineering, 24, 337-357 (1987).
26.
Zienkiewicz, OC, and Zhu, JZThe superconvergent patch recovery and a posteriori error estimates, Part 1: The recovery technique. International Journal of Numerical Methods in Engineering, 33, 1331-1364 ( 1992).
27.
Zienkiewicz, OC, and Zhu, JZThe superconvergent patch recovery and a posteriori error estimates, Part 2: Error estimates and adaptativity. International Journal of Numerical Methods in Engineering, 33, 1365-1382 (1992).
28.
Fraeijs de Veubeke , B. Displacement and equilibrium models in the finite element method, in ed. O. C. Zienkiewicz , Stress Analysis, John Wiley, New York, 1965.
29.
Debongnie, JF , Zhong, HG, and Beckers, P.Dual analysis with general boundary conditions . Computational Methods in Applied Mechanical Engineering , 122, 183-192 (1995).
30.
Ladevèze, P.Comparaisons de modèles de milieux continus. Thèse d’Etat, Université Pierre et Marie Curie, Paris, 1975.
31.
Ladevèze, P., Coffignal, G., and Pelle, JPAccuracy of elastoplastic and dynamic analysis , in ed. I. Babuska, J. Gago, E. Oliveira and O. C. Zienkiewicz, Accuracy Estimates and Adaptative Refinements in Finite Element Computations , pp. 181-203, John Wiley, New York, 1986.
32.
Ladevèze, P., Pelle, JP, and Rougeot, P.Error estimation and mesh optimization for classical finite element. Engineering Computation, 8, 69-80 (1991).
33.
Ladevèze, P.La maîtrise des modèles en mécanique des structures. Revue européenne des éléments finis, 1, 9-30 (1992).
34.
Ladevèze, P., and Moes, N.A new a posteriori error estimation for nonlinear time-dependent finite element analysis. Computational Methods in Applied Mechanical Engineering, 157, 45-68 (1997).
35.
Ladevèze, P., and Pelle, JPLa maîtrise du calcul en mécanique linéaire et non linéaire, Hermes Science , Paris, 2001.
36.
Ladevèze, P., and Florentin, E.Verification of stochastic models in uncertain environments using the constitutive relation error method. Computational Methods in Applied Mechanical Engineering, 196, 225-224 ( 2006).
37.
Ladevèze, P., Puel, G., Deraemaeker, A., and Romeuf, T.Validation of structural dynamics models containing uncertainties. Computational Methods in Applied Mechanical Engineering, 195, 373-393 (2006).
