We propose a rigorous variational formulation and an algorithm for solving the discretisation in time of the evolution problem for an implicit standard material, in terms of bipotentials.
BergaAde SaxcéG. Elastoplastic finite element analysis of soil problems with implicit standard material constitutive laws. Rev Eur des Eléments Finis1994; 3(3): 411–456.
2.
de SaxcéGFengZQ. New inequation and functional for contact with friction: The implicit standard material approach. Mech Struct and Mach1991; 19(3): 301–325.
3.
de SaxcéGBousshineL. On the extension of limit analysis theorems to the non-associated flow rules in soils and to the contact with Coulomb’s friction. In: Proc XI Polish Conference on Computer Methods in Mechanics, Kielce, 1993, vol. 2, pp.815–822.
4.
de SaxcéG. The bipotential method, a new variational and numerical treatment of the dissipative laws of materials. In: Proc 10th Int Conf on Mathematical and Computer Modelling and Scientific Computing, Boston, 1995.
5.
de SaxcéG. Une généralisation de l’inégalité de Fenchel et ses applications aux lois constitutives. C R Acad Sci, Paris, Sér II1992; 314: 125–129.
6.
BodovilléGde SaxcéG. Plasticity with non-linear kinematic hardening: Modelling and shakedown analysis by the bipotential approach. Eur J Mech, A/Solids2001; 20: 99–112.
7.
HjiajMBodovilléGde SaxcéG. Matériaux viscoplastiques et loi de normalité implicites. C R Acad Sci, Paris, Sér II, Fasc b, Méc Phys Astron2000; 328: 519–524.
8.
BodovilléG. On damage and implicit standard materials. C R Acad Sci, Paris, Sér II, Fasc b, Méc Phys. Astron1999; 327(8): 715–720.
9.
de SaxcéGBousshineL. Implicit standard materials. In: WeichertDMaierG (eds) Inelastic behaviour of structures under variable repeated loads, CISM Courses and Lectures 432. Vienna: Springer, 2002.
10.
ValléeCLerintiuCFortunéDBanMde SaxcéG. Hill’s bipotential. In: Mihailescu-SuliciuM (ed) New Trends in Continuum Mechanics, Theta Series in Advanced Mathematics. Bucharest: Theta Foundation, 2005, pp.339–351.
11.
BousshineLChaabaAde SaxcéG. Plastic limit load of plane frames with frictional contact supports. Int J Mech Sci2002; 44(11): 2189–2216.
12.
FengZ-QHjiajMde SaxcéGMrózZ. Effect of frictional anisotropy on the quasistatic motion of a deformable solid sliding on a planar surface. Comput Mech2006; 37: 349–361.
13.
FortinJHjiajMde SaxcéG. An improved discrete element method based on a variational formulation of the frictional contact law. Comput Geotech2002; 29(8): 609–640.
14.
HjiajMFengZ-Qde SaxcéGMrózZ. Three dimensional finite element computations for frictional contact problems with on-associated sliding rule. Int J Numer Methods Eng2004; 60(12): 2045–2076.
15.
de SaxcéGFengZ-Q. The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms. Math Comput1998; 28(4-8): 225–245.
16.
LabordePRenardY. Fixed points strategies for elastostatic frictional contact problems. Math Meth Appl Sci2008; 31: 415–441.
17.
BuligaMde SaxcéGValléeC. Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb’s dry friction law. J Convex Anal2010; 17(1): 81–94.
18.
BuligaMde SaxcéGValléeC. Existence and construction of bipotentials for graphs of multivalued laws. J Convex Anal2008; 15(1): 87–104.
19.
BuligaMde SaxcéGValléeC. Bipotentials for non monotone multivalued operators: Fundamental results and applications. Acta Appl Math2010; 110(2): 955–972.
20.
MateiANiculescuC. Weak solutions via bipotentials in mechanics of deformable solids. J Math Anal Appl2011; 379(1): 15–25.
21.
NayrolesB. Opérations algébriques en Mécanique des Structures. C R Acad Sci, Paris, Sér A1971; 273: 1075–1078.