In a previous paper, we proposed a symplectic version of the Brezis–Ekeland–Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non-associated plasticity. For this aim, we generalize the concept of bipotential to dynamical systems. The keystone idea is to define a symplectic bipotential. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis–Ekeland–Nayroles principle to the non-associated dissipative laws. We apply it to the non-associated plasticity and to the unilateral contact law with Coulomb’s dry friction.
SouriauJM. Structure of dynamical systems, a symplectic view of physics. New York: Birkhäuser, 1997.
3.
BarbarescoF. Symplectic foliation structures of non-equilibrium thermodynamics as dissipation model: application to metriplectic nonlinear lindblad quantum master equation. Entropy2022; 24: 1626.
4.
CoquinotBMorrisonPJ. A general metriplectic framework with application to dissipative extended magnetohydrodynamics. J Plasma Phys2020; 86: 835860302.
5.
MaterassiMMorrisonPJ. Metriplectic formalism: friction and much more, arXiv:1706.01455, 2017.
6.
MorrisonPJ. A paradigm for joined Hamiltonian and dissipative systems. Physica D1986; 18: 410.
7.
GrmelaMÖttingerHC. Dynamics and thermodynamics of complex fluids. I. development of a general formalism. Phys Rev E1997; 56: 6620–6632.
8.
ÖttingerHCGrmelaM. Dynamics and thermodynamics of complex fluids. II. illustrations of a general formalism. Phys Rev E1997; 56: 6633–6655.
9.
DuongMHPeletierMAZimmerJ. GENERIC formalism of a Vlasov–Fokker–Planck equation and connection to large-deviation principles. Nonlinearity2013; 26: 2951–2971.
10.
BrockettRW. Control theory and analytical mechanics. In: MartinCHermannR (eds) Geometric control theory, lie groups: History, frontiers and applications, vol. VII. Brookline: Mathematical Science Press, 1977, pp. 1–46.
11.
van der SchaftAJ. System theoretic properties of physical systems. Amsterdam: CWITract, 1984.
12.
MielkeATheilF. A mathematical model for rate-independent phase transformations with hysteresis. In: AlberHDBaleanRFarwigR (eds) Workshop on models of continuum mechanics in analysis and engineering. Düren: Shaker, 1999, pp. 117–129.
13.
MielkeA. Evolution in rate-independent systems. In: DafermosCFeireislE (eds) Handbook of differential equations: Evolutionary equations, vol. 2. Amsterdam: Elsevier, 2005, pp. 461–559.
14.
MielkeARoubíčekT. Rate-independent damage processes in nonlinear elasticity. Math Models Methods Appl Sci2006; 16: 177–209.
15.
BuligaM. Hamiltonian inclusions with convex dissipation with a view towards applications. Math Appl2009; 1: 228–235.
16.
BrezisHEkelandI. Un principe variationnel associé à certaines équations paraboliques. i. le cas indépendant du temps, ii. le cas dépendant du temps. C R Acad Sci Paris Ser A-B1976; 282: 971–974 and 1197–1198.
17.
NayrolesB. Deux théorèmes de minimum pour certains systèmes dissipatifs. C R Acad Sci Paris Ser A-B1976; 282: A1035–A1038.
18.
RiosH. étude de la question d’existence pour certains problèmes d’évolution par minimisation d’une fonctionnelle convexe. C R Acad Sci Paris Ser A-B1976; 283: A83–A86.
19.
RiosH. Une étude d’existence sur certains problèmes paraboliques. Ann Fac Sci Toulouse Math1979; 1: 235–255.
20.
MabroukM. Sur un problème d’évolution à données mesures; approche variationnelle. C R Acad Sci Paris Ser I Math1994; 318: 47–52.
21.
MabroukM. A variational approach for a semi-linear parabolic equation with measure data. Ann Fac Sci Toulouse Math2000; 9: 91–112.
22.
RoubíčekT. Direct method for parabolic problems. Adv Math Sci Appl2000; 10: 57–65.
23.
GhoussoubNMacCannRJ. A least action principle for steepest descent in non-convex landscape. Contemp Math2004; 362: 177–187.
24.
GhoussoubN. A theory of anti-selfdual Lagrangians: dynamical case. C R Acad Sci Paris Ser I2005; 340: 325–330.
25.
GhoussoubN. Anti-self-dual lagrangians: variational resolutions of non-self-adjoint equations and dissipative evolutions. Ann Inst H Poincare C Anal Non Lineaire2007; 24: 171–205.
26.
GhoussoubN. Self-dual partial differential systems and their variational principles. Springer, 2008.
27.
GhoussoubNMoameniA. Anti-symmetric Hamiltonians (II): variational resolutions for Navier–Stokes and other nonlinear evolutions. Ann Inst H Poincare C Anal Non Lineaire2009; 26: 223–255.
28.
VisintinA. Homogenization of the nonlinear Kelvin–Voigt model of visco-elasticity and of the Prager model of plasticity. Continuum Mech Thermodyn2006; 18: 223–252.
29.
VisintinA. Extension of the Brezis–Ekeland–Nayroles principle to monotone operators. Adv Math Sci Appl2008; 18: 633–650.
30.
VisintinA. Homogenization of nonlinear visco-elastic composites. J Math Pures Appl2008; 89: 477–504.
31.
VisintinA. Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl–Reuss model of elastoplasticity. Proc R Soc Edinb A2008; 138: 1–39.
32.
VisintinA. Structural stability of rate-independent nonpotential flows. Discrete Contin Dyn Syst Ser S2013; 6: 257–275.
33.
VisintinA. Variational formulation and structural stability of monotone equations. Calc Var Partial Differ Equ2013; 47: 273–317.
34.
CaoXOueslatiANguyenAD, et al. A symplectic Brezis–Ekeland–Nayroles principle for dynamic plasticity in finite strains. Int J Eng Sci2022; 183: 103791.
35.
StefanelliU. A variational principle for hardening elasto-plasticity. SIAM J Math Anal2008; 40: 623–652.
36.
BuligaMde SaxcéG. A symplectic Brezis–Ekeland–Nayroles principle. Math Mech Solids2016; 22: 1–15.
37.
CaoXOueslatiANguyenAD, et al. Numerical simulation of elasto-plastic problems by Brezis–Ekeland–Nayroles non-incremental variational principle. Comput Mech2020; 65: 1006–1018.
38.
CaoXOueslatiAde SaxcéG. A non-incremental approach for elastoplastic plates basing on the Brezis–Ekeland–Nayroles principle. Appl Math Model2021; 99: 359–379.
39.
CaoXOueslatiAShirafkanN, et al. A non-incremental numerical method for dynamic elastoplastic problems by the symplectic Brezis–Ekeland–Nayroles principle. Comput Methods Appl Mech Eng2021; 384: 11908.
40.
de SaxcéGFengZQ. New inequation and functional for contact with friction: the implicit standard material approach. Int J Mech Struct Mach1991; 19: 301–325.
41.
de SaxcéG. Une généralisation de l’inégalité de fenchel et ses applications aux lois constitutives. C R Acad Sci Paris Ser II1992; 314: 125–129.
42.
de SaxcéGBousshineL. Limit analysis theorems for the implicit standard materials: application to the unilateral contact with dry friction and the non associated flow rules in soils and rocks. Int J Mech Sci1998; 40(4): 387–398.
43.
ZouainNPontes FilhoIVaunatJ. Potentials for the modified Cam–Clay model. Eur J Mech A Solids2010; 29: 327–336.
44.
BodovilléGde SaxcéG. Plasticity with non linear kinematic hardening: modelling and shakedown analysis by the bipotential approach. Eur J Mech A Solids2001; 20: 99–112.
45.
BoubyCde SaxcéGTritschJB. On shakedown of structures under variable loads with a kinematic non linear and non associated hardening rule. In: WeichertDPonterA (eds) Limit states of materials and structures: Direct methods. Cham: Springer, 2009.
46.
BoubyCKondoDde SaxcéG. A comparative analysis of two formulations for non linear hardening plasticity models: application to shakedown analysis. Eur J Mech A Solids2015; 53: 48–61.
47.
MagnierVCharkalukEBoubyC, et al. Bipotential versus return mapping algorithms: implementation of non-associated flow rules. In: ToppingBHVMonteroGMontenegroR (eds) Proceedings of the 8th international conference on computational science and technology. Stirlingshire: Civil-Comp Press, 2006.
48.
HjiajMBodovilléGde SaxcéG. Matériaux viscoplastiques et loi de normalité implicites. C R Acad Sci Paris Ser IIb2000; 328: 519–524.
49.
BodovilléG. On damage and implicit standard materials. C R Acad Sci Paris Ser IIB1999; 327: 715–720.
50.
de SaxcéG. Implicit standard materials. In: WeichertDMaierG (eds) Inelastic behaviour of structures under variable repeated loads. CISM Courses Lectures. 432, Cham: Springer, 2002.
51.
ValléeVLerintiuCFortunéD, et al. A bipotential expressing simultaneous ordered spectral decomposition between stress and strain rate tensor. In: Mihăilescu-SuliciuM (ed.) New trends in continuum mechanics. Jhalariya, India: Theta, 2005, pp. 339–351.
52.
FitzpatrickS. Representing monotone operators by convex functions. In: Proceedings of the centre for mathematical analysis. Canberra, ACT, Australia: Australian National University, 1988, pp. 59–65.
53.
EkelandITemamR. Convex analysis and variational problems. Classics in Applied Mathematics. Philadelphia, PA: SIAM, 1999.
54.
BuligaMde SaxcéGValléeC. Existence and construction of bipotential for graphs of multivalued laws. J Convex Anal2008; 15: 87–104.
55.
BuligaMde SaxcéGValléeC. Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb’s dry friction law. J Convex Anal2010; 17: 81–94.
56.
VermeerPA. Non-associated plasticity for soils, concrete and rocks. In: HermannHJHoviJPLudingS (eds) Physics dry granular media. NATO Science Series E, vol. 350. Cham: Springer, 1998, pp. 163–196.
57.
DrescherADetournayE. Limit load in translational mechanisms for associative and nonassociative materials. Geotechnique1993; 43: 443–456.
58.
HamlaouiMOueslatiOde SaxcéG. A bipotential approach for the plastic limit load of strip footings with non-associated materials. Int J Non Linear Mech2017; 90: 1–10.
59.
De BorstRSabetSAHagemanT. Non-associated Drescher1993. Int J Mech Sci2022; 230: 107535.
60.
RudnickiJWRiceR. Conditions of the localization of the deformation in pressure-sensitive materials. J Mech Phys Solids1975; 23: 371–394.
61.
ZienkiewiczOCHumphesonCLewisRW. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique1975; 25: 671–689.
62.
ArmstrongPJFrederickCO. A mathematical representation of the multiaxial Bauschinger effects. Technical Report RD/B/N 731, C E G B, 1966
63.
MrózZ. On the description of anisotropic workhardening. J Mech Phys Solids1967; 15: 163–175.
64.
MrózZNorrisVAZienkiewiczOC. An anisotropic hardening model for soils and its application in cyclic loading. Int J Numer Methods Eng1978; 2: 203–221.
65.
de SaxcéG. A non incremental variational principle for brittle fracture. Int J Solids Struct2022; 252: 988.
