# American Institute of Mathematical Sciences

April  2013, 6(2): 567-590. doi: 10.3934/dcdss.2013.6.567

## Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact

 1 Conservatoire National des Arts et Métiers CNAM, 292 rue Saint-Martin, Département de Mathématiques (442), 75141 Paris, Cedex 03, France 2 Direction Technique et Scientifique, EGIS Industries, 4, rue Dolores Ibarruri, 93188 Montreuil Cedex, France

Received  July 2011 Revised  December 2011 Published  November 2012

Pseudo-potentials are very useful tools to define thermodynamically admissible constitutive rules. Bipotentials are convenient for numerical purposes, in particular for non-associative rules. Unfortunately, these functionals are not always easy to construct starting from a given constitutive law. This work proposes a procedure to find the pseudo-potentials and the bipotential starting from the usual description of a non-associative constitutive law. This method is applied to different non-associative plasticity models such as the Drucker-Prager model and the non-linear kinematic hardening model. The same procedure allows one to obtain the pseudo-potentials of an endochronic plasticity model. The pseudo-potentials for the contact problem with dissipation are constructed using the same ideas. For all these non-associative constitutive laws a bipotential is then automatically deduced.
Citation: Nelly Point, Silvano Erlicher. Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 567-590. doi: 10.3934/dcdss.2013.6.567
##### References:
 [1] P. Armstrong and C. Frederick, A mathematical representation of the multiaxial Bauschinger effect, G. E. G. B. Report RD/B/N 731, (1966). [2] Z. P. Bažant and P. D. Bath, Endochronic theory of inelasticity and failure of concrete, Journal of the Engineering Mechanics Division ASCE, 102 (1976), 701-722. [3] Z. P. Bažant and R. J. Krizek, Endochronic constitutive law for liquefaction of sand, Journal of the Engineering Mechanics Division ASCE, 102 (1976), 225-238. [4] Z. P. Bažant, Endochronic inelasticity and incremental plasticity, International Journal of Solids and Structures, 14 (1978), 691-714. doi: 10.1016/0020-7683(78)90029-X. [5] G. Bodovillé and G. de Saxcé., Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach, European Journal of Mechanics A/Solids, 20 (2001), 99-112. doi: 10.1016/S0997-7538(00)01109-8. [6] J. L. Chaboche, On some modifications of kinematic hardening to improve the description of ratchetting effects, International Journal of Plasticity, 7 (1991), 1-15. [7] I. F. Collins and G. T. Houlsby, Application of thermomechnical principles to the modelling of geotechnical materials, Proceedings of the Royal Society of London, Series A, 453 (1997), 1975-2001. [8] G. de Saxcé, A generalization of Fenchel's inequality and its applications to the constitutive laws, Comptes Rendus de l'Académie des Sciences, Série II, 314 (1992), 125-129. [9] G. de Saxcé and L. Bousshine, Limit analysis theorems for implicit standard materials: applications to the unilateral contact with dry friction and non-associated flow rules in soils and rocks, Int. J. Mech. Sci., 40 n4 (1998), 387-398. doi: 10.1016/S0020-7403(97)00058-1. [10] I. Einav, A. M. Puzrin and G. T. Houlsby, Numerical studies of hyperplasticity with single, multiple and a continuous field of yield surfaces, Int. J. Numer. Anal. Meth. Geomech., 27 (2003), 837-858. doi: 10.1002/nag.303. [11] M. A. Eisenberg and A. Phillips, A theory of plasticity with non-coincident yield and loading surfaces, Acta Mechanica, 11 (1971), 247-260. doi: 10.1007/BF01176559. [12] S. Erlicher and N. Point, Thermodynamic admissibility of Bouc-Wen type hysteresis models, Comptes Rendus Manique, 332 (2004), 51-57. doi: 10.1016/j.crme.2003.10.009. [13] 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). [14] 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. [15] S. Erlicher and O. S. Bursi, Bouc-Wen-type models with stiffness degradation: Thermodynamic analysis and applications, Journal of Engineering Mechanics ASCE, 134 (2008), 843-855. [16] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [17] M. Frémond, "Collisions," Edizioni del Dipartimento di Ingegneria Civile dell'Universita di Roma Tor Vergata, 2007. [18] B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Journal de Mécanique, 1 (1975), 39-63. (in French). [19] M. Hjiaj, J. Fortin and G. de Saxcé, A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the apex, International Journal of Engineering Science, 41 (2003), 1109-1143. doi: 10.1016/S0020-7225(02)00376-2. [20] G. T. Houlsby and A. M. Puzrin, A thermomechnical framework for constitutive models for rate-independent dissipative materials, Int. J. Plasticity, 16 (2000), 1017-1047. doi: 10.1016/S0749-6419(99)00073-X. [21] M. Jirek and Z. P. Bažant, "Inelastic Analysis of Structures," Wiley, Chichester, 2002. [22] J. Lemaitre and J.-L. Chaboche, "Mechanics of Solid Materials," Cambridge University Press, Cambridge,1990. [23] J. Lubliner, R. L. Taylor and F. Auricchio, A new model of generalized plasticity, International Journal of Solids and Structures, 30 (1993), 3171-3184. doi: 10.1016/0020-7683(93)90146-X. [24] J. Lubliner and R. L. Taylor, Two material models for cyclic plasticty: non linear kinematic hardening and generalized plasticity, International Journal of Plasticity, 11 (1995), 65-98. doi: 10.1016/0749-6419(94)00039-5. [25] D. G. Luenberger, "Linear and Nonlinear Programming," Addison-Wesley Publishing Company, Menlo Park, California,1984. [26] J. J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, Sie II, 271 (1970), 608-611. (in French). [27] 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. [28] A. M. Puzrin and G. T. Houlsby, A thermomechanical framework for rate-independent dissipative materials with internal functions, Int. Jour. of Plasticity, 17 (2001), 1147-1165. doi: 10.1016/S0749-6419(00)00083-8. [29] A. M. Puzrin and G. T. Houlsby, Fundamentals of kinematic hardening hyperplasticity, Int. Jour. of Solids and Structures, 38 (2001), 3771-3794. [30] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. [31] K. C. Valanis, A theory of viscoplasticity without a yield surface, Archiwum Mechaniki Stossowanej, 23 (1971), 517-551. [32] K. C. Valanis, Fundamental consequences of a new intrinsic time measure. Plasticity as a limit of the endochronic theory, Archiwum Mechaniki Stossowanej, 32 (1980), 171-191. [33] H. Ziegler, Discussion of some objections to thermodynamic orthogonality, Ingenieur-Archiv, 50 (1981), 149-164. doi: 10.1007/BF00536486. [34] H. Ziegler and C. Wehrli, The derivation of constitutive equations from the free energy and the dissipation function, Advances in Applied Mechanics, 25 (1987), 183-238. doi: 10.1016/S0065-2156(08)70278-3.

show all references

##### References:
 [1] P. Armstrong and C. Frederick, A mathematical representation of the multiaxial Bauschinger effect, G. E. G. B. Report RD/B/N 731, (1966). [2] Z. P. Bažant and P. D. Bath, Endochronic theory of inelasticity and failure of concrete, Journal of the Engineering Mechanics Division ASCE, 102 (1976), 701-722. [3] Z. P. Bažant and R. J. Krizek, Endochronic constitutive law for liquefaction of sand, Journal of the Engineering Mechanics Division ASCE, 102 (1976), 225-238. [4] Z. P. Bažant, Endochronic inelasticity and incremental plasticity, International Journal of Solids and Structures, 14 (1978), 691-714. doi: 10.1016/0020-7683(78)90029-X. [5] G. Bodovillé and G. de Saxcé., Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach, European Journal of Mechanics A/Solids, 20 (2001), 99-112. doi: 10.1016/S0997-7538(00)01109-8. [6] J. L. Chaboche, On some modifications of kinematic hardening to improve the description of ratchetting effects, International Journal of Plasticity, 7 (1991), 1-15. [7] I. F. Collins and G. T. Houlsby, Application of thermomechnical principles to the modelling of geotechnical materials, Proceedings of the Royal Society of London, Series A, 453 (1997), 1975-2001. [8] G. de Saxcé, A generalization of Fenchel's inequality and its applications to the constitutive laws, Comptes Rendus de l'Académie des Sciences, Série II, 314 (1992), 125-129. [9] G. de Saxcé and L. Bousshine, Limit analysis theorems for implicit standard materials: applications to the unilateral contact with dry friction and non-associated flow rules in soils and rocks, Int. J. Mech. Sci., 40 n4 (1998), 387-398. doi: 10.1016/S0020-7403(97)00058-1. [10] I. Einav, A. M. Puzrin and G. T. Houlsby, Numerical studies of hyperplasticity with single, multiple and a continuous field of yield surfaces, Int. J. Numer. Anal. Meth. Geomech., 27 (2003), 837-858. doi: 10.1002/nag.303. [11] M. A. Eisenberg and A. Phillips, A theory of plasticity with non-coincident yield and loading surfaces, Acta Mechanica, 11 (1971), 247-260. doi: 10.1007/BF01176559. [12] S. Erlicher and N. Point, Thermodynamic admissibility of Bouc-Wen type hysteresis models, Comptes Rendus Manique, 332 (2004), 51-57. doi: 10.1016/j.crme.2003.10.009. [13] 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). [14] 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. [15] S. Erlicher and O. S. Bursi, Bouc-Wen-type models with stiffness degradation: Thermodynamic analysis and applications, Journal of Engineering Mechanics ASCE, 134 (2008), 843-855. [16] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [17] M. Frémond, "Collisions," Edizioni del Dipartimento di Ingegneria Civile dell'Universita di Roma Tor Vergata, 2007. [18] B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés, Journal de Mécanique, 1 (1975), 39-63. (in French). [19] M. Hjiaj, J. Fortin and G. de Saxcé, A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the apex, International Journal of Engineering Science, 41 (2003), 1109-1143. doi: 10.1016/S0020-7225(02)00376-2. [20] G. T. Houlsby and A. M. Puzrin, A thermomechnical framework for constitutive models for rate-independent dissipative materials, Int. J. Plasticity, 16 (2000), 1017-1047. doi: 10.1016/S0749-6419(99)00073-X. [21] M. Jirek and Z. P. Bažant, "Inelastic Analysis of Structures," Wiley, Chichester, 2002. [22] J. Lemaitre and J.-L. Chaboche, "Mechanics of Solid Materials," Cambridge University Press, Cambridge,1990. [23] J. Lubliner, R. L. Taylor and F. Auricchio, A new model of generalized plasticity, International Journal of Solids and Structures, 30 (1993), 3171-3184. doi: 10.1016/0020-7683(93)90146-X. [24] J. Lubliner and R. L. Taylor, Two material models for cyclic plasticty: non linear kinematic hardening and generalized plasticity, International Journal of Plasticity, 11 (1995), 65-98. doi: 10.1016/0749-6419(94)00039-5. [25] D. G. Luenberger, "Linear and Nonlinear Programming," Addison-Wesley Publishing Company, Menlo Park, California,1984. [26] J. J. Moreau, Sur les lois de frottement, de plasticité et de viscosité, Comptes Rendus de l'Académie des Sciences, Sie II, 271 (1970), 608-611. (in French). [27] 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. [28] A. M. Puzrin and G. T. Houlsby, A thermomechanical framework for rate-independent dissipative materials with internal functions, Int. Jour. of Plasticity, 17 (2001), 1147-1165. doi: 10.1016/S0749-6419(00)00083-8. [29] A. M. Puzrin and G. T. Houlsby, Fundamentals of kinematic hardening hyperplasticity, Int. Jour. of Solids and Structures, 38 (2001), 3771-3794. [30] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, 1969. [31] K. C. Valanis, A theory of viscoplasticity without a yield surface, Archiwum Mechaniki Stossowanej, 23 (1971), 517-551. [32] K. C. Valanis, Fundamental consequences of a new intrinsic time measure. Plasticity as a limit of the endochronic theory, Archiwum Mechaniki Stossowanej, 32 (1980), 171-191. [33] H. Ziegler, Discussion of some objections to thermodynamic orthogonality, Ingenieur-Archiv, 50 (1981), 149-164. doi: 10.1007/BF00536486. [34] H. Ziegler and C. Wehrli, The derivation of constitutive equations from the free energy and the dissipation function, Advances in Applied Mechanics, 25 (1987), 183-238. doi: 10.1016/S0065-2156(08)70278-3.
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