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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

Nelly Point 1, and  Silvano Erlicher 2,

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.
Keywords: pseudo-potentials, convex analysis., Thermomechanics, bipotential.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
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, (). 

[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, (). 

[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, (). 

[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, (). 

[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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