# American Institute of Mathematical Sciences

2016, 9(2): 409-425. doi: 10.3934/dcdss.2016004

## Modelling contact with isotropic and anisotropic friction by the bipotential approach

 1 Laboratoire de Mécanique de Lille, UMR CNRS 8107, Université des Sciences et Technologies de Lille, bâtiment Boussinesq, Cité Scientifique, 59655 Villeneuve d'Ascq cedex

Received  May 2015 Revised  October 2015 Published  March 2016

Based on an extension of Fenchel's inequality, the bipotential approach is a non smooth mechanics tool used to model various non associative multivalued constitutive laws of dissipative materials (friction contact, soils, cyclic plasticity of metals, damage). Generally, such constitutive laws are given by a graph $M$. We propose a simple necessary and sufficient condition for the existence of a bipotential $b$ for which $M$ is the set of couples $(x,y)$ of dual variables such that $b(x,y) = \langle x,y \rangle$, and a method to construct such a bipotential by covering $M$ with cyclically monotone graphs which are not necessarily maximal (bipotential convex cover). As application, we show how to obtain the bipotential of the law of unilateral contact with Coulomb's friction by a bipotential convex cover. Introduced to extend the classical calculus of variation, the bipotential concept is also useful to construct numerical schemes for friction contact laws. In recents works, we extended the bipotential approach to a certain class of orthotropic frictional contact with a non-associated sliding rule proposed by Michałowski and Mróz. The bipotential suggests a predictor-corrector numerical scheme.
Citation: Géry de Saxcé. Modelling contact with isotropic and anisotropic friction by the bipotential approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 409-425. doi: 10.3934/dcdss.2016004
##### References:
 [1] G. Bodovillé, On damage and implicit standard materials,, C. R. Acad. Sci. Paris, 327 (1999), 715. [2] G. Bodovillé and G. de Saxcé, Plasticity with non linear kinematic hardening: Modelling and shakedown analysis by the bipotential approach,, Eur. J. Mech., 20 (2001), 99. [3] L. Bousshine, A. Chaaba and G. de Saxcé, Plastic limit load of plane frames with frictional contact supports,, Int. J. Mech. Sci., 44 (2002), 2189. doi: 10.1016/S0020-7403(02)00135-2. [4] M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Analysis, 15 (2008), 87. [5] M. Buliga, G. de Saxcé and C. Vallée, Bipotentials for non monotone multivalued operators: Fundamental results and applications,, Acta Applicandae Mathematicae, 110 (2010), 955. doi: 10.1007/s10440-009-9488-3. [6] M. Buliga, G. de Saxcé and C. Vallée, Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law,, J. Convex Analysis, 17 (2010), 81. [7] G. de Saxcé, Une généralisation de l'inégalité de Fenchel et ses applications aux lois constitutives,, C. R. Acad. Sci. Paris, 314 (1992), 125. [8] G. de Saxcé, The bipotential method, a new variational and numerical treatment of the dissipative laws of materials,, 10th Int. Conf. on Mathematical and Computer Modelling and Scientific Computing, (1995). [9] G. de Saxcé and L. Bousshine, Implicit standard materials,, D. Weichert G. Maier eds. Inelastic behaviour of structures under variable repeated loads, (2002). [10] G. de Saxcé and L. Bousshine, On the extension of limit analysis theorems to the non associated flow rules in soils and to the contact with Coulomb's friction,, XI Polish Conference on Computer Methods in Mechanics. Kielce, (1993), 815. [11] G. de Saxcé and Z. Q. Feng, New inequality and functional for contact friction: The implicit standard material approach,, Mechanics of Structures and Machines, 19 (1991), 301. doi: 10.1080/08905459108905146. [12] G. de Saxcé and Z. Q. Feng, The bi-potential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms,, Mathematical and Computer Modelling, 28 (1998), 225. doi: 10.1016/S0895-7177(98)00119-8. [13] W. Fenchel, On conjugate convex functions,, Canadian Journal of Mathematics, 1 (1949), 73. doi: 10.4153/CJM-1949-007-x. [14] Z.-Q. Feng, M. Hjiaj, G. de Saxcé and Z. Mróz, Effect of frictional anisotropy on the quasistatic motion of a deformable solid sliding on a planar surface,, Comput. Mech., 37 (2006), 349. doi: 10.1007/s00466-005-0674-5. [15] Z.-Q. Feng, M. Hjiaj, G. de Saxcé and Z. Mróz, Influence of frictional anisotropy on contacting surfaces during loading/unloading cycles,, International Journal of Non-Linear Mechanics, 41 (2006), 936. doi: 10.1016/j.ijnonlinmec.2006.08.002. [16] J. Fortin, M. Hjiaj and G. de Saxcé, An improved discrete element method based on a variational formulation of the frictional contact law,, Comput. Geotech., 29 (2002), 609. doi: 10.1016/S0266-352X(02)00016-2. [17] B. Halphen and S. Nguyen Quoc, Sur les matériaux standard généralisés,, C. R. Acad. Sci. Paris 14 (1975), 14 (1975), 39. [18] M. Hjiaj, G. Bodovillé and G. de Saxcé, Matériaux viscoplastiques et loi de normalité implicites,, C. R. Acad. Sci. Paris, 328 (2000), 519. doi: 10.1016/S1620-7742(00)00007-6. [19] M. Hjiaj, G. de Saxcé and Z. Mróz, A variational-inequality based formulation of the frictional contact law with a non-associated sliding rule,, European Journal of Mechanics A/Solids, 21 (2002), 49. doi: 10.1016/S0997-7538(01)01183-4. [20] M. Hjiaj, Z.-Q. Feng, G. de Saxcé and Z. Mróz, Three dimensional finite element computations for frictional contact problems with on-associated sliding rule,, Int. J. Numer. Methods Eng., 60 (2004), 2045. doi: 10.1002/nme.1037. [21] P. Laborde and Y. Renard, Fixed points strategies for elastostatic frictional contact problems,, Math. Meth. Appl. Sci., 31 (2008), 415. doi: 10.1002/mma.921. [22] R. Michałowski and Z. Mróz, Associated and non-associated sliding rules in contact friction problems,, Archives of Mechanics, 11 (1978), 259. [23] J. J. Moreau, Fonctionnelles Convexes,, Istituto Poligrafico e zecca dello stato, (2003). [24] Z. Mróz and S. Stupkiewicz, An anisotropic fricition and wear model,, International Journal of Solids and Structures, 31 (1994), 1113. [25] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1997). [26] C. Vallée, C. Lerintiu, D. Fortuné, M. Ban and G. de Saxcé, Hill's bipotential,, M. Mihailescu-Suliciu eds. New Trends in Continuum Mechanics, (2005), 339.

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##### References:
 [1] G. Bodovillé, On damage and implicit standard materials,, C. R. Acad. Sci. Paris, 327 (1999), 715. [2] G. Bodovillé and G. de Saxcé, Plasticity with non linear kinematic hardening: Modelling and shakedown analysis by the bipotential approach,, Eur. J. Mech., 20 (2001), 99. [3] L. Bousshine, A. Chaaba and G. de Saxcé, Plastic limit load of plane frames with frictional contact supports,, Int. J. Mech. Sci., 44 (2002), 2189. doi: 10.1016/S0020-7403(02)00135-2. [4] M. Buliga, G. de Saxcé and C. Vallée, Existence and construction of bipotentials for graphs of multivalued laws,, J. Convex Analysis, 15 (2008), 87. [5] M. Buliga, G. de Saxcé and C. Vallée, Bipotentials for non monotone multivalued operators: Fundamental results and applications,, Acta Applicandae Mathematicae, 110 (2010), 955. doi: 10.1007/s10440-009-9488-3. [6] M. Buliga, G. de Saxcé and C. Vallée, Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law,, J. Convex Analysis, 17 (2010), 81. [7] G. de Saxcé, Une généralisation de l'inégalité de Fenchel et ses applications aux lois constitutives,, C. R. Acad. Sci. Paris, 314 (1992), 125. [8] G. de Saxcé, The bipotential method, a new variational and numerical treatment of the dissipative laws of materials,, 10th Int. Conf. on Mathematical and Computer Modelling and Scientific Computing, (1995). [9] G. de Saxcé and L. Bousshine, Implicit standard materials,, D. Weichert G. Maier eds. Inelastic behaviour of structures under variable repeated loads, (2002). [10] G. de Saxcé and L. Bousshine, On the extension of limit analysis theorems to the non associated flow rules in soils and to the contact with Coulomb's friction,, XI Polish Conference on Computer Methods in Mechanics. Kielce, (1993), 815. [11] G. de Saxcé and Z. Q. Feng, New inequality and functional for contact friction: The implicit standard material approach,, Mechanics of Structures and Machines, 19 (1991), 301. doi: 10.1080/08905459108905146. [12] G. de Saxcé and Z. Q. Feng, The bi-potential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms,, Mathematical and Computer Modelling, 28 (1998), 225. doi: 10.1016/S0895-7177(98)00119-8. [13] W. Fenchel, On conjugate convex functions,, Canadian Journal of Mathematics, 1 (1949), 73. doi: 10.4153/CJM-1949-007-x. [14] Z.-Q. Feng, M. Hjiaj, G. de Saxcé and Z. Mróz, Effect of frictional anisotropy on the quasistatic motion of a deformable solid sliding on a planar surface,, Comput. Mech., 37 (2006), 349. doi: 10.1007/s00466-005-0674-5. [15] Z.-Q. Feng, M. Hjiaj, G. de Saxcé and Z. Mróz, Influence of frictional anisotropy on contacting surfaces during loading/unloading cycles,, International Journal of Non-Linear Mechanics, 41 (2006), 936. doi: 10.1016/j.ijnonlinmec.2006.08.002. [16] J. Fortin, M. Hjiaj and G. de Saxcé, An improved discrete element method based on a variational formulation of the frictional contact law,, Comput. Geotech., 29 (2002), 609. doi: 10.1016/S0266-352X(02)00016-2. [17] B. Halphen and S. Nguyen Quoc, Sur les matériaux standard généralisés,, C. R. Acad. Sci. Paris 14 (1975), 14 (1975), 39. [18] M. Hjiaj, G. Bodovillé and G. de Saxcé, Matériaux viscoplastiques et loi de normalité implicites,, C. R. Acad. Sci. Paris, 328 (2000), 519. doi: 10.1016/S1620-7742(00)00007-6. [19] M. Hjiaj, G. de Saxcé and Z. Mróz, A variational-inequality based formulation of the frictional contact law with a non-associated sliding rule,, European Journal of Mechanics A/Solids, 21 (2002), 49. doi: 10.1016/S0997-7538(01)01183-4. [20] M. Hjiaj, Z.-Q. Feng, G. de Saxcé and Z. Mróz, Three dimensional finite element computations for frictional contact problems with on-associated sliding rule,, Int. J. Numer. Methods Eng., 60 (2004), 2045. doi: 10.1002/nme.1037. [21] P. Laborde and Y. Renard, Fixed points strategies for elastostatic frictional contact problems,, Math. Meth. Appl. Sci., 31 (2008), 415. doi: 10.1002/mma.921. [22] R. Michałowski and Z. Mróz, Associated and non-associated sliding rules in contact friction problems,, Archives of Mechanics, 11 (1978), 259. [23] J. J. Moreau, Fonctionnelles Convexes,, Istituto Poligrafico e zecca dello stato, (2003). [24] Z. Mróz and S. Stupkiewicz, An anisotropic fricition and wear model,, International Journal of Solids and Structures, 31 (1994), 1113. [25] R. T. Rockafellar, Convex Analysis,, Princeton University Press, (1997). [26] C. Vallée, C. Lerintiu, D. Fortuné, M. Ban and G. de Saxcé, Hill's bipotential,, M. Mihailescu-Suliciu eds. New Trends in Continuum Mechanics, (2005), 339.
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