This article is devoted to the modeling of finite deformations of hyperelastic bodies under contact/impact conditions. A total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. A first order algorithm is applied to integrate the equations of motion. The contact problem is solved by the bi-potential method. For the finite element implementation, an explicit expression of the tangent operator for the hyperelastic model is derived. The classical Boussinesq–Love contact problem is first investigated numerically. A second example concerns the impact between two hyperelastic bodies in three-dimension.
Cet article est consacré à la modélisation de grandes déformations hyperélastiques en présence du contact et de l'impact. Une formulation en Lagrangien total est adoptée pour décrire le comportement avec non linéarités géométriques. Un schéma du premier ordre est appliqué pour intégrer les équations du mouvement. Le problème du contact est résolu à l'aide de la méthode du bi-potentiel. Pour l'implantation par éléments finis, une expression explicite de l'opérateur tangent est proposée dans le cas du modèle hyperélastique . Le problème classique du contact de Boussinesq–Love est traité numériquement. Un deuxième exemple concerne l'impact entre deux blocs hyperélastiques en trois dimensions.
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Mot clés : Mécanique des solides numérique, Elasticité isotrope, Contact–impact
Zhi-Qiang Feng 1; Claude Vallée 2
@article{CRMECA_2007__335_9-10_536_0,
author = {Zhi-Qiang Feng and Claude Vall\'ee},
title = {From {Boussinesq{\textendash}Love} contact to impact between hyperelastic bodies},
journal = {Comptes Rendus. M\'ecanique},
pages = {536--547},
publisher = {Elsevier},
volume = {335},
number = {9-10},
year = {2007},
doi = {10.1016/j.crme.2007.01.009},
language = {en},
}
Zhi-Qiang Feng; Claude Vallée. From Boussinesq–Love contact to impact between hyperelastic bodies. Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, pp. 536-547. doi : 10.1016/j.crme.2007.01.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.01.009/
[1] Contact Mechanics, Cambridge University Press, 1985
[2] Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Elements, SIAM, Philadelphia, 1988
[3] Finite Element Procedures in Contact–Impact Problems, Oxford University Press, 1993
[4] Computational Contact Mechanics, John Wiley & Sons, 2002
[5] Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, Springer-Verlag, 2002
[6] The bi-potential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms, Math. Comput. Modeling, Volume 28 (1998) no. 4–8, pp. 225-245 (Special issue: Recent Advances in Contact Mechanics)
[7] Solution of large deformation contact problems with friction between Blatz–Ko hyperelastic bodies, Int. J. Engrg. Sci., Volume 41 (2003), pp. 2213-2225
[8] Improved numerical dissipation for the time integration algorithms in structural dynamics, Earthquake Engrg. Struc. Dyn., Volume 5 (1977), pp. 283-292
[9] The non-smooth contact dynamics method, Comp. Meth. Appl. Mech. Engrg., Volume 177 (1999), pp. 235-257
[10] Z.-Q. Feng, B. Magnain, J.-M. Cros, P. Joli, Energy dissipation by friction in dynamic multibody contact problems, in: Z.-H. Yao, M.-W. Yuan, W.-X. Zhong (Eds.), Computational Mechanics, Beijing, China, Sept. 2004, WCCM VI in conjunction with APCOM04, Springer
[11] Finite deformations of Ogden's materials under impact loading, Int. J. Non-Linear Mech., Volume 41 (2006), pp. 575-585
[12] B. Magnain, Développement d'algorithmes et d'un code de calcul pour l'étude des problèmes de l'impact et du choc, PhD thesis, Université d'Evry – Val d'Essonne, 2006
[13] Modelling of safety glass for crash simulation, Comput. Mater. Sci., Volume 28 (2003), pp. 675-683
[14] Application of finite elastic theory to the deformation of rubbery materials, Trans. Soc. Rheology, Volume 6 (1962), pp. 223-251
[15] Non-Linear Elastic Deformations, Ellis Horwood, 1984
[16] Elasticité Tridimensionnelle, Collection RMA, Masson, 1985
[17] A three dimensional constitutive model for the large deformation stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, Volume 41 (1993), pp. 389-412
[18] A new constitutive relation for rubber, Rubber Chem. Technol., Volume 69 (1996), pp. 59-61
[19] Computational Inelasticity, Springer-Verlag, New York, 1998
[20] Finite thermoelasticity with limiting chain extensibility, J. Mech. Phys. Solids, Volume 51 (2003), pp. 1127-1146
[21] Nonlinear isotropic constitutive laws: choice of the three invariants, convex potentials and constitutive inequalities, Int. J. Engrg. Sci., Volume 37 (1999), pp. 1927-1941
[22] Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques, Gauthier–Villars, Paris, 1885
[23] Boussinesq's problem for a rigid cone, Quart. J. Math., Volume 10 (1939), p. 161
[24] A Treatise on the Mathematical Theory of Elasticity, University Press, Cambridge, 1952
[25] Rivlin's representation formula is ill-conceived for the determination of response functions via biaxial testing, J. Elasticity, Volume 70 (2003), pp. 129-147
[26] 2D or 3D frictional contact algorithms and applications in a large deformation context, Comm. Numer. Meth. Engrg., Volume 11 (1995), pp. 409-416
[27] New inequality and functional for contact with friction: The implicit standard material approach, Mech. Struct. Mach., Volume 19 (1991), pp. 301-325
[28] FER/impact: Logiciel de simulation numérique des problèmes d'impact, Rev. Europ. Mécanique Numérique, Volume 15 (2006), pp. 175-186
[29] http://gmfe16.cemif.univ-evry.fr:8080/~feng/FerImpact.html
[30] Improved implicit integrators for transient impact problems: dynamic frictional dissipation within an admissible conserving framework, Comp. Meth. Appl. Mech. Engrg., Volume 192 (2003), pp. 2223-2248
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