Quasistatic viscoelastic contact with friction and wear diffusion
Authors:
M. Shillor, M. Sofonea and J. J. Telega
Journal:
Quart. Appl. Math. 62 (2004), 379-399
MSC:
Primary 74M15; Secondary 74D10, 74H20, 74M10, 74R99
DOI:
https://doi.org/10.1090/qam/2054605
MathSciNet review:
MR2054605
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider a quasistatic problem of frictional contact between a deformable body and a moving foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated law of dry friction. The wear takes place on a part of the contact surface and its rate is described by the Archard differential condition. The main novelty in the model is the diffusion of the wear particles over the potential contact surface. Such phenomena arise in orthopaedic biomechanics where the wear debris diffuse and influence the properties of joint prosthesis and implants. We derive a weak formulation of the model which is given by a coupled system with an evolutionary variational inequality and a nonlinear evolutionary variational equation. We prove that, under a smallness assumption on some of the data, there exists a unique weak solution for the model.
- Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- Amina Amassad and Caroline Fabre, Existence for viscoplastic contact with Coulomb friction problems, Int. J. Math. Math. Sci. 32 (2002), no. 7, 411–437. MR 1942231, DOI https://doi.org/10.1155/S0161171202110179
- Amina Amassad and Mircea Sofonea, Analysis of a quasistatic viscoplastic problem involving Tresca friction law, Discrete Contin. Dynam. Systems 4 (1998), no. 1, 55–72. MR 1485363, DOI https://doi.org/10.3934/dcds.1998.4.55
- Lars-Erik Andersson, A quasistatic frictional problem with normal compliance, Nonlinear Anal. 16 (1991), no. 4, 347–369. MR 1093846, DOI https://doi.org/10.1016/0362-546X%2891%2990035-Y
- Kevin T. Andrews, M. Shillor, S. Wright, and A. Klarbring, A dynamic thermoviscoelastic contact problem with friction and wear, Internat. J. Engrg. Sci. 35 (1997), no. 14, 1291–1309. MR 1488812, DOI https://doi.org/10.1016/S0020-7225%2897%2987426-5
- K. T. Andrews, K. L. Kuttler, and M. Shillor, On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, European J. Appl. Math. 8 (1997), no. 4, 417–436. MR 1471601, DOI https://doi.org/10.1017/S0956792597003173
- Marius Cocu, Elaine Pratt, and Michel Raous, Formulation and approximation of quasistatic frictional contact, Internat. J. Engrg. Sci. 34 (1996), no. 7, 783–798. MR 1397607, DOI https://doi.org/10.1016/0020-7225%2895%2900121-2
D. Dowson (Ed.), Advances in Mechanical Tribology, Mechanical Engineering Publications, London, 1999.
- Weimin Han, Meir Shillor, and Mircea Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. Comput. Appl. Math. 137 (2001), no. 2, 377–398. MR 1865238, DOI https://doi.org/10.1016/S0377-0427%2800%2900707-X
- Weimin Han and Mircea Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM J. Numer. Anal. 38 (2000), no. 2, 556–579. MR 1770062, DOI https://doi.org/10.1137/S0036142998347309
- Weimin Han and Mircea Sofonea, Quasistatic contact problems in viscoelasticity and viscoplasticity, AMS/IP Studies in Advanced Mathematics, vol. 30, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002. MR 1935666
- Lars Johansson and Anders Klarbring, Thermoelastic frictional contact problems: modelling, finite element approximation and numerical realization, Comput. Methods Appl. Mech. Engrg. 105 (1993), no. 2, 181–210. MR 1220080, DOI https://doi.org/10.1016/0045-7825%2893%2990122-E
- N. Kikuchi and J. T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics, vol. 8, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. MR 961258
- A. Klarbring, A. Mikelić, and M. Shillor, Frictional contact problems with normal compliance, Internat. J. Engrg. Sci. 26 (1988), no. 8, 811–832. MR 958441, DOI https://doi.org/10.1016/0020-7225%2888%2990032-8
- A. Klarbring, A. Mikelić, and M. Shillor, The rigid punch problem with friction, Internat. J. Engrg. Sci. 29 (1991), no. 6, 751–768. MR 1107199, DOI https://doi.org/10.1016/0020-7225%2891%2990104-B
- J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal. 11 (1987), no. 3, 407–428. MR 881727, DOI https://doi.org/10.1016/0362-546X%2887%2990055-1
- J. A. C. Martins and Manuel D. P. Monteiro Marques (eds.), Contact mechanics, Solid Mechanics and its Applications, vol. 103, Kluwer Academic Publishers Group, Dordrecht, 2002. MR 1968651
M. Raous, M. Jean, and J. J. Moreau (Eds.), Contact Mechanics, Plenum Press, New York, 1995.
- M. Rochdi, M. Shillor, and M. Sofonea, Quasistatic viscoelastic contact with normal compliance and friction, J. Elasticity 51 (1998), no. 2, 105–126. MR 1664496, DOI https://doi.org/10.1023/A%3A1007413119583
J. Rojek and J. J. Telega, Contact problems with friction, adhesion and wear in orthopaedic biomechanics I: General developments, J. Theor. Appl. Mech. 39 (2001), 655–677.
J. Rojek, J. J. Telega, and S. Stupkiewicz, Contact problems with friction, adhesion and wear in orthopaedic biomechanics. II: Numerical implementation and application to implanted knee joints, J. Theor. Appl. Mech. 39 (2001), 680–706.
M. Shillor (Ed.), Recent advances in contact mechanics, Special issue of Math. Comput. Modelling, 28 (4–8) (1998).
- Meir Shillor and Mircea Sofonea, A quasistatic viscoelastic contact problem with friction, Internat. J. Engrg. Sci. 38 (2000), no. 14, 1517–1533. MR 1763038, DOI https://doi.org/10.1016/S0020-7225%2899%2900126-3
M. Shillor, M. Sofonea, and J. Telega, Models and Analysis of Quasistatic Contact, in preparation, 2004.
M. Shillor, M. Sofonea, and J. Telega, Analysis of viscoelastic contact with normal compliance friction and wear diffusion, C. R. Mecanique, 331 (2003), 395–400.
- Mircea Sofonea and Meir Shillor, Variational analysis of quasistatic viscoplastic contact problems with friction, Commun. Appl. Anal. 5 (2001), no. 1, 135–151. MR 1844677
- Mircea Sofonea, On a contact problem for elastic-viscoplastic bodies, Nonlinear Anal. 29 (1997), no. 9, 1037–1050. MR 1460437, DOI https://doi.org/10.1016/S0362-546X%2896%2900096-X
M. Sofonea and A. Matei, A fixed point result for operators defined on spaces of vector-valued continuous functions, Ann. Univ. Craiova, Series Math.-Info., to appear.
N. Strömberg, Continuum Thermodynamics of Contact, Friction and Wear, Ph.D. Thesis, Linköping University, Sweden, 1995.
- Niclas Strömberg, Lars Johansson, and Anders Klarbring, Derivation and analysis of a generalized standard model for contact, friction and wear, Internat. J. Solids Structures 33 (1996), no. 13, 1817–1836. MR 1392130, DOI https://doi.org/10.1016/0020-7683%2895%2900140-9
- Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497
A. Zmitrowicz (2001), Variational descriptions of wearing out solids and wear particles in contact mechanics, J. Theor. Appl. Mech. 39 (3), 791–808.
R. S. Adams, Sobolev Spaces, Academic Press, New York, 1975.
A. Amassad and C. Fabre, Existence for viscoplastic contact with Coulomb friction problems, Intl. J. Mathematics and Mathematical Sciences, to appear.
A. Amassad and M. Sofonea, Analysis of a quasistatic viscoplastic problem involving Tresca friction law, Discrete and Continuous Dynamical Systems, 4 (1998), 55–72.
L.-E. Andersson, A quasistatic frictional problem with normal compliance, Nonlinear Anal., 16 (1991), 347–370.
K. T. Andrews, A. Klarbring, M. Shillor, and S. Wright, A dynamic thermoviscoelastic contact problem with friction and wear, Int. J. Engng. Sci., 35 (1997), 1291–1309.
K. T. Andrews, K. L. Kuttler, and M. Shillor, On the dynamic behavior of a thermoviscoelastic body in frictional contact with a rigid obstacle, Euro. J. Appl. Math. 8 (1997), 417–436.
M. Cocu, E. Pratt, and M. Raous, Formulation and approximation of quasistatic frictional contact, Intl. J. Engng. Sci., 34 (1996), 783–798.
D. Dowson (Ed.), Advances in Mechanical Tribology, Mechanical Engineering Publications, London, 1999.
W. Han, M. Shillor, and M. Sofonea, Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, Journal of Computational and Applied Mathematics, 137 (2001), 377–398.
W. Han and M. Sofonea, Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM Journal of Numerical Analysis 38 (2000), 556–579.
W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, AMS/IP Studies in Advanced Mathematics, 30, American Mathematical Society—Intl. Press, Providence, 2002.
L. Johansson and A. Klarbring, Thermoelastic frictional contact problems: Modelling, finite element approximation and numerical realization, Comp. Meth. Appl. Mech. Eng., 105 (1993), 181–210.
N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, 1988.
A. Klarbring, A. Mikelic, and M. Shillor, Frictional contact problems with normal compliance, Intl. J. Engng. Sci., 26 (1988), 811–832.
A. Klarbring, A. Mikelic, and M. Shillor, The rigid punch problem with friction, Intl. J. Engng. Sci., 29 (1991), 751–768.
J. A. C. Martins and T. J. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear Anal., 11 (1987), 407–428.
J. A. C. Martins and Manuel D. P. Monteiro Marqnes (Eds.), Contact Mechanics, Kluwer Academic Publishers, Dordrecht, 2002.
M. Raous, M. Jean, and J. J. Moreau (Eds.), Contact Mechanics, Plenum Press, New York, 1995.
M. Rochdi, M. Shillor, and M. Sofonea, A quasistatic viscoelastic contact problem with normal compliance and friction, J. Elasticity, 51 (1998), 105–126.
J. Rojek and J. J. Telega, Contact problems with friction, adhesion and wear in orthopaedic biomechanics I: General developments, J. Theor. Appl. Mech. 39 (2001), 655–677.
J. Rojek, J. J. Telega, and S. Stupkiewicz, Contact problems with friction, adhesion and wear in orthopaedic biomechanics. II: Numerical implementation and application to implanted knee joints, J. Theor. Appl. Mech. 39 (2001), 680–706.
M. Shillor (Ed.), Recent advances in contact mechanics, Special issue of Math. Comput. Modelling, 28 (4–8) (1998).
M. Shillor and M. Sofonea, A quasistatic viscoelastic contact problem with friction, Int. J. Engng. Sci., 38 (2000), 1517–1533.
M. Shillor, M. Sofonea, and J. Telega, Models and Analysis of Quasistatic Contact, in preparation, 2004.
M. Shillor, M. Sofonea, and J. Telega, Analysis of viscoelastic contact with normal compliance friction and wear diffusion, C. R. Mecanique, 331 (2003), 395–400.
M. Sofonea and M. Shillor, Variational analysis of quasistatic viscoplastic contact problems with friction, Communications in Applied Analysis 5 (2001), 135–151.
M. Sofonea, On a contact problem for elastic-viscoplastic bodies, Nonlinear Anal., 29 (1997), 1037–1050.
M. Sofonea and A. Matei, A fixed point result for operators defined on spaces of vector-valued continuous functions, Ann. Univ. Craiova, Series Math.-Info., to appear.
N. Strömberg, Continuum Thermodynamics of Contact, Friction and Wear, Ph.D. Thesis, Linköping University, Sweden, 1995.
N. Strömberg, L. Johansson, and A. Klarbring, Derivation and analysis of a generalized standard model for contact friction and wear, Intl. J. Solids Structures, 33 (1996), 1817–1836.
E. Zeidler, Nonlinear Functional Analysis and its Applications, II/A: Linear Monotone Operators, Springer-Verlag, New York, 1990.
A. Zmitrowicz (2001), Variational descriptions of wearing out solids and wear particles in contact mechanics, J. Theor. Appl. Mech. 39 (3), 791–808.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74M15,
74D10,
74H20,
74M10,
74R99
Retrieve articles in all journals
with MSC:
74M15,
74D10,
74H20,
74M10,
74R99
Additional Information
Article copyright:
© Copyright 2004
American Mathematical Society