# American Institute of Mathematical Sciences

• Previous Article
Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations
• CPAA Home
• This Issue
• Next Article
Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis
January  2014, 13(1): 371-387. doi: 10.3934/cpaa.2014.13.371

## A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient

 1 Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan 2 Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309

Received  April 2013 Revised  June 2013 Published  July 2013

We consider a mathematical model that describes frictionless contact between a viscoplastic body and a deformable obstacle or foundation. The process is quasistatic and contact is modeled with the normal compliance with limited penetration condition, which has been introduced recently. Moreover, the contact stiffness coefficient is allowed to depend on the history of the contact process. We derive a variational formulation of the problem, which is in the form of a strongly nonlinear system coupling an integral equation and a time-dependent variational inequality. Then, we provide the analysis of the problem, which includes its unique weak solvability and the continuous dependence of the solution on the problem data. The proofs are based on results from the theory of history-dependent variational inequalities, on monotonicity and a fixed point argument.
Citation: Mircea Sofonea, Meir Shillor. A viscoplastic contact problem with a normal compliance with limited penetration condition and history-dependent stiffness coefficient. Communications on Pure & Applied Analysis, 2014, 13 (1) : 371-387. doi: 10.3934/cpaa.2014.13.371
##### References:
 [1] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance,, Quarterly Journal of Mechanics and Applied Mathematics, 65 (2012), 555. Google Scholar [2] M. Barboteu, A. Matei and M. Sofonea, On the behaviours of the solution of a viscoplastic contact problem,, Quarterly of Applied Mathematics, (). Google Scholar [3] N. Cristescu and I. Suliciu, "Viscoplasticity,", Martinus Nijhoff Publishers, (1982). Google Scholar [4] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Springer-Verlag, (1976). Google Scholar [5] C. Eck, J. Jarušek and M. Krbeč, "Unilateral Contact, Problems: Variational Methods and Existence Theorems, 270 (2005). Google Scholar [6] J. Jarušek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems,, Zeitschrift f\, 88 (2008), 3. Google Scholar [7] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance,, Int. J. Engng. Sci., 26 (1988), 811. Google Scholar [8] A. Klarbring, A. Mikelic and M. Shillor, On friction problems with normal compliance,, Nonlinear Analysis, 13 (1989), 935. Google Scholar [9] T. Laursen, "Computational Contact and Impact Mechanics,", Springer, (2002). Google Scholar [10] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws,, Nonlinear Analysis TMA, 11 (1987), 407. Google Scholar [11] J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena,, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527. Google Scholar [12] J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons,, Vehicle System Dynamics, 48 (2010), 675. doi: 10.1080/00423110903126478. Google Scholar [13] M. Shillor, M. Sofonea and J. Telega, "Models and Variational Analysis of Quasistatic Contact,", Lecture Notes in Physics {\bf655}, (2004). doi: 10.1007/b99799. Google Scholar [14] M. Sofonea and A. Matei, A mixed variational formulation for the Signorini frictionless problem in viscoplasticity,, Annals Univ. Ovidius Constanta, 12 (2004), 157. Google Scholar [15] M. Sofonea and A. Matei, "Mathematical Models in Contact Mechanics,", London Mathematical Society Lecture Note Series {\bf 398}, 398 (2012). doi: 10.1017/CBO9781139104166. Google Scholar [16] P. Wriggers, "Computational Contact Mechanics,", Wiley, (2002). doi: PMCid:PMC123642. Google Scholar

show all references

##### References:
 [1] M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance,, Quarterly Journal of Mechanics and Applied Mathematics, 65 (2012), 555. Google Scholar [2] M. Barboteu, A. Matei and M. Sofonea, On the behaviours of the solution of a viscoplastic contact problem,, Quarterly of Applied Mathematics, (). Google Scholar [3] N. Cristescu and I. Suliciu, "Viscoplasticity,", Martinus Nijhoff Publishers, (1982). Google Scholar [4] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,", Springer-Verlag, (1976). Google Scholar [5] C. Eck, J. Jarušek and M. Krbeč, "Unilateral Contact, Problems: Variational Methods and Existence Theorems, 270 (2005). Google Scholar [6] J. Jarušek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems,, Zeitschrift f\, 88 (2008), 3. Google Scholar [7] A. Klarbring, A. Mikelic and M. Shillor, Frictional contact problems with normal compliance,, Int. J. Engng. Sci., 26 (1988), 811. Google Scholar [8] A. Klarbring, A. Mikelic and M. Shillor, On friction problems with normal compliance,, Nonlinear Analysis, 13 (1989), 935. Google Scholar [9] T. Laursen, "Computational Contact and Impact Mechanics,", Springer, (2002). Google Scholar [10] J. A. C. Martins and J. T. Oden, Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws,, Nonlinear Analysis TMA, 11 (1987), 407. Google Scholar [11] J. T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena,, Computer Methods in Applied Mechanics and Engineering, 52 (1985), 527. Google Scholar [12] J. Piotrowski, Smoothing dry friction damping by dither generated in rolling contact of wheel and rail and its influence on ride dynamics of freight wagons,, Vehicle System Dynamics, 48 (2010), 675. doi: 10.1080/00423110903126478. Google Scholar [13] M. Shillor, M. Sofonea and J. Telega, "Models and Variational Analysis of Quasistatic Contact,", Lecture Notes in Physics {\bf655}, (2004). doi: 10.1007/b99799. Google Scholar [14] M. Sofonea and A. Matei, A mixed variational formulation for the Signorini frictionless problem in viscoplasticity,, Annals Univ. Ovidius Constanta, 12 (2004), 157. Google Scholar [15] M. Sofonea and A. Matei, "Mathematical Models in Contact Mechanics,", London Mathematical Society Lecture Note Series {\bf 398}, 398 (2012). doi: 10.1017/CBO9781139104166. Google Scholar [16] P. Wriggers, "Computational Contact Mechanics,", Wiley, (2002). doi: PMCid:PMC123642. Google Scholar
 [1] Xiaoliang Cheng, Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of two quasistatic history-dependent contact models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2425-2445. doi: 10.3934/dcdsb.2014.19.2425 [2] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 [3] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [4] María Teresa Cao-Rial, Peregrina Quintela, Carlos Moreno. Numerical solution of a time-dependent Signorini contact problem. Conference Publications, 2007, 2007 (Special) : 201-211. doi: 10.3934/proc.2007.2007.201 [5] Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621 [6] Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645 [7] Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 [8] Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523 [9] Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549 [10] Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 [11] Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133 [12] Yalin Zhang, Guoliang Shi. Continuous dependence of the transmission eigenvalues in one dimension. Inverse Problems & Imaging, 2015, 9 (1) : 273-287. doi: 10.3934/ipi.2015.9.273 [13] Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469 [14] Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419 [15] Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437 [16] P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677 [17] Stanislaw Migórski. Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1339-1356. doi: 10.3934/dcdsb.2006.6.1339 [18] Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557 [19] Pavel Krejčí, Thomas Roche. Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 637-650. doi: 10.3934/dcdsb.2011.15.637 [20] Ramon Quintanilla. Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 463-470. doi: 10.3934/dcdsb.2001.1.463

2018 Impact Factor: 0.925