December  2013, 6(6): 1587-1598. doi: 10.3934/dcdss.2013.6.1587

Dual formulation of a viscoplastic contact problem with unilateral constraint

1. 

Departement of Mathematics, University of Craiov, A.I. Cuza Street 13, 200585, Craiova

2. 

Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received  June 2012 Revised  September 2012 Published  April 2013

We consider a mathematical model which describes the contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the contact is frictionless and is modelled with unilateral constraint. We derive a variational formulation of the model which leads to a history-dependent quasivariational inequality for stress field, associated to a time-dependent convex. Then we prove the unique weak solvability of the model. The proof is based on an abstract existence and uniqueness result obtained in [11].
Citation: Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587
References:
[1]

M. Anders, "Dual-Dual Formulations for Frictional Contact Problems in Mechanics,", Ph.D thesis, (2011). Google Scholar

[2]

B. Awbi, M. Shillor and M. Sofonea, Dual formulation of a quasistatic viscoelastic contact problem with Tresca's friction law,, Applicable Analysis, 79 (2001), 1. doi: 10.1080/00036810108840949. Google Scholar

[3]

M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance,, Q. J. Mechanics Appl. Math., 65 (2012), 555. doi: 10.1093/qjmam/hbs016. Google Scholar

[4]

N. Cristescu and I. Suliciu, "Viscoplasticity,", Translated from the Romanian, 5 (1982). Google Scholar

[5]

W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,", AMS/IP Studies in Advanced Mathematics, 30 (2002). Google Scholar

[6]

I. Hlaváček, J. Haslinger, J. Nečas and J. Lovášek, "Solution of Variational Inequalities in Mechanics,", Translated from the Slovak by J. Jarník, 66 (1988). doi: 10.1007/978-1-4612-1048-1. Google Scholar

[7]

I. R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity,", Oxford Science Publications, (1993). Google Scholar

[8]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods,", SIAM Studies in Applied Mathematics, 8 (1988). Google Scholar

[9]

M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (2004). doi: 10.1007/b99799. Google Scholar

[10]

M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems,, Communications on Pure and Applied Analysis, 7 (2008), 645. doi: 10.3934/cpaa.2008.7.645. Google Scholar

[11]

M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics,, European Journal of Applied Mathematics, 22 (2011), 471. doi: 10.1017/S0956792511000192. Google Scholar

[12]

J. J. Telega, Topics on unilateral contact problems of elasticity and inelasticity,, in, (1988), 340. Google Scholar

show all references

References:
[1]

M. Anders, "Dual-Dual Formulations for Frictional Contact Problems in Mechanics,", Ph.D thesis, (2011). Google Scholar

[2]

B. Awbi, M. Shillor and M. Sofonea, Dual formulation of a quasistatic viscoelastic contact problem with Tresca's friction law,, Applicable Analysis, 79 (2001), 1. doi: 10.1080/00036810108840949. Google Scholar

[3]

M. Barboteu, A. Matei and M. Sofonea, Analysis of quasistatic viscoplastic contact problems with normal compliance,, Q. J. Mechanics Appl. Math., 65 (2012), 555. doi: 10.1093/qjmam/hbs016. Google Scholar

[4]

N. Cristescu and I. Suliciu, "Viscoplasticity,", Translated from the Romanian, 5 (1982). Google Scholar

[5]

W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,", AMS/IP Studies in Advanced Mathematics, 30 (2002). Google Scholar

[6]

I. Hlaváček, J. Haslinger, J. Nečas and J. Lovášek, "Solution of Variational Inequalities in Mechanics,", Translated from the Slovak by J. Jarník, 66 (1988). doi: 10.1007/978-1-4612-1048-1. Google Scholar

[7]

I. R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity,", Oxford Science Publications, (1993). Google Scholar

[8]

N. Kikuchi and J. T. Oden, "Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods,", SIAM Studies in Applied Mathematics, 8 (1988). Google Scholar

[9]

M. Shillor, M. Sofonea and J. J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (2004). doi: 10.1007/b99799. Google Scholar

[10]

M. Sofonea, C. Avramescu and A. Matei, A fixed point result with applications in the study of viscoplastic frictionless contact problems,, Communications on Pure and Applied Analysis, 7 (2008), 645. doi: 10.3934/cpaa.2008.7.645. Google Scholar

[11]

M. Sofonea and A. Matei, History-dependent quasi-variational inequalities arising in contact mechanics,, European Journal of Applied Mathematics, 22 (2011), 471. doi: 10.1017/S0956792511000192. Google Scholar

[12]

J. J. Telega, Topics on unilateral contact problems of elasticity and inelasticity,, in, (1988), 340. Google Scholar

[1]

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

[2]

Yunan Wu, Guangya Chen, T. C. Edwin Cheng. A vector network equilibrium problem with a unilateral constraint. Journal of Industrial & Management Optimization, 2010, 6 (3) : 453-464. doi: 10.3934/jimo.2010.6.453

[3]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[4]

Stanislaw Migórski, Anna Ochal, Mircea Sofonea. Analysis of a dynamic Elastic-Viscoplastic contact problem with friction. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 887-902. doi: 10.3934/dcdsb.2008.10.887

[5]

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

[6]

Nelly Point, Silvano Erlicher. Pseudo-potentials and bipotential: A constructive procedure for non-associated plasticity and unilateral contact. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 567-590. doi: 10.3934/dcdss.2013.6.567

[7]

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

[8]

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

[9]

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

[10]

Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116

[11]

Jeongho Ahn, David E. Stewart. A viscoelastic Timoshenko beam with dynamic frictionless impact. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 1-22. doi: 10.3934/dcdsb.2009.12.1

[12]

Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553

[13]

Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10

[14]

Samir Adly, Daniel Goeleven, Dumitru Motreanu. Periodic and homoclinic solutions for a class of unilateral problems. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 579-590. doi: 10.3934/dcds.1997.3.579

[15]

Simone Göttlich, Sebastian Kühn, Jan Peter Ohst, Stefan Ruzika, Markus Thiemann. Evacuation dynamics influenced by spreading hazardous material. Networks & Heterogeneous Media, 2011, 6 (3) : 443-464. doi: 10.3934/nhm.2011.6.443

[16]

Riccarda Rossi. Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1413-1466. doi: 10.3934/dcdss.2017075

[17]

Amina Amassad, Mircea Sofonea. Analysis of a quasistatic viscoplastic problem involving tresca friction law. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 55-72. doi: 10.3934/dcds.1998.4.55

[18]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85

[19]

Wenguo Shen. Unilateral global interval bifurcation for Kirchhoff type problems and its applications. Communications on Pure & Applied Analysis, 2018, 17 (1) : 21-37. doi: 10.3934/cpaa.2018002

[20]

François Gay-Balmaz, Tudor S. Ratiu. Clebsch optimal control formulation in mechanics. Journal of Geometric Mechanics, 2011, 3 (1) : 41-79. doi: 10.3934/jgm.2011.3.41

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]