# American Institute of Mathematical Sciences

May  2012, 11(3): 1185-1203. doi: 10.3934/cpaa.2012.11.1185

## Analysis of a contact problem for electro-elastic-visco-plastic materials

 1 Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585 Craiova, Romania 2 Departement of Mathematics, University of Craiov, A.I. Cuza Street 13, 200585, Craiova 3 Laboratoire de Mathématiques et Physique pour les Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan

Received  December 2010 Revised  September 2011 Published  December 2011

We consider a mathematical model which describes the quasistatic frictionless contact between a piezoelectric body and a foundation. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive, the material's behavior is described with an electro-elastic-visco-plastic constitutive law, the contact is modelled with normal compliance and finite penetration and the problem is studied in an unbounded interval of time. We derive a variational formulation of the problem and prove existence, uniqueness and regularity results. The proofs are based on recent results on history-dependent quasivariational inequalities obtained in [21].
Citation: Maria-Magdalena Boureanu, Andaluzia Matei, Mircea Sofonea. Analysis of a contact problem for electro-elastic-visco-plastic materials. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1185-1203. doi: 10.3934/cpaa.2012.11.1185
##### References:
 [1] M. Barboteu and M. Sofonea, Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation, Applied Mathematics and Computation, 215 (2009), 2978-2991. doi: 10.1016/j.amc.2009.09.045. [2] M. Barboteu and M. Sofonea, Modelling and analysis of the unilateral contact of a piezoelectric body with a conductive support, Journal of Mathematical Analysis and Applications, 358 (2009), 110-124. doi: 10.1016/j.jmaa.2009.04.030. [3] R.C. Batra and J.S. Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity, 38 (1995), 209-218. doi: 10.1007/BF00042498. [4] P. Bisegna, F. Lebon and F. Maceri, The unilateral frictional contact of a piezoelectric body with a rigid support, in "Contact Mechanics" (eds. J.A.C. Martins and Manuel D.P. Monteiro Marques), Kluwer, Dordrecht, (2002), 347-354. [5] T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electroelasticity, Memoirs on Differential Equations and Mathematical Physics, 10 (1997), 1-53. [6] N. Cristescu and I. Suliciu, "Viscoplasticity," Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982. [7] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," Studies in Advanced Mathematics 30, Americal Mathematical Society-International Press, 2002. [8] W. Han, Sofonea and K. Kazmi, Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3915-3926. doi: 10.1016/j.cma.2006.10.051. [9] I.R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity," Oxford University Press, Oxford, 1993. [10] J. Jarusek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift für Angewandte Matematik und Mechanik, ($ZAMM$) 88 (2008), 3-22. doi: 10.1002/zamm.200710360. [11] Z. Lerguet, M. Shillor and M. Sofonea, A frictional contact problem for an electro-viscoelastic body, Electronic Journal of Differential Equations, 170 (2007), 1-16. [12] F. Maceri and P. Bisegna, The unilateral frictionless contact of apiezoelectric body with a rigid support, Mathematical and Computer Modelling, 28 (1998), 19-28. doi: 10.1016/S0895-7177(98)00105-8. [13] R.D. Mindlin, Polarisation gradient in elastic dielectrics, International Journal of Solids and Structures, 4 (1968), 637-663. doi: 10.1016/0020-7683(68)90079-6. [14] R.D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, International Journal of Solids and Structures, 4 (1969), 1197-1213. doi: 10.1016/0020-7683(69)90053-5. [15] R.D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics, Journal of Elasticity, 4 (1972), 217-280. doi: 10.1007/BF00045712. [16] M. Shillor, M. Sofonea and J.J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes in Physics, 655, Springer, Berlin, 2004. [17] M. Sofonea and El H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Mathematical Modelling and Analysis, 9 (2004), 229-242. doi: 10.1080/13926292.2004.9637256. [18] M. Sofonea and El H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body, Advances in Mathematical Sciences and Applications, 14 (2004), 613-631. [19] L. Solymar and L.B. Au, "Solutions Manual for Lectures on the Electrical Properties of Materials," 5th Fifth Edition, Oxford University Press, Oxford, 1993. [20] 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), 645-658. doi: 10.3934/cpaa.2008.7.645. [21] M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in Contact Mechanics, European Journal of Applied Mathematics, 22 (2011), 471-491. doi: 10.1017/S0956792511000192. [22] R.A. Toupin, The elastic dielectrics, Journal of Rational Mechanics and Analysis, 5 (1956), 849-915. doi: 10.1512/iumj.1956.5.55033. [23] R.A. Toupin, A dynamical theory of elastic dielectrics, International Journal of Engineering Sciences, 1 (1963), 101-126. doi: 10.1016/0020-7225(63)90027-2. [24] W. Voigt, "Lehrbuch der Kristall-Physik," Teubner, Leipzig, 1910.

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##### References:
 [1] M. Barboteu and M. Sofonea, Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation, Applied Mathematics and Computation, 215 (2009), 2978-2991. doi: 10.1016/j.amc.2009.09.045. [2] M. Barboteu and M. Sofonea, Modelling and analysis of the unilateral contact of a piezoelectric body with a conductive support, Journal of Mathematical Analysis and Applications, 358 (2009), 110-124. doi: 10.1016/j.jmaa.2009.04.030. [3] R.C. Batra and J.S. Yang, Saint-Venant's principle in linear piezoelectricity, Journal of Elasticity, 38 (1995), 209-218. doi: 10.1007/BF00042498. [4] P. Bisegna, F. Lebon and F. Maceri, The unilateral frictional contact of a piezoelectric body with a rigid support, in "Contact Mechanics" (eds. J.A.C. Martins and Manuel D.P. Monteiro Marques), Kluwer, Dordrecht, (2002), 347-354. [5] T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electroelasticity, Memoirs on Differential Equations and Mathematical Physics, 10 (1997), 1-53. [6] N. Cristescu and I. Suliciu, "Viscoplasticity," Martinus Nijhoff Publishers, Editura Tehnica, Bucharest, 1982. [7] W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity," Studies in Advanced Mathematics 30, Americal Mathematical Society-International Press, 2002. [8] W. Han, Sofonea and K. Kazmi, Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3915-3926. doi: 10.1016/j.cma.2006.10.051. [9] I.R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity," Oxford University Press, Oxford, 1993. [10] J. Jarusek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems, Zeitschrift für Angewandte Matematik und Mechanik, ($ZAMM$) 88 (2008), 3-22. doi: 10.1002/zamm.200710360. [11] Z. Lerguet, M. Shillor and M. Sofonea, A frictional contact problem for an electro-viscoelastic body, Electronic Journal of Differential Equations, 170 (2007), 1-16. [12] F. Maceri and P. Bisegna, The unilateral frictionless contact of apiezoelectric body with a rigid support, Mathematical and Computer Modelling, 28 (1998), 19-28. doi: 10.1016/S0895-7177(98)00105-8. [13] R.D. Mindlin, Polarisation gradient in elastic dielectrics, International Journal of Solids and Structures, 4 (1968), 637-663. doi: 10.1016/0020-7683(68)90079-6. [14] R.D. Mindlin, Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, International Journal of Solids and Structures, 4 (1969), 1197-1213. doi: 10.1016/0020-7683(69)90053-5. [15] R.D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics, Journal of Elasticity, 4 (1972), 217-280. doi: 10.1007/BF00045712. [16] M. Shillor, M. Sofonea and J.J. Telega, "Models and Analysis of Quasistatic Contact," Lecture Notes in Physics, 655, Springer, Berlin, 2004. [17] M. Sofonea and El H. Essoufi, A piezoelectric contact problem with slip dependent coefficient of friction, Mathematical Modelling and Analysis, 9 (2004), 229-242. doi: 10.1080/13926292.2004.9637256. [18] M. Sofonea and El H. Essoufi, Quasistatic frictional contact of a viscoelastic piezoelectric body, Advances in Mathematical Sciences and Applications, 14 (2004), 613-631. [19] L. Solymar and L.B. Au, "Solutions Manual for Lectures on the Electrical Properties of Materials," 5th Fifth Edition, Oxford University Press, Oxford, 1993. [20] 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), 645-658. doi: 10.3934/cpaa.2008.7.645. [21] M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in Contact Mechanics, European Journal of Applied Mathematics, 22 (2011), 471-491. doi: 10.1017/S0956792511000192. [22] R.A. Toupin, The elastic dielectrics, Journal of Rational Mechanics and Analysis, 5 (1956), 849-915. doi: 10.1512/iumj.1956.5.55033. [23] R.A. Toupin, A dynamical theory of elastic dielectrics, International Journal of Engineering Sciences, 1 (1963), 101-126. doi: 10.1016/0020-7225(63)90027-2. [24] W. Voigt, "Lehrbuch der Kristall-Physik," Teubner, Leipzig, 1910.
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