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 & 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. 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. 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. 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, (2002), 347.

[5]

T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electroelasticity,, Memoirs on Differential Equations and Mathematical Physics, 10 (1997), 1.

[6]

N. Cristescu and I. Suliciu, "Viscoplasticity,", Martinus Nijhoff Publishers, (1982).

[7]

W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,", Studies in Advanced Mathematics \textbf{30}, 30 (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. doi: 10.1016/j.cma.2006.10.051.

[9]

I.R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity,", Oxford University Press, (1993).

[10]

J. Jarusek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems,, Zeitschrift für Angewandte Matematik und Mechanik, 88 (2008), 3. 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.

[12]

F. Maceri and P. Bisegna, The unilateral frictionless contact of apiezoelectric body with a rigid support,, Mathematical and Computer Modelling, 28 (1998), 19. 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. 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. doi: 10.1016/0020-7683(69)90053-5.

[15]

R.D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics,, Journal of Elasticity, 4 (1972), 217. doi: 10.1007/BF00045712.

[16]

M. Shillor, M. Sofonea and J.J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (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. 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.

[19]

L. Solymar and L.B. Au, "Solutions Manual for Lectures on the Electrical Properties of Materials,", 5th Fifth Edition, (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. 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. doi: 10.1017/S0956792511000192.

[22]

R.A. Toupin, The elastic dielectrics,, Journal of Rational Mechanics and Analysis, 5 (1956), 849. 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. doi: 10.1016/0020-7225(63)90027-2.

[24]

W. Voigt, "Lehrbuch der Kristall-Physik,", Teubner, (1910).

show all references

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. 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. 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. 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, (2002), 347.

[5]

T. Buchukuri and T. Gegelia, Some dynamic problems of the theory of electroelasticity,, Memoirs on Differential Equations and Mathematical Physics, 10 (1997), 1.

[6]

N. Cristescu and I. Suliciu, "Viscoplasticity,", Martinus Nijhoff Publishers, (1982).

[7]

W. Han and M. Sofonea, "Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity,", Studies in Advanced Mathematics \textbf{30}, 30 (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. doi: 10.1016/j.cma.2006.10.051.

[9]

I.R. Ionescu and M. Sofonea, "Functional and Numerical Methods in Viscoplasticity,", Oxford University Press, (1993).

[10]

J. Jarusek and M. Sofonea, On the solvability of dynamic elastic-visco-plastic contact problems,, Zeitschrift für Angewandte Matematik und Mechanik, 88 (2008), 3. 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.

[12]

F. Maceri and P. Bisegna, The unilateral frictionless contact of apiezoelectric body with a rigid support,, Mathematical and Computer Modelling, 28 (1998), 19. 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. 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. doi: 10.1016/0020-7683(69)90053-5.

[15]

R.D. Mindlin, Elasticity, piezoelasticity and crystal lattice dynamics,, Journal of Elasticity, 4 (1972), 217. doi: 10.1007/BF00045712.

[16]

M. Shillor, M. Sofonea and J.J. Telega, "Models and Analysis of Quasistatic Contact,", Lecture Notes in Physics, 655 (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. 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.

[19]

L. Solymar and L.B. Au, "Solutions Manual for Lectures on the Electrical Properties of Materials,", 5th Fifth Edition, (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. 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. doi: 10.1017/S0956792511000192.

[22]

R.A. Toupin, The elastic dielectrics,, Journal of Rational Mechanics and Analysis, 5 (1956), 849. 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. doi: 10.1016/0020-7225(63)90027-2.

[24]

W. Voigt, "Lehrbuch der Kristall-Physik,", Teubner, (1910).

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