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Antiplane shear deformation of piezoelectric bodies in contact with a conductive support

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Abstract

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive support. We model the material’s behavior with an electro-elastic constitutive law; the frictional contact is described with a boundary condition involving Clarke’s generalized gradient and the electrical condition on the contact surface is modelled using the subdifferential of a proper, convex and lower semicontinuous function. We derive a variational formulation of the model and then, using a fixed point theorem for set valued mappings, we prove the existence of at least one weak solution. Finally, the uniqueness of the solution is discussed; the investigation is based on arguments in the theory of variational-hemivariational inequalities.

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References

  1. Andrei I., Costea N.: Nonlinear hemivariational inequalities and applications to nonsmooth mechanics. Adv. Nonlinear Var. Inequal. 13(1), 1–17 (2010)

    Google Scholar 

  2. Barboteu M., Sofonea M.: Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support. J. Math. Anal. Appl. 358, 110–124 (2009)

    Article  Google Scholar 

  3. Batra R.C., Yang J.S.: Saint-Vernant’s principle in linear piezoelectricity. J. Elasticity 38, 209–218 (1995)

    Article  Google Scholar 

  4. Borrelli A., Horgan C.O., Patria M.C.: Saint-Vernant’s principle for antiplane shear deformations of linear piezoelectric materials. SIAM J. Appl. Math. 62, 2027–2044 (2002)

    Article  Google Scholar 

  5. Boureanu M., Matei A.: Weak solutions for antiplane models involving elastic materials with degeneracies. Z. Angew. Math. Phys. 61, 73–85 (2010)

    Article  Google Scholar 

  6. Brezis H.: Analyse Fonctionnelle: Théorie et Applications. Masson, Paris (1992)

    Google Scholar 

  7. Chan Y.-S., Paulino G.H., Fannjiang A.C.: The crack problem for nonhomogenuous materials under antiplane shear loading—a displacement based formulation. Int. J. Solids Struct. 38, 2989–3005 (2001)

    Article  Google Scholar 

  8. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    Google Scholar 

  9. Costea N.: Existence and uniquenes results for a class of quasi-hemivariational inequalities. J. Math. Anal. Appl. 373, 305–315 (2011)

    Article  Google Scholar 

  10. Costea N., Matei A.: Weak solutions for nonlinear antiplane problems leading to hemivariational inequalities. Nonlinear Anal. T.M.A. 72, 3669–3680 (2010)

    Article  Google Scholar 

  11. Costea N., Matei A.: Contact models leading to variational-hemivariational inequalities. J. Math. Anal. Appl. 386, 647–660 (2012)

    Article  Google Scholar 

  12. Costea N., Rădulescu V.: Existence results for hemivariational inequalities involving relaxed η− α monotone mappings. Commun. Appl. Anal. 13, 293–304 (2009)

    Google Scholar 

  13. Costea N., Rădulescu V.: Hartman-Stampacchia results for stably pseudomonotone operators and nonlinear hemivariational inequalities. Appl. Anal. 89(2), 175–188 (2010)

    Article  Google Scholar 

  14. Costea, N., Rădulescu, V.: Inequality problems of quasi-hemivariational type involving set-valued operators and a nonlinear term. J. Global Optim. (2011). doi:10.1007/s10898-011-9706-1

  15. Gilbert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity, Nonconvex Optimization Applications, Vol. 55. Kluwer, Dordrecht (2001)

    Book  Google Scholar 

  16. Hayes M.A., Saccomandi G.: Anti-plane shear motivations for viscoelastic Mooney-Rivlin materials. Quart. J. Mech. Appl. Math. 57, 379–392 (2004)

    Article  Google Scholar 

  17. Horgan C.O.: Anti-plane shear deformation in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995)

    Article  Google Scholar 

  18. Ikeda T.: Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990)

    Google Scholar 

  19. Lerguet Z., Shillor M., Sofonea M.: A frictional contact problem for an electro-viscoelastic body. Electron. J. Differ. Equ. 170, 1–16 (2007)

    Google Scholar 

  20. Matei A., Motreanu V.V., Sofonea M.: A quasistatic antiplane contact problem with slip dependent friction. Adv. Nonlinear Var. Inequal. 2, 1–21 (2001)

    Google Scholar 

  21. Migórski S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Cont. Dyn. Syst. Ser. B 6, 1339–1356 (2006)

    Article  Google Scholar 

  22. Migórski S., Ochal A., Sofonea M.: Weak solvability of antiplane frictional contact problems for elastic cylinders. Nonlinear Anal. R.W.A. 11(1), 172–183 (2010)

    Article  Google Scholar 

  23. Migórski S., Ochal A., Sofonea M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal. T.M.A. 70(10), 3738–3748 (2009)

    Article  Google Scholar 

  24. Motreanu D., Panagiotopoulos P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications, Nonconvex Optimization and its Applications, Vol. 29. Kluwer, Dordrecht (1999)

    Google Scholar 

  25. Motreanu D., Rădulescu V.: Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems. Kluwer, Dordrecht (2003)

    Google Scholar 

  26. Naniewicz Z., Panagiotopoulos P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)

    Google Scholar 

  27. Noor M.A., Noor K.I.: Iterative schemes for trifunction hemivariational inequalities. Optim. Lett. 5(2), 273–282 (2011)

    Article  Google Scholar 

  28. Panagiotopoulos P.D.: Hemivariational Inequalities: Applications to Mechanics and Engineering. Springer, Berlin (1993)

    Book  Google Scholar 

  29. Panagiotopoulos P.D.: Noconvex energy functions. Hemivariational inequalities and substationarity principles. Acta Mech. 42, 160–183 (1983)

    Google Scholar 

  30. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, Vol. 35. Springer, Berlin (2010)

    Book  Google Scholar 

  31. Patron V.Z., Kudryavtsev B.A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon & Breach, London (1988)

    Google Scholar 

  32. Sofonea M., Essoufi El H.: A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9, 229–242 (2004)

    Google Scholar 

  33. Sofonea M., Essoufi El H.: Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14, 613–631 (2004)

    Google Scholar 

  34. Sofonea M., Niculescu C., Matei A.: An antiplane contact problem for viscoelastic materials with long-term memory. Math. Model. Anal. 11, 212–228 (2006)

    Google Scholar 

  35. Sofonea M., Matei A.: Variational Inequalities with Applications. A study of Antiplane Frictional Contact Problems, Advances in Mechanics and Mathematics, Vol. 18. Springer, New York (2009)

    Google Scholar 

  36. Tarafdar E.: A fixed point theorem equivalent to the Fan–Knaster–Kuratowski–Mazurkiewicz theorem. J. Math. Anal. Appl. 128(2), 475–479 (1987)

    Article  Google Scholar 

  37. Yang J., Yang J.S.: An Introduction to the Theory of Piezoelectricity. Springer, New York (2005)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Ionicǎ Andrei, Nicuşor Costea & Andaluzia Matei

  2. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700, Bucharest, Romania

    Nicuşor Costea

Authors
  1. Ionicǎ Andrei
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  2. Nicuşor Costea
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  3. Andaluzia Matei
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Correspondence to Nicuşor Costea.

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Andrei, I., Costea, N. & Matei, A. Antiplane shear deformation of piezoelectric bodies in contact with a conductive support. J Glob Optim 56, 103–119 (2013). https://doi.org/10.1007/s10898-011-9815-x

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