Variational analysis for some frictional contact problems
DOI:
https://doi.org/10.5269/bspm.v38i7.44258Abstract
We consider a class of evolutionary variational problems which describes the static frictional contact between a piezoelectric body and a conductive obstacle. The formulation is in a form of coupled system involving the displacement and electric potentiel fieelds. We provide the existence of unique weak solution of the problems. The proof is based on the evolutionary variational inequalities and Banach's xed point theorem.
References
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https://doi.org/10.1137/S0036142998347309
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https://doi.org/10.1007/978-3-642-51677-1
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https://doi.org/10.1023/A:1007413119583
17. Sofonea, M., Essoufi, El H. , A piezoelectric contact problem with slip dependent coefficient of friction, A Mathematical Modelling and Analysis, 9, 229-242, (2004).
https://doi.org/10.3846/13926292.2004.9637256
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19. Rochdi, M., Shillor, M., Sofonea, M., Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51, 105-126, (1998).
https://doi.org/10.1023/A:1007413119583
20. Shillor M., Sofonea M., Telega J.J., Models and Variational Analysis of Quasistatic Contact, Lecture Notes Phys., 655, Springer, Berlin, (2004).
https://doi.org/10.1007/b99799
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https://doi.org/10.1512/iumj.1956.5.55033
22. Weimin, H., Sofonea, M., Kamran, K., Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Computer Methods in Applied Mechanics and Engineering, Vol. 196, 3915-3926, (2007).
https://doi.org/10.1016/j.cma.2006.10.051
https://doi.org/10.1007/978-94-017-1154-8_37
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)
https://doi.org/10.1016/j.jmaa.2009.04.030
3. Dieudonne, J., Foundations of Modern Analysis, Academic Press, (1960).
4. Duvaut G., Lions, J. L., Les in'equations en m'ecanique et physique, Dunod, Paris, (1972).
5. Han, W., Sofonea, M., Evolutionary variational inequalities arising in viscoelastic contact problems, SIAM Journal of Numerical Analysis, 38, 556-579, (2000).
https://doi.org/10.1137/S0036142998347309
6. Han, W., Sofonea, M., Quasistatic Contact Problems inViscoelasticity and Viscoplasticity. In: Studies in Advanced Mathematics, vol. 30. American Mathematical Society, RI - Intl. Press, Providence, Sommerville, MA., (2002).
https://doi.org/10.1090/amsip/030
7. Lerguet, Z., Shillor, M., Sofonea, M., A frictional contact problem for an electro-viscoelastic body, Electronic Journal of Differential Equations, 170,1-17, (2007).
8. Lerguet, Z., Zellagui, Z., Benseridi, H., Drabla, S., Variational analysis of an electro viscoelastic contact problem with friction, Journal of the Association of Arab Universities for Basic and Applied Sciences,14, 93-100, (2013)
https://doi.org/10.1016/j.jaubas.2012.10.001
9. Maceri, F., Bisenga, P., The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. Comp. Modelling, 28, 19-28, (1998).
https://doi.org/10.1016/S0895-7177(98)00105-8
10. Mindlin, R. D., Elasticity; piezoelasticity and cristal lattice dynamics, J, of Elasticity, 4, 217-280, (1972).
https://doi.org/10.1007/BF00045712
11. Necas, J., On regularity of solutions to nonlinear variational inequalities for second order elleptic systems, Rec. Mat. Series VI, 8, 481, (1975).
12. Naniewicz, Z., Panagiotopoulos, P.D., Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, Basel, Hong Kong, (1995).
13. Oden, J.T., Kikuchi, N., Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Engng. Sci., 18, 10, 1171-1284, (1980).
https://doi.org/10.1016/0020-7225(80)90111-1
14. Ouafik,Y., A piezoelectric body in frictional contact, Bull. Math. Soc. Sc.Roumanie, 84(96), 2, 233-242, (2005).
15. Panagiotopoulos, P.D., Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin, (1993).
https://doi.org/10.1007/978-3-642-51677-1
16. Rochdi M, Shillor M, Sofonea M, Quasistatic viscoelastic contact compliance and friction, Journal of elasticity 51 (1998) 105-126
https://doi.org/10.1023/A:1007413119583
17. Sofonea, M., Essoufi, El H. , A piezoelectric contact problem with slip dependent coefficient of friction, A Mathematical Modelling and Analysis, 9, 229-242, (2004).
https://doi.org/10.3846/13926292.2004.9637256
18. Sofonea, M., Essoufi, El H.; A quasistatic frictional contact of a viscoelastic piezoelectric body, Adv, Math. Sci. Appl., 14, (2004), 613-631.
19. Rochdi, M., Shillor, M., Sofonea, M., Quasistatic viscoelastic contact with normal compliance and friction, Journal of Elasticity, 51, 105-126, (1998).
https://doi.org/10.1023/A:1007413119583
20. Shillor M., Sofonea M., Telega J.J., Models and Variational Analysis of Quasistatic Contact, Lecture Notes Phys., 655, Springer, Berlin, (2004).
https://doi.org/10.1007/b99799
21. Toupin R. A., The elastics dielectrics, J. Rat., Mech. Analysis, 5, 849-915, (1956).
https://doi.org/10.1512/iumj.1956.5.55033
22. Weimin, H., Sofonea, M., Kamran, K., Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Computer Methods in Applied Mechanics and Engineering, Vol. 196, 3915-3926, (2007).
https://doi.org/10.1016/j.cma.2006.10.051
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2019-10-13
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