Previous |  Up |  Next


Full entry | Fulltext not available (moving wall 24 months)      Feedback
weak solution; variational formulation; antiplane shear deformation; electro-viscoelastic material; Tresca's friction; fixed point; variational inequality
We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent variational equation for the potential field. Then we prove the existence of a unique weak solution to the model. Moreover, the proof is based on arguments of evolution equations and on the Banach fixed-point theorem.
[1] Andreu, F., Mazón, J. M., Sofonea, M.: Entropy solutions in the study of antiplane shear deformations for elastic solids. Math. Models Methods Appl. Sci. 10 (2000), 99-126. DOI 10.1142/S0218202500000082 | MR 1750246 | Zbl 1077.74583
[2] Batra, R. C., Yang, J. S.: Saint-Venant's principle in linear piezoelectricity. J. Elasticity 38 (1995), 209-218. DOI 10.1007/BF00042498 | MR 1336038 | Zbl 0828.73061
[3] Bisegna, P., Lebon, F., Maceri, F.: The unilateral frictional contact of a piezoelectric body with a rigid support. Contact Mechanics J. A. C. Martins et al. Proc. of the 3rd Contact Mechanics International Symposium, Praia da Consolação, 2001 Solic. Mech. Appl. 103, Kluwer Academic Publishers, Dordrecht (2002), 347-354. MR 1968676 | Zbl 1053.74583
[4] Borrelli, A., Horgan, C. O., Patria, M. C.: Saint-Venant's principle for antiplane shear deformations of linear piezoelectric materials. SIAM J. Appl. Math. 62 (2002), 2027-2044. DOI 10.1137/S0036139901392506 | MR 1918305 | Zbl 1047.74019
[5] Campillo, M., Dascalu, C., Ionescu, I. R.: Instability of a periodic system of faults. Geophys. J. Int. 159 (2004), 212-222. DOI 10.1111/j.1365-246X.2004.02365.x
[6] Campillo, M., Ionescu, I. R.: Initiation of antiplane shear instability under slip dependent friction. J. Geophys. Res. 102 (1997), 363-371. DOI 10.1029/97JB01508
[7] Denkowski, Z., Migórski, S., Ochal, A.: A class of optimal control problems for piezoelectric frictional contact models. Nonlinear Anal., Real World Appl. 12 (2011), 1883-1895. MR 2781904 | Zbl 1217.49008
[8] Hoarau-Mantel, T.-V., Matei, A.: Analysis of a viscoelastic antiplane contact problem with slip-dependent friction. Int. J. Appl. Math. Comput. Sci. 12 (2002), 51-58. MR 1905993 | Zbl 1038.74032
[9] Horgan, C. O.: Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37 (1995), 53-81. DOI 10.1137/1037003 | MR 1327716 | Zbl 0824.73018
[10] Horgan, C. O.: Recent developments concerning Saint-Venant's principle: a second update. Appl. Mech. Rev. 49 (1996), 101-111. DOI 10.1115/1.3101961
[11] Horgan, C. O., Miller, K. L.: Antiplane shear deformations for homogeneous and inhomogeneous anisotropic linearly elastic solids. J. Appl. Mech. 61 (1994), 23-29. DOI 10.1115/1.2901416 | MR 1266833 | Zbl 0809.73016
[12] Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990).
[13] Ionescu, I. R., Dascalu, Ch., Campillo, M.: Slip-weakening friction on a periodic system of faults: Spectral analysis. Z. Angew. Math. Phys. 53 (2002), 980-995. DOI 10.1007/PL00012624 | MR 1963548 | Zbl 1014.35068
[14] Ionescu, I. R., Wolf, S.: Interaction of faults under slip-dependent friction. Nonlinear eigenvalue analysis. Math. Methods Appl. Sci. 28 (2005), 77-100. DOI 10.1002/mma.550 | MR 2105794 | Zbl 1062.86006
[15] Lerguet, Z., Shillor, M., Sofonea, M.: A frictional contact problem for an electro-viscoelastic body. Electron. J. Differ. Equ. (electronic only) 2007 (2007), 16 pages. Zbl 1139.74041
[16] Maceri, F., Bisegna, P.: The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28 (1998), 19-28. DOI 10.1016/S0895-7177(98)00105-8 | MR 1616376 | Zbl 1126.74392
[17] Matei, A., Motreanu, V. V., Sofonea, M.: A quasistatic antiplane contact problem with slip dependent friction. Adv. Nonlinear Var. Inequal. 4 (2001), 1-21. MR 1830622 | Zbl 1205.74132
[18] Migórski, S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discrete Contin. Dyn. Syst., Ser. B 6 (2006), 1339-1356. DOI 10.3934/dcdsb.2006.6.1339 | MR 2240746 | Zbl 1109.74039
[19] Migórski, S., Ochal, A., Sofonea, M.: Modeling and analysis of an antiplane piezoelectric contact problem. Math. Models Methods Appl. Sci. 19 (2009), 1295-1324. DOI 10.1142/S0218202509003796 | MR 2555472
[20] Migórski, S., Ochal, A., Sofonea, M.: Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70 (2009), 3738-3748. DOI 10.1016/ | MR 2504461 | Zbl 1159.74026
[21] Migórski, S., Ochal, A., Sofonea, M.: Weak solvability of antiplane frictional contact problems for elastic cylinders. Nonlinear Anal., Real World Appl. 11 (2010), 172-183. MR 2570537 | Zbl 1241.74029
[22] Migórski, S., Ochal, A., Sofonea, M.: Analysis of a piezoelectric contact problem with subdifferential boundary condition. Proc. R. Soc. Edinb., Sect. A, Math. 144 (2014), 1007-1025. DOI 10.1017/S0308210513000607 | MR 3265542 | Zbl 1306.49014
[23] Patron, V. Z., Kudryavtsev, B. A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon & Breach, London (1988).
[24] Sofonea, M., Dalah, M., Ayadi, A.: Analysis of an antiplane electro-elastic contact problem. Adv. Math. Sci. Appl. 17 (2007), 385-400. MR 2374134 | Zbl 1131.74036
[25] Sofonea, M., Essoufi, El H.: Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004), 613-631. MR 2111832 | Zbl 1078.74036
[26] Sofonea, M., Niculescu, C. P., Matei, A.: An antiplane contact problem for viscoelastic materials with long-term memory. Math. Model. Anal. 11 (2006), 213-228. MR 2231211 | Zbl 1104.74049
[27] Zhou, Z.-G., Wang, B., Du, S.-Y.: Investigation of antiplane shear behavior of two collinear permeable cracks in a piezoelectric material by using the nonlocal theory. J. Appl. Mech. 69 (2002), 388-390. DOI 10.1115/1.1445144 | Zbl 1110.74805
Partner of
EuDML logo