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Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity. (English) Zbl 1194.74210
Summary: A dynamic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle is numerically studied in this paper. The contact is modelled using the normal compliance contact condition and the linear electro-viscoelastic constitutive law is employed to simulate the piezoelectric effects. The variational formulation is a coupled system composed of a parabolic nonlinear variational equation for the velocity field and a linear variational equation for the electric potential. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the accuracy and the performance of the algorithm.

##### MSC:
 74M15 Contact in solid mechanics 74F15 Electromagnetic effects in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74D05 Linear constitutive equations for materials with memory
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##### References:
 [1] Angelov, T.A., On a rolling problem with damage and wear, Mech. res. commun., 26, 281-286, (1999) · Zbl 0945.74660 [2] Armero, F.; Petocz, E., A new class of conserving algorithms for dynamic contact problems, Comput. methods appl. mech. engrg., 158, 269-300, (1998) · Zbl 0954.74055 [3] Batra, R.C.; Yang, J.S., Saint-venant’s principle in linear piezoelectricity, J. elast., 38, 209-218, (1995) · Zbl 0828.73061 [4] Barboteu, M.; Fernández, J.R.; Ouafik, Y., Numerical analysis of two frictionless elastic-piezoelectric contact problems, J. math. anal. appl., 339, 905-917, (2008) · Zbl 1127.74028 [5] M. Barboteu, J.R. Fernández, Y. Ouafik, Numerical analysis of a frictionless viscoelastic piezoelectric contact problem, M2AN Math. Model. Numer. Anal. (in press). · Zbl 1142.74029 [6] Bisegna, P.; Lebon, F.; Maceri, F., The unilateral frictional contact of a piezoelectric body with a rigid support, contact mechanics (praia da consolao, 2001), Solid mech. appl., 103, 347-354, (2002) · Zbl 1053.74583 [7] Ciarlet, P.G., The finite element method for elliptic problems, (), 17-352 · Zbl 0198.14601 [8] Duvaut, G.; Lions, J.L., Inequalities in mechanics and physics, (1976), Springer Verlag Berlin · Zbl 0331.35002 [9] Fan, H.; Sze, K.-Y.; Yang, W., Two dimensional contact of a piezoelectric half space, Int. J. solids struct., 33, 9, 1305-1315, (1996) [10] Fisher, K.A.; Wriggers, P., Frictionless 2D contact formulations for finite deformations based on the mortar method, Comput. mech., 36, 226-244, (2005) · Zbl 1102.74033 [11] Fisher-Cripps, A.C., Introduction to contact mechanics, Mechanical engineering series, (2000), Springer [12] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer New York · Zbl 0575.65123 [13] Hilber, H.; Hughes, T.; Taylor, R., Improved numerical dissipation for time integration algorithms in structural dynamics, Earth engrg. struct. dyn., 5, 283-292, (1977) [14] Han, W.; Sofonea, M.; Kazmi, K., Analysis and numerical solution of a frictionless contact problem for electro-elastic-visco-plastic materials, Comput. methods appl. mech. engrg., 196, 3915-3926, (2007) · Zbl 1173.74362 [15] Han, W.; Shillor, M.; Sofonea, M., Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage, J. comput. appl. math., 137, 377-398, (2001) · Zbl 0999.74087 [16] Han, W.; Sofonea, M., Quasistatic contact problems in viscoelasticity and viscoplasticity, (2002), American Mathematical Society - International Press · Zbl 1013.74001 [17] Hartmann, S.; Brunssen, S.; Ramm, E.; Wohlmuth, W., Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy, Int. J. numer. methods engrg., 70, 883-912, (2007) · Zbl 1194.74218 [18] Hüeber, S.; Matei, A.; Wohlmuth, B.I., A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity, Bull. math. soc. sci. math. roumanie, 48, 96, 209-232, (2005) · Zbl 1105.74028 [19] Ideka, T., Fundamentals of piezoelectricity, (1990), Oxford University Press Oxford [20] Jing, Y.; Luo, J.; Yi, X.; Gu, X., Design and evaluation of PZT thin-film micro-actuator for hard disk drives, Sens. actuat. A: phys., 116, 2, 329-335, (2004) [21] Jun, S.; Zhaowei, Z., Finite element analysis of a IBM suspension integrated with a PZT microactuator, Sens. actuat. A: phys., 100, 2-3, 257-263, (2002) [22] Klarbring, A.; Mikelić, A.; Shillor, M., Frictional contact problems with normal compliance, Int. J. engrg. sci., 26, 811-832, (1988) · Zbl 0662.73079 [23] Kuttler, K.L.; Shillor, M., Regularity of solutions to a dynamic frictionless contact problem with normal compliance, Nonlinear anal., 59, 1063-1075, (2004) · Zbl 1085.74035 [24] Laursen, T.A., Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis, (2003), Springer [25] Laursen, T.A.; Chawla, V., Design of energy conserving algorithms for frictionless dynamic contact problems, Int. J. numer. methods engrg., 40, 863-886, (1997) · Zbl 0886.73067 [26] Maceri, F.; Bisegna, B., The unilateral frictionless contact of a piezoelectric body with a rigid support, Math. comput. modell., 28, 19-28, (1998) · Zbl 1126.74392 [27] Martins, J.A.C.; Oden, J.T., Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, Nonlinear anal., 11, 3, 407-428, (1987) · Zbl 0672.73079 [28] Mindlin, R.D., Polarisation gradient in elastic dielectrics, Int. J. solids struct., 4, 637-663, (1968) · Zbl 0159.57001 [29] Mindlin, R.D., Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films, Int. J. solids struct., 4, 1197-1213, (1969) [30] Mindlin, R.D., Elasticity, piezoelasticity and crystal lattice dynamics, J. elast., 4, 217-280, (1972) [31] Morro, A.; Straughan, B., A uniqueness theorem in the dynamical theory of piezoelectricity, Math. methods appl. sci., 14, 5, 295-299, (1991) · Zbl 0725.73023 [32] Muradova, A.D.; Stavroulakis, G.E., A unilateral contact model with buckling in von Kármán plates, Nonlinar anal. real world appl., 8, 1261-1271, (2007) · Zbl 1114.74043 [33] Ouafik, Y., A piezoelectric body in frictional contact, Bull. math. soc. sci. math. roumanie, 48, 96, 233-242, (2005) · Zbl 1114.74038 [34] Puso, M.A., A 3D mortar method for solid mechanics, Int. J. numer. methods engrg., 59, 1107-1118, (2004) [35] Sauer, R.A.; Li, S., A contact mechanics model for quasi-continua, Int. J. numer. methods engrg., 71, 931-962, (2007) · Zbl 1194.74226 [36] Sofonea, M.; Essoufi, E.-H., Quasistatic frictional contact of a viscoelastic piezoelectric body, Adv. math. sci. appl., 14, 1, 25-40, (2004) [37] Sofonea, M.; Ouafik, Y., A piezoelectric contact problem with normal compliance, Appl. math., 32, 425-442, (2005) · Zbl 1138.74372 [38] Solberg, J.M.; Jones, R.E.; Papadopoulos, P., A family of simple two-pass dual formulations for the finite element solution of contact problems, Comput. methods appl. mech. engrg., 196, 202-782, (2007) · Zbl 1120.74835 [39] Toupin, R.A., The elastic dielectrics, J. rational mech. anal., 5, 849-915, (1956) · Zbl 0072.23803 [40] Toupin, R.A., A dynamical theory of elastic dielectrics, Int. J. engrg. sci., 1, 101-126, (1963) [41] Toupin, R.A., Stress tensors in elastic dielectrics, Arch. rational mech. anal., 5, 440-452, (1960) · Zbl 0113.23502 [42] Turbé, N.; Maugin, G.A., On the linear piezoelectricity of composite material, Math. methods appl. sci., 14, 6, 403-412, (1991) · Zbl 0731.73071 [43] Wriggers, P., Computational contact mechanics, (2002), Wiley
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