In this work, we consider the quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelastic constitutive law is employed to model the piezoelectric material and the normal compliance condition is used to model the contact. The variational formulation is derived in a form of a coupled system for the displacement and electric potential fields. 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 an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.
Mots-clés : piezoelectricity, viscoelasticity, normal compliance, error estimates, numerical simulations
@article{M2AN_2008__42_4_667_0, author = {Barboteu, Mikael and Fern\'andez, Jose Ramon and Ouafik, Youssef}, title = {Numerical analysis of a frictionless viscoelastic piezoelectric contact problem}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {667--682}, publisher = {EDP-Sciences}, volume = {42}, number = {4}, year = {2008}, doi = {10.1051/m2an:2008022}, mrnumber = {2437778}, zbl = {1142.74029}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008022/} }
TY - JOUR AU - Barboteu, Mikael AU - Fernández, Jose Ramon AU - Ouafik, Youssef TI - Numerical analysis of a frictionless viscoelastic piezoelectric contact problem JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 667 EP - 682 VL - 42 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008022/ DO - 10.1051/m2an:2008022 LA - en ID - M2AN_2008__42_4_667_0 ER -
%0 Journal Article %A Barboteu, Mikael %A Fernández, Jose Ramon %A Ouafik, Youssef %T Numerical analysis of a frictionless viscoelastic piezoelectric contact problem %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 667-682 %V 42 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008022/ %R 10.1051/m2an:2008022 %G en %F M2AN_2008__42_4_667_0
Barboteu, Mikael; Fernández, Jose Ramon; Ouafik, Youssef. Numerical analysis of a frictionless viscoelastic piezoelectric contact problem. ESAIM: Modélisation mathématique et analyse numérique, Volume 42 (2008) no. 4, pp. 667-682. doi : 10.1051/m2an:2008022. http://www.numdam.org/articles/10.1051/m2an:2008022/
[1] Solution of frictional contact problems by an EBE preconditioner. Comput. Mech. 20 (1997) 370-378. | Zbl
, and ,[2] Finite element approximation of piezoelectric plates. Internat. J. Numer. Methods Engrg. 50 (2001) 1469-1499. | MR | Zbl
, and ,[3] Numerical analysis of two frictionless elastic-piezoelectric contact problems. J. Math. Anal. Appl. 339 (2008) 905-917. | MR | Zbl
, and ,[4] Saint-Venant's principle in linear piezoelectricity. J. Elasticity 38 (1995) 209-218. | MR | Zbl
and ,[5] The unilateral frictional contact of a piezoelectric body with a rigid support, in Contact mechanics (Praia da Consolação, 2001), Solid Mech. Appl. 103, Kluwer Acad. Publ., Dordrecht (2002) 347-354. | MR | Zbl
, and ,[6] The finite element method for elliptic problems, in Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions Eds., North Holland (1991) 17-352. | MR | Zbl
,[7] Inequalities in Mechanics and Physics. Springer Verlag, Berlin (1976). | MR | Zbl
and ,[8] A frictionless contact problem for elastic-viscoplastic materials with normal compliance: Numerical analysis and computational experiments. Numer. Math. 90 (2002) 689-719. | MR | Zbl
, and ,[9] Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). | MR | Zbl
,[10] Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society-International Press (2002). | MR | Zbl
and ,[11] Variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction and damage. J. Comput. Appl. Math. 137 (2001) 377-398. | MR | Zbl
, and ,[12] 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 (2005) 209-232. | MR | Zbl
, and ,[13] Fundamentals of Piezoelectricity. Oxford University Press, Oxford (1990).
,[14] Frictional contact problems with normal compliance. Internat. J. Engrg. Sci. 26 (1988) 811-832. | MR | Zbl
, and ,[15] The unilateral frictionless contact of a piezoelectric body with a rigid support. Math. Comput. Modelling 28 (1998) 19-28. | MR | Zbl
and ,[16] Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11 (1987) 407-428. | MR | Zbl
and ,[17] Polarisation gradient in elastic dielectrics. Internat. J. Solids Structures 4 (1968) 637-663. | Zbl
,[18] Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Internat. J. Solids Structures 5 (1969) 1197-1213.
,[19] Elasticity, piezoelasticity and crystal lattice dynamics. J. Elasticity 4 (1972) 217-280.
,[20] A uniqueness theorem in the dynamical theory of piezoelectricity. Math. Methods Appl. Sci. 14 (1991) 295-299. | MR | Zbl
and ,[21] A piezoelectric body in frictional contact. Bull. Math. Soc. Sci. Math. Roumanie 48 (2005) 233-242. | MR | Zbl
,[22] Quasistatic frictional contact of a viscoelastic piezoelectric body. Adv. Math. Sci. Appl. 14 (2004) 25-40. | MR | Zbl
and ,[23] A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal. 9 (2004) 229-242. | MR | Zbl
and ,[24] A piezoelectric contact problem with normal compliance. Appl. Math. 32 (2005) 425-442. | EuDML | MR | Zbl
and ,[25] The elastic dielectrics. J. Rational Mech. Anal. 5 (1956) 849-915. | MR | Zbl
,[26] Stress tensors in elastic dielectrics. Arch. Rational Mech. Anal. 5 (1960) 440-452. | MR | Zbl
,[27] A dynamical theory of elastic dielectrics. Internat. J. Engrg. Sci. 1 (1963) 101-126. | MR
,[28] On the linear piezoelectricity of composite materials. Math. Methods Appl. Sci. 14 (1991) 403-412. | MR | Zbl
and ,[29] Computational Contact Mechanics. Wiley-Verlag (2002). | Zbl
,Cited by Sources: