In this paper, we study, from both variational and numerical points of view, a dynamic contact problem between a viscoelastic-viscoplastic piezoelectric body and a deformable obstacle. The contact is modelled using the classical normal compliance contact condition. The variational formulation is written as a nonlinear ordinary differential equation for the stress field, a nonlinear hyperbolic variational equation for the displacement field and a linear variational equation for the electric potential field. An existence and uniqueness result is proved using Gronwall’s lemma, adequate auxiliary problems and fixed-point arguments. Then, fully discrete approximations are introduced using an Euler scheme and the finite element method, for which some a priori error estimates are derived, leading to the linear convergence of the algorithm under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to show the accuracy of the algorithm and the behaviour of the solution.
Accepté le :
DOI : 10.1051/m2an/2016027
Mots-clés : Viscoelasticity, viscoplasticity, piezoelectricity, existence and uniqueness, a priori error estimates, numerical simulations
@article{M2AN_2017__51_2_565_0,
author = {Campo, Marco and Fern\'andez, Jose R. and Rodr{\'\i}guez-Ar\'os, \'Angel D. and Rodr{\'\i}guez, Jos\'e M.},
title = {Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {565--586},
publisher = {EDP-Sciences},
volume = {51},
number = {2},
year = {2017},
doi = {10.1051/m2an/2016027},
mrnumber = {3626411},
zbl = {1398.74033},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016027/}
}
TY - JOUR AU - Campo, Marco AU - Fernández, Jose R. AU - Rodríguez-Arós, Ángel D. AU - Rodríguez, José M. TI - Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 565 EP - 586 VL - 51 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016027/ DO - 10.1051/m2an/2016027 LA - en ID - M2AN_2017__51_2_565_0 ER -
%0 Journal Article %A Campo, Marco %A Fernández, Jose R. %A Rodríguez-Arós, Ángel D. %A Rodríguez, José M. %T Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 565-586 %V 51 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016027/ %R 10.1051/m2an/2016027 %G en %F M2AN_2017__51_2_565_0
Campo, Marco; Fernández, Jose R.; Rodríguez-Arós, Ángel D.; Rodríguez, José M. Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 565-586. doi : 10.1051/m2an/2016027. http://archive.numdam.org/articles/10.1051/m2an/2016027/
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