Geodesic
Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem
Campo, Marco1 ; Fernández, Jose R.2 ; Rodríguez-Arós, Ángel D.3 ; Rodríguez, José M.3
1 Centro Universitario de la Defensa, Escuela Naval Militar, Plaza de España s/n, 36920 Marín, Spain.
2 Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain.
3 Departamento de Métodos Matemáticos e Representación, Universidade de A Coruña, A Coruña, Spain.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 565-586

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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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016027
Classification : 65M12, 65M15, 65M25, 65M60
Keywords: Viscoelasticity, viscoplasticity, piezoelectricity, existence and uniqueness, a priori error estimates, numerical simulations

Campo, Marco 1 ; Fernández, Jose R. 2 ; Rodríguez-Arós, Ángel D. 3 ; Rodríguez, José M. 3

1 Centro Universitario de la Defensa, Escuela Naval Militar, Plaza de España s/n, 36920 Marín, Spain.
2 Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain.
3 Departamento de Métodos Matemáticos e Representación, Universidade de A Coruña, A Coruña, Spain. 
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     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},
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     zbl = {1398.74033},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016027/}
}
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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

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