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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2016027/
Thick obstacle problems with dynamic adhesive contact. ESAIM: M2AN 42 (2008) 1021–1045. | DOI | Numdam | MR | Zbl
,Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. 29 (2009) 43–71. | DOI | MR | Zbl
and ,Analysis of nonlinear thermo-viscoelastic-viscoplastic contacts. Int. J. Engrg. Sci. 78 (2014) 1–17. | DOI | MR | Zbl
, and ,Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support. J. Math. Anal. Appl. 358 (2009) 110–124. | DOI | MR | Zbl
and ,A class of evolutionary variational inequalities with applications in viscoelasticity. Math. Models Methods Appl. Sci. 15 (2005) 1595–1617. | DOI | MR | Zbl
, and ,Numerical analysis of a frictionless viscoelastic piezoelectric contact problem. ESAIM: M2AN 42 (2008) 667–682. | DOI | Numdam | MR | Zbl
, and ,Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity. Comput. Methods Appl. Mech. Engrg. 197 (2008) 3724–3732. | DOI | MR | Zbl
, and ,Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM J. Numer. Anal. 53 (2015) 527–550. | DOI | MR | Zbl
, , , and ,V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei. Bucharest-Noordhoff, Leyden (1976). | MR | Zbl
Mathematical analysis of a viscoelastic problem with temperature-dependent coefficients. I. Existence and uniqueness. Math. Methods Appl. Sci. 30 (2007) 1545–1568. | DOI | MR | Zbl
, and ,Saint-Venant’s principle in linear piezoelectricity. J. Elasticity 38 (1995) 209–218. | DOI | MR | Zbl
and ,Unilateral dynamic contact of two viscoelastic beams. Quart. Appl. Math. 69 (2011) 477–507. | DOI | MR | Zbl
and ,A dynamic viscoelastic contact problem with normal compliance and damage. Finite Elem. Anal. Des. 42 (2005) 1–24. | DOI | MR
, , and ,Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Engrg. 196 (2006) 476–488. | DOI | MR | Zbl
, , , and ,A viscoelastic viscoplastic constitutive model including mechanical degradation: uniaxial transient finite element formulation at finite strains and application to space truss structures. Appl. Math. Model. 39 (2015) 1725–1739. | DOI | MR | Zbl
, and ,A nonlinear viscoelastic-plastic rheological model for rocks based on fractional derivative theory, Int. J. Mod. Phys. B 27 (2013) 1350149. | DOI | MR | Zbl
, , , , ,P.G. Ciarlet, Basic error estimates for elliptic problems. In: Vol. II of Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions. (1993) 17–351. | MR | Zbl
Existence of solutions of a dynamic Signorini’s problem with nonlocal friction in viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 1099–1109. | DOI | MR | Zbl
,Analysis of a class of implicit evolution inequalities associated to dynamic contact problems with friction. Int. J. Eng. Sci. 328 (2000) 1534–1549. | MR | Zbl
and ,Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body. Z. Angew. Math. Phys. 57 (2006) 523–546. | DOI | MR | Zbl
and ,A dynamic contact problem involving a Timoshenko beam model. Appl. Numer. Math. 63 (2013) 117–128. | DOI | MR | Zbl
and ,Dynamic contact of a beam against rigid obstacles: convergence of a velocity-based approximation and numerical results. Nonlinear Anal. Real World Appl. 22 (2015) 520-536. | DOI | MR | Zbl
and ,G. Duvaut and J.L. Lions, Inequalities in mechanics and physics. Springer Verlag, Berlin (1976). | MR | Zbl
C. Eck, J. Jarusek and M. Krbec, Unilateral contact problems. Variational methods and existence theorems. Vol. 270 of Pure Appl. Math. Chapman & Hall/CRC, Boca Raton (2005). | MR | Zbl
Some qualitative results on the dynamic viscoelasticity of the Reissner-Mindlin plate model. Quart. J. Mech. Appl. Math. 57 (2004) 59–78. | DOI | MR | Zbl
and ,An a posteriori error analysis for dynamic viscoelastic problems. ESAIM: M2AN 45 (2011) 925-945. | DOI | Numdam | MR | Zbl
and ,A viscoelastic-viscoplastic constitutive equation and its finite element implementation. Comput. Struct. 17 (1983) 499–509. | DOI | Zbl
and ,New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl
,Numerical aspects associated with the implementation of a finite strain, elasto-viscoelastic-viscoplastic constitutive theory in principal stretches. Int. J. Numer. Methods Engrg. 83 (2010) 366–402. | DOI | MR | Zbl
and ,T. Ideka, Fundamentals of piezoelectricity. Oxford University Press. Oxford (1990).
Dynamic contact problems with slip dependent friction in viscoelasticity. Int. J. Appl. Math. Comput. Sci. 12 (2002) 71–80. | MR | Zbl
and ,Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Models Methods Appl. Sci. 9 (1999) 11–34. | DOI | MR | Zbl
and ,A time-integration method for the viscoelastic-viscoplastic analyses of polymers and finite element implementation. Int. J. Numer. Methods Engrg. 79 (2009) 550–575. | DOI | MR | Zbl
and ,Frictional contact problems with normal compliance. Int. J. Engrg. Sci. 26 (1988) 811–832. | DOI | MR | Zbl
, and ,Dynamic friction contact problem with general normal and friction laws. Nonlinear Anal. 28 (1997) 559–575. | DOI | MR | Zbl
,Dynamic bilateral contact with discontinuous friction coefficient, Nonlinear Anal. 45 (2001) 309–327. | DOI | MR | Zbl
and ,Dynamic contact problem for viscoelastic piezoelectric materials with slip dependent friction. Nonlinear Anal. 71 (2009) 1414–1424. | DOI | MR | Zbl
and ,Hertzian contact of anisotropic piezoelectric bodies. J. Elasticity 84 (2006) 153–166. | DOI | MR | Zbl
and ,Modeling of nonlinear viscoelastic-viscoplastic frictional contact problems. Int. J. Engrg. Sci. 74 (2014) 103–117. | DOI | MR | Zbl
, and ,Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11 (1987) 407–428. | DOI | MR | Zbl
and ,Analysis of a dynamic contact problem for electro-viscoelastic cylinders. Nonlinear Anal. 73 (2010) 1221–1238. | DOI | MR | Zbl
, and ,Coupled viscoelastic-viscoplastic modeling of homogeneous and isotropic polymers: numerical algorithm and analytical solutions. Comput. Methods Appl. Mech. Engrg. 200 (2011) 3381–3394. | DOI | MR | Zbl
, and ,Polarisation gradient in elastic dielectrics. Int. J. Solids Struct. 4 (1968) 637–663. | DOI | Zbl
,Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Internat. J. Solids Struct. 4 (1969) 1197–1213. | DOI
,A uniqueness theorem in the dynamical theory of piezoelectricity. Math. Methods Appl. Sci. 14 (1991) 295–299. | DOI | MR | Zbl
and ,M. Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage. Vol. 276 of Pure Appl. Math. Chapman & Hall/CRC, Boca Raton (2006). | MR | Zbl
Stress tensors in elastic dielectrics. Arch. Rational Mech. Anal. 5 (1960) 440–452. | DOI | MR | Zbl
,A dynamical theory of elastic dielectrics. Int. J. Engrg. Sci. 1 (1963) 101–126. | DOI | MR
,Cité par Sources :