A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation

• Volume: 66, Issue: 1, page 87-113
• ISSN: 0862-7940

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Abstract

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We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the normal compliance condition with finite penetration, the regularized Coulomb law, and the regularized electrical conductivity condition. The existence and uniqueness results are provided using the theory of variational inequalities and Schauder's fixed-point theorem. We also prove that the solution of the latter problem converges towards that of the former as the friction and electrical conductivity coefficients converge towards zero. The numerical solutions of the problems are achieved by using a successive iteration technique; their convergence is also established. The numerical treatment of the contact condition is realized using an Augmented Lagrangian type formulation that leads us to use Uzawa type algorithms. Numerical experiments are performed to show that the numerical results are consistent with the theoretical analysis.

How to cite

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Benkhira, El-Hassan, Fakhar, Rachid, and Mandyly, Youssef. "A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation." Applications of Mathematics 66.1 (2021): 87-113. <http://eudml.org/doc/296989>.

@article{Benkhira2021,
abstract = {We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the normal compliance condition with finite penetration, the regularized Coulomb law, and the regularized electrical conductivity condition. The existence and uniqueness results are provided using the theory of variational inequalities and Schauder's fixed-point theorem. We also prove that the solution of the latter problem converges towards that of the former as the friction and electrical conductivity coefficients converge towards zero. The numerical solutions of the problems are achieved by using a successive iteration technique; their convergence is also established. The numerical treatment of the contact condition is realized using an Augmented Lagrangian type formulation that leads us to use Uzawa type algorithms. Numerical experiments are performed to show that the numerical results are consistent with the theoretical analysis.},
author = {Benkhira, El-Hassan, Fakhar, Rachid, Mandyly, Youssef},
journal = {Applications of Mathematics},
keywords = {piezoelectric body; nonlinear elastic constitutive Hencky's law; normal compliance contact condition; Coulomb's friction law; iteration method; augmented Lagrangian; Uzawa block relaxation},
language = {eng},
number = {1},
pages = {87-113},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation},
url = {http://eudml.org/doc/296989},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Benkhira, El-Hassan
AU - Fakhar, Rachid
AU - Mandyly, Youssef
TI - A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 87
EP - 113
AB - We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the normal compliance condition with finite penetration, the regularized Coulomb law, and the regularized electrical conductivity condition. The existence and uniqueness results are provided using the theory of variational inequalities and Schauder's fixed-point theorem. We also prove that the solution of the latter problem converges towards that of the former as the friction and electrical conductivity coefficients converge towards zero. The numerical solutions of the problems are achieved by using a successive iteration technique; their convergence is also established. The numerical treatment of the contact condition is realized using an Augmented Lagrangian type formulation that leads us to use Uzawa type algorithms. Numerical experiments are performed to show that the numerical results are consistent with the theoretical analysis.
LA - eng
KW - piezoelectric body; nonlinear elastic constitutive Hencky's law; normal compliance contact condition; Coulomb's friction law; iteration method; augmented Lagrangian; Uzawa block relaxation
UR - http://eudml.org/doc/296989
ER -

References

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1. Barboteu, M., Sofonea, M., Analysis and numerical approach of a piezoelectric contact problem, Ann. Acad. Rom. Sci., Math. Appl. 1 (2009), 7-30. (2009) Zbl1426.74233MR2660410
2. Barboteu, M., Sofonea, M., 10.1016/j.amc.2009.09.045, Appl. Math. Comput. 215 (2009), 2978-2991. (2009) Zbl1381.74161MR2563414DOI10.1016/j.amc.2009.09.045
3. Benkhira, E.-H., Fakhar, R., Mandyly, Y., 10.1080/01630563.2019.1600546, Numer. Funct. Anal. Optim. 40 (2019), 1291-1314. (2019) Zbl1419.35059MR3949123DOI10.1080/01630563.2019.1600546
4. Brézis, H., 10.5802/aif.280, Ann. Inst. Fourier 18 (1968), 115-175 French. (1968) Zbl0169.18602MR0270222DOI10.5802/aif.280
5. Essoufi, E.-H., Benkhira, E.-H., Fakhar, R., 10.4208/aamm.09-m0980, Adv. Appl. Math. Mech. 2 (2010), 355-378. (2010) Zbl1262.74017MR2610419DOI10.4208/aamm.09-m0980
6. Essoufi, E.-H., Fakhar, R., Koko, J., 10.1080/01630563.2015.1078812, Numer. Funct. Anal. Optim. 36 (2015), 1533-1558. (2015) Zbl1333.74081MR3432896DOI10.1080/01630563.2015.1078812
7. Han, W., 10.1007/b101775, Advances in Mechanics and Mathematics 8, Springer, New York (2005). (2005) Zbl1081.65065MR2101057DOI10.1007/b101775
8. Haslinger, J., Mäkinen, R., 10.1007/bf01742734, Struct. Optim. 4 (1992), 133-141. (1992) DOI10.1007/bf01742734
9. Lerguet, Z., Shillor, M., Sofonea, M., A frictional contact problem for an electro-viscoelastic body, Electron. J. Differ. Equ. 2007 (2007), Paper No. 170, 16 pages. (2007) Zbl1139.74041MR2366063
10. Ouafik, Y., Contribution à l'étude mathématique et numérique des structures piézoélectriques encontact, Ph.D. Dissertation, Perpignan University, Perpignan (2007), Available at https://tel.archives-ouvertes.fr/tel-00192884 French. (2007)

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