Applications of Mathematics, first online, pp. 1-27

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

El-Hassan Benkhira, Rachid Fakhar, Youssef Mandyly

Received August 6, 2019.   Published online May 14, 2020.

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.
Keywords:  piezoelectric body; nonlinear elastic constitutive Hencky's law; normal compliance contact condition; Coulomb's friction law; iteration method; augmented Lagrangian; Uzawa block relaxation
Classification MSC:  35J87, 74C05, 49J40, 47J25, 74S05, 65N55, 37M05
DOI:  10.21136/AM.2020.0195-19

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Affiliations:   El-Hassan Benkhira, University Moulay Ismaïl, ESTM, Laboratory LEM2A, BP 3103, Toulal-Meknès, Morocco, e-mail:; Rachid Fakhar, Youssef Mandyly (corresponding author), University Sultan Moulay Slimane, Laboratory LS3M, 25000 Khouribga, Morocco, e-mail:,

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