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.

中文翻译：

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非线性压电材料在与导电基础摩擦接触过程中的收敛结果和数值研究

我们考虑两个静态问题，它们描述了压电体与障碍物（即所谓的基础）之间的接触。假设材料的本构关系是电弹性的，并且涉及非线性弹性本构亨基定律。在第一个问题中，假设接触是无摩擦的，基础是不导电的，而在第二个问题中，假设接触是摩擦的，基础是导电的。接触建模为具有有限穿透的正常柔顺条件、正则化库仑定律和正则化电导率条件。存在性和唯一性结果是使用变分不等式理论和 Schauder 不动点定理提供的。我们还证明，当摩擦系数和电导率系数趋向于零时，后一个问题的解决方案会向前者的解决方案收敛。问题的数值解是通过使用逐次迭代技术实现的；它们的收敛性也成立。接触条件的数值处理是使用增强拉格朗日型公式实现的，该公式导致我们使用 Uzawa 型算法。进行了数值实验，表明数值结果与理论分析一致。接触条件的数值处理是使用增强拉格朗日型公式实现的，该公式导致我们使用 Uzawa 型算法。进行了数值实验，表明数值结果与理论分析一致。接触条件的数值处理是使用增强拉格朗日型公式实现的，该公式导致我们使用 Uzawa 型算法。进行了数值实验，表明数值结果与理论分析一致。