Solvability of a dynamic contact problem between a piezoelectric body and a conductive foundation |
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Authors: | Mikael Barboteu Mircea Sofonea |
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Institution: | Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan, France |
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Abstract: | We consider a mathematical model which describes the dynamic process of contact between a piezoelectric body and an electrically conductive foundation. We model the material’s behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the normal compliance condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then, under a smallness assumption on the data, we prove the existence of a unique weak solution to the model. We also investigate the behavior of the solution with respect the electric data on the contact surface and prove a continuous dependence result. Then, we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the contact by using a penalized approach and a version of Newton’s method. We implement this scheme in a numerical code and, in order to verify its accuracy, we present numerical simulations in the study of two-dimensional test problems. These simulations provide a numerical validation of our continuous dependence result and illustrate the effects of the conductivity of the foundation, as well. |
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Keywords: | Piezoelectric material Frictionless contact Normal compliance Monotone operator Fixed point Weak solution Penalization method Finite element method Newton method Numerical simulations |
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