Existence, Uniqueness, and Regularity Results for Piezoelectric Systems

Abstract

We investigate the time-harmonic piezoelectric system (a system coupling the elasticity system with the full Maxwell's equations) in polyhedral domains of the space. Existence and uniqueness results of weak solutions are proved in different cases. We describe the corner and edge singularities of that system and deduce some regularity results. 1. Introduction. Smart structures made of piezoelectric and/or piezomagnetic materials are gaining attention in applications since they are able to transform the energy from one type to another (magnetic, electric, and mechanical), allowing them to be used as sensors and/or actuators. Commonly used piezoelectric materials are ceramics and quartz. The mathematical model of this system starts to be well established [2, 8, 14, 24, 26] and corresponds to a coupling between the elasticity system and Maxwell's equations (see below). A full mathematical analysis is not yet done, except in some particular cases [13, 19]. Namely, in these two works the electric field E is assumed to be curl free, i.e., E = ∇ϕ, where ϕ is an electric potential and a two-dimensional reduction is made. In [13], existence and uniqueness results in smooth domains are obtained using integral equations, while in [19] a variational formulation in polygonal domains is given and two-dimensional singularities are briefly described. On the other hand, there exists an extensive list of papers from mechanics literature describing singularities of some particular piezoelectric materials with a plane crack [25, 27, 30] or along wedges [29]. But to our knowledge, an exact description of corner/edge singularities of the general piezoelectric system in three-dimensional polyhedral domains is not yet obtained. Such a description is very important since piezoelectric ceramics are very brittle, and therefore their fracture behavior must be understood. The knowledge of such singularities also has numerical implications, such as convergence speed. This paper has, therefore, the following goals: We present a general piezoelectric system, which includes standard models of ceramics like the P ZT or the BaT iO 3. We further develop some variational formulations which are the natural ones because they lead to solutions in the energy spaces (here called weak solutions). We prove existence and uniqueness results of weak solutions of the time-harmonic system in two different cases: the case when the magnetic permeability matrix is positive definite (BaT iO 3) and the case when the magnetic permeability matrix is zero (P ZT). In that second case, we even give two different formulations and show …

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