Asymptotic modeling of thin piezoelectric plates

Abstract

Abstract. We study the asymptotic behavior of a three dimensional flat, heterogeneous and anisotropic piezoelectric body when its thickness -seen as a parameter -goes to zero. Depending on the type of the electrical loading two models are obtained which are related to the plate used as a sensor or as an actuator. These models are explicitly derived in any piezoelectric crystal symmetry class. For some of them, a striking structural switch-off of the piezoelectric effect occurs. The static case is solved through a unifying approach using techniques of singular perturbation while the transient situation is formulated in terms of evolution equations in Hilbert spaces of possible states with finite electromechanical energy, so that the study of these transient problems are easily deduced from the static case through the Trotter theory of convergence of semi-groups of operators acting on variable spaces.

Asymptotic modeling of piezoelectric plates with electric field gradient

“…We show that depending on the type of electric loading, three different models indexed by p appear at the limit. This result extends our previous study in [8] and shows that gradient theory broadens the understanding of sensors and actuators. When p = 2 and p = 3 we are able to express the constitutive laws as a Schur complement of the second order piezoelectric tensor (see (2) and (11)).…”

“…-a 3 involves mechanical terms only, -α 3 never vanishes, -b 2 vanishes for the crystalline classes m, 32, 422, 6, 622 and 6m2, as in first order piezoelectricity (see [8]), -α 2 always vanishes except for the classes 3, 32 and 3m, -β 2 always vanishes except for the class m, -when p = 2, there is a complete electromechanical decoupling ( b 2 = α 2 = β 2 = 0) for the classes 422, 6, 622 and 6m2, nevertheless the operators a 2 , c 2 and γ 2 involve a mixture of elastic, dielectric and coupling coefficients. In these cases, M ε…”

“…If p = 2, 3 it is possible to generalize the method described in [8]. The key point of this method is to observe that some components of k p (ε, s p (ε)) and of M k p (ε, s p (ε)) have vanishing limits when ε goes to 0 so that it suggests a suitable orthogonal decomposition of H in the following subspaces: …”

“…For p ≥ 2, using suitable test functions in P(ε, Ω) p as in [8], we get (M k p ) − p = 0. Taking test functions such that k p (ε, r) = k(r) 0 p we conclude that s p is the unique solution of P(Ω) p so that the whole sequence s p (ε) converges.…”

Mathematical modeling of piezomagnetoelectric thin plates

“…Moreover the role that plays the crystal class of the genuine material constituting the device on its behavior has never been achieved, at the authors knowledge. Indeed, a rigorous theoretical investigation of such structures is quite recent (see [10], [11], [12], [15] and references quoted therein). Here, taking advantage of the multiscaled shape of MEMS plates of electromagneticoelastic nature, we present the results of their rigorous mathematical derivation that has been carried out in [13].…”