Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate

Abstract

The aim of this paper is to derive a reduced model for a piezoelectric plate and to study its actuator and sensor capabilities. In the first part, we focus on the asymptotic modeling for thin plates formed by stacking layers of different piezoelectric materials. In the asymptotic model, the mechanical and electric unknowns are shown to be partly decoupled. In the second part, we study the actuator and sensor capabilities of this model. We use two discrete non-differentiable multi-objective optimization problems, which are solved by genetic algorithms. Several numerical results are reported.

Asymptotic modeling of thin piezoelectric plates

“…These facts, presented in [17] and [19], already noted under stronger symmetry hypothesis in [10], [15] and [9], have also been observed latter in [5].…”

Frictional contact of an anisotropic piezoelectric plate

“…This happens since the equations in (3.39a) become independent from each other since the friction terms containing μ vanish, since M ebci αβ (u) = M ebci αβ (ζ 3 ) only depends on ζ 3 and N ebci αβ (u) = N ebci αβ (ζ 1 , ζ 2 ) only depends on (ζ 1 , ζ 2 ), see also [8,11].…”

“…Note that the proof of the previous theorem is technically more involved than in the case without contact (see [8,11,28]). In particular, to identify the asymptotic model (3.9) the derivation of explicit formulas for κ = (κ ij ) is crucial: in the case without contact weak convergences of the sequences (u h ), (ϕ h ) and (κ h (u h )) is sufficient, while for the contact case it is necessary to prove the strong convergences (see Step 3).…”

“…It was observed that, if no contact and friction conditions have to be taken into account, for certain problems the mechanical and the electric parts of the equations decouple in an asymptotic process [8,11,26,28] (see [1,7,23,25,33] for related results where only partial or no decoupling occurs). In the presence of contact and friction it is not at all obvious if similar results hold true.…”

“…We mention [26] for piezoelectric plates including magnetic effects, [28] for piezoelectric thin plates with homogeneous isotropic elasticity coefficients, [8,11,12,23,25,33] for anisotropic piezoelectric plates and rods, [7] for geometrically nonlinear thin piezoelectric shells, and [27] for the modelling of eigenvalue problems for thin piezoelectric shells. However, these papers do not take into account the effects of possible contact or friction with a rigid foundation; nevertheless for elastic rods and shells, one finds asymptotic frictionless contact models in [32] and [20], respectively.…”