Keywords: piezoelectric, laminates, mixed variational formulation, capacitance.

The aim of the present work is to discuss in details an issue which is often neglected in beam models of piezoelectric laminates: the influence of hypotheses on cross-sectional transverse normal stress and strain on the derivation of the constitutive properties of one-dimensional models from the three-dimensional geometric and material data. Beam models presented in literature usually retain classical assumptions for single-layered elastic structure by assuming that either transversal normal stress or transversal normal deformations are vanishing layer by layer. However, as it shown by finite element simulations based on a 3D model, these hypotheses are not verified in piezoelectric laminates with thickness-polarized piezoelectric layers. Indeed, when a potential difference is applied between the electrodes of a piezoelectric layer, it naturally tends to isotropically extend (or shrink) in the plane orthogonal to the polarization axis. Such a in-plane-isotropic deformation is in competition with Poisson effect in elastic layer, which associates axial extension to transversal shrinking. When piezoelectric and elastic layers are bonded together to form a laminated piezoelectric beam, this contrasting behaviour is conciliated by the appearance of non-negligible normal stresses both in the axial direction and transverse direction. Beam model based on standard assumptions completely neglect transverse interactions between different layers. As a consequence significant errors are introduced in the deduction of one-dimensional constitutive properties from the three-dimensional ones.

To overcome this issue a novel model of laminated piezoelectric beams including transverse interactions between different layers is proposed. It includes both sensory and actuation effects and is based on the following hypotheses: i) equivalent single-layer Euler-Bernoulli kinematics; ii) layerwise linear distribution of mechanical stress with non-vanishing transverse normal stress; iii) layerwise linear distribution of electric potential; iv) layerwise constant distribution of the electric displacement. In particular, the distribution of transverse normal stress is determined by imposing that its force and moment resultant over the beam cross- section are vanishing. The model is derived by adopting a mixed variational formulation and by imposing the further constraints on transverse stress through Lagrange multiplier method.

Numerical comparisons between the proposed modelling approach, standard ones and FEM simulations are presented. In particular, for a sandwich piezoelectric beam and for a two-layer beam, the estimated behaviour of the beam constitutive coefficients in function of the thickness ratio between elastic and piezoelectric layers is compared. Moreover, the main features of the proposed model are highlighted by presenting the through-the-thickness distribution of the 3D state fields associated to beam-axis deformations and applied voltage. Close agreement with numerical simulation based on three-dimensional model is shown. As main peculiarity, the proposed beam model is able to coherently estimate the equivalent piezoelectric capacitance and transverse normal stress and strain distribution, independently of the thickness ratio between elastic and piezoelectric layers.

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