A nonlinear electromechanical model for partially debonded thin-sheet piezoelectric actuators

Abstract

Piezoelectric thin sheets/films hold a key position in the design of some advanced electromechanical structures, such as integrated actuators, energy harvesting devices and programmable structures. The development of new advanced materials, such as metamaterials, has generated a growing interest in utilizing the nonlinear response of piezoelectric elements to achieve new desired features. Nonetheless, there has been a lack of exploration into the modelling and analysis of thin-sheet piezoelectric actuators undergoing nonlinear deformation. The current paper presents a nonlinear model of a thin-sheet piezoelectric actuator partially bonded to a linear elastic, electrically insulative host. The formulation of the problem takes into account the associated effects of the debonding and the adhesive layer on the response of the piezoelectric structure. The thin-sheet actuator is assumed as a generalized electro-elastic Euler–Bernoulli beam, characterized by the piezoelectric effect, axial and bending deformation and nonlinear deflection in the debonded part. The developed nonlinear integral equations in terms of interfacial stresses are solved using Chebyshev polynomials to determine the deformation of the actuator and the local stress fields. This nonlinear model enables the analysis of the electromechanical behaviour of actuators subjected to different in-plane electric loading, considering a range of geometries. The bending moment, axial force, displacement and interfacial stress distribution along the actuator are carefully evaluated. The current study creates valuable insights into the behaviour of such actuators with nonlinear deformation, facilitating the enhancement of actuator performance for applications in specific metamaterials.

Appendix A: Effective material constants

The mechanical and electrical properties of piezoceramic materials can be described fully by the equation of motion

$$\begin{aligned} \sigma _{ji,j}+f_i=\rho \ddot{u}_i \end{aligned}$$

(53)

Gauss’ law

$$\begin{aligned} D_{i,i}=0 \end{aligned}$$

(54)

and the constitutive equations

$$\begin{aligned} \{\sigma \}=[c]\{\epsilon \}-[e]\{E\},\ \ \ \ \{\mathrm{{d}}\}=[e]\{\epsilon \}+[\epsilon ]\{E\} \end{aligned}$$

(55)

where

$$\begin{aligned} \epsilon _{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}),\ \ E_i=-V_{,i} \end{aligned}$$

(56)

In these equations, $\{\sigma \}$ and $\{\epsilon \}$ are the stress and strain fields, $f_i$ and $\rho$ are the body force and the mass density, while $\{\mathrm{{d}}\}$, $\{E\}$ and V represent the electric displacement, the electric field intensity and the potential, respectively. [c] is a matrix containing the elastic stiffness parameters for a constant electric potential, [e] represents a tensor containing the piezoelectric constants and $[\epsilon ]$ represents the dielectric constants for zero strains.

According to the electro-elastic line actuator model, the effective material constants of the actuator the model under plane strain condition is given by

$$\begin{aligned} \begin{array}{ll} E^p=c_{11}-\displaystyle {\frac{c_{13}^2}{c_{33}}},\ \ \ \ e_{31}^p=e_{31}-e_{33}\displaystyle {\frac{c_{13}}{c_{33}}},\ \ \ \epsilon _{33}^p=\epsilon _{33}+\displaystyle {\frac{e^2_{33}}{c_{33}}}&\ \end{array} \end{aligned}$$

(57)

where the direction of polarization is designated as being the z-axis.