Nonlinear behaviour of piezoceramics under weak electric fields

Abstract

Piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak electric fields, they exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In order to model such nonlinearities, a nonlinear electric enthalpy density function (using quadratic and cubic terms) valid for a general 3-D piezoelectric continuum has been proposed in this work. Linear (i.e. proportional) and nonlinear damping models have also been proposed. The coupled nonlinear finite element equations have been derived using variational formulation. The classical linearization technique has been used to derive the linearized stiffness and damping matrices which helps in assembling the nonlinear matrices and solution of resulting nonlinear equation. The general 3-D finite element formulation is discussed in this paper. In a companion paper by Samal et al., numerical results on various typical examples are shown to match very well with the experimental observations.

An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

“…For repeated indices, Einstein's summation convention is implied in the tensorial expressions. In order to model the nonlinearities in a coupled piezoelectric medium, the linear electric enthalpy density function was modified by Samal et al (2006a) incorporating higher order terms (viz., quadratic and cubic in strain and electric field vector) in the energy expression of the coupled piezoelectric domain and hence is written as…”

“…From the experiments carried out at different electric field magnitudes, it was observed that the damping properties of the material depends upon the magnitude of electric field excitation and the dependence is not linear (as opposed to the linear viscous damping assumption in many analyses) especially for soft PZTs. In order to better describe the damping in the coupled piezoelectric domain, a generalized nonlinear damping model was developed by Samal et al (2006a), which considers linear as well as higher order time derivatives of both strain and electric field vector in the virtual work expression by damping forces dW D and the expression for dW D , is written as (Samal et al, 2006a)…”

“…von Wagner (2003) and Neumann (2002) also predicted the amplitude of displacement and current responses of piezo-rods of different diameters using a 1-D nonlinear formulation which used the nonlinear damping theory (for deriving the virtual work done by the dissipative damping forces). Samal et al (2006a) developed a generalized 3-D formulation for modeling the nonlinear behavior of piezoelectric materials under weak electric fields using a 3-D nonlinear electric enthalpy density function expression and nonlinear dissipative work expression dW D in the extended Hamilton's principle. Samal et al (2006a) also developed a finite element (FE) solution procedure for the above generalized nonlinear model and applied it to solve different problems in the 2-D and 3-D domain and compared the FE results with those of experiment (Samal et al, 2006b).…”

“…In this paper, a closed form solution for displacement and current responses of a simple rectangular geometry has been developed in 3-D domain based on the nonlinear electric enthalpy density function and virtual work (due to nonlinear damping forces) of Samal et al (2006a). Experiments have been conducted on a rectangular piezoceramic slab and the displacement and current responses have been compared with those of closed form solution.…”

Explicit finite-difference time-domain scheme for the simulation of 1-3 piezoelectric effect in axisymmetrical configurations

“…The authors are conscious that the stated problem has been approached successfully using FEM (finite element method) schemes [12][13][14][15]; however, the ease of model construction and availability of direct time domain calculations when using FDTD models justifies their introduction for many particular case studies. In fact, in echoimpulse techniques and directional radiation problems, the time domain knowledge of physical variables is a priority, and it is difficult to obtain them from the inverse-Fourier transform of numerically-obtained eigenvalues.…”