Nonlinear behaviour of piezoceramics under weak electric fields: Part-I: 3-D finite element formulation

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.