Nonlinear effects of piezoceramics excited by weak electric fields

There is a wide range of nonlinear effects which can be observed in piezoceramics. One example is the well-known butterfly hysteresis behavior for large stresses and strong electric fields. For small stresses and weak electric fields, piezoceramics are usually described by linear constitutive equations around an operating point in the butterfly hysteresis curve. Nevertheless, typical nonlinear effects can be observed when piezoceramic actuators and structures with embedded piezoceramics are excited in resonance, even if the stresses and the electric field remain small. This was observed and described, e. g., by Beige and Schmidt [1, 2], who investigated longitudinal rod vibrations using the piezoelectric 31-effect. They modeled these nonlinearities using higher-order quadratic and cubic elastic and electric terms. Typical nonlinear effects, e. g., dependence of the resonance frequency on the amplitude, superharmonics in spectra and a nonlinear relation between excitation voltage and vibration amplitude were also observed by von Wagner et al. [3] in piezo-beam systems. In this paper, the work is extended to longitudinal vibrations of piezoceramic rods using the piezoelectric 33-effect. The experiments with piezoelectric cylinders PIC 181 manufactured by PI-Ceramic, clearly exhibited not only nonlinearities of the Duffing type, but also quadratic nonlinearities. These nonlinearities are modeled using an extended electric enthalpy density, including nonlinear quadratic and cubic elastic terms, piezoelectric terms and dielectric terms. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. Simple rod models for slender cylinders are used for the description of the piezoceramics. The equations of motion are solved using perturbation techniques. Then, 'nonlinear' parameters can be identified, and the numerical results are compared to those obtained experimentally. The nonlinear effects described in this paper may have strong influence on the relation between excitation voltage and response amplitude whenever piezoceramic actuators and structures are excited at resonance.