Nonlinear forced vibrations of a hysteretic bar: Revisited

An earlier analytical model describing forced longitudinal vibrations of a bar with quadratic hysteretic nonlinearity is revised. To this end, the nonlinear constitutive there introduced is corrected and the origin of the errors is explained. The new resulting equation of motion, which includes also linear dissipative forces, is solved by means of a perturbation approach which makes use of the modulus defect of the material as a perturbation parameter. The experimentally observed linear dependence on the excitation amplitude of both resonance frequency shift and nonlinear attenuation is recovered. Similarly, the model predicts the generation of only odd harmonics with amplitude proportional to the square of the source strength. The solution includes the effects of the material hysteretic nonlinearity both in the time and in the spatial dependence of the vibration patterns, the latter feature being neglected by all other models. The relevance of this improvement resides in the potential the present model offers to deal with damage confined in narrow regions containing stress concentrators responsible for crack initiation. Furthermore, it is argued that such an approach may allow modeling of recent advanced imaging methods of nonlinear crack-like defects, which exploit the break-down of reciprocity in material systems with damage. ► Forced vibrations of a hysteretic bar are described analytically. ► Correction of an earlier analytical model is presented. ► The spatial component of the analytical solution includes the nonlinear effects. ► Experimentally observed nonlinear dispersion/attenuation is correctly predicted.