Solutions of Non-Classical Nonlinear Acoustics Wave Equation in a Conical Bar with Micro-Cracks

Non-classical nonlinear acoustic wave (NNAW) plays an important role in the evaluation of micro-cracked solids. In the present work, the characteristics of NNAW in a conical bar embedded with micro-cracks was investigated by modifying the stress-strain relationship with hysteresis and discrete memory characteristics. Considering the variation of the bar's cross section to be insignificant, a one-dimensional model was developed and the general wave equation was modified accordingly. Analytical analysis of the model was then achieved via the perturbation theory, which showed that accurate prediction of the crack location could be accomplished by analyzing the longitudinal displacement fields across the cracked area. Numerical simulations were carried out to demonstrate the effectivity of the model. The results showed that: (a) the presence of the micro-cracks induced non-classical nonlinear wave distortion, while the longitudinal strain was discontinuous on the two sides of the crack; (b) only the odd-order harmonics were generated, whose displacement amplitudes were highly related to the position and width of the micro-cracked region, and was also dependent on the strength parameter of the non-classical nonlinearity; (c) the displacement amplitudes of the third harmonic components was larger than other higher order harmonics, and could be employed to discern and locate the micro-cracked region.