Nonlinear Bulk Wave Propagation in a Material with Randomly Distributed Symmetric and Asymmetric Hysteretic Nonlinearity

Abstract

The nonlinear ultrasonic technique is an effective nondestructive testing technique for the detection of very small early-stage damages in solids that often remain insensitive to linear ultrasonic techniques. Interaction of a single frequency (f) ultrasonic wave with early-stage complex non-linear micro-defects such as dislocations, grains, micro-cracks, and micro-pores, etc., generates ultrasonic waves with higher harmonics (2f, 3f, 4f, 5f). In theoretical and computational studies, early-stage damages are modeled as homogeneous nonlinear material models like quadratic, cubic, and hysteretic nonlinearities. Interaction of single frequency wave with quadratic nonlinearity generates both odd and even harmonics but in the case of cubic and symmetric hysteretic nonlinearity, only odd harmonics generated due to material nonlinearity are observed. Symmetric hysteretic nonlinearity shows symmetric hysteretic force versus displacement hysteretic curves. In practice, the early-stage damages are highly localized and randomly distributed, and hysteretic. The objective of this investigation here is to study the interaction of a single frequency and two-frequency (one-way two-wave mixing) ultrasonic waves with the randomly distributed hysteretic nonlinear local damages. As the theoretical studies in hysteretic nonlinearities are challenging, here a numerical study of such a complex nature of wave propagation is considered. Symmetric and asymmetric hysteretic nonlinearities are considered independently. A one-dimensional spatial domain discretized as a long-chain of springmass elements with a random distribution of hysteretic spring elements. For symmetric hysteretic element, famously used Bouc-Wen model implemented and for asymmetric hysteretic element recently proposed Generalized Bouc-Wen model is implemented to capture asymmetric hysteretic force versus displacement nature of hysteretic curves. A single-frequency Gaussian pulse is sent from the left end of the spatial domain and the time responses are recorded at the one-fifth of total length and right end of the spatial domain. The same computational experiments repeated for ten different cases of randomly distributed symmetric and asymmetric hysteretic elements each and independently. In both the symmetric and asymmetric cases, harmonically scattered waves from randomly distributed local nonlinearities with sufficiently less amplitude than the input wave are observed. The frequency response of the recorded waves shows  only odd harmonics in the case of randomly distributed symmetric hysteretic nonlinear damages, but in the case of asymmetric hysteretic nonlinear damages, both the odd and harmonics are observed. The evolving symmetric and asymmetric hysteretic curves are observed in the case of symmetric and asymmetric hysteretic nonlinear damages respectively. To understand the one-way two-wave mixing phenomenon in thesehysteretic nonlinearities, a Gaussian pulse with two input frequencies is sent from the left end of the spatial domain. Due to mixing in symmetric hysteretic damage, sum and difference frequencies corresponding to only odd harmonics of input frequency along with the odd original odd harmonics are seen. In asymmetric hysteretic damage cases, sum and difference frequencies corresponding to both the odd and even harmonics are observed along with the original odd and even harmonics. In mixing significant amount of input energy is supplied to the possible frequency combinations present near the first few odd and/or even harmonics.