Wave-mixing due to two consecutive short pulses in Reid's pinched hysteretic material

Abstract

Early-stage degradation of metals due to fatigue includes the generation of dislocation substructures and their accumulation near grain boundaries produces local plasticity and micro-cracks. In theoretical, computational, and experimental nonlinear ultrasonics studies, the early stage damages are assumed as quadratic, cubic, and hysteretic nonlinearities. Interaction of a single frequency ultrasonic wave with hysteretic nonlinearity generates only odd harmonics (3f, 5f, 7f,…). A single pulse containing two input frequencies (f1, f2) passed through hysteretic material generates sum (f1 + f2, 2f1+ f2, f1 + 2f2,….) and difference frequencies (f1 - f2, 2f1 - f2, f1 - 2f2,….) along with the corresponding odd harmonics (3f1, 5f1, 7f1,…, 3f2, 5f2, 7f2,…). In this computational study, wave mixing due to two consecutive short pulses is implemented to generate only a specific set of mixed frequencies. A spatial domain is discretized as a long chain of a spring-mass system with a hysteretic spring element modeled as Reid’s hysteresis model. Care is taken to ensure spatial resolution and time stepping to achieve stability and convergence of the study based on the CFL condition. The proposed approach to solve this problem is very simple in understanding and takes less computational resources and computational time. Two consecutive Gaussian pulses of two frequencies (f1, f2) and amplitude ratio (A1 = 6A2) are sent through one end of a long chain with a specified time delay between two pulses. Wave mixing due to the memory effect generates only a specific sum (3f1 + f2) and (f1 + 3f2), and difference (3f1 - f2) frequencies which are not seen in the case of a single pulse with two input frequencies. Also, the corresponding odd harmonics (3f1, 5f1, 3f2, 7f1) of both the input frequencies are observed. The dominant sum and difference frequencies are sensitive to the amplitudes (A1, A2), input frequencies (f1, f2), and the time delay between two pulses. The hysteretic loops observed in various combinations are significantly different (triangular and pinched, translated triangular and pinched, and non-triangular and non-pinched or open) than each other showing their unique characteristics for both the Gaussian and sinusoidal input pulses. This study shows the effectiveness of the considered wave-mixing technique for the quantification of early-stage hysteretic damage in actual practice using nonlinear ultrasonics.