# Reflection of Nonlinear Waves in Reid's Hysteretic Material: A Numerical Perspective

## Palavras-chave:

nonlinear waves, wave-mixing, pinched hysteresis, nonlinear ultrasonics## Resumo

Nonlinear ultrasonics is effective in characterizing early-stage damages in solids. Interaction of a single

frequency (f) elastic wave with early-stage damages like dislocation substructures, micro-cracks, and micro-voids,

etc., generates higher harmonics (2f, 3f, 4f, 5f,..). In theoretical and computational studies early-stage damages are

modeled as nonlinear material models. Material models such as quadratic, cubic, and hysteretic nonlinearities are

commonly implemented in nonlinear wave propagation studies. To understand the interaction of the ultrasonic

wave with micro-cracks a pinched hysteretic nonlinearity looks the best fit as it can capture the nonlinear contact

mechanisms like opening and closing of micro-cracks which is also known as crack clapping and sliding at the

interfaces. One dimensional spatial domain is discretized as a long chain of spring-mass elements. Reid’s pinched

hysteretic elements are used in a long chain of spring-mass elements for the numerical study of the nonlinear wave

propagation through symmetric hysteretic material. Interaction of a single frequency elastic wave with Reid’s

symmetric hysteretic nonlinearity generates only odd harmonics (3f, 5f, 7f,...). Nonlinear reflected waves from

both the free and fixed end cases contain only odd harmonics. After reflection, nonlinear wave transfers energy

from 5th and 7th harmonics to 3rd harmonics. Pinched hysteretic loops are observed corresponding to both the

incident and reflected wave. The pinching at the origin of the hysteretic loops gets opened due to the reflection of

nonlinear waves. Evolving pinched hysteretic loops are observed due to Gaussian pulse as an input pulse whereas

repetitive pinched hysteretic loops are observed due to sine pulse as an input pulse. In one-way two-wave mixing,

both incident and reflected waves from free and fixed ends contain sum and difference frequency harmonics along

with the corresponding odd harmonics of input frequencies. Reflected waves transfer energy from the frequency

combinations present near 5th harmonics to frequency combinations present near 3rd harmonics. Minor hysteretic

loops due to wave mixing are observed within the major pinched hysteresis loops. As this numerical study is simple

in understanding, formulation, and implementation, it will help to solve inverse problems in nonlinear waves with

less computational resources and within a short time.