Reflection of Nonlinear Waves in Reid's Hysteretic Material: A Numerical Perspective — CILAMCE-PANACM 2021 Proceedings Paper

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 — also known as crack clapping and sliding at the interfaces.</p><p dir="ltr">A 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 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 free and fixed end cases contain only odd harmonics. After reflection, the nonlinear wave transfers energy from the 5th and 7th harmonics to the 3rd harmonic.

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 a Gaussian pulse as the input pulse, whereas repetitive pinched hysteretic loops are observed due to a sine pulse as the input pulse.</p><p dir="ltr">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 the 5th harmonics to frequency combinations present near the 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 fewer computational resources and within a short time.</p><p dir="ltr"><br></p><p dir="ltr"><b>Key Contributions:</b></p><ul><li>Numerical investigation of nonlinear wave reflection in Reid's pinched hysteretic material</li><li>Comparison of free-end and fixed-end boundary conditions</li><li>Demonstration of odd-only harmonic generation and energy transfer from 5th/7th harmonics to 3rd harmonic upon reflection</li><li>Analysis of pinched hysteretic loop evolution and opening under incident and reflected waves</li><li>One-way two-wave mixing behavior with sum/difference frequencies and minor loop formation</li><li>Computationally efficient framework suitable for inverse problems in nonlinear ultrasonics</li></ul><p dir="ltr"><br></p><p dir="ltr"><b>Methodology:</b> A one-dimensional domain is discretized as a long chain of 800 spring-mass elements with masses m₁ = m₂ = … = m and stiffness k₁ = k₂ = … = k.

Hysteretic nonlinearity is added in parallel to linear springs using Reid's hysteretic element. Material properties: m = 1.3e-6 kg, k = 38.5 MN/m, k_H = 38.5 kN/m, η = 0.3. Input pulses include single-frequency sine, mixed sine, single-frequency Gaussian, and mixed Gaussian pulses.

Free-end and fixed-end boundary conditions are studied separately. MATLAB built-in ODE solver is used, with time-stepping based on the CFL condition. Incident and reflected waves are independently analyzed for frequency response and hysteretic curves at the 400th mass.</p><p dir="ltr"><br></p><p dir="ltr"><b>Results:</b> Single-frequency excitation produces only odd harmonics in both incident and reflected waves.

Upon reflection, energy transfers from the 5th and 7th harmonics to the 3rd harmonic. Pinched hysteretic loops open at the center after reflection, generating three closed symmetric loops. At the fixed end, reflected hysteretic loops are rotated 90° due to the 180° phase change.

In one-way two-wave mixing (f₁ = 0.06 MHz, f₂ = 0.1 MHz, A₁/A₂ = 3), sum and difference frequencies appear alongside odd harmonics, but only those whose net difference is an odd multiplier. Amplitude ratios (R/I) at the 3rd harmonic are ~3.5× the fundamental ratio, at the 7th harmonic ~1.5×, and at the 5th harmonic ~0.5×.</p><p dir="ltr"><br></p><p dir="ltr"><b>Conclusions:</b> The study demonstrates that the simple formulation, fewer computational resources, and short computational time make this numerical approach useful for solving nonlinear inverse problems in nonlinear ultrasonics for in-situ and ex-situ structural health monitoring.

Initial quick computational studies help select input pulse amplitude and frequency combinations and target possible frequency combinations for nonlinear pulse-echo experiments in pinched hysteretic materials.</p><p dir="ltr"><br></p><p dir="ltr"><b>Publication details:</b></p><ul><li><b>Proceedings:</b> Proceedings of the XLII Ibero-Latin-American Congress on Computational Methods in Engineering and III Pan-American Congress on Computational Mechanics</li><li><b>Publisher:</b> ABMEC-IACM (Associação Brasileira de Métodos Computacionais em Engenharia — International Association for Computational Mechanics)</li><li><b>Year:</b> 2021</li><li><b>Place:</b> Rio de Janeiro, Brazil</li><li><b>Conference dates:</b> November 9–12, 2021</li></ul><p dir="ltr"><br></p><p dir="ltr"><b>Acknowledgements (from paper):</b> Supported by the Acoustical Society of America (ASA) through the CIRE student grant.

Discussions with Saurabh Biswas are gratefully acknowledged.</p><p dir="ltr"><br></p><p dir="ltr"><b>Research portfolio:</b> <a href="sites.google.com/view/pravinkumarghodake/research" target="_blank" rel="noreferrer">sites.google.com/view/pravinkumarghodake/research</a></p><p dir="ltr"><br></p><p dir="ltr"><b>Keywords:</b> nonlinear waves, wave-mixing, pinched hysteresis, nonlinear ultrasonics, Reid's hysteretic material, contact acoustic nonlinearity, harmonic generation, energy transfer, spring-mass chain model, one-way two-wave mixing, structural health monitoring, CILAMCE 2021, PANACM 2021, proceedings paper</p>