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Reflection of Nonlinear Waves in Reid's Hysteretic Material: A Numerical Perspective — CILAMCE-PANACM 2021 Proceedings Paper Pravinkumar Ghodake (3701242) Engineering education Mechanical engineering asset management Mechanical engineering not elsewhere classified Acoustics and noise control (excl. architectural acoustics) Acoustics and acoustical devices; waves Solid mechanics Theoretical and applied mechanics Computational complexity and computability Computational modelling and simulation in earth sciences Numerical and computational mathematics not elsewhere classified Data visualisation and computational (incl. parametric and generative) design Numerical modelling and mechanical characterisation Applications in physical sciences Other physical sciences not elsewhere classified Nonlinear optics and spectroscopy Numerical analysis Numerical computation and mathematical software Numerical solution of differential and integral equations Engineering design Computer aided design Design practice and methods Industrial and product design Design not elsewhere classified Design history, theory and criticism Models and simulations of design Theory and design of materials Granular mechanics CILAMCE 2021 PANACM 2021 nonlinear waves wave-mixing pinched hysteresis nonlinear ultrasonics Reid's hysteretic material hysteretic damping crack clapping contact acoustic nonlinearity (CAN) micro-cracks dislocation substructures Dislocation substructures micro-voids early-stage damage harmonic generation odd harmonics energy transfer spring-mass chain model one-way two-wave mixing sum and difference frequencies minor hysteresis loops reflection of nonlinear waves free end fixed end structural health monitoring Structural health monitoring Aircraft availability Structural health monitoring; Interfacial debonding; ACTs; Vibration-based structural health monitoring, data-driven methods, thermal response, bridges Structural health monitoring. Structural Health Monitoring (SHM) structural health monitoring (SHM) Structural health monitoring (SHM) Structural Health Monitoring STRUCTURAL HEALTH MONITORING Structural health monitoring Structural health monitoring systems Structural Health Monitoring Sensors Structural Health Monitoring (SHM). Structural health monitoring; Hybrid ceramic bearings; Systematic review; Meta-Analysis non-destructive evaluation non-destructive evaluation inspection Non-destructive evaluation (NDE) Non-Destructive Evaluation (NDE) Non-destructive Evaluation Non-Destructive Evaluation Non-destructive evaluation proceedings paper solid mechanics solid mechanics problems Solid mechanics and dynamics Solid mechanics Solid Mechanics Pravinkumar Ghodake Pravinkumar Ramchandra Ghodake Design Optimization Design & Technology Generative design research Elastodynamics simulations This item is a peer-reviewed proceedings paper presented by Pravinkumar R. Ghodake (Department of Mechanical Engineering, IIT Bombay) at the XLII Ibero-Latin-American Congress on Computational Methods in Engineering (CILAMCE 2021) and the III Pan-American Congress on Computational Mechanics (PANACM 2021), held in Rio de Janeiro, Brazil, November 9–12, 2021. The paper was published in the official conference proceedings by ABMEC-IACM. Title of contribution: "Reflection of Nonlinear Waves in Reid's Hysteretic Material: A Numerical Perspective" Abstract: 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 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 — also known as crack clapping and sliding at the interfaces.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.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. Key Contributions:<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> Methodology: 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. Results: 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×. Conclusions: 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. Publication details:<ul><li>Proceedings: Proceedings of the XLII Ibero-Latin-American Congress on Computational Methods in Engineering and III Pan-American Congress on Computational Mechanics</li><li>Publisher: ABMEC-IACM (Associação Brasileira de Métodos Computacionais em Engenharia — International Association for Computational Mechanics)</li><li>Year: 2021</li><li>Place: Rio de Janeiro, Brazil</li><li>Conference dates: November 9–12, 2021</li></ul> Acknowledgements (from paper): Supported by the Acoustical Society of America (ASA) through the CIRE student grant. Discussions with Saurabh Biswas are gratefully acknowledged. Research portfolio: <a href="https://sites.google.com/view/pravinkumarghodake/research" target="_blank" rel="noreferrer">https://sites.google.com/view/pravinkumarghodake/research</a> Keywords: 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 2026-09-30T05:05:20Z Text Conference contribution 10.6084/m9.figshare.34029873.v1 https://figshare.com/articles/conference_contribution/Reflection_of_Nonlinear_Waves_in_Reid_s_Hysteretic_Material_A_Numerical_Perspective_CILAMCE-PANACM_2021_Proceedings_Paper/34029873 CC BY 4.0