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Effect of Loss in Local Stiffness on Harmonic Scattering of Longitudinal Wave from a Quadratically Nonlinear Local Damage — ISTAM 2021 Conference Paper Pravinkumar Ghodake (3701242) Engineering education Applications in physical sciences Other physical sciences not elsewhere classified Acoustics and noise control (excl. architectural acoustics) Acoustics and acoustical devices; waves Solid mechanics Applied mathematics not elsewhere classified Mechanical engineering asset management Numerical modelling and mechanical characterisation Mechanical engineering not elsewhere classified Theoretical and applied mechanics Nonlinear optics and spectroscopy Numerical analysis Numerical computation and mathematical software Numerical solution of differential and integral equations Numerical and computational mathematics not elsewhere classified Computational complexity and computability Computational modelling and simulation in earth sciences Data visualisation and computational (incl. parametric and generative) design Engineering design Computer aided design Design practice and methods Design history, theory and criticism Industrial and product design Design not elsewhere classified Models and simulations of design Theory and design of materials nonlinear ultrasonics Nonlinear ultrasonics Nonlinear Ultrasonics harmonic scattering harmonic scattering intensities Harmonic Scattering Angle-resolved Harmonic Scattering loss in local stiffness quadratically nonlinear material longitudinal wave longitudinal wave velocity Longitudinal waves longitudinal waves compensatory waves elastodynamic reciprocity energy transfer Energy transfer -- Testing Energy Transfer Assisted Amplified Exciplex Emission Energy Transfer BehaviorA novel higher harmonics Higher harmonics higher harmonics generation first harmonic first harmonic frequency first harmonic method first harmonic alone second harmonic second harmonic based Second Harmonic Generation Macromolecular interactions second harmonic amplitudes second harmonic efficiencies Second Harmonic Generation Response... Second Harmonic Generation Guided Raman Spectroscopy Second Harmonic Generation Nonlinear Second Harmonic Generating Nanoparticles second harmonic frequency Second Harmonic Generation Pattern Second Harmonic Generation Efficient frequency conversion techniques early-stage damage Early-Stage Damage Detection micro-cracks micro-voids dislocation breakaways damage quantification damage quantification law Damage quantification Damage Quantification structural health monitoring Structural health monitoring; Hybrid ceramic bearings; Systematic review; Meta-Analysis Structural Health Monitoring (SHM). Structural health monitoring systems Structural health monitoring Aircraft availability Structural health monitoring; Interfacial debonding; ACTs; Vibration-based Structural Health Monitoring Sensors 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 non-destructive evaluation inspection Non-destructive evaluation (NDE) Non-Destructive Evaluation (NDE) Non-destructive Evaluation Non-Destructive Evaluation non-destructive evaluation Non-destructive evaluation analytical solution Analytical solution parameters Analytical solutions analytical solutions Analytical Solution Analytical solution Perturbation method; perturbation method (MHPM) Perturbation Methods power spectrum techniques power spectrum of frequency bands power spectrum density analysis Power spectrum analysis Power Spectrum Analysis Power Spectrum inverse problems of PDEs INVERSE PROBLEMS Inverse problems (Differential equations) -- Numerical solutions ISTAM 2021 Solid Mechanics Pravinkumar Ghodake Pravinkumar Ramchandra Ghodake Design Optimization Design & Technology Generative design research Elastodynamics simulations This conference contribution is a research paper presented by Pravinkumar Ghodake (Department of Mechanical Engineering, IIT Bombay) at the 66th Congress of the Indian Society of Theoretical and Applied Mechanics (ISTAM 2021). The work derives analytical theoretical solutions to demonstrate the critical sensitivity of wave fields to local stiffness reductions caused by micro-void accumulation and dislocation breakaways. Title of contribution: "Effect of Loss in Local Stiffness on Harmonic Scattering of Longitudinal Wave from a Quadratically Nonlinear Local Damage" Abstract: The amplitudes of the higher harmonics of transmitted longitudinal waves in a material with uniformly distributed early-stage damages can be correlated to the intensity of the early-stage damage present inside the material. In practical applications, materials are under fatigue loading resulting in highly local early-stage damages. To understand the interaction of the longitudinal wave with such local nonlinear elastic material, Wang and Achenbach [1] assumed a cubically nonlinear local material without loss in the local stiffness of a nonlinear material. They obtained theoretical solutions for the backscattered and forward-scattered harmonic waves by considering the continuity of stress and displacement at the interfaces, harmonic generation in nonlinear material, the concept of compensatory waves, and using the reciprocity theorem of elastodynamics.In practice, loss in local stiffness due to damage mechanisms such as micro-cracks, micro-voids, and breakaway of multiple dislocations and dislocation substructures also need to be considered. In this study, the importance and sensitivity of loss in local stiffness on harmonic scattering is demonstrated through obtaining theoretical solutions using continuity conditions at interfaces, harmonic generation in quadratically nonlinear material, and the concept of compensatory waves. Harmonically backscattered waves from local quadratically nonlinear damages are highly sensitive to loss in local stiffness and early-stage damage sizes. Energy transfer between 1st and 2nd harmonics due to loss in local stiffness and damage sizes (0–20 mm) is explained in detail by considering power spectrum responses of the obtained theoretical solutions. This formulation can be used to increase the robustness of inverse problems in nonlinear wave propagation studies proposed to find the loss in local stiffness, the intensity of early-stage damages, damage sizes, and the position of local damages. Key Contributions:<ul><li>Analytical solutions for harmonically scattered waves from quadratically nonlinear local damage with loss in local stiffness</li><li>Demonstration of energy transfer between 1st and 2nd harmonics due to damage size and stiffness loss</li><li>Power spectrum analysis of backscattered and forward-scattered waves</li><li>Framework for improving inverse problem robustness in nonlinear wave propagation</li></ul> Methodology: A one-dimensional domain is considered for wave propagation. Highly local early-stage damage is modeled as a quadratically nonlinear material with Young's modulus E₂, while the remaining domain is modeled as a linear material with Young's modulus E₁. The ratio E₂/E₁ ~ 0.9–1 represents the slight loss in local stiffness due to microscale damage mechanisms. An approximate analytical solution for wave propagation of a single-frequency wave in quadratically nonlinear material is obtained using the regular perturbation method. The concept of compensatory waves, proposed by Wang and Achenbach, is implemented to derive displacement fields on both sides of the interface, considering impedance mismatch. Displacement and stress continuity equations are applied at the interfaces to obtain expressions for the amplitudes of compensatory waves. Maplesoft software is used to carry out symbolic calculations. Results:<ul><li>At no loss in local stiffness (γ² = 1), only the 2nd harmonic of the backscattered wave is observed, while both 1st and 2nd harmonics are present in the forward-scattered wave</li><li>Increasing γ² shows a nonlinear decrease in amplitudes of 1st and 2nd harmonics of backscattered waves and a linear decrease in amplitudes of forward-scattered waves</li><li>With increase in damage size, a slight decrease in energy of 1st harmonics is observed, whereas a nonlinear increase in 2nd harmonic amplitude shows energy transfer from 1st to 2nd harmonics</li><li>The sinusoidal response of backscattered wave amplitudes indicates energy transfer between 1st and 2nd harmonics at known damage sizes</li></ul> Conclusions: The importance of loss in local stiffness in harmonic scattering is demonstrated through theoretical solutions obtained using the concept of compensatory waves. Backscattered harmonic waves from quadratically nonlinear material are highly sensitive to loss in local stiffness and local early-stage damage sizes. Energy transfer between 1st and 2nd harmonics, mainly due to damage sizes and loss in local stiffness, is explained in detail. This formulation can be used to increase the robustness of inverse problems in nonlinear wave propagation studies. Loss in local stiffness can be related to the density and aspect ratio of micro-cracks from different micromechanics models. References: [1] Y. Wang and J. D. Achenbach, "Reflection of ultrasound from a region of cubic material nonlinearity due to harmonic generation," Acta Mechanica, vol. 229, no. 2, pp. 763–778, 2018. Published record (ResearchGate): <a href="https://www.researchgate.net/publication/356916787_Effect_of_Loss_in_Local_Stiffness_on_Harmonic_Scattering_of_Longitudinal_Wave_from_a_Quadratically_Nonlinear_Local_Damage" target="_blank" rel="noreferrer">https://www.researchgate.net/publication/356916787_Effect_of_Loss_in_Local_Stiffness_on_Harmonic_Scattering_of_Longitudinal_Wave_from_a_Quadratically_Nonlinear_Local_Damage</a> 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 ultrasonics, harmonic scattering, loss in local stiffness, quadratically nonlinear material, longitudinal wave, compensatory waves, elastodynamic reciprocity, energy transfer, early-stage damage, structural health monitoring, non-destructive evaluation, damage quantification, analytical solution, ISTAM 2021 2026-09-30T03:41:42Z Text Conference contribution 10.6084/m9.figshare.34028808.v1 https://figshare.com/articles/conference_contribution/Effect_of_Loss_in_Local_Stiffness_on_Harmonic_Scattering_of_Longitudinal_Wave_from_a_Quadratically_Nonlinear_Local_Damage_ISTAM_2021_Conference_Paper/34028808 CC BY 4.0