Effect of Loss in Local Stiffness on Harmonic Scattering of Longitudinal Wave from a Quadratically Nonlinear Local Damage — ISTAM 2021 Conference Paper

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.</p><p dir="ltr"><br></p><p dir="ltr"><b>Key Contributions:</b></p><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><p dir="ltr"><br></p><p dir="ltr"><b>Methodology:</b><br>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.</p><p><br></p><p dir="ltr"><br></p><p dir="ltr"><b>Results:</b></p><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><p dir="ltr"><br></p><p dir="ltr"><b>Conclusions:</b><br>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.</p><p dir="ltr"><br></p><p><br></p><p dir="ltr"><br></p><p dir="ltr"><b>References:</b><br>[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.</p><p dir="ltr"><br></p><p dir="ltr"><b>Published record (ResearchGate):</b> <a href="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">researchgate.net/publication/356916787_Effect_of_Loss_in_Local_Stiffness_on_Harmonic_Scattering_of_Longitudinal_Wave_from_a_Quadratically_Nonlinear_Local_Damage</a><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><br></p><p dir="ltr"><br></p><p dir="ltr"><b>Keywords:</b> 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</p>