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Multiple Scattering of Elastic Waves in Polycrystals

Anubhav Roy, Christopher M. Kube

TL;DR

This work extends elastodynamic multiple-scattering theory for polycrystals by solving the Dyson equation under second-order smoothing approximation (SOSA), thereby incorporating recurrent scattering that can revisit grains. The authors derive SOSA self-energies alongside the conventional FOSA terms, valid across Rayleigh, stochastic, and geometric regimes, and demonstrate how to compute attenuation and phase-velocity dispersion from the resulting dispersion relations. Numerical results for iron and lithium show SOSA increases attenuation and modifies dispersion, especially for transverse waves, with Li (high anisotropy) exhibiting the strongest SOSA effects and closer agreement to finite-element predictions. The study provides a foundational step toward more accurate predictions of wave propagation in polycrystals and opens avenues for extending the framework to more complex microstructures and Bethe–Salpeter based backscatter analyses, with potential impact on nondestructive evaluation and materials characterization.

Abstract

Elastic waves that propagate in polycrystalline materials attenuate due to scattering of energy out of the primary propagation direction in addition to becoming dispersive in their group and phase velocities. Attenuation and dispersion are modeled through multiple scattering theory to describe the mean displacement field or the mean elastodynamic Green's function. The Green's function is governed by the Dyson equation and was solved previously (Weaver, 1990) by truncating the multiple scattering series at first-order, which is known as the first-order smoothing approximation (FOSA). FOSA allows for multiple scattering but places a restriction on the scattering events such that a scatterer can only be visited once during a particular multiple scattering process. In other words, recurrent scattering between two scatterers is not permitted. In this article, the Dyson equation is solved using the second-order smoothing approximation (SOSA). The SOSA permits scatterers to be visited twice during the multiple scattering process and, thus, provides a more complete picture of the multiple scattering effects on elastic waves. The derivation is valid at all frequencies spanning the Rayleigh, stochastic, and geometric scattering regimes without additional approximations that limit applicability in strongly scattering cases (like the Born approximation). The importance of SOSA is exemplified through analyzing specific weak and strongly scattering polycrystals. Multiple scattering effects contained in SOSA are shown to be important at the beginning of the stochastic scattering regime and are particularly important for transverse (shear) waves. This step forward opens the door for a deeper fundamental understanding of multiple scattering phenomena in polycrystalline materials.

Multiple Scattering of Elastic Waves in Polycrystals

TL;DR

This work extends elastodynamic multiple-scattering theory for polycrystals by solving the Dyson equation under second-order smoothing approximation (SOSA), thereby incorporating recurrent scattering that can revisit grains. The authors derive SOSA self-energies alongside the conventional FOSA terms, valid across Rayleigh, stochastic, and geometric regimes, and demonstrate how to compute attenuation and phase-velocity dispersion from the resulting dispersion relations. Numerical results for iron and lithium show SOSA increases attenuation and modifies dispersion, especially for transverse waves, with Li (high anisotropy) exhibiting the strongest SOSA effects and closer agreement to finite-element predictions. The study provides a foundational step toward more accurate predictions of wave propagation in polycrystals and opens avenues for extending the framework to more complex microstructures and Bethe–Salpeter based backscatter analyses, with potential impact on nondestructive evaluation and materials characterization.

Abstract

Elastic waves that propagate in polycrystalline materials attenuate due to scattering of energy out of the primary propagation direction in addition to becoming dispersive in their group and phase velocities. Attenuation and dispersion are modeled through multiple scattering theory to describe the mean displacement field or the mean elastodynamic Green's function. The Green's function is governed by the Dyson equation and was solved previously (Weaver, 1990) by truncating the multiple scattering series at first-order, which is known as the first-order smoothing approximation (FOSA). FOSA allows for multiple scattering but places a restriction on the scattering events such that a scatterer can only be visited once during a particular multiple scattering process. In other words, recurrent scattering between two scatterers is not permitted. In this article, the Dyson equation is solved using the second-order smoothing approximation (SOSA). The SOSA permits scatterers to be visited twice during the multiple scattering process and, thus, provides a more complete picture of the multiple scattering effects on elastic waves. The derivation is valid at all frequencies spanning the Rayleigh, stochastic, and geometric scattering regimes without additional approximations that limit applicability in strongly scattering cases (like the Born approximation). The importance of SOSA is exemplified through analyzing specific weak and strongly scattering polycrystals. Multiple scattering effects contained in SOSA are shown to be important at the beginning of the stochastic scattering regime and are particularly important for transverse (shear) waves. This step forward opens the door for a deeper fundamental understanding of multiple scattering phenomena in polycrystalline materials.
Paper Structure (12 sections, 100 equations, 4 figures, 2 tables)

This paper contains 12 sections, 100 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Elastodynamic Dyson equation
  3. Dyson equation solutions in the wavenumber domain
  4. First-order smoothing approximation
  5. Second-order smoothing approximation
  6. Results
  7. First-order smoothing approximation
  8. Second-order smoothing approximation
  9. Conclusions and Discussion
  10. Expression for the covariance tensor
  11. CPV integrals
  12. Coefficients $\Omega^{PQUV}_{n_1n_2n_3}$

Figures (4)

  • Figure 1: Equivalence of the UT of Stanke and Kino and FOSA for all frequencies in the three scattering regimes in which the Rayleigh and geometric scattering display asymptotic behavior at low and high $2\ell p_0$ values, with the stochastic regime being between. Results are shown for low, moderate, and strongly scattering materials of aluminum, iron, and lithium, respectively.
  • Figure 2: Attenuation results for (a) longitudinal waves in lithium, (b) transverse waves in lithium, (c) longitudinal waves in iron, and (d) transverse waves in iron. Phase velocity dispersion for (e) longitudinal waves in lithium, (f) transverse waves in lithium, (g) longitudinal waves in iron, and (h) transverse waves in iron. The phase velocity dispersion is normalized with respect to the phase velocities of a non-scattering reference medium to isolate the influence of scattering.
  • Figure 3: The effects of including the SOSA terms is studied as a ratio to the corresponding FOSA estimates of attenuation and wavespeed for longitudinal and transverse wave propagation in lithium and iron.
  • Figure : Graphical Abstract: Pictorial representation for the mean Green's function with first- and second-order scattering effects, which includes recurrent scattering from grains.