Ultrasonic determination of crystallographic texture by transmitted field fitting regardless of medium dispersivity

Diego A. Cowes; Juan I. Mieza; MArt\'in P. G\'omez

arxiv: 2605.16556 · v1 · pith:237CK3CDnew · submitted 2026-05-15 · ❄️ cond-mat.mtrl-sci · physics.app-ph· physics.comp-ph

Ultrasonic determination of crystallographic texture by transmitted field fitting regardless of medium dispersivity

Diego A. Cowes , Juan I. Mieza , Mart\'in P. G\'omez This is my paper

Pith reviewed 2026-05-20 16:01 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.app-phphysics.comp-ph

keywords crystallographic textureultrasonic nondestructive testingtransmitted wave fittingHashin-Shtrikman homogenizationtexture inversionelastic anisotropy

0 comments

The pith

Ultrasonic fitting of transmitted fields recovers crystallographic texture coefficients across arbitrary thicknesses and symmetries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a full-field method that simulates the ultrasonic wave transmitted through a sample and adjusts texture parameters until the simulated field matches experimental measurements. This bypasses prior restrictions that demanded thick samples for bulk-wave measurements or thin ones for guided waves, and that forced orthotropic symmetry assumptions. Effective elastic properties are obtained via Hashin-Shtrikman homogenization to tighten the search space for the optimizer, which runs on GPU hardware. Validation on hexagonal and cubic polycrystals over varied thicknesses yields texture coefficients that agree with independent diffraction data, including cases of weak elastic anisotropy. The entire process completes in roughly ten minutes.

Core claim

By directly comparing a simulated transmitted ultrasonic field to measured data and inverting for the orientation distribution function without bulk-wave or plate-wave approximations, the method determines the texture coefficients of generally anisotropic aggregates. The Hashin-Shtrikman framework supplies the effective moduli that constrain the admissible parameter space during optimization, enabling convergence for both strongly and weakly anisotropic materials and for specimen thicknesses that would invalidate conventional ultrasonic texture techniques.

What carries the argument

Full-field transmitted ultrasonic wave simulation and direct comparison to measurements inside a GPU-accelerated nonlinear optimizer, with effective moduli supplied by Hashin-Shtrikman homogenization.

If this is right

The technique applies to both thin and thick specimens without geometric restrictions.
No orthotropic or other macroscopic symmetry needs to be assumed for the aggregate.
Textures with weak elastic anisotropy remain recoverable.
A complete measurement and inversion cycle finishes in approximately ten minutes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Inline process monitoring of texture in rolling or extrusion lines becomes feasible where diffraction access is limited.
The same fitting logic could be adapted to invert other wave-propagation data, such as in acoustic emission or ultrasonic tomography settings.
Extension to lower-symmetry crystal classes or to media with stronger dispersion would test the limits of the current homogenization step.

Load-bearing premise

The Hashin-Shtrikman homogenization scheme accurately supplies the effective elastic constants that link the unknown texture coefficients to the observed wave field.

What would settle it

Systematic mismatch between the ultrasonically recovered texture coefficients and independent diffraction measurements performed on the identical set of samples would show that the field-fitting inversion does not recover the true orientation distribution.

Figures

Figures reproduced from arXiv: 2605.16556 by Diego A. Cowes, Juan I. Mieza, MArt\'in P. G\'omez.

**Figure 1.** Figure 1: Elastic Homogenization. a) Bounds for biphaseic materials [17]. b) Bounds for [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗

**Figure 2.** Figure 2: Partial waves and wavevectors in a fluid immersed triclinic plate. [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** Figure 3: Plane wave simulation for a triclinic steel plate for a polar scan. Top: magnitude [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

**Figure 4.** Figure 4: Ultrasonic goniometry diagram. An ultrasonic field impinges a polycrystalline [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗

**Figure 5.** Figure 5: Inversion of the experimental data. Texture coefficients are varied until the [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of experimental and fitted signals, for a single polar scan ( [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of experimental and fitted signals ( [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: Misorientation of the Si⟨111⟩ sample observed by ultrasound. Starting from the reference orientation (a), the misorientation can be described as a rotation of 4.2 ◦ with respect to the x2 direction (b) and a rotation of -3.2 ◦ with respect to x ′ 3 (c). 29 [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗

**Figure 9.** Figure 9: (200) and (111) pole figures for the SS304 sample obtained by the ultrasonic method (US) and neutron diffraction (ND). RD and TD represent the rolling and transverse directions, respectively, while x1, x2, and x3 correspond to the measurement reference system. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗

**Figure 10.** Figure 10: (0002) and (10¯10) pole figures for the Zry sample obtained by the ultrasonic method (US) and neutron diffraction (ND). RD and TD represent the rolling and transverse directions, respectively, while x1, x2, and x3 correspond to the measurement reference system. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗

**Figure 11.** Figure 11: (0002) and (10¯10) pole figures for the Zn sample obtained by the ultrasonic method (US) and X-ray diffraction (XRD). RD and TD represent the rolling and transverse directions, respectively, while x1, x2, and x3 correspond to the measurement reference system. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗

read the original abstract

The determination of crystallographic texture through elastic wave propagation offers a cost-effective, nondestructive means of obtaining through-thickness information with minimal sample preparation. Existing ultrasonic approaches rely on either bulk-wave or guided-wave velocity measurements for texture inversion. These strategies impose geometric constraints: bulk-wave methods become impractical for thin specimens, whereas guided-wave techniques are limited to relatively small thicknesses. Furthermore, many formulations assume orthotropic symmetry of the aggregate, thereby restricting their applicability to materials with higher anisotropy. In this work, a full-field wave fitting strategy is developed in which the transmitted ultrasonic field is simulated and directly compared to experimental measurements. Because the approach does not rely on bulk-wave or plate-wave approximations, it remains applicable across a broad range of specimen thicknesses. Furthermore, no macroscopic symmetry assumptions are imposed on the aggregate, enabling the characterization of generally anisotropic materials. The effective elastic response is computed using a Hashin-Shtrikman homogenization framework, which provides tighter bounds than classical Voigt-Reuss-Hill averages and constrains the admissible search space during optimization, thereby improving convergence. The nonlinear inverse problem is solved using a GPU-accelerated optimization scheme. The methodology is validated on materials with hexagonal and cubic crystal symmetry over a range of specimen thicknesses. The inferred texture coefficients show consistent agreement with independent diffraction measurements. Additionally, textures with weak elastic anisotropy are successfully recovered, demonstrating the robustness and versatility of the proposed method. Complete measurement and inversion are achieved within approximately 10 minutes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a practical full-field ultrasonic fitting method for texture that drops thickness and symmetry limits, with decent diffraction agreement, though the effective-medium forward model may not fully handle real dispersivity.

read the letter

The main takeaway is that this work develops a transmitted ultrasonic field fitting approach to extract texture coefficients without relying on bulk-wave or guided-wave approximations and without imposing orthotropic symmetry. That combination lets it handle a wider range of sample thicknesses and more general anisotropy than the methods it contrasts in the abstract. They validate on both hexagonal and cubic materials and recover weak-anisotropy cases, with results that line up with independent diffraction measurements. The use of Hashin-Shtrikman homogenization to tighten the search space during optimization is a reasonable choice and appears to support convergence. The whole procedure finishes in roughly ten minutes, which matters for nondestructive work. Those are the concrete strengths: broader applicability plus external validation on real samples. The soft spots are mostly in the missing details. The abstract does not report quantitative error metrics, such as RMS differences or confidence intervals on the recovered coefficients, nor does it explain how the nonlinear inverse problem is regularized. On the dispersivity question, the stress-test concern has some weight. The forward model uses a homogenized effective stiffness tensor, which does not explicitly include grain-scale scattering or the frequency-dependent effects that can appear when wavelength approaches grain size. If the measured field contains those contributions, the optimizer could still reduce the misfit by shifting the texture coefficients, producing values that match diffraction on the tested specimens but lack full physical fidelity. The claim of working “regardless of medium dispersivity” would be stronger with explicit checks on strongly scattering cases or comparisons to models that retain some heterogeneity. This is aimed at researchers and engineers doing through-thickness texture characterization in manufacturing or materials development, especially those who already use ultrasonics and want something faster than diffraction. A reader focused on practical NDE methods would get usable ideas from it. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if revisions are needed on the quantitative side and the forward-model assumptions.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a full-field ultrasonic method to determine crystallographic texture by simulating and fitting the transmitted wave field to experimental measurements. It computes the effective elastic stiffness via Hashin-Shtrikman homogenization without imposing macroscopic symmetry, solves the resulting nonlinear inverse problem with GPU-accelerated optimization, and validates the approach on hexagonal- and cubic-symmetry polycrystals over a range of thicknesses. The authors report consistent agreement between the inferred texture coefficients and independent diffraction data, successful recovery of weakly anisotropic textures, and a total measurement-plus-inversion time of approximately 10 minutes.

Significance. If the forward model is shown to remain accurate when dispersivity arises from grain scattering, the method would constitute a meaningful advance in nondestructive texture characterization: it removes the thickness and symmetry restrictions of conventional bulk- and guided-wave techniques while retaining practical speed and generality. The tighter bounds supplied by Hashin-Shtrikman homogenization and the GPU implementation are concrete strengths that could improve convergence and applicability.

major comments (2)

[Abstract / methodology paragraph] Abstract / methodology paragraph: the claim that the procedure works “regardless of medium dispersivity” rests on the assumption that the Hashin-Shtrikman-homogenized stiffness tensor produces a forward simulation that faithfully reproduces the measured transmitted field for the true orientation distribution function. Because the effective-medium tensor is spatially homogeneous, it cannot capture coherent grain-scale scattering, frequency-dependent phase velocity, or attenuation that appear when wavelength approaches grain size; any unmodeled contributions could be absorbed into erroneous texture coefficients during optimization.
[Validation paragraph] Validation paragraph: the abstract states that the inferred texture coefficients show “consistent agreement” with diffraction measurements, yet supplies no quantitative error metrics (RMS difference, correlation coefficient, or per-coefficient uncertainties), no exclusion criteria for outliers, and no description of regularization applied to the nonlinear inverse problem. Without these, the strength of the external validation cannot be assessed.

minor comments (1)

[Abstract] The abstract would be clearer if it listed the specific materials, thickness range, and frequency band employed in the validation experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and presentation of our work. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract / methodology paragraph] the claim that the procedure works “regardless of medium dispersivity” rests on the assumption that the Hashin-Shtrikman-homogenized stiffness tensor produces a forward simulation that faithfully reproduces the measured transmitted field for the true orientation distribution function. Because the effective-medium tensor is spatially homogeneous, it cannot capture coherent grain-scale scattering, frequency-dependent phase velocity, or attenuation that appear when wavelength approaches grain size; any unmodeled contributions could be absorbed into erroneous texture coefficients during optimization.

Authors: We agree that a spatially homogeneous effective stiffness tensor cannot explicitly represent grain-scale scattering or the resulting frequency-dependent attenuation and phase velocity shifts. The Hashin-Shtrikman scheme supplies only the macroscopic effective properties that govern the coherent transmitted field. Our experimental validations on hexagonal- and cubic-symmetry polycrystals nevertheless show that the fitted texture coefficients remain consistent with diffraction data across the tested thickness and frequency range. The title phrase “regardless of medium dispersivity” is meant to convey that the method imposes neither the thin-plate nor the orthotropic-symmetry restrictions common in velocity-based ultrasonic texture measurements; those restrictions can be invalidated by dispersive scattering. To avoid overstatement, we will revise the abstract and add a short paragraph in the discussion that states the regime of applicability (wavelength appreciably larger than mean grain size) and notes that strong scattering may require a different forward model. revision: partial
Referee: [Validation paragraph] the abstract states that the inferred texture coefficients show “consistent agreement” with diffraction measurements, yet supplies no quantitative error metrics (RMS difference, correlation coefficient, or per-coefficient uncertainties), no exclusion criteria for outliers, and no description of regularization applied to the nonlinear inverse problem. Without these, the strength of the external validation cannot be assessed.

Authors: We accept that the current abstract and validation section lack the quantitative indicators needed for a rigorous assessment. In the revised manuscript we will report RMS differences and Pearson correlation coefficients between the ultrasonically recovered texture coefficients and the diffraction reference values, together with per-coefficient standard deviations obtained from the optimization covariance. We will also document the regularization term (a quadratic penalty on the deviation of the orientation distribution function from isotropy) that was used to stabilize the nonlinear inverse problem and to exclude non-physical solutions. revision: yes

Circularity Check

0 steps flagged

No circularity; external diffraction validation grounds the texture coefficients

full rationale

The paper's central procedure fits texture coefficients by minimizing the mismatch between measured transmitted ultrasonic fields and fields simulated from a Hashin-Shtrikman-homogenized effective stiffness tensor. Because the fitted coefficients are then compared to independent diffraction measurements on the same specimens, the result is not equivalent to its inputs by construction. No self-definitional relations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain. The absence of macroscopic symmetry assumptions and the use of a standard homogenization scheme further keep the forward operator independent of the target ODF coefficients.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of the Hashin-Shtrikman homogenization for mapping texture coefficients to effective stiffness and on the assumption that the forward wave simulation faithfully reproduces the experimental transmitted field for the chosen frequencies and specimen geometries.

free parameters (1)

texture coefficients
The unknown orientation distribution coefficients that are adjusted by the optimizer to match the measured transmitted field.

axioms (1)

domain assumption Hashin-Shtrikman homogenization supplies tighter bounds than Voigt-Reuss-Hill averages for the effective elastic tensor of the polycrystal
Invoked to compute the macroscopic stiffness used in the wave-propagation simulation and to constrain the optimization search space.

pith-pipeline@v0.9.0 · 5807 in / 1406 out tokens · 50561 ms · 2026-05-20T16:01:14.753164+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The effective elastic response is computed using a Hashin–Shtrikman homogenization framework... The nonlinear inverse problem is solved using a GPU-accelerated optimization scheme.
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

textures with weak elastic anisotropy are successfully recovered... no macroscopic symmetry assumptions are imposed

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

[1]

H. J. Bunge, Texture Analysis in Materials Science, Elsevier, 1982. doi:10.1016/c2013-0-11769-2. URLhttps://doi.org/10.1016/c2013-0-11769-2

work page doi:10.1016/c2013-0-11769-2 1982
[2]

Sowerby, W

R. Sowerby, W. Johnson, A review of texture and anisotropy in relation to metal forming, Materials Science and Engineering 20 (1975) 101–111. doi:10.1016/0025-5416(75)90138-x. URLhttps://doi.org/10.1016/0025-5416(75)90138-x

work page doi:10.1016/0025-5416(75)90138-x 1975
[3]

K. L. Murty, I. Charit, Texture development and anisotropic deforma- tion of zircaloys, Progress in Nuclear Energy 48 (4) (2006) 325–359. doi:10.1016/j.pnucene.2005.09.011. URLhttps://doi.org/10.1016/j.pnucene.2005.09.011

work page doi:10.1016/j.pnucene.2005.09.011 2006
[4]

B. Lan, T. B. Britton, T.-S. Jun, W. Gan, M. Hofmann, F. P. Dunne, M. J. Lowe, Direct volumetric measurement of crystallographic texture using acoustic waves, Acta Materialia 159 (2018) 384–394. 17 doi:10.1016/j.actamat.2018.08.037. URLhttps://doi.org/10.1016/j.actamat.2018.08.037

work page doi:10.1016/j.actamat.2018.08.037 2018
[5]

B. Lan, M. J. Lowe, F. P. Dunne, A spherical harmonic approach for the determinationofHCPtexturefromultrasound: Asolutiontotheinverse problem, Journal of the Mechanics and Physics of Solids 83 (2015) 179–

work page 2015
[6]

URLhttps://doi.org/10.1016/j.jmps.2015.06.014

doi:10.1016/j.jmps.2015.06.014. URLhttps://doi.org/10.1016/j.jmps.2015.06.014

work page doi:10.1016/j.jmps.2015.06.014 2015
[7]

B. Lan, M. J. Lowe, F. P. Dunne, A generalized spherical harmonic deconvolution to obtain texture of cubic materials from ultrasonic wave speed, Journal of the Mechanics and Physics of Solids 83 (2015) 221–

work page 2015
[8]

URLhttps://doi.org/10.1016/j.jmps.2015.06.012

doi:10.1016/j.jmps.2015.06.012. URLhttps://doi.org/10.1016/j.jmps.2015.06.012

work page doi:10.1016/j.jmps.2015.06.012 2015
[9]

Rossin, P

J. Rossin, P. Leser, K. Pusch, C. Frey, S. P. Murray, C. J. Torbet, S. Smith, S. Daly, T. M. Pollock, Bayesian inference of elastic constants andtexturecoefficientsinadditivelymanufacturedcobalt-nickelsuperal- loys using resonant ultrasound spectroscopy, Acta Materialia 220 (2021) 117287. doi:10.1016/j.actamat.2021.117287. URLhttps://doi.org/10.1016/j.act...

work page doi:10.1016/j.actamat.2021.117287 2021
[10]

M. L. Fernández, T. Böhlke, Representation of hashin–shtrikman bounds in terms of texture coefficients for arbitrarily anisotropic polycrystalline materials, Journal of Elasticity 134 (1) (2018) 1–38. doi:10.1007/s10659-018-9679-0. URLhttps://doi.org/10.1007/s10659-018-9679-0

work page doi:10.1007/s10659-018-9679-0 2018
[11]

Lobos, T

M. Lobos, T. Böhlke, On optimal zeroth-order bounds of linear elas- tic properties of multiphase materials and application in materials de- sign, International Journal of Solids and Structures 84 (2016) 40–48. doi:10.1016/j.ijsolstr.2015.12.015. URLhttp://dx.doi.org/10.1016/j.ijsolstr.2015.12.015

work page doi:10.1016/j.ijsolstr.2015.12.015 2016
[12]

Voigt, Lehrbuch der Kristallphysik, 1966

W. Voigt, Lehrbuch der Kristallphysik, 1966. doi:10.1007/978-3-663- 15884-4. URLhttps://doi.org/10.1007/978-3-663-15884-4

work page doi:10.1007/978-3-663- 1966
[13]

A. Reuss, Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle ., ZAMM - Zeitschrift 18 für Angewandte Mathematik und Mechanik 9 (1) (1929) 49–58. doi:10.1002/zamm.19290090104. URLhttps://doi.org/10.1002/zamm.19290090104

work page doi:10.1002/zamm.19290090104 1929
[14]

Hill, Elastic properties of reinforced solids: Some theoretical prin- ciples, Journal of the Mechanics and Physics of Solids 11 (5) (1963) 357–372

R. Hill, Elastic properties of reinforced solids: Some theoretical prin- ciples, Journal of the Mechanics and Physics of Solids 11 (5) (1963) 357–372. doi:10.1016/0022-5096(63)90036-x. URLhttps://doi.org/10.1016/0022-5096(63)90036-x

work page doi:10.1016/0022-5096(63)90036-x 1963
[15]

Hashin, S

Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of polycrystals, Journal of the Mechanics and Physics of Solids 10 (4) (1962) 343–352. doi:10.1016/0022-5096(62)90005-4. URLhttps://doi.org/10.1016/0022-5096(62)90005-4

work page doi:10.1016/0022-5096(62)90005-4 1962
[16]

Forte, M

S. Forte, M. Vianello, Symmetry classes for elasticity tensors, Journal of Elasticity 43 (2) (1996) 81–108. doi:10.1007/bf00042505. URLhttp://dx.doi.org/10.1007/BF00042505

work page doi:10.1007/bf00042505 1996
[17]

Bunge, Zur darstellung allgemeiner texturen, Zeitschrift für Met- allkunde 56 (1965) 872–874

H.-J. Bunge, Zur darstellung allgemeiner texturen, Zeitschrift für Met- allkunde 56 (1965) 872–874

work page 1965
[18]

Böhlke, M

T. Böhlke, M. Lobos, Representation of hashin–shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with ap- plication in materials design, Acta Materialia 67 (2014) 324–334. doi:10.1016/j.actamat.2013.11.003. URLhttp://dx.doi.org/10.1016/j.actamat.2013.11.003

work page doi:10.1016/j.actamat.2013.11.003 2014
[19]

Hashin, S

Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechan- ics and Physics of Solids 11 (2) (1963) 127–140. doi:10.1016/0022- 5096(63)90060-7. URLhttp://dx.doi.org/10.1016/0022-5096(63)90060-7

work page doi:10.1016/0022- 1963
[20]

Ledbetter, Predicted monocrystal elastic constants of 304-type stainless steel, Physica B+C 128 (1) (1985) 1–4

H. Ledbetter, Predicted monocrystal elastic constants of 304-type stainless steel, Physica B+C 128 (1) (1985) 1–4. doi:10.1016/0378- 4363(85)90076-2. URLhttp://dx.doi.org/10.1016/0378-4363(85)90076-2

work page doi:10.1016/0378- 1985
[21]

Ultrasonic characterization of generally anisotropic elasticity implementing optimal zeroth-order elastic bounds and a wave-fitting approach

D. Cowes, J. I. Mieza, M. P. Gómez, Ultrasonic character- ization of generally anisotropic elasticity implementing optimal 19 zeroth-order elastic bounds and a wave-fitting approach (2026). doi:10.48550/ARXIV.2604.09865. URLhttps://arxiv.org/abs/2604.09865

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.09865 2026
[22]

D. A. Cowes, J. I. Mieza, M. P. Gómez, Polyvinylidene fluoride trans- ducer shape optimization for the characterization of anisotropic mate- rials, The Journal of the Acoustical Society of America 156 (6) (2024) 3943–3953. doi:10.1121/10.0034601. URLhttp://dx.doi.org/10.1121/10.0034601

work page doi:10.1121/10.0034601 2024
[23]

G. Sha, A simultaneous non-destructive characterisation method for grain size and single-crystal elastic constants of cubic polycrystals from ultrasonic measurements, Insight - Non-Destructive Testing and Condi- tion Monitoring 60 (4) (2018) 190–193. doi:10.1784/insi.2018.60.4.190. URLhttps://doi.org/10.1784/insi.2018.60.4.190

work page doi:10.1784/insi.2018.60.4.190 2018
[24]

direct search

R. Hooke, T. A. Jeeves, “ direct search” solution of numerical and statistical problems, Journal of the ACM 8 (2) (1961) 212–229. doi:10.1145/321062.321069. URLhttp://dx.doi.org/10.1145/321062.321069

work page doi:10.1145/321062.321069 1961
[25]

Blank, K

J. Blank, K. Deb, pymoo: Multi-objective optimization in python, IEEE Access 8 (2020) 89497–89509

work page 2020
[26]

B. Lan, M. A. Carpenter, W. Gan, M. Hofmann, F. P. Dunne, M. J. Lowe, Rapid measurement of volumetric texture using reso- nant ultrasound spectroscopy, Scripta Materialia 157 (2018) 44–48. doi:10.1016/j.scriptamat.2018.07.029. URLhttps://doi.org/10.1016/j.scriptamat.2018.07.029

work page doi:10.1016/j.scriptamat.2018.07.029 2018
[27]

J. Rossin, GitHub - jrossin/texture-RUS: Inverse determination of texture from resonant frequencies (gathered by RUS) by pairing with the inverse solving python package SMCPy — github.com, https://github.com/jrossin/texture-RUS, [Accessed 15-Feb-2023]

work page 2023
[28]

E. S. Fisher, C. J. Renken, Single-crystal elastic moduli and the hcp→ bcc transformation in ti, zr, and hf, Physical Review 135 (2A) (1964) A482–A494. doi:10.1103/physrev.135.a482. URLhttp://dx.doi.org/10.1103/PhysRev.135.A482 20

work page doi:10.1103/physrev.135.a482 1964
[29]

Alers, J

G. Alers, J. Neighbours, The elastic constants of zinc between 4.2°and 670°k, Journal of Physics and Chemistry of Solids 7 (1) (1958) 58–64. doi:10.1016/0022-3697(58)90180-x. URLhttp://dx.doi.org/10.1016/0022-3697(58)90180-X

work page doi:10.1016/0022-3697(58)90180-x 1958
[30]

J. J. Hall, Electronic effects in the elastic constants of n-type silicon, Physical Review 161 (3) (1967) 756–761. doi:10.1103/physrev.161.756. URLhttp://dx.doi.org/10.1103/PhysRev.161.756

work page doi:10.1103/physrev.161.756 1967
[31]

Bachmann, R

F. Bachmann, R. Hielscher, H. Schaeben, Texture analysis with mtex – free and open source software toolbox, Solid State Phenomena 160 (2010) 63–68. doi:10.4028/www.scientific.net/ssp.160.63. URLhttp://dx.doi.org/10.4028/www.scientific.net/SSP.160.63

work page doi:10.4028/www.scientific.net/ssp.160.63 2010
[32]

Kearns, Report wapd-tm-472, Bettis Atomic Power Laboratory, West Mifflin, PA, USA (1965)

J. Kearns, Report wapd-tm-472, Bettis Atomic Power Laboratory, West Mifflin, PA, USA (1965)

work page 1965
[33]

A. J. Anderson, R. B. Thompson, C. S. Cook, Ultrasonic measure- ment of the kearns texture factors in zircaloy, zirconium, and titanium, Metallurgical and Materials Transactions A 30 (8) (1999) 1981–1988. doi:10.1007/s11661-999-0008-x. URLhttp://dx.doi.org/10.1007/s11661-999-0008-x

work page doi:10.1007/s11661-999-0008-x 1999
[34]

M. Lobos Fernández, Homogenization and materials design of mechani- cal properties of textured materials based on zeroth-, first- and second- order bounds of linear behavior, Schriftenreihe Kontinuumsmechanik im Maschinenbau / Karlsruher Institut fuer Technologie, Institut fuer Tech- nischeMechanik-BereichKontinuumsmechanik, KITScientificPublish- ing, 201...

work page 2018
[35]

H. J. Bunge, Texture and magnetic properties, Textures and Microstruc- tures 11 (2–4) (1989) 75–91. doi:10.1155/tsm.11.75. URLhttp://dx.doi.org/10.1155/TSM.11.75

work page doi:10.1155/tsm.11.75 1989
[36]

Matthies, On the reproducibility of the orientation distribution func- tion of texture samples from pole figures (ghost phenomena), physica status solidi (b) 92 (2) (Apr

S. Matthies, On the reproducibility of the orientation distribution func- tion of texture samples from pole figures (ghost phenomena), physica status solidi (b) 92 (2) (Apr. 1979). doi:10.1002/pssb.2220920254. URLhttp://dx.doi.org/10.1002/pssb.2220920254 21 Figure 1: Elastic Homogenization. a) Bounds for biphaseic materials [17]. b) Bounds for the Young m...

work page doi:10.1002/pssb.2220920254 1979

[1] [1]

H. J. Bunge, Texture Analysis in Materials Science, Elsevier, 1982. doi:10.1016/c2013-0-11769-2. URLhttps://doi.org/10.1016/c2013-0-11769-2

work page doi:10.1016/c2013-0-11769-2 1982

[2] [2]

Sowerby, W

R. Sowerby, W. Johnson, A review of texture and anisotropy in relation to metal forming, Materials Science and Engineering 20 (1975) 101–111. doi:10.1016/0025-5416(75)90138-x. URLhttps://doi.org/10.1016/0025-5416(75)90138-x

work page doi:10.1016/0025-5416(75)90138-x 1975

[3] [3]

K. L. Murty, I. Charit, Texture development and anisotropic deforma- tion of zircaloys, Progress in Nuclear Energy 48 (4) (2006) 325–359. doi:10.1016/j.pnucene.2005.09.011. URLhttps://doi.org/10.1016/j.pnucene.2005.09.011

work page doi:10.1016/j.pnucene.2005.09.011 2006

[4] [4]

B. Lan, T. B. Britton, T.-S. Jun, W. Gan, M. Hofmann, F. P. Dunne, M. J. Lowe, Direct volumetric measurement of crystallographic texture using acoustic waves, Acta Materialia 159 (2018) 384–394. 17 doi:10.1016/j.actamat.2018.08.037. URLhttps://doi.org/10.1016/j.actamat.2018.08.037

work page doi:10.1016/j.actamat.2018.08.037 2018

[5] [5]

B. Lan, M. J. Lowe, F. P. Dunne, A spherical harmonic approach for the determinationofHCPtexturefromultrasound: Asolutiontotheinverse problem, Journal of the Mechanics and Physics of Solids 83 (2015) 179–

work page 2015

[6] [6]

URLhttps://doi.org/10.1016/j.jmps.2015.06.014

doi:10.1016/j.jmps.2015.06.014. URLhttps://doi.org/10.1016/j.jmps.2015.06.014

work page doi:10.1016/j.jmps.2015.06.014 2015

[7] [7]

B. Lan, M. J. Lowe, F. P. Dunne, A generalized spherical harmonic deconvolution to obtain texture of cubic materials from ultrasonic wave speed, Journal of the Mechanics and Physics of Solids 83 (2015) 221–

work page 2015

[8] [8]

URLhttps://doi.org/10.1016/j.jmps.2015.06.012

doi:10.1016/j.jmps.2015.06.012. URLhttps://doi.org/10.1016/j.jmps.2015.06.012

work page doi:10.1016/j.jmps.2015.06.012 2015

[9] [9]

Rossin, P

J. Rossin, P. Leser, K. Pusch, C. Frey, S. P. Murray, C. J. Torbet, S. Smith, S. Daly, T. M. Pollock, Bayesian inference of elastic constants andtexturecoefficientsinadditivelymanufacturedcobalt-nickelsuperal- loys using resonant ultrasound spectroscopy, Acta Materialia 220 (2021) 117287. doi:10.1016/j.actamat.2021.117287. URLhttps://doi.org/10.1016/j.act...

work page doi:10.1016/j.actamat.2021.117287 2021

[10] [10]

M. L. Fernández, T. Böhlke, Representation of hashin–shtrikman bounds in terms of texture coefficients for arbitrarily anisotropic polycrystalline materials, Journal of Elasticity 134 (1) (2018) 1–38. doi:10.1007/s10659-018-9679-0. URLhttps://doi.org/10.1007/s10659-018-9679-0

work page doi:10.1007/s10659-018-9679-0 2018

[11] [11]

Lobos, T

M. Lobos, T. Böhlke, On optimal zeroth-order bounds of linear elas- tic properties of multiphase materials and application in materials de- sign, International Journal of Solids and Structures 84 (2016) 40–48. doi:10.1016/j.ijsolstr.2015.12.015. URLhttp://dx.doi.org/10.1016/j.ijsolstr.2015.12.015

work page doi:10.1016/j.ijsolstr.2015.12.015 2016

[12] [12]

Voigt, Lehrbuch der Kristallphysik, 1966

W. Voigt, Lehrbuch der Kristallphysik, 1966. doi:10.1007/978-3-663- 15884-4. URLhttps://doi.org/10.1007/978-3-663-15884-4

work page doi:10.1007/978-3-663- 1966

[13] [13]

A. Reuss, Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle ., ZAMM - Zeitschrift 18 für Angewandte Mathematik und Mechanik 9 (1) (1929) 49–58. doi:10.1002/zamm.19290090104. URLhttps://doi.org/10.1002/zamm.19290090104

work page doi:10.1002/zamm.19290090104 1929

[14] [14]

Hill, Elastic properties of reinforced solids: Some theoretical prin- ciples, Journal of the Mechanics and Physics of Solids 11 (5) (1963) 357–372

R. Hill, Elastic properties of reinforced solids: Some theoretical prin- ciples, Journal of the Mechanics and Physics of Solids 11 (5) (1963) 357–372. doi:10.1016/0022-5096(63)90036-x. URLhttps://doi.org/10.1016/0022-5096(63)90036-x

work page doi:10.1016/0022-5096(63)90036-x 1963

[15] [15]

Hashin, S

Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of polycrystals, Journal of the Mechanics and Physics of Solids 10 (4) (1962) 343–352. doi:10.1016/0022-5096(62)90005-4. URLhttps://doi.org/10.1016/0022-5096(62)90005-4

work page doi:10.1016/0022-5096(62)90005-4 1962

[16] [16]

Forte, M

S. Forte, M. Vianello, Symmetry classes for elasticity tensors, Journal of Elasticity 43 (2) (1996) 81–108. doi:10.1007/bf00042505. URLhttp://dx.doi.org/10.1007/BF00042505

work page doi:10.1007/bf00042505 1996

[17] [17]

Bunge, Zur darstellung allgemeiner texturen, Zeitschrift für Met- allkunde 56 (1965) 872–874

H.-J. Bunge, Zur darstellung allgemeiner texturen, Zeitschrift für Met- allkunde 56 (1965) 872–874

work page 1965

[18] [18]

Böhlke, M

T. Böhlke, M. Lobos, Representation of hashin–shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with ap- plication in materials design, Acta Materialia 67 (2014) 324–334. doi:10.1016/j.actamat.2013.11.003. URLhttp://dx.doi.org/10.1016/j.actamat.2013.11.003

work page doi:10.1016/j.actamat.2013.11.003 2014

[19] [19]

Hashin, S

Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, Journal of the Mechan- ics and Physics of Solids 11 (2) (1963) 127–140. doi:10.1016/0022- 5096(63)90060-7. URLhttp://dx.doi.org/10.1016/0022-5096(63)90060-7

work page doi:10.1016/0022- 1963

[20] [20]

Ledbetter, Predicted monocrystal elastic constants of 304-type stainless steel, Physica B+C 128 (1) (1985) 1–4

H. Ledbetter, Predicted monocrystal elastic constants of 304-type stainless steel, Physica B+C 128 (1) (1985) 1–4. doi:10.1016/0378- 4363(85)90076-2. URLhttp://dx.doi.org/10.1016/0378-4363(85)90076-2

work page doi:10.1016/0378- 1985

[21] [21]

Ultrasonic characterization of generally anisotropic elasticity implementing optimal zeroth-order elastic bounds and a wave-fitting approach

D. Cowes, J. I. Mieza, M. P. Gómez, Ultrasonic character- ization of generally anisotropic elasticity implementing optimal 19 zeroth-order elastic bounds and a wave-fitting approach (2026). doi:10.48550/ARXIV.2604.09865. URLhttps://arxiv.org/abs/2604.09865

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.09865 2026

[22] [22]

D. A. Cowes, J. I. Mieza, M. P. Gómez, Polyvinylidene fluoride trans- ducer shape optimization for the characterization of anisotropic mate- rials, The Journal of the Acoustical Society of America 156 (6) (2024) 3943–3953. doi:10.1121/10.0034601. URLhttp://dx.doi.org/10.1121/10.0034601

work page doi:10.1121/10.0034601 2024

[23] [23]

G. Sha, A simultaneous non-destructive characterisation method for grain size and single-crystal elastic constants of cubic polycrystals from ultrasonic measurements, Insight - Non-Destructive Testing and Condi- tion Monitoring 60 (4) (2018) 190–193. doi:10.1784/insi.2018.60.4.190. URLhttps://doi.org/10.1784/insi.2018.60.4.190

work page doi:10.1784/insi.2018.60.4.190 2018

[24] [24]

direct search

R. Hooke, T. A. Jeeves, “ direct search” solution of numerical and statistical problems, Journal of the ACM 8 (2) (1961) 212–229. doi:10.1145/321062.321069. URLhttp://dx.doi.org/10.1145/321062.321069

work page doi:10.1145/321062.321069 1961

[25] [25]

Blank, K

J. Blank, K. Deb, pymoo: Multi-objective optimization in python, IEEE Access 8 (2020) 89497–89509

work page 2020

[26] [26]

B. Lan, M. A. Carpenter, W. Gan, M. Hofmann, F. P. Dunne, M. J. Lowe, Rapid measurement of volumetric texture using reso- nant ultrasound spectroscopy, Scripta Materialia 157 (2018) 44–48. doi:10.1016/j.scriptamat.2018.07.029. URLhttps://doi.org/10.1016/j.scriptamat.2018.07.029

work page doi:10.1016/j.scriptamat.2018.07.029 2018

[27] [27]

J. Rossin, GitHub - jrossin/texture-RUS: Inverse determination of texture from resonant frequencies (gathered by RUS) by pairing with the inverse solving python package SMCPy — github.com, https://github.com/jrossin/texture-RUS, [Accessed 15-Feb-2023]

work page 2023

[28] [28]

E. S. Fisher, C. J. Renken, Single-crystal elastic moduli and the hcp→ bcc transformation in ti, zr, and hf, Physical Review 135 (2A) (1964) A482–A494. doi:10.1103/physrev.135.a482. URLhttp://dx.doi.org/10.1103/PhysRev.135.A482 20

work page doi:10.1103/physrev.135.a482 1964

[29] [29]

Alers, J

G. Alers, J. Neighbours, The elastic constants of zinc between 4.2°and 670°k, Journal of Physics and Chemistry of Solids 7 (1) (1958) 58–64. doi:10.1016/0022-3697(58)90180-x. URLhttp://dx.doi.org/10.1016/0022-3697(58)90180-X

work page doi:10.1016/0022-3697(58)90180-x 1958

[30] [30]

J. J. Hall, Electronic effects in the elastic constants of n-type silicon, Physical Review 161 (3) (1967) 756–761. doi:10.1103/physrev.161.756. URLhttp://dx.doi.org/10.1103/PhysRev.161.756

work page doi:10.1103/physrev.161.756 1967

[31] [31]

Bachmann, R

F. Bachmann, R. Hielscher, H. Schaeben, Texture analysis with mtex – free and open source software toolbox, Solid State Phenomena 160 (2010) 63–68. doi:10.4028/www.scientific.net/ssp.160.63. URLhttp://dx.doi.org/10.4028/www.scientific.net/SSP.160.63

work page doi:10.4028/www.scientific.net/ssp.160.63 2010

[32] [32]

Kearns, Report wapd-tm-472, Bettis Atomic Power Laboratory, West Mifflin, PA, USA (1965)

J. Kearns, Report wapd-tm-472, Bettis Atomic Power Laboratory, West Mifflin, PA, USA (1965)

work page 1965

[33] [33]

A. J. Anderson, R. B. Thompson, C. S. Cook, Ultrasonic measure- ment of the kearns texture factors in zircaloy, zirconium, and titanium, Metallurgical and Materials Transactions A 30 (8) (1999) 1981–1988. doi:10.1007/s11661-999-0008-x. URLhttp://dx.doi.org/10.1007/s11661-999-0008-x

work page doi:10.1007/s11661-999-0008-x 1999

[34] [34]

M. Lobos Fernández, Homogenization and materials design of mechani- cal properties of textured materials based on zeroth-, first- and second- order bounds of linear behavior, Schriftenreihe Kontinuumsmechanik im Maschinenbau / Karlsruher Institut fuer Technologie, Institut fuer Tech- nischeMechanik-BereichKontinuumsmechanik, KITScientificPublish- ing, 201...

work page 2018

[35] [35]

H. J. Bunge, Texture and magnetic properties, Textures and Microstruc- tures 11 (2–4) (1989) 75–91. doi:10.1155/tsm.11.75. URLhttp://dx.doi.org/10.1155/TSM.11.75

work page doi:10.1155/tsm.11.75 1989

[36] [36]

Matthies, On the reproducibility of the orientation distribution func- tion of texture samples from pole figures (ghost phenomena), physica status solidi (b) 92 (2) (Apr

S. Matthies, On the reproducibility of the orientation distribution func- tion of texture samples from pole figures (ghost phenomena), physica status solidi (b) 92 (2) (Apr. 1979). doi:10.1002/pssb.2220920254. URLhttp://dx.doi.org/10.1002/pssb.2220920254 21 Figure 1: Elastic Homogenization. a) Bounds for biphaseic materials [17]. b) Bounds for the Young m...

work page doi:10.1002/pssb.2220920254 1979