# Understanding effective elastic constants and elastic symmetry in RUS: from single crystals to textured polycrystal

Florian Le Bourdais, Nicolas Leymarie, Audrey Gardahaut

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Florian Le Bourdais, Nicolas Leymarie, Audrey Gardahaut. Understanding effective elastic constants and elastic symmetry in RUS: from single crystals to textured polycrystal. ICOTOM 2024 - 20th international conference on textures of materials, Jun 2024, Metz, France. cea-05522648

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# On the relationship between microstructure and elastic constants measured by Resonant Ultrasonic Spectroscopy: from single crystals to additively manufactured polycrystals

Florian Le Bourdais¹, a, Nicolas Leymarie ¹, Audrey Gardahaut¹

¹ Université Paris-Saclay, CEA, List, F-91120, Palaiseau, France

aflorian.lebourdais@cea.fr

**Abstract.** In this paper, we report our recent progress regarding the development of a resonant ultrasound spectroscopy (RUS) probing system and associated tools at CEA List. These tools allow to accurately characterize the full elastic tensor for metallic materials of various shapes and microstructures.

We first briefly present the fundamentals of RUS: forward models used to compute the resonance frequencies, inversion algorithms. We discuss in particular the choice of elastic symmetry used during inversion for finding numerical values of the elastic constants. We then present applications of our setup to use-cases where RUS measurements can provide several types of information. We first discuss a case study on a 316L single crystal with lattice orientation differing from the sample axes and the sensitivity of RUS to this aspect. We then show how this sensitivity translates to a 316L polycrystal. Our last sample study discusses anisotropic elasticity evidence and relation to EBSD texture measured on a 316L wire arc additive manufacturing sample. Finally, we will review the models linking microstructure and effective elastic constants measured by RUS and point out research questions at the intersection of resonant ultrasound and classical metallic characterization techniques of texture such as EBSD and XRD.

**Keywords:** RUS, ultrasound, resonance, non-destructive testing, additive manufacturing, elasticity tensor, characterization, microstructure, anisotropy.

**Corresponding author:** Florian Le Bourdais, florian.lebourdais@cea.fr

## 1. Introduction

Elastic characterization, i.e. knowledge of elastic constants, plays an important role in the design of ultrasonic non-destructive testing (UT-NDT) methods. While characterizing elasticity is relatively easy with an isotropic material, where only two elastic constants are needed to describe propagation of longitudinal and shear waves (typically $c_{11}$ and $c_{44}$ or Young modulus and Poisson coefficient), things get more complicated in textured polycrystals. In that case, elastic symmetry is often not known before in-depth characterization and numerical values of elastic constants are usually known only imprecisely. Uncertainty about elastic constants of a material has adverse effects on inspection results and will therefore pose difficulty for UT-NDT methods that aim at certifying produced materials and parts. Therefore, as part of CEA's mission to develop UT-NDT methods, we have started working on a measurement method called Resonant Ultrasonic Spectroscopy (RUS) that aims at providing elastic constants on a wide range of materials, in particular those of interest in metal additive manufacturing (AM). In this work, we first present fundamentals about this elastic characterization technique and review the literature for elastic symmetry models of additive manufacturing samples. We then present
case studies ranging from single crystals to additively manufactured textured polycrystals to show
how the RUS measurement reflects the underlying texture. Finally, we present the models found in
the literature that allow interpreting elastic constants determined by RUS as texture components
typically measured by EBSD in the texture community and discuss how RUS could become a
valuable addition to the texture characterization toolbox.

**2. Characterization fundamentals using RUS technique**

The term RUS was originally coined in the 1990s by authors at Los Alamos National Laboratory [1] [2]. This method involves two steps: first, the physical measurement of the vibration spectrum of a sample to determine its resonance frequencies, and second, solving an inverse problem to find the best match between the measured resonance frequencies and those predicted by a forward model. This process is schematically shown in Figure 1.

<figure><img src="image_2.png" /><figcaption>Figure 1. Schematic overview of RUS as implemented in this paper. Figure inspired by [3].</figcaption></figure>

The outcome of the RUS procedure is, with a known geometry and density, the numerical values of
the elastic constants. These result from the solution of the optimization problem resulting either from
the minimization of the “classical” RUS cost function $\chi^2 = \sum_{i=1}^{N} \frac{(f_i(C_{ij}) - f_i^{exp})^2}{(f_i^{exp})^2}$ [1] or the
maximization of the posterior probability in the Bayesian inversion context [4] [5]. To determine
these constants, an elastic symmetry has to be chosen a priori before running the inversion procedure.

**3. Choice of elastic symmetry for RUS inversion in additive manufacturing**

As mentioned before, the elastic symmetry is parametrized by two coefficients in the isotropic case. The number of coefficients increases to 3 in the cubic case, 9 in the orthotropic case and 21 in the most general triclinic symmetry case. Based on the fit quality (i.e. as measured by the cost function), this choice can then be iterated until a model is found that fits the measured data well. RUS has originally been developed to characterize single crystal materials, for which crystal symmetry is known. However, practitioners have long been using it on polycrystals, where it is often not clear a priori, which elastic symmetry correctly describes the sample under analysis. As written before, metal samples encountered in UT-NDT usually behave isotropically in terms of wave propagation at the frequencies of interest, but this is effectively the result of the random distribution of grains as well as their small size compared to ultrasonic wavelengths in the frequency range used to inspect them. The question then arises as to what happens when grain sizes are not small anymore with respect to interrogation wavelength and what if the distribution of orientations or grain sizes is not random but has been determined by the fabrication process. This can happen in metal AM or welding as the thermal gradient leading to solidification of the molten metal is repeatedly oriented in the same
direction of space, *e.g.* the build plate (AM case) or the nearest base metal (weld case), leading to *texture*. A survey of the literature shows that most authors working on polycrystals seem to have implicitly answered the question of the elastic symmetry choice by “whatever symmetry works well in practice”, without precisely discussing the underlying issue of how to represent polycrystal elasticity correctly based on the underlying microstructure. Rahman *et al.* [6] tried isotropic, cubic, hexagonal, and rotated orthotropic symmetry to represent Inconel 718 elastic properties, without providing a convincing argument as to what the aggregate symmetry should be. Rossin *et al.* [7] chose an orthotropic representation combined with 3 Euler angles to represent the texture of Laser Powder Bed Fusion (L-PBF) manufactured Inconel 625. They compare the obtained constants to those from Voigt-Reuss-Hill (VRH) homogenization from EBSD data and find agreement to within 1%. In subsequent studies [8] [9], Rossin *et al.* used a tensorial representation of the ODF coefficients to produce second order Hashin-Shtrikman bounds of elastic constants that represent the homogenized material using 9 texture coefficients and 3 single crystal elastic constants. Malashin *et al.* [10] model various nickel alloys including Inconel 718 as orthotropic elastic symmetry and find convincing ultrasonic results. Our own work on AlSi7Mg0.6 aluminum alloys [11] uses isotropic symmetry to represent elasticity, justified by empirical low fit error in the RUS process and a comparison with elastic constants obtained under orthotropic symmetry hypothesis. As this short review shows, a variety of proposed symmetries exist, but these works do not show any complete theoretical argument as to how to choose the symmetry, even though some authors come close and discuss the physical origins of AM texture due to solidification, lasing strategy and the global heat flow to the build plate, *e.g.* [7].

In the next section, we will review some of our own results to better inform the choice of elastic symmetry, by reviewing single crystal elasticity as well as polycrystal elasticity RUS results.

## 4. Selected RUS results: from single crystals to additively manufactured polycrystals

As pointed out in the previous section, it is important to understand the microstructure of the samples under analysis to correctly choose elastic symmetry for the RUS inversion. An illuminating example to understand the sensitivity of RUS to texture is the results obtained with a single crystal of 316L austenitic stainless steel. This study has been published in [12]. The sample consists of a thin plate single crystal as shown in Figure 2 a). It should be noted that the crystal lattice 100 orientations do not coincide with the sample reference axes. This disorientation leads to inversely determined elastic constants expressed in the sample axes that have many nonzero values instead of taking the usual cubic shape, see Figure 2 b). Additionally, the inverted orientation parameters, expressed in Bunge Euler angles as (-0.213°, 34.789°, 44.912°) match the known manufacturer values indicated by lattice directions in Figure 2 a). The link between the elastic tensor $c_{ijkl}$ expressed in the crystal reference frame and the “effective” one in a rotated frame $d_{pqrs}$ is given by the expression $d_{pqrs} = \sum_{ijkl} R_{pi}R_{qj}R_{rk}R_{sl}c_{ijkl}$ where $R_{ij}$ is the rotation matrix describing the geometric transformation between frames.

An important conclusion from this study is that for a single grain, the resonance frequencies are sensitive to orientations of the underlying lattice. Hence, it can be reasoned that for polycrystals, this sensitivity will lead to effective texture detection.
<figure><img src="image_3.png" /><figcaption>Figure 2. 316L single crystal a) picture and schematic diagram of three lattice directions, b) elastic constants obtained and c) RUS inversion result.</figcaption></figure>

Indeed, we have confirmed this sensitivity for polycrystals in two cases. The first case is that of samples cut from a welded block of, again, 316L stainless steel wire. It is known that the weld has elongated grains that align with the thermal gradient direction in the welding mould. A qualitative confirmation of the sensitivity to the texture is found when measuring the first resonance frequency for the six samples cut at different angles from the main block; see Figure 3 a) and b). This is further confirmed by EBSD analysis of orientation maps shown in Figure 3 c) and d) which clearly indicate the approximately 30° degree inclination of grains. It is interesting to note that the relationship between the cut angle and the resonant frequency is monotonic. Quantitative interpretation of the relationship could probably be achieved to determine the directions of larger and smaller stiffness due to the shift to higher frequencies of the first resonance frequency.
<figure><img src="image_4.png" /><figcaption>Figure 3. Results from the 316L weld polycrystal mould and samples cut at six different angles. a) Schematic representation of samples, b) first resonance frequency measured for each sample, c) EBSD orientation map from 30° cut sample and d) pole figures for 30° cut sample.</figcaption></figure>

This case study then fulfills the hope that was expressed based on the single crystal case results. However, it is not so straightforward to understand the reason for these effects since the samples are both textured and exhibit large elongated grains and these two effects can compound or interact in to produce these results. This study was partially published in [13] (in French).

A further example, again involving the same material, 316L stainless steel, was encountered while working on Wire Arc Additive Manufacturing (WAAM) samples manufactured during the Grade2XL project. The sample was cut from a larger block and analysed using RUS, see Figure 4 a) (a full study is to be submitted soon on this topic). Two models were developed to fit the resonance spectrum from the samples. The first one under the orthotropic symmetry hypothesis, the second one making use of the sample orientations measured using EBSD at University of Ghent, as shown in Figure 4 b). The second model consists in computing the elastic constants using the Hill average [14] [15] based on single crystal constants and the per-pixel Bunge Euler angle orientations. The texture-based model, once the single crystal constants have been optimized based on the resonant frequencies, finds that the best elastic description is one that exhibits symmetry quite close to orthotropic symmetry, with additions of nonzero constants outside of the orthotropic shape of relatively small numerical values (i.e. below 15 GPa). The effective orthotropic model finds similar orders of magnitude for the elastic constants. In particular the structure of the $c_{44}$, $c_{55}$, $c_{66}$ triplet values is identical with $c_{55}$ being almost half of the values of $c_{44}$, $c_{66}$, indicating less shear stiffness in direction 2, i.e. the build direction, in which the grains are elongated as shown in the orientation map.
<figure><img src="image_5.png" /><figcaption>Figure 4. Results from the WAAM manufactured sample of the Grade2XL study. a) Sample cutting location, b) EBSD orientation map, c) inversion using effective orthotropic model and its d) elastic constants, e) inversion using EBSD-texture model and its f) elastic constants.</figcaption></figure>

The numerical values of the elastic constants hide another aspect of the symmetry of the sample. Solving the Christoffel equation [16], we can determine the elastic wave speeds in any spatial direction based on the elastic constants. If we do so, we can try to relate the symmetry of wave speeds obtained to the pole figures associated to the orientation map in Figure 4. Figure 5 a) shows the wave speeds obtained for three planes defined by normal vectors [1 0 0], [0 1 0] and [0 0 1]. These wave speeds are then shown in their geometric context using parallel projection in b) alongside the (100) pole figures obtained from the EBSD measurement previously discussed.

<figure><img src="image_6.png" /><figcaption>Figure 5. a) Elastic wave speeds obtained by solving the Christoffel equation using the elastic constants from the orthotropic model for three orthogonal planes, b) parallel projection of the wave speeds on their respective planes and c) (100) pole figure based on EBSD measurement.</figcaption></figure>
The pole figure in Figure 5 can be interpreted as a fibre texture [17] since (100) orientations can be found mostly along the build direction and can be found with almost equal probability in the plane orthogonal to the build direction. The ultrasonic velocities deduced from the elastic constants found from RUS reflect this description. Velocities found in the plane normal to the build direction show almost no variations with angle in the plane, contrary to the velocities in other planes. This seems to indicate that transversely isotropic elastic symmetry with *c* axis parallel to build direction could also have been used in the RUS inversion, “forcing” the symmetry observed in the inversion data. To conclude this section, the results shown here emphasize that in the case of the anisotropic 316L WAAM sample, the elastic constants encode the same kind of symmetry than the aggregate orientation measurements, hinting at the fact that quantitative models could certainly link the texture to the elastic constants.

## 5. Linking texture and elastic constants measured using RUS: models and possibilities from the literature

In fact, authors have already established several ways to relate effective elastic constants to quantitative texture description. We have found four possibilities described in the literature, including the Voigt-Reuss-Hill approach already discussed in the last section. They are summarized in Table 1. The elastic self-consistent model described in [18] is based on the use of the Eshelby inclusion problem solution and takes into account average grain morphology, texture description and allow for a direct computation of aggregate elastic constants, without the possibility to invert the relationship. The Hashin-Shtrikman 2nd order bounds already discussed in section 3 used by Rossin *et al.* [8] do not take into account gran morphology but allow going back and forth between aggregate elastic constants and texture. Finally, the spherical convolution model by Lan *et al.* [19] [20] [21] allows relating acoustic velocities of the single crystal to the polycrystal and has been shown to be able to compute meaningful textures when compared to neutron diffraction [20] [21].

<table><caption>Table 1. Summary of the three models from the literature for polycrystal elastic constant calculation using texture.</caption><thead><tr><td>Model</td><td>Inputs</td><td>Accounts for grain orientations</td><td>Accounts for grain shapes</td><td>Can deduce texture from effective elastic constants?</td></tr></thead><tbody><tr><td>Voigt-Reuss-Hill averaging</td><td>Single crystal elastic constants, texture description (Euler angles for each pixel)</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>Elastic self-consistent [18]</td><td>Single crystal elastic constants, grain descriptions (Euler angles, shape)</td><td>Yes</td><td>Yes</td><td>No</td></tr><tr><td>Tensorial form of the Hashin-Shtrikman bounds [8]</td><td>Single crystal elastic constants, 4th order texture coefficients</td><td>Yes</td><td>No</td><td>Yes</td></tr><tr><td>Spherical convolution between single crystal wave speeds and texture [20]</td><td>Single crystal elastic constants to compute wave speeds, texture description as ODF coefficients</td><td>Yes</td><td>No</td><td>Yes</td></tr></tbody></table>

As can be seen in Table 1, there are some differences between models, in terms of inputs and capabilities to be useful in terms of application to RUS elastic constants. Based on our work and aims, it seems that the most interesting property for applications in ultrasonic RUS characterization is the
possibility to recover texture information. In that case, the most potent models are those used by
Rossin et al. [8] and Lan et al. [20]. Both are based on different hypotheses and could thus lead to
different results. We plan to evaluate these two models based on our data to find out whether these
differences are significant and how well they compare to EBSD data.

**6. Conclusions**

In this paper, we have reviewed the progress made in our group in the development of a resonant
ultrasound spectroscopy system for elastic characterization of metallic materials and have presented
several case studies that show how RUS measured elastic constants reflect the underlying texture of
the material. The study of single crystal elasticity has been used to understand the sensitivity of RUS
to crystal lattice orientation, and this sensitivity has been confirmed for polycrystals with textured
microstructures. Thus, it can be said that the RUS measurement provides elastic constants that encode
the effective sample texture. We have found several models in the literature that allow to go further
and interpret these elastic constants as ODF texture coefficients, typically measured by EBSD pole
figure inversion in the community. Our work will continue with the aim of establishing a bridge
between RUS and EBSD methods for texture characterization.

**7. Acknowledgments**

This project has received funding from the European Union's Horizon 2020 research and innovation
programme under grant agreement no 862017.

We thank Anne-Sophie Thorr (Naval Group) for providing the coupons part of the study within the
Grade2XL project as well as helpful build information. We thank Mahdi Mahmoudiniya and Leo
Kestens (Ghent University) for performing the ESBD measurements.
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