# Softening of Shear Elastic Coefficients in Shape Memory Alloys Near the Martensitic Transition: A Study by Laser-Based Resonant Ultrasound Spectroscopy

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- a Cu-Al-Ni single crystal (sample dimensions in austenite 3.05 × 2.03 × 1.29 mm ${}^{3}$, sample orientation in austenite approximately $\left[1\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[1\phantom{\rule{0.166667em}{0ex}}\overline{1}\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[0\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\right]$, mass density 6.78 g·cm ${}^{-3}$). This sample was characterized with cooling from 330 K towards the martensite start temperature (${M}_{\mathrm{S}}$, 299 K). To discern sufficient details of the behavior near the transition temperature, the measurement was done with a 0.5 K step (temperature control ±0.01 K) and the internal friction parameter ${Q}^{-1}$ was evaluated at each temperature. Six lowest resonant modes were used for the determination of ${c}^{\prime}$ and the analysis of the internal friction;
- a Ni-Mn-Ga single crystal (sample dimensions in austenite 3.41 × 2.80 × 0.92 mm ${}^{3}$ at 335 K, sample orientation in austenite approximately $\left[1\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[0\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\right]$, mass density 8.12 g·cm ${}^{-3}$). This sample was characterized with cooling from above the Curie temperature (${T}_{\mathrm{C}}$, 372 K) towards ${M}_{\mathrm{S}}=328$ K, where the sample transformed into 10 M modulated martensite [26] (a thermally induced mixture of variants). Afterwards, an external magnetic field was used for martensite reorientation [27,28], and the sample was set in a single (tetragonal) variant state with the $c-$ axis that was oriented along the shortest edge. Note that the 10 M martensite is slightly monoclinic, so this state was still a mixture of monoclinic variants sharing a common $c-$ axis, but such a microstructure can be understood as a single variant in the tetragonal approximation [29]. The sample in this condition was then characterized with heating from 289 K until the sample transformed back to austenite (austenite start temperature, ${A}_{\mathrm{S}}\approx {M}_{\mathrm{S}}$). This procedure was repeated second time, with the $c-$ axis set along the longest edge of the sample. For the sample in austenite, ten lowest resonant modes were used for ${c}^{\prime}$ determination. However, for temperatures that were below ${M}_{\mathrm{S}}$, the strong internal friction in 10 M martensite did not allow for reliable detection and identification of more than two or three resonant modes. This was insufficient for the reliable determination of any of the elastic coefficients of the (effectively tetragonal) mixture of monoclinic martensites. Hence, instead, we analyzed the RUS data assuming again cubic symmetry, which resulted in some effective ${c}^{\prime}$ for the martensite. Although this effective elastic constant has no direct relation to any shearing mode of the 10 M modulated lattice, its temperature evolution provides a qualitative information on the shear instability in martensite (see e.g., [30,31] for similar approaches);
- another Ni-Mn-Ga single crystal (dimensions in martensite 4.92 × 4.65 × 3.40 mm ${}^{3}$, sample orientation approximately ${\left[0\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\right]}_{\mathrm{bct}}\perp {\left[0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0\right]}_{\mathrm{bct}}\perp {\left[1\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}0\right]}_{\mathrm{bct}}$ for the dominant variant, mass density 8.12 g·cm ${}^{-3}$). At the room temperature, this sample was in tetragonal non-modulated (NM) martensite phase, consisting of a mixture of all three variants with one orientation being dominant. The sample was characterized with heating from the room temperature over the reverse transition temperature ${A}_{\mathrm{S}}\approx 400$ K. In this case, the ${A}_{\mathrm{S}}$ temperature was above the Curie point (373 K), i.e., the transition occurred in the paramagnetic state. Similarly as in the case of sample No.2, an effective ${c}^{\prime}$ coefficient was used for characterizing the shear softening in non-modulated martensite below ${A}_{\mathrm{S}}$; and,
- a Ni-Ti sample (sample dimensions in austenite 3.48 × 3.16 × 2.78 mm ${}^{3}$, sample orientation in austenite approximately $\left[1\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0\right]\perp \left[0\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\right]$, mass density 6.50 g·cm ${}^{-3}$). This sample was characterized with cooling from 293 K towards the vicinity of the austenite→R-phase transition temperature ${R}_{\mathrm{S}}\approx 270$ K. Twenty lowest resonant modes were traced in the measured temperature interval. Unlike samples Nos. 1, 2, and 3, the Ni-Ti sample was heterogeneous, consisting of a Ni-Ti matrix with finely dispersed precipitates. This means that the observed behavior (the elastic constants, the transformation temperatures, etc.) resulted from an interplay between the lattice (phonon mediated) instability of the matrix and the local stress fields that come from the precipitates. The discussion of such phenomena falls beyond the scope of this paper; here we will treat the measured elastic coefficients and their temperature dependencies as sufficiently representing the studied material at the macro-scale.

## 3. Results and Discussion

#### 3.1. Cu-Al-Ni

#### 3.2. Ni-Mn-Ga

#### 3.3. Ni-Ti

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Delaey, L. Diffusionless transformations. In Phase Transformations in Materials; Kostorz, G., Ed.; Wiley-VCH: Weinheim, Germany, 2001; pp. 583–654. [Google Scholar]
- Nakanishi, N. Elastic constants as they relate to lattice properties and martensite formation. Prog. Mater. Sci.
**1980**, 24, 143–265. [Google Scholar] [CrossRef] - Lüthi, B. Physical Acoustics in the Solid State; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Otsuka, K.; Ren, X. Physical metallurgy of Ti-Ni-based shape memory alloys. Prog. Mater. Sci.
**2005**, 50, 511–678. [Google Scholar] [CrossRef] - Chernenko, V.A.; Pons, J.; Seguí, C.; Cesari, E. Premartensitic phenomena and other phase transformations in Ni-Mn-Ga alloys studied by dynamical mechanical analysis and electron diffraction. Acta Mater.
**2002**, 50, 53–60. [Google Scholar] [CrossRef] - Muto, S.; Oshima, R.; Fujita, F.E. Elastic softening and elastic strain energy consideration in the f.c.c.-f.c.t. transformation of FePd alloys. Acta Metall. Mater.
**1990**, 38, 685–694. [Google Scholar] [CrossRef] - Stipcich, M.; Mañosa, L.; Planes, A.; Morin, M.; Zarestky, J.; Lograsso, T.; Stassis, C. Elastic constants of Ni-Mn-Ga magnetic shape memory alloys. Phys. Rev. B Condens. Matter Mater. Phys.
**2004**, 70, 054115. [Google Scholar] [CrossRef][Green Version] - Worgull, J.; Petti, E.; Trivisonno, J. Behavior of the elastic properties near an intermediate phase transition in Ni
_{2}MnGa. Phys. Rev. B Condens. Matter Mater. Phys.**1996**, 54, 15695–15699. [Google Scholar] [CrossRef] - Musgrave, M.J.P. Crystal Acoustics; Holden Day: San Francisco, CA, USA, 1965. [Google Scholar]
- Ledbetter, H.M. Sound velocities and elastic-constant averaging for polycrystalline copper. J. Phys. D Appl. Phys.
**1980**, 13, 1879–1884. [Google Scholar] [CrossRef] - Seiner, H.; Kopeček, J.; Sedlák, P.; Bodnárová, L.; Landa, M.; Sedmák, P.; Heczko, O. Microstructure, martensitic transformation and anomalies in c′-softening in Co-Ni-Al ferromagnetic shape memory alloys. Acta Mater.
**2013**, 61, 5869–5876. [Google Scholar] [CrossRef] - Zoubková, K. Elastic Response of Ferromagnetic Shape Memory Alloys in the Vicinity of the Critical Point. Master’s Thesis, Czech Technical University, Prague, Czech Republic, 2018. [Google Scholar]
- Papadakis, E.P.; Lerch, T.P. Pulse superposition, pulse-echo overlap and related techniques. In Handbook of Elastic Properties of Solids, Liquids and Gases, 1–4; Levy, M., Bass, H., Stern, R., Eds.; Academic Press: San Diego, CA, USA, 2001; pp. 39–66. [Google Scholar]
- Migliori, A.; Sarrao, J.L.; Visscher, W.M.; Bell, T.M.; Lei, M.; Fisk, Z.; Leisure, R.G. Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids. Phys. B Phys. Condens. Matter
**1993**, 183, 1–24. [Google Scholar] [CrossRef][Green Version] - Leisure, R.G.; Willis, F.A. Resonant ultrasound spectroscopy. J. Phys. Condens. Matter
**1997**, 9, 6001–6029. [Google Scholar] [CrossRef] - Hirao, M.; Ogi, H. EMATs for Science and Industry: Noncontacting Ultrasonic Measurements; Kluwer Academic Publishers: London, UK, 2003. [Google Scholar]
- Vasil’ev, A.N.; Kokorin, V.V.; Savchenko, Y.I.; Chernenko, V.A. The magnetoelastic properties of a Ni
_{2}MnGa single crystal. Sov. Phys. JETP**1990**, 71, 803–805. [Google Scholar] - Landa, M.; Sedlák, P.; Seiner, H.; Heller, L.; Bicanová, L.; Šittner, P.; Novák, V. Modal resonant ultrasound spectroscopy for ferroelastics. Appl. Phys. A Mater. Sci. Process.
**2009**, 96, 557–567. [Google Scholar] [CrossRef] - Hurley, D.H.; Reese, S.J.; Park, S.K.; Utegulov, Z.; Kennedy, J.R.; Telschow, K.L. In situ laser-based resonant ultrasound measurements of microstructure mediated mechanical property evolution. J. Appl. Phys.
**2010**, 107, 063510. [Google Scholar] [CrossRef][Green Version] - Wolfe, J.P. Imaging Phonons (Acoustic Wave Propagation in Solids); Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Sedlák, P.; Seiner, H.; Zídek, J.; Janovská, M.; Landa, M. Determination of All 21 Independent Elastic Coefficients of Generally Anisotropic Solids by Resonant Ultrasound Spectroscopy: Benchmark Examples. Exp. Mech.
**2014**, 54, 1073–1085. [Google Scholar] [CrossRef] - Sato, S.; Inakagi, K.; Gusev, V.E.; Wright, O.B. Resonant ultrasound spectroscopy using optical excitation and detection. AIP Conf. Proc.
**1999**, 463, 424–426. [Google Scholar] - Zadler, B.J.; Le Rousseau, J.H.L.; Scales, J.A.; Smith, M.L. Resonant ultrasound spectroscopy: Theory and application. Geophys. J. Int.
**2004**, 156, 154–169. [Google Scholar] [CrossRef] - Sumino, Y.; Ohno, I.; Goto, T.; Kumazawa, M. Measurement of elastic constants and internal frictions on single-crystal mgo by rectangular parallelepiped resonance. J. Phys. Earth
**1976**, 24, 263–273. [Google Scholar] [CrossRef][Green Version] - Seiner, H.; Sedlák, P.; Koller, M.; Landa, M.; Ramírez, C.; Osendi, M.I.; Belmonte, M. Anisotropic elastic moduli and internal friction of graphene nanoplatelets/silicon nitride composites. Compos. Sci. Technol.
**2013**, 75, 93–97. [Google Scholar] [CrossRef] - Straka, L.; Drahokoupil, J.; Pacherová, O.; Fabiánová, K.; Kopecký, V.; Seiner, H.; Hänninen, H.; Heczko, O. The relation between lattice parameters and very low twinning stress in Ni50Mn25+xGa25-x magnetic shape memory alloys. Smart Mater. Struct.
**2016**, 25, 025001. [Google Scholar] [CrossRef][Green Version] - Sozinov, A.; Likhachev, A.A.; Lanska, N.; Söderberg, O.; Ullakko, K.; Lindroos, V.K. Stress- and magnetic-field-induced variant rearrangement in Ni-Mn-Ga single crystals with seven-layered martensitic structure. Mater. Sci. Eng. A
**2004**, 378, 399–402. [Google Scholar] [CrossRef] - Scheerbaum, N.; Heczko, O.; Liu, J.; Hinz, D.; Schultz, L.; Gutfleisch, O. Magnetic field-induced twin boundary motion in polycrystalline Ni-Mn-Ga fibres. New J. Phys.
**2008**, 10, 073002. [Google Scholar] [CrossRef][Green Version] - Straka, L.; Heczko, O.; Seiner, H.; Lanska, N.; Drahokoupil, J.; Soroka, A.; Fähler, S.; Hänninen, H.; Sozinov, A. Highly mobile twinned interface in 10 M modulated Ni-Mn-Ga martensite: Analysis beyond the tetragonal approximation of lattice. Acta Mater.
**2011**, 59, 7450–7463. [Google Scholar] [CrossRef] - Zhao, P.; Dai, L.; Cullen, J.; Wuttig, M. Magnetic and elastic properties of Ni49.0Mn23.5 Ga27.5 premartensite. Metall. Mater. Trans. A Phys. Metall. Mater. Sci.
**2007**, 38, 745–751. [Google Scholar] [CrossRef] - Dai, L.; Cui, J.; Wuttig, M. Elasticity of Austenitic and Martensitic NiMnGa. Proc. SPIE Int. Soc. Opt. Eng.
**2003**, 5053, 595–602. [Google Scholar] - Ogi, H.; Sato, K.; Asada, T.; Hirao, M. Complete mode identification for resonance ultrasound spectroscopy. J. Acoust. Soc. Am.
**2002**, 112, 2553–2557. [Google Scholar] [CrossRef] - Recarte, V.; Pérez-Landazábal, J.I.; Nó, M.L.; San Juan, J. Study by resonant ultrasound spectroscopy of the elastic constants of the β phase in Cu-Al-Ni shape memory alloys. Mater. Sci. Eng. A
**2004**, 370, 488–491. [Google Scholar] [CrossRef] - Landa, M.; Sedlák, P.; Šittner, P.; Seiner, H.; Heller, L. On the evaluation of temperature dependence of elastic constants of martensitic phases in shape memory alloys from resonant ultrasound spectroscopy studies. Mater. Sci. Eng. A
**2008**, 481–482, 567–573. [Google Scholar] [CrossRef] - Landa, M.; Sedlák, P.; Šittner, P.; Seiner, H.; Novák, V. Temperature dependence of elastic properties of cubic and orthorhombic phases in Cu-Al-Ni shape memory alloy near their stability limits. Mater. Sci. Eng. A
**2007**, 462, 320–324. [Google Scholar] [CrossRef] - Sugimoto, K.; Mori, T.; Otsuka, K.; Shimizu, K. Simultaneous measurements of internal friction, young’s modulus and shape change associated with thermoelastic martensite transformation in CuAlNi single crystals. Scr. Metall.
**1974**, 8, 1341–1348. [Google Scholar] [CrossRef] - Sedlák, P.; Seiner, H.; Landa, M.; Novák, V.; Šittner, P.; Mañosa, L. Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Mater.
**2005**, 53, 3643–3661. [Google Scholar] [CrossRef] - Seiner, H.; Sedlak, P.; Landa, M. Shape recovery mechanism observed in single crystals of Cu-Al-Ni shape memory alloy. Phase Transitions
**2008**, 81, 537–551. [Google Scholar] [CrossRef] - Vronka, M.; Seiner, H.; Heczko, O. Temperature dependence of twinning stress–Analogy between Cu–Ni–Al and Ni–Mn–Ga shape memory single crystals. Philos. Mag.
**2017**, 97, 1479–1497. [Google Scholar] [CrossRef] - Seiner, H.; Heczko, O.; Sedlák, P.; Bodnárová, L.; Novotný, M.; Kopeček, J.; Landa, M. Combined effect of structural softening and magneto-elastic coupling on elastic coefficients of NiMnGa austenite. J. Alloy. Compd.
**2013**, 577, S131–S135. [Google Scholar] [CrossRef] - Seiner, H.; Sedlák, P.; Bodnárová, L.; Drahokoupil, J.; Kopecký, V.; Kopeček, J.; Landa, M.; Heczko, O. The effect of antiphase boundaries on the elastic properties of Ni-Mn-Ga austenite and premartensite. J. Phys. Condens. Matter
**2013**, 25, 425402. [Google Scholar] [CrossRef] [PubMed] - Söderberg, O.; Straka, L.; Novák, V.; Heczko, O.; Hannula, S.-P.; Lindroos, V.K. Tensile/compressive behaviour of non-layered tetragonal Ni52.8Mn25.7Ga21.5 alloy. Mater. Sci. Eng. A
**2004**, 386, 27–33. [Google Scholar] [CrossRef] - Sozinov, A.; Lanska, N.; Soroka, A.; Zou, W. 12% magnetic field-induced strain in Ni-Mn-Ga-based non-modulated martensite. Appl. Phys. Lett.
**2013**, 102, 021902. [Google Scholar] [CrossRef] - Bodnárová, L.; Zelený, M.; Sedlák, P.; Straka, L.; Heczko, O.; Sozinov, A.; Seiner, H. Switching the soft shearing mode orientation in Ni-Mn-Ga non-modulated martensite by Co and Cu doping. Smart Mater. Struct.
**2020**, 29, 045022. [Google Scholar] [CrossRef] - Heczko, O.; Seiner, H.; Sedlák, P.; Kopeček, J.; Kopecký, V.; Landa, M. Resonant ultrasound spectroscopy—A tool to probe magneto-elastic properties of ferromagnetic shape memory alloys. Eur. Phys. J. B
**2013**, 86, 62. [Google Scholar] [CrossRef] - James, R.D.; Zhang, Z. A Way to Search for Multiferroic Materials with “Unlikely” Combinations of Physical Properties. Springer Ser. Mater. Sci.
**2005**, 79, 159–175. [Google Scholar] - Cui, J.; Chu, Y.S.; Famodu, O.O.; Furuya, Y.; Hattrick-Simpers, J.; James, R.D.; Ludwig, A.; Thienhaus, S.; Wuttig, M.; Zhang, Z.; et al. Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width. Nat. Mater.
**2006**, 5, 286–290. [Google Scholar] [CrossRef] - Seiner, H.; Kopecký, V.; Landa, M.; Heczko, O. Elasticity and magnetism of Ni
_{2}MnGa premartensitic tweed. Phys. Status Solidi (B) Basic Res.**2014**, 251, 2097–2103. [Google Scholar] [CrossRef] - Ren, X.; Miura, N.; Zhang, J.; Otsuka, K.; Tanaka, K.; Koiwa, M.; Suzuki, T.; Chumlyakov, Y.I.; Asai, M. A comparative study of elastic constants of Ti-Ni based alloys prior to martensitic transformation. Mater. Sci. Eng. A
**2001**, 312, 196–206. [Google Scholar] [CrossRef] - Brill, T.M.; Mittelbach, S.; Assmus, W.; Mullner, M.; Luthi, B. Elastic properties of NiTi. J. Phys. Condens. Matter
**1991**, 3, 9621–9627. [Google Scholar] [CrossRef] - Mercier, O.; Melton, K.N.; Gremaud, G.; Hägi, J. Single-crystal elastic constants of the equiatomic NiTi alloy near the martensitic transformation. J. Appl. Phys.
**1980**, 51, 1833–1834. [Google Scholar] [CrossRef] - Ren, X.; Otsuka, K. The role of softening in elastic constant c
_{44}in martensitic transformation. Scr. Mater.**1998**, 38, 1669–1675. [Google Scholar] [CrossRef] - Šittner, P.; Landa, M.; Lukáš, P.; Novák, V. R-phase transformation phenomena in thermomechanically loaded NiTi polycrystals. Mech. Mater.
**2006**, 38, 475–492. [Google Scholar] [CrossRef]

**Figure 1.**Simulated errors in ${c}^{\prime}$ resulting from a small misorentation of the sample: (

**a**) methods based on time-of-flight measurements; (

**b**) resonant ultrasound spectroscopy. The misorientation angle $\psi $ is defined in the sketch on the very left.

**Figure 2.**Evolution of the elastic constant ${c}^{\prime}$ (

**a**) and the internal friction parameter (

**b**) for the CuAlNi single crystal.

**Figure 3.**Evolution of the elastic constant ${c}^{\prime}$ for the Ni-Mn-Ga samples: (

**a**) sample No.2 with the transition temperature below the Curie point ${T}_{C}$; (

**b**) sample No.3 with the transition temperature above the curie point. In (

**a**), the constants ${c}^{\prime}$ for two tetragonal variants of martensite V1 and V2 are effective elastic constants only, obtained under the assumption that the material has a cubic symmetry (see the text for more details); the dash-dot line is the extrapolation of the behavior of austenite above the Curie point. In (

**b**), the ${c}^{\prime}$ below ${A}_{S}$ is again an effective parameter, representing the initial mixture of variants. Note the different temperature scales between (

**a**,

**b**).

**Figure 4.**(

**a**) the misorientation between the sample edges (${y}_{1,\dots ,3}$) and the principal $\langle 1\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}0\rangle $ directions. The map plotted on the unit sphere is a map of a function used for determination of the symmetry planes of the material for the measured 21 (triclinic) elastic constants, as introduced in [21]. In particular, the sharp minima (blue spots) correspond to directions perpendicular to the mirror planes. (

**b**) the resulting evolution of the elastic constants.

**Table 1.**Chemical compositions of the studied samples and the corresponding transition temperatures determined by differential scanning calorimetry (if not stated otherwise).

Sample No. | Alloy | Composition (at.%) | ${\mathit{A}}_{\mathbf{S}}$ [K] | ${\mathit{A}}_{\mathbf{F}}$ [K] | ${\mathit{M}}_{\mathbf{S}}$ [K] | ${\mathit{M}}_{\mathbf{F}}$ [K] | ${\mathit{R}}_{\mathbf{S}}$ [K] | ${\mathit{R}}_{\mathbf{F}}$ [K] |
---|---|---|---|---|---|---|---|---|

1 | Cu-Al-Ni | Cu ${}_{69.4}$ Al ${}_{27.2}$ Ni ${}_{3.4}$ | 395 | 314 | 299 | 284 | - | - |

2 | Ni-Mn-Ga | Ni ${}_{50.0}$ Mn ${}_{28.9}$ Ga ${}_{21.1}$ | 328 | 329 | 328 | 326 | - | - |

3 | Ni-Mn-Ga | Ni ${}_{50.5}$ Mn ${}_{30.4}$ Ga ${}_{19.1}$ | 400 ${}^{a}$ | 420 ${}^{a}$ | - | - | - | - |

4 | Ni-Ti | multiphase ${}^{b}$ | 240 | 286 | 232 | 226 | ∼270 | 245 |

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**MDPI and ACS Style**

Sedlák, P.; Janovská, M.; Bodnárová, L.; Heczko, O.; Seiner, H.
Softening of Shear Elastic Coefficients in Shape Memory Alloys Near the Martensitic Transition: A Study by Laser-Based Resonant Ultrasound Spectroscopy. *Metals* **2020**, *10*, 1383.
https://doi.org/10.3390/met10101383

**AMA Style**

Sedlák P, Janovská M, Bodnárová L, Heczko O, Seiner H.
Softening of Shear Elastic Coefficients in Shape Memory Alloys Near the Martensitic Transition: A Study by Laser-Based Resonant Ultrasound Spectroscopy. *Metals*. 2020; 10(10):1383.
https://doi.org/10.3390/met10101383

**Chicago/Turabian Style**

Sedlák, Petr, Michaela Janovská, Lucie Bodnárová, Oleg Heczko, and Hanuš Seiner.
2020. "Softening of Shear Elastic Coefficients in Shape Memory Alloys Near the Martensitic Transition: A Study by Laser-Based Resonant Ultrasound Spectroscopy" *Metals* 10, no. 10: 1383.
https://doi.org/10.3390/met10101383