# 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

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**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