# Thermodynamic Analysis of Anomalous Shape of Stress–Strain Curves for Shape Memory Alloys

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{A}as well as [110]

_{A}directions in a Ni

_{49}Fe

_{18}Ga

_{27}Co

_{6}single crystal, was normal as well as anomalous, respectively). It is often observed that the transformation is not complete [1,7,8,9], but there are also examples when the whole sample has been transformed [4,10]. Furthermore, the strain recovery of the stress induced martensite to austenite during heating can be very fast (burst-like recovery). For instance, it was observed that the DSC peak of this recovery was only about 10

^{−3}–10

^{−5}degree wide (instead of the usual 1–50 K wide transitions at typical 1–10 K/min rates) and was accompanied with jumping of the sample as well as with audible click [1,2,3,4,5,6,7,8,9,10]. In addition, the DSC peak appeared at higher temperature (by 10–60 K higher) than the corresponding peak of the reverse transformation measured after thermally induced cycling. This indicates that the martensite structure formed during stress induced changes is more stable than the thermally induced one. Both the martensite stabilization, manifested in the shift of the DSC peak to higher temperatures during heating, and the anomalous stress–strain curves, are interpreted by the presence/competition of two different martensitic structural modifications [2,3,9]. These can be denoted as M

_{1}and M

_{2}structures, respectively. For instance, in Ni

_{49}Fe

_{18}Ga

_{27}Co

_{6}single crystal M

_{1}and M

_{2}were identified as twinned 14 M martensite, as well as L1

_{o}detwinned tetragonal martensite, respectively [3], or in Cu–Al–Ni alloys as β′ (18 R monoclinic) and/or twinned γ′ (2 H orthorhombic) as well as detwinned γ′ phases, respectively [7,8,11]. It can be added that the appearance of stress drops/jumps or the macroscopic jump of the sample itself can cause difficulties in experiments. For instance, in ref. [2] it was observed that the DSC peak of the thermal recovery showed an anomalous shift to lower temperatures with increasing heating rates. In [7] it was mentioned that the slow motion of the crosshead of the machine did not go down so fast and a stress drop could be registered or that the testing machine can be slightly deformed itself. However, in these papers the above effects were carefully handled, and it can be concluded that the observations summarized above on the anomalous stress–strain curves and on the burst-like recovery are real effects.

_{A}and S

_{M}are the stiffness of the austenite and martensite, respectively. However, this conclusion contradicts to a set of experimental results where anomalous stress–strain curves were observed for ${S}_{A}-{S}_{M}<0$ [3,7,9].

## 2. Model Calculations

#### 2.1. The Local Equilibrium Formalism

_{c}is the change in the chemical free energy per unit volume and its derivative can be written as

_{M}− s

_{A}(∆s < 0) as well as Δu = u

_{M}− u

_{A}= Δu (<0) are the entropy and internal energy changes per unit volume, respectively, and they are independent of ξ. T and σ are the temperature and stress as well as ε

^{tr}is the transformation strain, which is positive for tension.

^{tr}, is constant and independent of ξ (i.e., the derivative of Equation (3) is zero). However, in general ε

^{tr}can depend on the T and/or σ values since in Equation (3) the stress-term has a tensor character [24,25]. Thus, even for the application of uniaxial stress (which leads to a scalar term, as in Equation (3)), ε

^{tr}in principle can depend on T and σ too. In case of formation and growth of two martensite structural modifications, it can also depend on the volume fraction of one of these, η(ξ) = η(T,σ) = V

_{M}

_{2}/V

_{M}(V

_{M}= V

_{M}

_{1}+ V

_{M}

_{2}, V = V

_{M}+ V

_{A}), [24,25,26]. The η dependence of ε

^{tr}can be given by the relation [24,25,26]:

^{tr}(η) = ε

_{1}+ (ε

_{2}− ε

_{1})η

_{1}and ε

_{2}are the transformation strains when fully one certain martensite structure forms (at η ≅ 0 and η ≅ 1, respectively). Of course, the details can be very complex and the value of ε can also be different for twinned or detwinned martensite variants [27,28,29]. As a consequence, η can have a ξ-dependence, η(ξ), which explains the ξ-dependence of ε

^{tr}: ε

^{tr}(η(ξ)). Furthermore, ε

_{i}, due to the different temperature dependence of the elastic moduli of the austenite and martensite phases, can have a direct temperature dependence too [22,30,31]). Thus, considerations on the ξ-dependence of the actual strain, $\epsilon \left(\xi \right)$ (which should be distinguished from the transformation strain, ${\epsilon}^{tr}$ can be important for estimation of the σ(ε) curve (see also below). It is worth mentioning that our description, in order to concentrate on the interpretation of the main important features, in its form is a simplified one, and is applicable indeed for single crystals (with two martensite modifications), where the effects summarized in the introduction were observed. Of course, for more complex systems (e.g., for polycrystalline materials with numerous martensite variants) one would need a more sophisticated treatment, such as the phase field method in the form of the well-known Ginzburg–Landau theory (see e.g., [32]).

#### 2.2. Expressions for the $\sigma \left(\xi \right)$ $\left(\sigma \left(\epsilon \right)\right)$ Functions

^{tr}depends on ξ, and we have from (2) and (3)

^{MA}as well as

^{AM}will be used only if a comparison of the forward and reverse transformations is made.

^{tr}).

## 3. Discussion

#### 3.1. Expressions for the Widths of Transformations—Investigation of the Stability during Phase Transformation

#### 3.1.1. Meaning of the Elastic and Dissipative Terms

_{f}(>0) originates from the frictional-type motion of the interfaces and can be supposed that it is proportional to ξ, while ${D}_{n}$ (>0) is due to the nucleation energy. In the simple case when a large number of martensite nuclei form (smooth transformation [22]) ${D}_{n}$ can also be approximately a monotonic linear function of ξ and thus

_{n}(ξ) can be a complicated (step-wise) function of ξ (see also below).

#### 3.1.2. Stress–Strain Loops

#### 3.1.3. Thermal Hysteresis Loops

#### 3.2. Growth of Two Martensite Modifications

#### 3.3. Nucleation Difficulties

_{1}martensite modification nucleates first and grows and at a certain transformed volume fraction, ξ

_{c}(~ε

_{c}) the second martensite M

_{2}nucleates. Suppose that M

_{2}is more stable than M

_{1}, in accordance with the higher temperature of burst-like MA transformation and, as above, ${\epsilon}_{2}={\epsilon}^{tr2}>{\epsilon}_{1}={\epsilon}^{tr1}$. Under the same assumptions as Equation (14) was obtained, we can write for the slopes of the linearized ${\sigma}_{1}\left(\xi \right)$ and ${\sigma}_{2}\left(\xi \right)$ functions

_{2}the slope is approximately zero. Thus, the slope of ${\sigma}_{1}\left(\xi \right)$ is larger. In addition, consider the case when ${\sigma}_{Ms1}<$ ${\sigma}_{Ms2}$ i.e., if

_{o}

_{2}is large enough. Indeed, it can be the case if we assume that the nucleation of M

_{2}is more difficult than the nucleation of M

_{1}. For instance, it is well-known in the literature [33,34] that there can be a competition between habit-plane variants, which can easily nucleate from austenite (and accompanied with smaller transformation strain), and oriented martensites. The nucleation of the latter one is more difficult because of crystallographic compatibility problems, depending on the orientation relationships, and the formation and growth of which can be accompanied with smaller accumulated stresses (see e.g., [9,34,35,36]). The formation of M

_{2}at a certain critical stress/strain can happen either from the not transformed yet austenite or from the growing M

_{1}(see e.g., [34]).

_{c}(~ε

_{c}), at which $\sigma \left({\xi}_{c}\right)={\sigma}_{Ms2},$ and thus M

_{2}can nucleate which leads to a stress drop. For the estimation of the stress drop we can use the results of [22]. It was obtained, for the stress induced austenite to one-martensite-modification transformation, that (due to the transformation strain) there should exists an overall decrease in stress. This decrease, as compared to the pure elastic contribution to the stress–strain curve, is proportional to the product of the martensite volume fraction, $\xi $, the transformation strain, ${\epsilon}^{tr}$, and the effective stiffness in the two-phase region, S(ξ), as compared to the pure elastic contribution to the stress–strain curve: $\Delta \sigma \cong -\xi {\epsilon}^{tr}S\left(\xi \right)$ ($\xi $ corresponds to the parameter $\xi \cong \frac{{n}_{z}\left(\epsilon \right)}{{N}_{y}}$ used in [22] and $S\left(\xi \right)=\frac{{S}_{A}{S}_{M}}{{S}_{M}+\xi \left({S}_{A}-{S}_{M}\right)}$). Thus, the stress drop due to the nucleation of M

_{2}martensite at ξ

_{c}is proportional to ${\epsilon}_{2}^{tr}-{\epsilon}_{1}^{tr}$:

_{1}start stress then after the stress drop, the austenite + M

_{1}two-phase system will be elastically deformed until the nucleation of M

_{1}martensite happens again and the process is repeated. In this (i) case the slope after the stress drop will be the similar as the initial slope of the stress–strain curve, with the effective stiffness S(ξ

_{c}). In the opposite case, (ii), after the stress drop there is a further growth of M

_{1}(with a similar course of the stress–strain curve as observed during the first stage of the process before the stress-drop) until a new M

_{2}nucleates and the process is repeated. Both can be observed in experiments: (i) see Figure 1b of [2] where the segment of the σ~ε curve after the stress-drop is similar to the initial part, or (ii) in Figure 1a,b in [1] as examples. Figure 1 in [1] also illustrates that with decreasing test temperature there is a transition from (i) to (ii). Regarding the unloading process under compression, with decreasing stress the M

_{2}is stable well below the austenite start stress of M

_{1}, ${\sigma}_{as1}$, (even below a certain temperature M

_{2}remains stable at zero stress). Thus, retwinning of M

_{2}(i.e., the formation of M

_{1}from M

_{2}) can start at a certain low stress level and this leads to the expansion of the sample (${\epsilon}_{2}>{\epsilon}_{1}$) and a stress jump can appear (see Figure 1b). Further details, like the course of the curve during the stress-drops before reaching the minimum, or where and how M

_{2}nucleates can only be explained on the basis of microscopic investigations (see e.g., [35,36,37]) and out of the scope of thermodynamic considerations used here.

#### 3.4. About the Burst-like Recovery

_{2}: once the twinning is achieved an “easy way” is opened for transformation to the M

_{1}or austenite phase(s). Thus, the observed overheating is the consequence of the (nucleation) barrier against re-twinning. We can obtain an approximate expression for the shift of the forward transformation peak of the burst-like recovery as compared to the same thermally induced DSC peak, ΔT, using the results described in the preceding sections. ΔT, can be given as the difference of the peak temperatures, T

_{p}, during heating:

_{2}/A transformation and for this $\left({e}_{12}-{e}_{o2}\right)\cong 0$ (see (21)), as well as for two martensite variants we can also assume that the difference of the equilibrium transformation temperatures ($\Delta {T}_{o}={T}_{o2}-{T}_{o1})$ and the transformation entropy is approximately zero ($\Delta {s}_{1}\cong \Delta {s}_{2}=\Delta s$):

_{i}is the nucleation energy for the heating process. Thus, the shift of the transformation peak is a measure of the change in the nucleation energy: $\Delta D={D}_{2}-{D}_{1}.$ As a numerical example we can use the results of [40], where it was obtained that ΔT was about 36 K and using the transformation entropy in this alloy, 0.75 J/mol K [40], we find that ${D}_{2}-{D}_{1}\cong 27\mathrm{J}/\mathrm{mol}$. This can be compared to the half of the dissipative energy per one thermal cycle, 2.8 J/mol [41], which shows that the nucleation energy for M

_{2}can be about an order of magnitude larger than for M

_{1}.

## 4. Conclusions

_{s}≅ A

_{f}) can be obtained during burst-like thermal recovery, indicating that the second derivative of the elastic energy is approximately zero in this case. It is also shown that the stress–strain loops for smooth transformations are also stable if only one type of martensite growths. Instability can appear, i.e., the overall slope of the $AM$ branch can be negative, if nucleation and growth of two martensite modifications (variants) takes place and if the second (more stable) one is the final product with ${\epsilon}_{2}^{tr}>{\epsilon}_{1}^{tr}.$ It is found that local stress-drops, Δσ, on the stress–strain curve can appear if the nucleation of the second martensite is difficult and the presence of few local stress-drops alone can also result in an overall negative slope of the σ~ε curve. Depending on the magnitude of Δσ, the course of the stress–strain curve after the stress drop is similar to the initial (elastic) part or to the part belonging to A/M

_{1}transformation. It is illustrated that shift of the temperature of the thermal recovery of M

_{2}is a measure of the change in the nucleation energy $\Delta D={D}_{2}-{D}_{1}$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Stress–strain curves schematically: (

**a**) typical loop with positive slope of the uploading branch, (

**b**) anomalous loop with negative overall slope and stress drops on it.

**Figure 2.**Schematic σ(ξ) hysteresis loop for ε

^{tr}= const. > 0. Since ε = ξε

^{tr}, with the actual value of ε, this plot corresponds to the σ versus ε plots.

**Figure 3.**Schematic ${\sigma}_{1}\left(\xi \right)$ and ${\sigma}_{1}\left(\xi \right)$ functions, and the stress drop, $\Delta \sigma $ at the critical volume fraction where the second martensite nucleates (see also the text).

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**MDPI and ACS Style**

Beke, D.L.; Kamel, S.M.; Daróczi, L.; Tóth, L.Z.
Thermodynamic Analysis of Anomalous Shape of Stress–Strain Curves for Shape Memory Alloys. *Materials* **2022**, *15*, 9010.
https://doi.org/10.3390/ma15249010

**AMA Style**

Beke DL, Kamel SM, Daróczi L, Tóth LZ.
Thermodynamic Analysis of Anomalous Shape of Stress–Strain Curves for Shape Memory Alloys. *Materials*. 2022; 15(24):9010.
https://doi.org/10.3390/ma15249010

**Chicago/Turabian Style**

Beke, Dezső L., Sarah M. Kamel, Lajos Daróczi, and László Z. Tóth.
2022. "Thermodynamic Analysis of Anomalous Shape of Stress–Strain Curves for Shape Memory Alloys" *Materials* 15, no. 24: 9010.
https://doi.org/10.3390/ma15249010