We are IntechOpen, the world's leading publisher of Open Access books Built by scientists, for scientists

3,650+

Open access books available

114,000+

International authors and editors

118M+

Downloads

151 Countries delivered to Our authors are among the

Top 1%

most cited scientists

12.2% Contributors from top 500 universities

Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI)

## Interested in publishing with us? Contact book.department@intechopen.com

Numbers displayed above are based on latest data collected. For more information visit www.intechopen.com

## **Meet the editor**

Dr. Farzad Ebrahimi received his MSc and PhD degrees in Mechanical Engineering, with a specialization in applied design, from the University of Tehran, Iran, in 2009. He joined the department of mechanical engineering of Imam Khomeini International University as an assistant professor in 2010.

He is involved in several international journals as editor and reviewer. He serves on the editorial board of the "Journal of advances in mechanical engineering." He is the author of books *Smart FG Plates* and *Progress in Analysis of FG Structures*, Nova Science Publishers, NY. He also served as the editor of the book *Nanocomposites - New Trends and Developments*, InTechOpen Publisher. His research interests focus on areas of smart materials and structures, nanostructures, vibration nanocomposites, and composite materials and structures, and he has published several researches in these fields.

## Contents

**Preface XI**


Atta Muhammad Nizamani, Jawaid Daudpoto and Muhammad Ali Nizamani

## Preface

Shape memory alloys (SMAs) are metals that "remember" their original shapes and exhibit two very unique properties: pseudo-elasticity and the shape memory effect. SMAs have drawn significant attention and interest in recent years in a broad range of commercial ap‐ plications, due to their unique and superior properties; this commercial development has been supported by fundamental and applied research studies.

This book is a result of contributions of experts from international scientific community working in different aspects of shape memory alloys and reports on the state-of-the-art re‐ search and development findings on this topic through original and innovative research studies. Through its five chapters, the reader will have access to works related to ferromag‐ netic SMAs, while it introduces some specific applications like development of faster SMA actuators and application of nanostructural SMAs in medical devices.

The book contains up-to-date publications of leading experts, and the edition is intended to furnish valuable recent information to the professionals involved in shape memory alloys' analysis and applications. The text is addressed not only to researchers but also to professio‐ nal engineers, students, and other experts in a variety of disciplines, both academic and in‐ dustrial, seeking to gain a better understanding of what has been done in the field recently and what kind of open problems are in this area.

I hope that readers will find the book useful and inspiring by examining the recent develop‐ ments in shape memory alloys.

Lastly, I would like to thank all the authors for their excellent contributions in different areas covered by this book and the InTechOpen team, especially the publishing process manager Ms. Romina Rovan, for their support and patience during the whole process of creating this book.

> **Dr. Farzad Ebrahimi** Department of Mechanical Engineering Imam Khomeini International University, Qazvin, Iran

**Shape Memory Alloys: Fundamentals**

## **Ferromagnetic Shape Memory Alloys: Foams and Microwires**

Xuexi Zhang and Mingfang Qian

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69702

#### **Abstract**

Ferromagnetic shape memory alloys exhibit martensite transformation (MT) and magnetic transition and thus may be actuated by thermal and magnetic fields. The working frequency of these alloys may be higher than conventional shape memory alloys, such as Ni-Ti, because the magnetic field may operate at higher frequency. This chapter focuses on some fundamental topics of these multifunctional materials, including the composition-structure relationship, the synthesis of the foams and microwires, the martensite transformation and magnetic transition characters, the properties (magneticfield-induced strain (MFIS), magnetocaloric effects (MCEs), shape memory effects, and superelastic effects), and applications. The improvement of the magnetic-field-induced strain due to the reduced constraint of twin boundary motion caused by grain boundaries in polycrystalline Ni-Mn-Ga foams and the size effects of the superelasticity and magnetocaloric properties in Ni-Mn-X (X = In, Sn, Sb) microwires are detailed and addressed.

**Keywords:** ferromagnetic shape memory alloys (FSMAs), Ni-Mn-Ga alloys, foams, microwires, martensite transformation, ferromagnetic-field-induced strain (MFIS), magnetocaloric effects (MCEs)

#### **1. Introduction**

Ferromagnetic shape memory Heusler alloys, such as Ni-Mn-Ga and Ni-Mn-X (X = In, Sn, Sb), are receiving increasing attentions due to their multifunctional properties, that is, magnetic-field-induced strain (MFIS), magnetocaloric effect (MCE), magnetoresistance, etc. Before 1996, some works concern the martensite transformation (MT) of Ni-Mn-Ga alloys [1–3]. The relationship between the composition on martensite and magnetic transformation temperatures has also been revealed [4]. In 1996, Ullakko et al. published the first paper on the MFIS

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

of single-crystalline Ni-Mn-Ga alloys [5]. Since then, high MFIS of 6.4% has been found in Ni-Mn-Ga alloys with 5M martensite structure [6] and of 10% with 5M martensite structure [7]. By allying a high magnetic field, the single-crystalline Ni-Mn-Ga alloy with non-modulate (NM) martensite structure may show a giant MFIS as high as 12% [8]. In 2007, Müllner et al. published their first paper on Ni-Mn-Ga foam and found that 0.24% MFIS may be achieved in the foams with single-model pore architecture [9]. By introducing secondary pores, thus forming dual-pore architecture, the foams may exhibit a MFIS of 8.7% after suitable thermal-magnetic training, which approaches the theoretical limit of single-crystalline Ni-Mn-Ga alloys with 14M martensite structure [10]. In 2008, Scheerbaum et al. [11] produced Ni-Mn-Ga fibers with diameter ~60–100 μm, which showed a 1% MFIS upon applying an external magnetic field of 2 T. Since 2012, researchers in Harbin Institute of Technology make use of the so-called melt extraction method to fabricate microwires on a large quantity [12, 13]. The obtained microwire has a naked surface, which can be directly used in MEMS or NEMS, and acts as building blocks for composites and complex-shaped components. Systematic research works have also been carried out to reveal the superelasticity [14, 15], shape memory [12, 16], and MCE properties [17–19].

Ni-Mn-X (X = In, Sn, Sb) alloys, another important ferromagnetic shape memory alloy family, attracted attention of the scientific society since Kainuma et al. [20] reported the giant stress output in Ni-Mn-In-Co alloys. On contrary to Ni-Mn-Ga alloys, these Ni-Mn-X alloys exhibit paramagnetic/antiferromagnetic martensite and ferromagnetic austenite. As a result, a bias magnetic field may stabilize the austenite phase, which is responsible for the shift of martensite transformation temperatures to lower temperature. Such metamagnetic structural transition from the paramagnetic/antiferromagnetic martensite to ferromagnetic austenite under a bias magnetic field may produce giant MCE [21].

This chapter describes the synthesis, processing, and properties (especially the MIFS and MCE) of these alloys, with emphasis on Ni-Mn-Ga foams and microwires. The up-to-date results, mainly reported in the past decades, will be covered. The aim of this chapter is to provide researchers an overview of the background, current status, and future development of the ferromagnetic shape memory alloys.

#### **2. Composition-structure relationship in ferromagnetic shape memory alloys**

#### **2.1. Relationship between composition (***e***/***a***) and transformation temperatures**

Off-stoichiometric Ni-Mn-Ga ferromagnetic shape memory alloys (FSMAs) have been attracted much attention because of their multiplicity of functional properties like excellent magnetic-field-induced strains (MFIS) [6, 7, 10], magnetocaloric effect (MCE) [22–24], conventional/magnetic shape memory effects (SME) [12], etc. The martensite phase or the martensite transformation (MT) temperatures (martensite start and finish temperature *Ms* , *Mf* , and the austenite start and finish temperature *As* , *Af* ) of the alloys are closely related to their functional properties. For instance, the Ni-Mn-Ga alloys show MFIS only in the martensitic state owing to the reorientation of the twin boundaries in an applied magnetic field. As for the MCE, the effective refrigeration working temperature interval (WTI) of the cooling system is within the MT range. For Ni-Mn-Ga alloys, the MT temperatures of the alloy are particularly sensitive to their compositions [4].

of single-crystalline Ni-Mn-Ga alloys [5]. Since then, high MFIS of 6.4% has been found in Ni-Mn-Ga alloys with 5M martensite structure [6] and of 10% with 5M martensite structure [7]. By allying a high magnetic field, the single-crystalline Ni-Mn-Ga alloy with non-modulate (NM) martensite structure may show a giant MFIS as high as 12% [8]. In 2007, Müllner et al. published their first paper on Ni-Mn-Ga foam and found that 0.24% MFIS may be achieved in the foams with single-model pore architecture [9]. By introducing secondary pores, thus forming dual-pore architecture, the foams may exhibit a MFIS of 8.7% after suitable thermal-magnetic training, which approaches the theoretical limit of single-crystalline Ni-Mn-Ga alloys with 14M martensite structure [10]. In 2008, Scheerbaum et al. [11] produced Ni-Mn-Ga fibers with diameter ~60–100 μm, which showed a 1% MFIS upon applying an external magnetic field of 2 T. Since 2012, researchers in Harbin Institute of Technology make use of the so-called melt extraction method to fabricate microwires on a large quantity [12, 13]. The obtained microwire has a naked surface, which can be directly used in MEMS or NEMS, and acts as building blocks for composites and complex-shaped components. Systematic research works have also been carried out to reveal the superelasticity [14, 15], shape memory [12, 16],

Ni-Mn-X (X = In, Sn, Sb) alloys, another important ferromagnetic shape memory alloy family, attracted attention of the scientific society since Kainuma et al. [20] reported the giant stress output in Ni-Mn-In-Co alloys. On contrary to Ni-Mn-Ga alloys, these Ni-Mn-X alloys exhibit paramagnetic/antiferromagnetic martensite and ferromagnetic austenite. As a result, a bias magnetic field may stabilize the austenite phase, which is responsible for the shift of martensite transformation temperatures to lower temperature. Such metamagnetic structural transition from the paramagnetic/antiferromagnetic martensite to ferromagnetic austenite under a

This chapter describes the synthesis, processing, and properties (especially the MIFS and MCE) of these alloys, with emphasis on Ni-Mn-Ga foams and microwires. The up-to-date results, mainly reported in the past decades, will be covered. The aim of this chapter is to provide researchers an overview of the background, current status, and future development

**2. Composition-structure relationship in ferromagnetic shape memory** 

Off-stoichiometric Ni-Mn-Ga ferromagnetic shape memory alloys (FSMAs) have been attracted much attention because of their multiplicity of functional properties like excellent magnetic-field-induced strains (MFIS) [6, 7, 10], magnetocaloric effect (MCE) [22–24], conventional/magnetic shape memory effects (SME) [12], etc. The martensite phase or the martensite

properties. For instance, the Ni-Mn-Ga alloys show MFIS only in the martensitic state owing

, *Mf*

) of the alloys are closely related to their functional

, and the

**2.1. Relationship between composition (***e***/***a***) and transformation temperatures**

transformation (MT) temperatures (martensite start and finish temperature *Ms*

, *Af*

and MCE properties [17–19].

4 Shape Memory Alloys - Fundamentals and Applications

bias magnetic field may produce giant MCE [21].

of the ferromagnetic shape memory alloys.

austenite start and finish temperature *As*

**alloys**

Here, we focused our content on Ni-Mn-Ga alloys due to the space limit. For the stoichiometric Ni<sup>2</sup> MnGa, the MT temperatures are lower than room temperature (RT), ~185 K; thus, no martensite exists at RT. By varying the composition of 5 at.%, a huge *Ms* change from 154 to 458 K can be obtained [4, 25–30]. The effect of composition on MT temperatures of the Ni-Mn-Ga alloys was studied by Chernenko et al. at 1995 [4]. A general view of transformation temperature shifting as a function of the Ni/Mn/Ga element content was summarized as (1) at a constant value of Mn content, Ga addition lowers *Ms* temperature, (2) Mn addition (instead of Ga) at constant Ni concentration increases *Ms* , and (3) substitution of Ni atoms by Mn at constant Ga content results in alloys with lower *Ms* .

Furthermore, based on the *Ms* and transformation enthalpy (Δ*H*), Ni-Mn-Ga alloys were classified into three groups, as demonstrated in **Table 1** [4, 31, 32], where *Tc* stands for the magnetic transformation of Curie temperature of the alloy.

Thereafter, more systematic and quantized studies have been performed [25–30]. The effects of composition on temperature and Δ*H* can be formulated by the linear regression listed as follows [30]:

$$\mathcal{M}\_{\circ}\text{ (K)} = 25.44\text{Ni} \text{ (at. }\% \text{)} - 4.86\text{Mn} \text{ (at. }\% \text{)} - 38.83\text{Ga} \text{ (at. }\% \text{)}\tag{1}$$

$$
\Delta\mathsf{H} \text{ (J/g)} = 0.72\text{Ni (at. }\% \text{)} - 0.16\text{Mn (at. }\% \text{)} - 1.23\text{Ga (at. }\% \text{)}\tag{2}
$$

The number of valence electrons per atom (*e*/*a*) for Ni, Mn, and Ga atoms is 10(3d9.94s0.1), 7(3d5 4s<sup>2</sup> ), and 10(4s<sup>2</sup> 4p1 ), respectively [26]; thus, the valence electron concentration of the alloy can be defined as

\$\mu\$

 $\nu\_{\mu} = \nu\_{\nu} + \nu\_{\mu} + \nu\_{\nu} + \dots + \nu\_{\mu}\$$ 
 $\text{can be defined as}$ 

$$\mathbf{e} \Big| \mathbf{a} = \frac{10 \,\text{N}\_{\text{act}} + 7 \,\text{M}\_{\text{act}} + 3 \,\text{Ga}\_{\text{act}}}{\text{N}\_{\text{act}} + \text{M}\_{\text{act}} + \text{Ga}\_{\text{act}}}\tag{3}$$

The relationship between *Ms* temperature and *e*/*a* is summarized and presented in **Figure 1**. Statistical analysis shows a relatively large standard deviation of 44.7 K and a maximum error of 140.2 K for the extreme case, which implies that the MT may also be very sensitive to the microstructure of the alloy, that is, internal stress and the degree of the atomic order [14].


**Table 1.** Classification of Ni-Mn-Ga alloys according to *Ms* and enthalpy (Δ*H*) [4, 17].

**Figure 1.** The relationship between martensite transformation temperatures and valence electron concentration of Ni-Mn-Ga alloy [29].

Based on **Figure 1**, the effects of *e*/*a* on *Ms* temperature can be formulated by the linear regression, as listed in Eq. (4), indicating that the MT temperature increases with increasing *e*/*a*:

$$\mathcal{M}\_s = 702.5 \text{(e.o.g)} - 5067 \tag{4}$$

Besides, *Tc* varies less with composition variation (Mn, 20–35 at.%; Ga, 16–17 at.%) than *Ms* for Ni-Mn-Ga alloys [17, 29]. The effect of composition on the *Tc* of Ni-Mn-Ga alloy mainly relies on the Mn-Mn distance since the ferromagnetism mainly depends on the Mn-Mn atomic interaction [33]. The stoichiometric Ni2 MnGa alloy possesses the strongest ferromagnetic interaction between the neighboring Mn-Mn atoms with *Tc* ~ 376 K. Either with the concentration or expansion of the unit cell, the Mn-Mn nearest distance may be decreased; thus, the ferromagnetic interaction between neighboring Mn-Mn atoms can be weakened, which gives rise to a slight decrease of *Tc* [34].

Above all, the tunable transformation temperatures lead to multifunctional properties in Ni-Mn-Ga alloys, which will be demonstrated in details in Section 5.

#### **2.2. Phase and structure of the martensite**

Ni-Mn-Ga alloys exhibit a chemical ordering transition (a kind of second-order phase transition) from partially ordered high-temperature B2 phase to chemical ordered L2<sup>1</sup> phase during cooling. In the L2<sup>1</sup> phase, the Ni, Mn, and Ga atoms occupy the specific sites of the crystal lattice, as shown in **Figure 2**.

The L2<sup>1</sup> phase has a cubic face-centered lattice. On the other hand, Ni2 MnGa alloy displays martensite transformation (a kind of first-order phase transition). The martensite phase may show various stacking sequences, that is, a modulate structure, which thus creates martensite structure, such as five-layer modulate (5M) martensite phase (**Figure 3**), seven-layer modulate martensite phase (7M, **Figure 4**), and non-modulate (NM) martensite phase (**Figure 5**). The kind of modulation mainly depends on the composition of the alloy. These different martensite structures exhibit different twinning stresses, which has a large effect on the magneticfield-induced strain and will be summarized in detail in Section 5.1.2.

**Figure 2.** L2<sup>1</sup> crystal structure of the Ni<sup>2</sup> MnGa alloys.

#### **3. Synthesis of ferromagnetic Ni-Mn-Ga foams and microwires**

#### **3.1. Ni-Mn-Ga foams**

Based on **Figure 1**, the effects of *e*/*a* on *Ms*

6 Shape Memory Alloys - Fundamentals and Applications

interaction [33]. The stoichiometric Ni2

rise to a slight decrease of *Tc*

Ni-Mn-Ga alloys [17, 29]. The effect of composition on the *Tc*

interaction between the neighboring Mn-Mn atoms with *Tc*

[34].

Besides, *Tc*

Ni-Mn-Ga alloy [29].

temperature can be formulated by the linear regres-

MnGa alloy possesses the strongest ferromagnetic

of Ni-Mn-Ga alloy mainly relies

~ 376 K. Either with the concen-

for

sion, as listed in Eq. (4), indicating that the MT temperature increases with increasing *e*/*a*:

= 702.5(/) − 5067 (4)

**Figure 1.** The relationship between martensite transformation temperatures and valence electron concentration of

on the Mn-Mn distance since the ferromagnetism mainly depends on the Mn-Mn atomic

tration or expansion of the unit cell, the Mn-Mn nearest distance may be decreased; thus, the ferromagnetic interaction between neighboring Mn-Mn atoms can be weakened, which gives

varies less with composition variation (Mn, 20–35 at.%; Ga, 16–17 at.%) than *Ms*

Ferromagnetic Ni-Mn-Ga foams can produce large magnetic-field-induced strain of ~2.0–8.7% due to the reduction of constraints imposed by grain boundaries and the formation of bamboo


**Figure 3.** (a) 5M crystal structure of the Ni<sup>2</sup> MnGa alloys and (b) simulated diffraction pattern of 5M martensite along crystal zone axis of [0 1 0].


**Figure 4.** (a) 7M crystal structure of the Ni<sup>2</sup> MnGa alloys and (b) simulated diffraction pattern of 7M martensite along crystal zone axis of [0 1 0].

**Figure 5.** (a) NM crystal structure of the Ni<sup>2</sup> MnGa alloys and simulated diffraction pattern of NM martensite along crystal zone axis of (b) [0 1 0] and (c) [0 0 1].

grains in the struts [10]. In this section, the synthesis of the Ni-Mn-Ga foams with single-pore and dual-pore distribution is demonstrated in detail [9, 35].

The casting replication method, that is, using liquid metal infiltration of a preform of ceramic space holder powder [36, 37], was used to prepare the Ni-Mn-Ga foams. Sodium aluminate (NaAlO2 ) was used as space holder in this case due to its high melting temperature ~1650°C, excellent chemical stability with molten metals, and good solubility in acid. In order to create the single-pore and dual-pore distribution, different sizes of NaAlO2 powders were prepared as follows: (1) purchased NaAlO2 powders was cold pressed at 125 MPa and sintered at 1500°C for 3 h in the air; (2) the sintered body was broken up with a mortar and pestle into powders; and (3) the resulting powder was sieved into different size ranges: R1 for singlepore foam and R2 (coarse) and R3 (fine) for dual-pore foam. The specific size of the powder was tuned due to the requirement of the foam porosity.

For single-pore specimen, the R1 powder was directly poured into an alumina crucible and slightly trapped to the designed height. For dual-pore specimen, the coarse and fine powders were poured alternatively (layer by layer) in the crucible filled with acetone: the coarse powder was first poured in a small batch, followed by a small batch of fine powder to settle in the space between the coarse ones of the previous batch. After that, the crucible was heated to evaporate the acetone. Both crucibles were heated at 1500°C for 3 h in the air to create necks between powders for the infiltration of the Ni-Mn-Ga liquid metal and the formation of open pores.

For infiltration process, the ingot was placed on top of the sintered powder preform and then heated to melt the Ni-Mn-Ga alloy under vacuum in the furnace. After melting the alloy, high-purity Ar gas was introduced in the furnace at a pressure of 1.34 atm. to squeeze the molten alloy into the preform, and the temperature was then cooled to room temperature at 7°C/min.

The removal of the NaAlO2 is critical for the preparation of the Ni-Mn-Ga foams. For the single-pore foam, a 10% HCl solution was used. Ten percent of HCl solution not only solve the NaAlO2 but also solve the Ni-Mn-Ga alloy, thus, thinned the struts. In this case, for the dual-pore foam, the thin struts around the fine powders would be dissolved after long exposure to the 10% HCl solution. Thus, a two-step method was applied: (1) removal of the coarse powders without alloy dissolution and (2) removal of the remaining fine powders as well as some thinning of the alloy. The acid for the first step should dissolve NaAlO2 but not the alloy. The acid for the second step should rapidly dissolve NaAlO2 while slowly dissolving the alloy only to open small fenestrations between the fine NaAlO2 powders. The mass loss of the bulk nonporous Ni-Mn-Ga alloy vs. time plots is shown in **Figure 6a**, which are linear for the acids of 10% HCl, 20% HCl, and 34% H2 SO<sup>4</sup> . The slopes corresponding to dissolution rates are 16 and 5 mg/m<sup>2</sup> min in 10% HCl and 34% H2 SO<sup>4</sup> , respectively, while triple the rate when doubled the concentration to 20% HCl.

As a result, 34% H2 SO<sup>4</sup> and 10% HCl were selected for the two steps, respectively. **Figure 6b** plots the mass loss vs. time for a foam sample with dual-size NaAlO2 powder immersed in 34% H2 SO<sup>4</sup> . After 2000 min, only 87% of the NaAlO2 was removed from the sample. Thereafter, a similar foam sample was firstly immersed in a 34% H2 SO<sup>4</sup> solution for 645 min (corresponding

grains in the struts [10]. In this section, the synthesis of the Ni-Mn-Ga foams with single-pore

The casting replication method, that is, using liquid metal infiltration of a preform of ceramic space holder powder [36, 37], was used to prepare the Ni-Mn-Ga foams. Sodium aluminate

excellent chemical stability with molten metals, and good solubility in acid. In order to cre-

at 1500°C for 3 h in the air; (2) the sintered body was broken up with a mortar and pestle into powders; and (3) the resulting powder was sieved into different size ranges: R1 for singlepore foam and R2 (coarse) and R3 (fine) for dual-pore foam. The specific size of the powder

) was used as space holder in this case due to its high melting temperature ~1650°C,

powders were pre-

powders was cold pressed at 125 MPa and sintered

MnGa alloys and simulated diffraction pattern of NM martensite along

MnGa alloys and (b) simulated diffraction pattern of 5M martensite along

MnGa alloys and (b) simulated diffraction pattern of 7M martensite along

and dual-pore distribution is demonstrated in detail [9, 35].

was tuned due to the requirement of the foam porosity.

pared as follows: (1) purchased NaAlO2

**Figure 5.** (a) NM crystal structure of the Ni<sup>2</sup>

crystal zone axis of (b) [0 1 0] and (c) [0 0 1].

**Figure 3.** (a) 5M crystal structure of the Ni<sup>2</sup>

8 Shape Memory Alloys - Fundamentals and Applications

**Figure 4.** (a) 7M crystal structure of the Ni<sup>2</sup>

crystal zone axis of [0 1 0].

crystal zone axis of [0 1 0].

ate the single-pore and dual-pore distribution, different sizes of NaAlO2

(NaAlO2

**Figure 6.** (a) Plot of mass loss vs. time in bulk Ni-Mn-Ga alloy immersed in 10% HCl, 20% HCl, and 34% H2 SO<sup>4</sup> ; (b) plot of mass loss vs. time for foam with dual-size NaAlO2 powders in 34% H2 SO<sup>4</sup> ; and (c) plot of foam porosity vs. time for foam with dual-size NaAlO2 powders immersed in 10% HCl after 645 min in 34% H2 SO<sup>4</sup> [35].

to the rapid dissolution stage in **Figure 6b**) and then transferred to the 10% HCl solution. A porosity of 55% was measured after 1140 min in 10% HCl, which was higher than the original NaAlO2 fraction of 45%, indicating the full removal of the NaAlO2 and partial dissolve of the alloy, as shown in **Figure 6c**. Further immersion increased the porosity due to the dissolve of the alloy only.

The morphologies of the fabricated single-pore and dual-pore foams are presented in **Figures 7** and **8**. In foams containing dual pores (**Figure 7b**), the alloy between large pores, which is solid in the single-pore foams (**Figure 7a**), contains small pores, thus producing many small nodes and struts. The three-dimensional architecture of both foams can be seen more vivid in **Figure 8**.

#### **3.2. Ni-Mn-Ga microwires**

Metallic microwires can be produced by a variety of methods, that is, melt extraction, Taylor method, melt-spinning, in-rotating water spinning, etc. Details of the fabrication method regarding the fabrication of ferromagnetic wires have been demonstrated by Peng et al. [38]. In recent years, the Taylor-Ulitovsky method was widely used for many metals and alloys. In this process, a metallic ingot is put in a glass tube and melted by induction heating. The glass tube was chosen to have relatively higher melting point than the ingot. Then, the glass tube is softened due to its contact with the molten metal, and it can be drawn. Recently, this method has been modified and applied to create glass-coated Ni-Mn-Ga microwires with equiaxed cross section, as shown in **Figure 9** [39, 40].

In the meantime, melt extraction has been established as a cost-effective and highly efficient method for the production of intrinsically brittle Ni-Mn-Ga alloys on a large scale [13]. The rapid solidification rate during melt extraction is suitable for obtaining refined microstructure, extended elemental solubility, and reduced elemental segregation [41].

A schematic melt extraction facility is displayed in **Figure 10a**. The microwire preparation process consists of the following steps: (1) the Ni-Mn-Ga ingot was inserted into a ceramic

**Figure 7.** Optical micrographs of polished cross section of Ni-Mn-Ga foams. Metal struts (S) and nodes (N) appear bright, and the pores appear dark. Foams with (a) single and (b) dual-pore size distribution are presented [35].

**Figure 8.** SEM micrographs of cut and etched surface of Ni-Mn-Ga foams showing three-dimensional structure and connectivity of pores. (a) Single- and (b) dual-pore foams are presented [35].

**Figure 9.** SEM image of representative Ni-Mn-Ga microwires created by the Taylor method [40].

to the rapid dissolution stage in **Figure 6b**) and then transferred to the 10% HCl solution. A porosity of 55% was measured after 1140 min in 10% HCl, which was higher than the original

alloy, as shown in **Figure 6c**. Further immersion increased the porosity due to the dissolve of

The morphologies of the fabricated single-pore and dual-pore foams are presented in **Figures 7** and **8**. In foams containing dual pores (**Figure 7b**), the alloy between large pores, which is solid in the single-pore foams (**Figure 7a**), contains small pores, thus producing many small nodes and struts. The three-dimensional architecture of both foams can be seen more vivid

Metallic microwires can be produced by a variety of methods, that is, melt extraction, Taylor method, melt-spinning, in-rotating water spinning, etc. Details of the fabrication method regarding the fabrication of ferromagnetic wires have been demonstrated by Peng et al. [38]. In recent years, the Taylor-Ulitovsky method was widely used for many metals and alloys. In this process, a metallic ingot is put in a glass tube and melted by induction heating. The glass tube was chosen to have relatively higher melting point than the ingot. Then, the glass tube is softened due to its contact with the molten metal, and it can be drawn. Recently, this method has been modified and applied to create glass-coated Ni-Mn-Ga microwires with equiaxed

In the meantime, melt extraction has been established as a cost-effective and highly efficient method for the production of intrinsically brittle Ni-Mn-Ga alloys on a large scale [13]. The rapid solidification rate during melt extraction is suitable for obtaining refined microstruc-

A schematic melt extraction facility is displayed in **Figure 10a**. The microwire preparation process consists of the following steps: (1) the Ni-Mn-Ga ingot was inserted into a ceramic

**Figure 7.** Optical micrographs of polished cross section of Ni-Mn-Ga foams. Metal struts (S) and nodes (N) appear bright, and the pores appear dark. Foams with (a) single and (b) dual-pore size distribution are presented [35].

ture, extended elemental solubility, and reduced elemental segregation [41].

and partial dissolve of the

fraction of 45%, indicating the full removal of the NaAlO2

NaAlO2

the alloy only.

in **Figure 8**.

**3.2. Ni-Mn-Ga microwires**

10 Shape Memory Alloys - Fundamentals and Applications

cross section, as shown in **Figure 9** [39, 40].

**Figure 10.** Scheme of the melt extraction process and the obtained Ni-Mn-Ga microwires. (a) Schematic illustration of the melt extraction setup, (b) macroscopic morphology, (c) SEM image, and (d) diameter distribution [13, 18].

crucible and placed into a chamber, (2) the chamber was evacuated to 10−3 Pa and then filled with 50 Pa Ar gas, (3) the top part of the ingot was induction heated, thus forming a melting pool in the crucible, (4) the molten phase was driven at a feed rate (*Vm*) of 40–120 μm/s toward a rotating wheel with wheel rotation velocity (*Vw*) ranging from 13 to 25 m/s, and (5) the molten alloy was extracted out by the wheel [9]. Typical macroscopic and SEM morphologies of melt-extracted Ni-Mn-Ga microwires are presented in **Figure 10b** and **c**. Microwires with fairly uniform diameter ranging from 45 to 65 μm (**Figure 10d**) were prepared on a large scale with optimized processing parameters: wheel velocity of 23 m/s, feed rate of 60–90 μm/s, and heating power of 20 kW [13].

The microstructural evolution during the melt extraction process, that is, the nucleation behavior of the molten alloy, the unique grain growth behavior, the grain distribution, and the texture, has been systematically studied [13]. As shown in **Figure 11**, microwires with singular nucleation (earlier stage) and dual nucleation (later stage) sites were found at different stages of the preparation: at the beginning of the melt extraction process, the wheel tip temperature is low, and crystallization of the molten alloy nucleates at the wheel tip. Subsequently, the molten alloy nucleates at the two sides of the wheel tip when the wheel tip temperature increased.

Unique grain growth behavior was observed along the radial direction, creating columnar grains growing along the crystal <0 0 1> direction with a fanlike texture in the cross section of the microwire, as shown in **Figure 11**. Finally, at the later stage of the crystallization, the texture evolved into <0 0 1> crystal orientation perpendicular to the flattened surface of the microwire, as shown in **Figure 12**. On this occasion, assuming 5M martensite was formed in the microwires, only variants with *a*-axis [1 0 0] or *c*-axis [0 0 1] perpendicular to the flattened part could exist. The reduced number of twin variants may reduce the incompatibility and thus favor the MFIS.

**Figure 11.** SEM images (a, c, e, g) and EBSD orientation maps (b, d, f, h) of the cross section (a–d) and the longitudinal section (e–h) of the melt-extracted Ni-Mn-Ga microwires [13].

**Figure 12.** (a) SEM image, (b) EBSD orientation map, and (c) discrete inverse pole figure with respect to (b) of the meltextracted Ni-Mn-Ga microwires at later stage [13].

#### **4. Martensite transformation of ferromagnetic shape memory alloys**

#### **4.1. First-order martensite transformation**

crucible and placed into a chamber, (2) the chamber was evacuated to 10−3 Pa and then filled with 50 Pa Ar gas, (3) the top part of the ingot was induction heated, thus forming a melting pool in the crucible, (4) the molten phase was driven at a feed rate (*Vm*) of 40–120 μm/s toward a rotating wheel with wheel rotation velocity (*Vw*) ranging from 13 to 25 m/s, and (5) the molten alloy was extracted out by the wheel [9]. Typical macroscopic and SEM morphologies of melt-extracted Ni-Mn-Ga microwires are presented in **Figure 10b** and **c**. Microwires with fairly uniform diameter ranging from 45 to 65 μm (**Figure 10d**) were prepared on a large scale with optimized processing parameters: wheel velocity of 23 m/s, feed rate of 60–90 μm/s, and

The microstructural evolution during the melt extraction process, that is, the nucleation behavior of the molten alloy, the unique grain growth behavior, the grain distribution, and the texture, has been systematically studied [13]. As shown in **Figure 11**, microwires with singular nucleation (earlier stage) and dual nucleation (later stage) sites were found at different stages of the preparation: at the beginning of the melt extraction process, the wheel tip temperature is low, and crystallization of the molten alloy nucleates at the wheel tip. Subsequently, the molten alloy nucleates at the two sides of the wheel tip when the wheel tip

Unique grain growth behavior was observed along the radial direction, creating columnar grains growing along the crystal <0 0 1> direction with a fanlike texture in the cross section of the microwire, as shown in **Figure 11**. Finally, at the later stage of the crystallization, the texture evolved into <0 0 1> crystal orientation perpendicular to the flattened surface of the microwire, as shown in **Figure 12**. On this occasion, assuming 5M martensite was formed in the microwires, only variants with *a*-axis [1 0 0] or *c*-axis [0 0 1] perpendicular to the flattened part could exist. The reduced number of twin variants may reduce the incompatibility and

**Figure 11.** SEM images (a, c, e, g) and EBSD orientation maps (b, d, f, h) of the cross section (a–d) and the longitudinal

section (e–h) of the melt-extracted Ni-Mn-Ga microwires [13].

heating power of 20 kW [13].

12 Shape Memory Alloys - Fundamentals and Applications

temperature increased.

thus favor the MFIS.

#### *4.1.1. Conventional martensite trasnformation*

Upon cooling, like in steel, ferromagnetic shape memory alloy undergoes a diffusionless phase transformation from the austenite state to the martensite state, named martensite transformation (MT). Owing to the diffusionless character of the transformation, a slight change of the position of the atoms in cubic austenite phase during cooling leads to a tetragonal distortion of the unit cell.

As was mentioned in Section 2.2, Heusler alloys possess two important phases: the cubic austenite phase and the martensite phase (with different lattice structures). The MT process can be illustrated based on **Figure 13** [17]: upon cooling from an austenite state, the sample starts to form martensite at the martensite start temperature, *Ms* . The sample is fully transformed to martensite state below the martensite finish temperature, *Mf* . During the reverse transformation when the sample heats from a fully martensite state, the austenite starts to form above the austenite start temperature, *As* , and is completely transformed to austenite above the austenite finish temperature, *Af* .

As shown in the DSC curve of **Figure 13** (upper), the transition from the austenite to martensite is an exothermic reaction, which means that heat is released for this process and the enthalpy change of the system is negative. On the other hand, the reverse transformation from martensite to austenite is an endothermic reaction with a positive enthalpy change. Besides, during DSC measurements, the temperatures at maximum endothermic and exothermic heat flow are defined as *Ap* and *Mp* , respectively. While as shown in the magnetization-temperature (*M*-*T*) curves of **Figure 13** (lower), the *Ap* and *Mp* are derived from the peak values of the first derivative plot of the curves during heating and cooling, respectively (shown in the inset). Martensite transformation is a first-order phase transformation (FOT). A FOT is usually accompanied by an inevitable transformation hysteresis, which can be interpreted by |*Ap* − *Mp* | or (*As* − *Ms* + *Af* − *Mf* )/2.

**Figure 13.** Heating and cooling of DSC curves (upper) and *M-T* curves at low magnetic field of ferromagnetic shape memory alloys [17].

The MT temperatures are very sensitive to the composition for the alloys. For Ni-Mn-Ga and other Ni-Mn-X alloys, the composition dependence of the MT temperatures has been demonstrated in Section 2.1 of the present chapter.

#### *4.1.2. Premartensite transformation*

Premartensite transformation (PMT), a weak FOT of the austenite into a micromodulated premartensite phase prior to the martensite transformation itself, has been observed in Ni-Mn-Ga alloys with *Tc* well above the MT temperature, that is, mainly in Ni-Mn-Ga alloys from Group I (**Table 1**) with near-stoichiometric composition [31, 42]. A PMT at ~260 K has been found in Ni<sup>2</sup> MnGa alloys prior to its MT [43], with the austenite phase transformed to an orthogonal three-layer modulated structure, which is similar as the 5M and 7M modulated structure [44].

Existence of unusual physical property change has been found around the PMT temperature during cooling. Firstly, before the PMT, the samples started to exhibit a drastic increase of elastic modulus [32]. Furthermore, the internal friction is increasing from zero to a maximum in the premartensite phase and decreases again in martensite [31]. This sudden soft mode phonon freezing can only be observed in the premartensite phase before the MT; thus, this transformation is defined as the PMT.

#### *4.1.3. Intermartensite transformation*

Intermartensite transformation (IMT), a structural transformation from one martensite to another, has been found in high-temperature Ni-Mn-Ga alloys, that is, mainly in Ni-Mn-Ga alloys from Group III (**Table 1**). Upon cooling in these samples, the austenite phase changes to a 5M then 7M and then non-modulated phase or to 7M and then non-modulated phase or directly from austenite to non-modulated phase [32]. While upon heating, only a phase transformation from the non-modulated martensite to austenite was observed [45]. On the other hand, the IMT can also be induced with increasing stress following the same transformation routes as is induced with decreasing temperature [32].

#### **4.2. Second-order magnetic transformation**

The increased distance between the Mn atoms in the Ni-Mn-Ga L2<sup>1</sup> structure changes the Mn-Mn exchange interaction from antiferromagnetic of pure Mn to ferromagnetic [33]. Therefore, a magnetic transformation is taking place at the Curie temperature (*Tc* ) from a ferromagnetic phase to a paramagnetic phase upon heating. The magnetic transformation is a second-order transformation (SOT) with no latent heat associated with the phase transformation and can be easily detected from the M-T curves with a drastic falling of the magnetization. The *Tc* of the Heusler alloys is also influenced by the composition and, thus, can be tuned. In our previous work, the MT temperature was increased, and the *Tc* was lowered after Cu doping in Ni-Mn-Ga alloy, leading to an overlap of the martensite and magnetic transformation, that is, magneto-structural coupling, in the microwire. The magneto-structural coupling enhanced the MCE of the alloy [17].

#### **5. Properties and application of ferromagnetic shape memory alloys**

#### **5.1. Magnetic-field-induced strain (MFIS)**

#### *5.1.1. Overview of MFIS*

The MT temperatures are very sensitive to the composition for the alloys. For Ni-Mn-Ga and other Ni-Mn-X alloys, the composition dependence of the MT temperatures has been demon-

**Figure 13.** Heating and cooling of DSC curves (upper) and *M-T* curves at low magnetic field of ferromagnetic shape

Premartensite transformation (PMT), a weak FOT of the austenite into a micromodulated premartensite phase prior to the martensite transformation itself, has been observed in

from Group I (**Table 1**) with near-stoichiometric composition [31, 42]. A PMT at ~260 K has

an orthogonal three-layer modulated structure, which is similar as the 5M and 7M modulated

Existence of unusual physical property change has been found around the PMT temperature during cooling. Firstly, before the PMT, the samples started to exhibit a drastic increase of elastic modulus [32]. Furthermore, the internal friction is increasing from zero to a maximum in the premartensite phase and decreases again in martensite [31]. This sudden soft mode phonon freezing can only be observed in the premartensite phase before the MT; thus, this

Intermartensite transformation (IMT), a structural transformation from one martensite to another, has been found in high-temperature Ni-Mn-Ga alloys, that is, mainly in Ni-Mn-Ga alloys from Group III (**Table 1**). Upon cooling in these samples, the austenite phase changes to a 5M then 7M and then non-modulated phase or to 7M and then non-modulated phase or directly from austenite to non-modulated phase [32]. While upon heating, only a phase

well above the MT temperature, that is, mainly in Ni-Mn-Ga alloys

MnGa alloys prior to its MT [43], with the austenite phase transformed to

strated in Section 2.1 of the present chapter.

14 Shape Memory Alloys - Fundamentals and Applications

*4.1.2. Premartensite transformation*

transformation is defined as the PMT.

*4.1.3. Intermartensite transformation*

Ni-Mn-Ga alloys with *Tc*

been found in Ni<sup>2</sup>

memory alloys [17].

structure [44].

Magnetic-field-induced strain (MFIS) comes from twin boundary motion under the application of a magnetic field, which is driven by the magnetostress produced by high magnetocrystalline anisotropy of the Ni-Mn-Ga alloy and its low twining stress [35], as schematically illustrated in **Figure 14** [46]. The MFIS property is useful for actuation and sensing purposes [47]. MFIS was firstly reported in 1996 by Ullakko et al. [5]. A strain of 0.19% under a magnetic field of 0.43 T was obtained from a Ni2 MnGa sample at 265 K. From then on, the interest of the MFIS in FSMAs grew rapidly all over the world. Besides, many different compositions, such as Ni-Fe-Ga [48], Ni-Mn-In [49], Co-Ni-Al [50], Co [51], Fe [52], and rare earth [53]-doped Ni-Mn-Ga alloys, have also been investigated.

So far, large MFIS (~1–10%) has been achieved only for Ni-Mn-Ga single crystals [6, 7]. The most representative one was reported in 2000, when Murray et al. [6] achieved a 6% MFIS in a 5M single-crystalline Ni-Mn-Ga alloy at room temperature under a magnetic field of 0.62 T. During the experiment, different stresses were applied to restore the MFIS and to measure the magnetostress. After that, in 2002, Sozinov et al. [54] published a giant MFIS of about 9.5% in 7M Ni-Mn-Ga single crystals at ambient temperature in a magnetic field of less than 1 T. However, the fabrication of single crystals is difficult because of sever segregation, low growth speed, and high cost. On the other hand, polycrystalline Ni-Mn-Ga alloys may be produced with much lower cost, but their MFIS is vanishingly small (<0.01%) because of the low mobility of twin boundaries constrained by grain boundaries [55, 56]. Many works have been

**Figure 14.** Schematic illustration of rotation of magnetic moments and twin boundary motion in tetragonal FSMAs under magnetic fields [46].

carried out to enhance their MFIS up to 1% by introducing strong textures and subsequent training in coarse-grained polycrystalline Ni-Mn-Ga alloys [57–60]. Recently, considering the hindering effect of the grain boundaries on the twin boundary motion, approaches regarding reduction of grain boundaries, that is, by producing a porous material (foam) or reducing sample size, have been carried out, which will be discussed in detail in Sections 5.1.2 and 5.1.3.

#### *5.1.2. MFIS in Ni-Mn-Ga foams*

As mentioned earlier, fine-grained polycrystalline Ni-Mn-Ga alloys are easier to fabricate but with vanishingly small strain due to the constraints provided by the grain boundaries. Introducing porosity in Ni-Mn-Ga alloys not only reduces the constraints imposed by grain boundaries but also maintains the ease of processing associated with casting polycrystalline Ni-Mn-Ga. After certain grain growth heat treatment, the twins can span between the pores, as shown in **Figure 15** [10]. As a result, the twin boundaries can move as freely as in singlecrystalline bulk material within the grains.

Research work regarding the MFIS in Ni-Mn-Ga foams was firstly reported by the research group of Müllner et al. [9]. With 76% open porosity, the foam displayed a fully reversible MFIS as large as 0.12% with excellent stability over 25 million magnetomechanical cycles. Thereafter, the same group further increased the MFIS to 2.0–8.7% in dual-pore Ni-Mn-Ga foams (**Figures 7b** and **8b**) with 62% porosity after thermo-magneto-mechanical cycling training [10]. These obtained strains are much larger than those of any polycrystalline and comparable to those of single crystals. Different from bulk single or polycrystalline alloy, these open-porosity foams allow fluid flow, making them potentially useful as micro-pumps and magnetocaloric materials [10].

#### *5.1.3. MFIS in Ni-Mn-Ga microwires*

carried out to enhance their MFIS up to 1% by introducing strong textures and subsequent training in coarse-grained polycrystalline Ni-Mn-Ga alloys [57–60]. Recently, considering the hindering effect of the grain boundaries on the twin boundary motion, approaches regarding reduction of grain boundaries, that is, by producing a porous material (foam) or reducing sample size, have been carried out, which will be discussed in detail in Sections 5.1.2 and 5.1.3.

**Figure 14.** Schematic illustration of rotation of magnetic moments and twin boundary motion in tetragonal FSMAs

As mentioned earlier, fine-grained polycrystalline Ni-Mn-Ga alloys are easier to fabricate but with vanishingly small strain due to the constraints provided by the grain boundaries. Introducing porosity in Ni-Mn-Ga alloys not only reduces the constraints imposed by grain boundaries but also maintains the ease of processing associated with casting polycrystalline Ni-Mn-Ga. After certain grain growth heat treatment, the twins can span between the pores, as shown in **Figure 15** [10]. As a result, the twin boundaries can move as freely as in single-

Research work regarding the MFIS in Ni-Mn-Ga foams was firstly reported by the research group of Müllner et al. [9]. With 76% open porosity, the foam displayed a fully reversible MFIS as large as 0.12% with excellent stability over 25 million magnetomechanical cycles. Thereafter, the same group further increased the MFIS to 2.0–8.7% in dual-pore Ni-Mn-Ga foams (**Figures 7b** and **8b**) with 62% porosity after thermo-magneto-mechanical cycling training [10]. These obtained strains are much larger than those of any polycrystalline and comparable to those of single crystals. Different from bulk single or polycrystalline alloy, these open-porosity foams allow fluid flow, making them potentially useful as micro-pumps and

*5.1.2. MFIS in Ni-Mn-Ga foams*

under magnetic fields [46].

16 Shape Memory Alloys - Fundamentals and Applications

magnetocaloric materials [10].

crystalline bulk material within the grains.

Constraint of twin boundary caused by grain boundary is three dimensional in bulk alloys and two dimensional in thin films. It can be further reduced to one dimensional in microwires. Furthermore, by reducing sample size, small-sized Ni-Mn-Ga alloys show low inertia [61], low eddy current loss at high frequency [62], high magnetocrystalline anisotropy, and work output [63]. Recently, oligocrystalline microwires with bamboo or near-bamboo structures have attracted much interests owning to a less constrained environment and more free surface [64–66], which makes them the closest approximation to single crystals. The micrographs of Ni-Mn-Ga and Cu-Al-Ni oligocrystalline microwires are presented in **Figures 16** and **17**, respectively. Martensite plates spanning across the wire diameter can be observed [65, 66].

However, because of the lack of textures, only a subset of grains was prone to favorably oriented to show a detectable MFIS. Approximately 1% MFIS was found in a not-constrained and randomly textured Ni-Mn-Ga without bias stress by magnetizing the microwire parallel and perpendicular to the wire axis up to 2 T [11]. Recently, a 1 T rotating magnetic field caused the Ni-Mn-Ga microwire to bend to a curvature corresponding to a surface strain of 1.5% [40]. Mechanical or the combined thermo-magneto-mechanical training in martensite state may effectively lower the twining stress, to form preferential oriented variants [9, 58], and, thus, may in favor of achieving a high MFIS in oligocrystalline microwires. The realization of large MFIS in one-dimensional microwires may provide a wide prospect in the application of micro-actuators and sensors.

**Figure 16.** SEM micrographs of (a) free surface and (b) cross section of Ni-Mn-Ga oligocrystalline microwires prepared by melt extraction and subsequent heat treatment [66].

**Figure 17.** (a–c) Micrographs of Cu-Al-Ni oligocrystalline microwires prepared by Taylor method and subsequent heat treatment. (d and e) Montaged optical micrographs of the longitudinal section of the wire [65].

#### **5.2. Magnetocaloric effect (MCE)**

#### *5.2.1. Overview of MCE*

Magnetic refrigeration, based on the magnetocaloric effect (MCE), has attracted interests as a potential alternative to well-established compression-evaporation technique for room temperature refrigeration because of the compactness, high efficiency, and environmental friendship. To characterize a MCE, the adiabatic temperature change (Δ*Tad*), the magnetic entropy change (Δ*SM*), and the relative cooling power (*RCP*) should be measured. The most recently researched ambient magnetic refrigeration materials (MCM) mainly include La(Fe,Si)13 based [67–70] and Gd5 (Si,Ge)<sup>4</sup> based [67, 71] alloys (contain rare-earth elements) and MnAs based [67, 72–74], MnFe(P,X) (X = As, Ge, Si) [75, 76], Fe-Rh [77], NiMn based [78–82] alloys (rare-earth free). These types of materials undergo a first-order magnetic phase transition (FOMT), exhibit large hystereses, and show a large value of Δ*SM*. For example, with a magnetic field change Δ*H* = 5 T, Δ*SM* = 35–50 J/kg K for MnAs-based [73], Gd5 (Si,Ge)<sup>4</sup> -based [67], and La(Fe,Si)13-based [83] alloys and Δ*SM* = 35–40 J/kg K for Ni-Mn-In-Co [84] and Ni-Mn-Sn [85] alloys were obtained. For some MCE, the low working temperature span (Δ*TFWHM*) and high thermal and magnetic hysteresis loss [86, 87] limit their applications. MCM with reduced dimensions (particle, microwire, film, or foam) [88, 89] have been proposed to broaden the Δ*TFWHM* by preparing the alloy with gradient composition distribution state. In addition, deduction in dimensions is an effect way to decrease the hysteresis loss by reducing internal stress and the constraints between grains attributed to the high surface-area-to-volume ratio [19].

#### *5.2.2. MCE in Ni-Mn-based alloys*

**5.2. Magnetocaloric effect (MCE)**

by melt extraction and subsequent heat treatment [66].

18 Shape Memory Alloys - Fundamentals and Applications

Magnetic refrigeration, based on the magnetocaloric effect (MCE), has attracted interests as a potential alternative to well-established compression-evaporation technique for room temperature refrigeration because of the compactness, high efficiency, and environmental friendship. To characterize a MCE, the adiabatic temperature change (Δ*Tad*), the magnetic entropy change (Δ*SM*), and the relative cooling power (*RCP*) should be measured. The most recently researched ambient magnetic refrigeration materials (MCM) mainly include La(Fe,Si)13 based [67–70] and

**Figure 17.** (a–c) Micrographs of Cu-Al-Ni oligocrystalline microwires prepared by Taylor method and subsequent heat

treatment. (d and e) Montaged optical micrographs of the longitudinal section of the wire [65].

**Figure 16.** SEM micrographs of (a) free surface and (b) cross section of Ni-Mn-Ga oligocrystalline microwires prepared

based [67, 71] alloys (contain rare-earth elements) and MnAs based [67, 72–74],

*5.2.1. Overview of MCE*

Gd5

(Si,Ge)<sup>4</sup>

Very recently, we reported magnetocaloric effects of high-content Fe-doped Ni44.9Fe4.3Mn38.3Sn12.5 polycrystalline microwires prepared by a melt extraction technique [18], as displayed in **Figure 18**. Under a magnetic field of 20 kOe, the Δ*Sm* peak value, related to the first-order martensite transformation (FOMT) of the present Ni44.9Fe4.3Mn38.3Sn12.5 microwires, reaches 3.0 J/kg K, which is comparable to the Ni44Fe6 Mn40Sn10 ribbons (1.8 J/kg K under 20 kOe) [90] and Ni51.6Mn32.9Sn15.5 films (1.5 J/kg K under 10 kOe) [91]. The second-order martensite transformation (SOMT) related to Δ*Sm* of −3.7 J/kg K and Δ*TFWHM* of ~85 K under 5 T is comparable to that of Ni48.8Mn26.7Ga20.8Cu3.7 microwires (−8.3 J/kg K with Δ*TFWHM* of 13 K under 5 T) [19] and Ni53.5Mn23.8Ga22.7 films (−8.5 J/kg K with Δ*TFWHM* of 17 K under 6 T) [92]. The studied Ni44.9Fe4.3Mn38.3Sn12.5 microwires, exhibiting large surface-area-to-volume ratio, giant MCE property, and low cost, may act as potential magnetic refrigerants.

As shown in **Figure 19a**, the Δ*Sm* of Ni48Mn26Ga19.5Fe6.5 microwires [19] with a diameter of ~50 μm shifted from positive to negative to the applied field is higher than 0.5 T. These interesting positive-to-negative Δ*Sm* transition behaviors have also been found in Ni50.1Mn20.7Ga29.6 single crystal at 0.8 T [93] and Ni54.5Fe1 Mn20Ga24.5 polycrystalline alloy at 0.5 T [94]. The maximum negative Δ*Sm* is ~4.7 J/kg K under 5 T corresponding to magnetic-field-induced structural transition from martensite to austenite phase. At 20 kOe, Δ*Sm* in the microwires is 1.71 J/kg K, which is comparable to Ni-Fe-Mn-Ga alloy (2.1 J/kg K) [94]. Moreover, the value of Δ*Sm* observed at magnetic transition under 3 T is 2.05 J/kg K in Fe-doped Ni54.5Fe1 Mn20Ga24.5 microwires. Compared to the un-doped Ni50.95Mn25.45Ga23.6 microwires, the Δ*Sm* is about three times higher under the same magnetic field [95].

It can be seen from **Figure 19b** that the annealed Ni48.8Mn26.7Ga20.8Cu3.7 microwires [87] exhibited magneto-structural coupling and wide martensitic transformation temperature range, which contribute to a Δ*Sm* of 8.3 J/kg K with a wide Δ*TFWHM* of 13 K under a magnetic field of 5 T. The obtained RC in Ni-Mn-Ga-Cu microwire (78.0 J/kg) is comparable with those of Ni-Mn-based alloys (70–115 J/kg) [96] and superior to those of Ni-Mn-Ga-Cu bulk alloys (72–75 J/kg) [97–99]. On the other hand, when compared to Gd [100] or LaFe13−*<sup>x</sup>* Si*x* [101] alloys, the Ni-Mn-Ga-Cu microwires are rare-earth free and thus cost-effective, which helps for the practical applications.

**Figure 18.** The Δ*Sm* as a function of external magnetic field in the present Ni44.9Fe4.3Mn38.3Sn12.5 microwires and some other Ni-Mn-Z-based (Z = Sn, Sb, In, Ga) Heusler compounds in the forms of ribbons (R), thin films (F), and microwires (W) [18].

**Figure 19.** Entropy changes as a function of temperature for (a) Ni48Mn26Ga19.5Fe6.5 [19] and (b) Ni48.8Mn26.7Ga20.8Cu3.7 [17] microwires under different magnetic fields.

#### **5.3. Shape memory and superelasticity**

In Sections 5.1 and 5.2, Ni-Mn-Ga alloy has been demonstrated to exhibit high MFIS and excellent MCE properties both driven by external magnetic field. Actually, due to the thermoelastic MT, Ni-Mn-Ga alloys also show well-pronounced thermal field-induced superelasticity (SE, stress-induced MT at austenite state and recovered upon unloading) [45], one-way shape memory effect (OWSME, deformed under stress at martensitic state and recovered during heating), and two-way shape memory effect (TWSME, shape change between martensite and austenite states continuously upon heating and cooling) [8, 9], which is similar to traditional SMAs, such as Ni-Ti.

So far, OWSME strain up to ~6.1% [102] and SE ~6% [45] has been achieved in Ni-Mn-Ga single crystals under compression mode. TWSME induced by training has reached 9% in single-crystalline Ni-Mn-Ga alloy [8] under tension mode. On the other hand, related reports for SME and SE properties for bulk polycrystalline Ni-Mn-Ga alloys are relatively seldom and lower than those of single crystals [102, 103]. Besides, due to the intrinsic brittleness of the alloy, previous reports are mostly focused on the compression processes than on the tension mode.

#### *5.3.1. Shape memory effect and superelasticity in Ni-Mn-Ga microwires*

Due to the transcrystalline fracture tendency of polycrystalline Ni-Mn-Ga alloys, efforts have been made to enhance their ductility. Microstructure refinement and sample size reduction have been proven to lower the brittleness [6, 9]. Recently, small-sized Ni-Mn-Ga microwires have been made by rapid solidification methods involving Taylor method and melt extraction technique to investigate the SME and SE properties [12, 14, 39, 40]. A large reversible SE strain of 10.9% has been reported in polycrystalline Ni-Mn-Ga glass-coated wires tested in tension mode [39]. The Taylor method as applied to Ni-Mn-Ga microwire fabrication allows production of wires with a uniform circular cross section while with lower efficiency compared with the melt extraction technique. In this section, the SME and SE in melt-extracted microwires are displayed based on our previous work.

Ni-Mn-Ga microwires prepared by melt extraction technique exhibited a higher reversible SE strain and recoverable SME strain compared with bulk parent alloys [12]. **Figure 20** displays the typical recoverable SME curve of as-extracted Ni-Mn-Ga microwire, and A–C demonstrates the SME strain regarding twin boundary motion after unloading (~1.5%). Unlike single crystals, the twin boundary motion process did not show a stress plateau upon loading (A–B) due to the refinement of the grains.

Owing to the rapid solidification process during fabrication, high internal stresses, reduced degree of atomic order, and other defects may affect the SME and SE properties in the asextracted microwires. A stepwise chemical ordering annealing was carried out, and the effect of annealing on the SE behavior was investigated in our previous work [14]. SE comparison before and after annealing is shown in **Figure 21**. Annealing decreases the stress-induced MT (SIM) stress and the hysteresis and improves the reversibility during superelastic cycling.

Furthermore, polycrystalline Ni-Mn-Ga microwires exhibit higher-temperature dependences compared with Ni-Mn-Ga single crystals and conventional superelastic alloys (**Figure 22**), which are considered to be related to the small grains achieved by melt extraction. Besides, the temperature dependences of the microwires are lowered after annealing (inset II), that is, the slope (*dσAs*/*dT*) of the annealed microwire, 15.6 MPa/K, is smaller than that of as-extracted one, 17.5 MPa/K, revealing an easier reverse transformation process in the annealed microwire.

**5.3. Shape memory and superelasticity**

20 Shape Memory Alloys - Fundamentals and Applications

microwires under different magnetic fields.

[18].

tional SMAs, such as Ni-Ti.

In Sections 5.1 and 5.2, Ni-Mn-Ga alloy has been demonstrated to exhibit high MFIS and excellent MCE properties both driven by external magnetic field. Actually, due to the thermoelastic MT, Ni-Mn-Ga alloys also show well-pronounced thermal field-induced superelasticity (SE, stress-induced MT at austenite state and recovered upon unloading) [45], one-way shape memory effect (OWSME, deformed under stress at martensitic state and recovered during heating), and two-way shape memory effect (TWSME, shape change between martensite and austenite states continuously upon heating and cooling) [8, 9], which is similar to tradi-

**Figure 19.** Entropy changes as a function of temperature for (a) Ni48Mn26Ga19.5Fe6.5 [19] and (b) Ni48.8Mn26.7Ga20.8Cu3.7 [17]

**Figure 18.** The Δ*Sm* as a function of external magnetic field in the present Ni44.9Fe4.3Mn38.3Sn12.5 microwires and some other Ni-Mn-Z-based (Z = Sn, Sb, In, Ga) Heusler compounds in the forms of ribbons (R), thin films (F), and microwires (W)

So far, OWSME strain up to ~6.1% [102] and SE ~6% [45] has been achieved in Ni-Mn-Ga single crystals under compression mode. TWSME induced by training has reached 9% in Above all, melt-extracted Ni-Mn-Ga microwires after annealing exhibit lower SE stresses, lower-temperature dependences, and higher SE reversibility. Besides, the preparation is convenient and efficient. Given these properties, Ni-Mn-Ga microwire is expected to be used for practical applications, such as superelastic materials, micro-actuator materials, and microsensor materials in various fields.

**Figure 20.** Tensile stress-strain curves of as-extracted Ni-Mn-Ga microwires at martensite state.

**Figure 21.** Tensile stress-strain curves of (a) as-extracted and (b) annealed Ni-Mn-Ga microwires at temperature levels (*Ttest* − *Ms* ) approximately from 0 to 15°C at austenite state (equilibrate at 80°C and then cool down to *Ttest*). *ε*irr, *ε*se, and *ε*el stand for the irreversible strain (recovered upon heating), strain recovery upon reverse transformation, and elastic recovery, respectively [14].

#### *5.3.2. Shape memory effect and superelasticity in Ni-Mn-Ga-Fe microwires*

The ductility of Ni-Mn-Ga alloys can be enhanced by the fourth element doping [104–106]. Among various doping elements, Fe has attracted many attentions [107, 108]. In our previous work, SME and SE behaviors of Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* (*x* = 1–6%) microwires (diameter ~30–40 μm)

**Figure 22.** SIM critical stress (*σMs*) vs. temperature plots of the Ni-Mn-Ga microwires and other alloys [14].

were investigated [16]. OWSME and TWSME curves of the Ni49.7Mn25Ga19.8Fe5.5 microwires are shown in **Figure 23a** and **b**, respectively. An OWSME strain of ~1.0% was induced by stress at its martensite state under tension mode in the microwire and completely recovered upon heating to its austenite state (**Figure 23a**). With respect to the TWSME, the application in sensors is generally used under certain applied stress in thermal cycle. Different external stresses were applied during the thermal cycling; fully recoverable strain was obtained in the

**Figure 23.** Tensile stress-strain curves of Ni49.7Mn25Ga19.8Fe5.5 microwires. (a) OWSME and (b) TWSME [16].

*5.3.2. Shape memory effect and superelasticity in Ni-Mn-Ga-Fe microwires*

**Figure 20.** Tensile stress-strain curves of as-extracted Ni-Mn-Ga microwires at martensite state.

22 Shape Memory Alloys - Fundamentals and Applications

work, SME and SE behaviors of Ni50Mn25Ga25−*<sup>x</sup>*

(*Ttest* − *Ms*

recovery, respectively [14].

The ductility of Ni-Mn-Ga alloys can be enhanced by the fourth element doping [104–106]. Among various doping elements, Fe has attracted many attentions [107, 108]. In our previous

**Figure 21.** Tensile stress-strain curves of (a) as-extracted and (b) annealed Ni-Mn-Ga microwires at temperature levels

) approximately from 0 to 15°C at austenite state (equilibrate at 80°C and then cool down to *Ttest*). *ε*irr, *ε*se, and *ε*el stand for the irreversible strain (recovered upon heating), strain recovery upon reverse transformation, and elastic

Fe*<sup>x</sup>*

(*x* = 1–6%) microwires (diameter ~30–40 μm)

Ni49.7Mn25Ga19.8Fe5.5 microwire (**Figure 23b**). The TWSME strain of the microwire increased from 0.84% at 156 MPa to 1.504% at 468 MPa.

The SE behavior of the Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* (*x* = 4.5) was studied due to its favorable MT temperature near RT. The SE tensile stress-strain curves obtained at various temperatures in Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* (*x* = 4.5) are demonstrated in **Figure 24**.

The SE of Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* (*x* = 4.5) microwires shows similar behavior as that of Ni-Mn-Ga microwires [9]. The maximum strain recovery rates achieved in Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* (*x* = 4.5) are 94 and 90%, respectively, which was higher than that in as-extracted Ni-Mn-Ga microwires while lower than Ni-Mn-Ga microwires after chemical ordering annealing.

#### **5.4. Application of ferromagnetic shape memory alloys**

Ni-Mn-Ga alloys may act as actuator by making the use of the large MFIS and magnetic shape memory effect (MSM) or as sensor related to the dependence of the magnetization change under an external compressive stress. For MFIS/MSM, the working frequency is high because they are driven by external magnetic fields. This character is superior to the traditional shape memory alloys, such as Ni-Ti, which are driven by temperature change and thus only work at much smaller frequency. Some typical examples of the actuators created based on the ferromagnetic shape memory alloys are shown in **Figures 25**–**27**.

**Figure 24.** Tensile stress-strain curves obtained at various temperatures in Ni50Mn25Ga25−*<sup>x</sup>* Fe*<sup>x</sup>* microwires. (a) *x* = 4 and (b) *x* = 5 [15].

**Figure 25.** MSM-spring actuator (MAGNETOSHAPE® by ETO MAGNETIC GmbH) [109].

**Figure 25** demonstrates the most straightforward MSM-spring actuator: the elongation is obtained by a magnetic field perpendicular to the motion, while contraction is obtained thanks to the elastic force of a spring. The magnetic field induces a magnetostress in the MSM alloy. In the spring actuator, such a magnetic force must work always against the elastic force of the spring, which is bigger at bigger strain [109].

Ni49.7Mn25Ga19.8Fe5.5 microwire (**Figure 23b**). The TWSME strain of the microwire increased

perature near RT. The SE tensile stress-strain curves obtained at various temperatures in

94 and 90%, respectively, which was higher than that in as-extracted Ni-Mn-Ga microwires

Ni-Mn-Ga alloys may act as actuator by making the use of the large MFIS and magnetic shape memory effect (MSM) or as sensor related to the dependence of the magnetization change under an external compressive stress. For MFIS/MSM, the working frequency is high because they are driven by external magnetic fields. This character is superior to the traditional shape memory alloys, such as Ni-Ti, which are driven by temperature change and thus only work at much smaller frequency. Some typical examples of the actuators created based on the ferromagnetic

(*x* = 4.5) was studied due to its favorable MT tem-

Fe*<sup>x</sup>*

microwires. (a) *x* = 4 and

Fe*<sup>x</sup>*

(*x* = 4.5) are

(*x* = 4.5) microwires shows similar behavior as that of Ni-Mn-Ga

Fe*<sup>x</sup>*

microwires [9]. The maximum strain recovery rates achieved in Ni50Mn25Ga25−*<sup>x</sup>*

while lower than Ni-Mn-Ga microwires after chemical ordering annealing.

**Figure 24.** Tensile stress-strain curves obtained at various temperatures in Ni50Mn25Ga25−*<sup>x</sup>*

**Figure 25.** MSM-spring actuator (MAGNETOSHAPE® by ETO MAGNETIC GmbH) [109].

(*x* = 4.5) are demonstrated in **Figure 24**.

from 0.84% at 156 MPa to 1.504% at 468 MPa.

Fe*<sup>x</sup>*

**5.4. Application of ferromagnetic shape memory alloys**

shape memory alloys are shown in **Figures 25**–**27**.

The SE behavior of the Ni50Mn25Ga25−*<sup>x</sup>*

24 Shape Memory Alloys - Fundamentals and Applications

Fe*<sup>x</sup>*

The SE of Ni50Mn25Ga25−*<sup>x</sup>*

Ni50Mn25Ga25−*<sup>x</sup>*

(b) *x* = 5 [15].

**Figure 26a** presents another important MSM actuator: push-push or multistable actuator [109]. It is composed of two MSM units which arranged antagonistically. In this case, one element acts as a load for another one. The movement in both directions can be controlled magnetically. There is the moving rod in yellow on the top. **Figure 26b** shows an application of a push-push actuator: the MSM device is used to move a mirror on the top and redirect an optical signal.

For FSMA thin films, a novel actuation mechanism has been developed, which makes use of both the ferromagnetic transition and the martensitic transformation. The mechanism is illustrated in **Figure 27** for a Ni-Mn-Ga bending actuator placed in the inhomogeneous magnetic field of a miniature permanent magnet.

**Figure 26.** (a) Example of MSM push-push actuator (MAGNETOSHAPE® by ETO MAGNETIC GmbH). (b) Optical switch actuated by a push-push actuator (Lappeenranta University of Technology, Finland) [109].

**Figure 27.** Prototype of a FSMA micro-actuator for control of a micro-mirror [110].

Depending on the temperature of the micro-actuator, either magnetic or shape recovery forces occur in opposite directions, while the corresponding biasing forces remain small. Thus, an almost perfect antagonism can be realized in a single component part. By applying an alternating electrical current, a periodic oscillation of the beam can be excited. This motion can be used to control the deflection of a micro-mirror attached to the front end of the actuator. For instance, a micro-scanner prototype has been developed based on the actuation mechanism (see **Figure 27**). The overall dimensions are 7 mm × 2 mm × 5 mm.

The micro-pump shown in **Figure 28**, developed in Peter Müllner's lab at Boise State University, may deliver sub-microliter volumes of dugs directly to specific regions of the brain. The micropump was small and robust and can be placed on a head stage on a rat so that drugs can be delivered and brain activity can be monitored while the rat is moving about [111].

**Figure 29** shows a prototype of an energy harvester realized by former Adaptamat [109]. A repeated application of tension and compression force deforms the MSM material and induces a voltage in the coil. Such a voltage can be used to supply energy to a small load. Theoretically, the maximum energy per volume unit that can be extracted from an MSM alloy is equal to the work output, that is, about 150 kJ/m<sup>3</sup> .

In Ni-Mn-Ga alloys, the first-order martensite transformation (FOMT) and second-order magnetic transition (SOMT) produce traditional MCE [112], as displayed in **Figure 30a**. On the other hand, in Ni-Mn-Z-based (Z = Sn, In, Sb) alloys, both direct and inverse MCE (see **Figure 30b**) may be created. The inverse MCE originates from a metamagnetic structural

**Figure 28.** Micro-pump created based on Ni-Mn-Ga alloys [111].

**Figure 29.** Energy harvester based on MSM alloys developed by Adaptamat [109].

Ferromagnetic Shape Memory Alloys: Foams and Microwires http://dx.doi.org/10.5772/intechopen.69702 27

**Figure 30.** Schematic illustration of (a) traditional magnetocaloric effect [112] and (b) inverse magnetocaloric effect [21].

transition from the paramagnetic/antiferromagnetic martensite to ferromagnetic austenite under a bias magnetic field [21], while the direct MCE is attributed to the magnetic transition of the austenite phase around its Curie point [113]. The first-order martensite transformation (FOMT) is responsible for such inverse MCE, which usually exhibits a huge Δ*Sm* and a large adiabatic temperature change Δ*Tad* but rather low Δ*TFWHM* and high hysteresis loss [86]. Ni-Mn-In-Co alloys may also produce giant stress output through magnetic-field-induced martensite transformation [20].

#### **6. Conclusions remarks**

Depending on the temperature of the micro-actuator, either magnetic or shape recovery forces occur in opposite directions, while the corresponding biasing forces remain small. Thus, an almost perfect antagonism can be realized in a single component part. By applying an alternating electrical current, a periodic oscillation of the beam can be excited. This motion can be used to control the deflection of a micro-mirror attached to the front end of the actuator. For instance, a micro-scanner prototype has been developed based on the actuation mechanism

The micro-pump shown in **Figure 28**, developed in Peter Müllner's lab at Boise State University, may deliver sub-microliter volumes of dugs directly to specific regions of the brain. The micropump was small and robust and can be placed on a head stage on a rat so that drugs can be

**Figure 29** shows a prototype of an energy harvester realized by former Adaptamat [109]. A repeated application of tension and compression force deforms the MSM material and induces a voltage in the coil. Such a voltage can be used to supply energy to a small load. Theoretically, the maximum energy per volume unit that can be extracted from an MSM alloy

In Ni-Mn-Ga alloys, the first-order martensite transformation (FOMT) and second-order magnetic transition (SOMT) produce traditional MCE [112], as displayed in **Figure 30a**. On the other hand, in Ni-Mn-Z-based (Z = Sn, In, Sb) alloys, both direct and inverse MCE (see **Figure 30b**) may be created. The inverse MCE originates from a metamagnetic structural

.

delivered and brain activity can be monitored while the rat is moving about [111].

(see **Figure 27**). The overall dimensions are 7 mm × 2 mm × 5 mm.

is equal to the work output, that is, about 150 kJ/m<sup>3</sup>

26 Shape Memory Alloys - Fundamentals and Applications

**Figure 28.** Micro-pump created based on Ni-Mn-Ga alloys [111].

**Figure 29.** Energy harvester based on MSM alloys developed by Adaptamat [109].

The martensite transformation temperature of FSMAs is sensitive to the composition, while the magnetic transition temperature is less sensitive to the composition. Some empirical formula has been summarized to build the relationship between the transformation temperatures and compositions. In addition, the martensite and austenite crystal structures as well as the magnetic properties of these alloys have been extensively investigated.

Single-crystalline FSMAs, such as Ni-Mn-Ga alloys, have been widely studied because the high MFIS is usually generated in the single-crystalline alloys since low resistant to the twin boundary motion. The compositional segregation during the growth of single-crystalline alloys has to be carefully controlled in order to fabricate the alloy ingots with repeatable MFIS. On the other hand, polycrystalline alloys may be fabricated by low-cost methods, such as casting. The polycrystalline Ni-Mn-Ga foams produced by replication casting may generate MFIS as high as 8.7% after suitable training.

Polycrystalline microwire may be synthesized on a large scale by melt extraction, which exhibits pronounced properties, such as MFIS, magnetic entropy change, and superelasticity. Small-sized materials, such as powders, microwires, ribbons, and films, exhibit giant-specific surface area, that is, surface-area-to-volume ratio, which is responsible for the reduced constraints to the twin boundary motion, enhanced heat exchange efficiency, and improved magnetic refrigeration hysteresis loss. The underlying mechanisms between the material size and MCE properties need to be further studied.

The martensite transformation in FSMAs may be induced by external heat change, similar to conventional shape memory alloys, such as Ni-Ti, as well as by external magnetic field. As the operation frequency of an external magnetic field can be much higher than a heat field, FSMAs can work at much higher frequencies than conventional Ni-Ti alloys. The high MFIS produced in single-crystalline Ni-Mn-Ga bulk alloys and polycrystalline foams may find application in high-efficient actuators. On the other hand, the magnetization property change of a FSMA occurs under an external mechanical straining, such as compression. By measuring the magnetization or the induction voltage, the FSMAs may act as sensors, such as force, position, or acceleration sensors.

FSMAs attract much attention in the recent years as high-efficient magnetic refrigeration materials. Significant conventional or inverse magnetocaloric effects have been investigated during the martensite and magnetic transformations. For the alloys (i.e., Ni-Mn-Ga alloys) in which the martensite and magnetic transformations produce the same sign of the magnetic entropy change, the creation of the partial and full magneto-structural coupling states by compositional tuning may be adopted to optimize the magnetic entropy change and working temperature interval.

#### **Author details**

Xuexi Zhang\* and Mingfang Qian

\*Address all correspondence to: xxzhang@hit.edu.cn

School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, China

#### **References**


[5] Ullakko K, Huang JK, Kantner C, OHandley RC, Kokorin VV. Large magnetic-fieldinduced strains in Ni<sup>2</sup> MnGa single crystals. Applied Physics Letters. 1996;**69**:1966

straints to the twin boundary motion, enhanced heat exchange efficiency, and improved magnetic refrigeration hysteresis loss. The underlying mechanisms between the material size and

The martensite transformation in FSMAs may be induced by external heat change, similar to conventional shape memory alloys, such as Ni-Ti, as well as by external magnetic field. As the operation frequency of an external magnetic field can be much higher than a heat field, FSMAs can work at much higher frequencies than conventional Ni-Ti alloys. The high MFIS produced in single-crystalline Ni-Mn-Ga bulk alloys and polycrystalline foams may find application in high-efficient actuators. On the other hand, the magnetization property change of a FSMA occurs under an external mechanical straining, such as compression. By measuring the magnetization or the induction voltage, the FSMAs may act as sensors, such as

FSMAs attract much attention in the recent years as high-efficient magnetic refrigeration materials. Significant conventional or inverse magnetocaloric effects have been investigated during the martensite and magnetic transformations. For the alloys (i.e., Ni-Mn-Ga alloys) in which the martensite and magnetic transformations produce the same sign of the magnetic entropy change, the creation of the partial and full magneto-structural coupling states by compositional tuning may be adopted to optimize the magnetic entropy change and working

School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, China

[1] Chernenko VA, Vitenko IN. Structural characterization and properties of the Ni2

ribbon transforming martensitically. Materials Science Forum. 1994;**166-169**:439

[2] Kokorin VV, Chernenko VA, Valkov VI, Konoplyuk SM, Khapalyuk EA. Magnetic trans-

[3] Martynov VV. X-ray diffraction study of thermally and stress-induced phase transformations in single crystalline Ni-Mn-Ga alloys. Journal de Physique IV. **1995**;5:91

[4] Chernenko VA, Cesari E, Kokorin VV, Vitenko IN. The development of new ferromagnetic shape-memory alloys in Ni-Mn-Ga system. Scripta Metallurgica et Materialia.

MnGa compounds. Fizika Tverdogo Tela. 1995;**37**:3718

MnGa

MCE properties need to be further studied.

28 Shape Memory Alloys - Fundamentals and Applications

force, position, or acceleration sensors.

temperature interval.

**Author details**

**References**

formation in Ni<sup>2</sup>

1995;**33**:1239

Xuexi Zhang\* and Mingfang Qian

\*Address all correspondence to: xxzhang@hit.edu.cn


[33] Ayuela A, Enkovaara J, Ullakko K, Nieminen RM. Structural properties of magnetic Heusler alloys. Journal of Physics: Condensed Matter. 1999;**11**:2017

[19] Liu Y, Zhang X, Xing D, Shen H, Chen D, Liu J, Sun J. Magnetocaloric effect (MCE) in melt-extracted Ni-Mn-Ga-Fe Heusler microwires. Journal of Alloys and Compounds.

[20] Kainuma R, Imano Y, Ito W, Sutou Y, Morito H, Okamoto S, Kitakami O, Oikawa K, Fujita A, Kanomata T, Ishida K. Magnetic-field-induced shape recovery by reverse phase

[21] Liu J, Gottschall T, Skokov KP, Moore JD, Gutfleisch O. Giant magnetocaloric effect

[22] Pasquale M, Sasso CP, Lewis LH, Giudici L, Lograsso T, Schlagel D. Magnetostructural transition and magnetocaloric effect in Ni55Mn20Ga25 single crystals. Physical Review B.

[23] Li Z, Zhang Y, Sánchez-Valdés CF, Sánchez Llamazares JL, Esling C, Zhao X, Zuo L. Giant magnetocaloric effect in melt-spun Ni-Mn-Ga ribbons with magneto-multistruc-

[24] Li Z, Xu K, Zhang Y, Tao C, Zheng D, Jing C. Two successive magneto-structural transformations and their relation to enhanced magnetocaloric effect for Ni55.8Mn18.1Ga26.1

[25] Vasil'Ev AN, Bozhko AD, Khovailo VV, Dikshtein IE, Shavrov VG, Buchelnikov VD, Matsumoto M, Suzuki S, Takagi T, Tani J. Structural and magnetic phase transitions in

[26] Chernenko VA. Compositional instability of beta-phase in Ni-Mn-Ga alloys. Scripta

[27] Ullakko K, Ezer Y, Sozinov A, Kimmel G, Yakovenko P, Lindroos VK. Magnetic-fieldinduced strains in polycrystalline Ni-Mn-Ga at room temperature. Scripta Materialia.

[28] Murray SJ, Farinelli M, Kantner C, Huang JK, Allen SM, O'Handley RC. Field-induced strain under load in Ni-Mn-Ga magnetic shape memory materials. Journal of Applied

[29] Jin X, Marioni M, Bono D, Allen SM, O Handley RC, Hsu TY. Empirical mapping of Ni-Mn-Ga properties with composition and valence electron concentration. Journal of

[30] Wu SK, Yang ST. Effect of composition on transformation temperatures of Ni-Mn-Ga

[31] Segui C, Cesari E, Pons J, Chernenko V. Internal friction behaviour of Ni-Mn-Ga.

[32] Chernenko VA, Pons J, Segu C, Cesari E. Premartensitic phenomena and other phase transformations in Ni-Mn-Ga alloys studied by dynamical mechanical analysis and elec-

driven by structural transitions. Nature Materials. 2012;**11**:620

tural transformation. Applied Physics Letters. 2014;**104**:44101

shape-memory alloys Ni2+xMn1−xGa. Physical Review B. 1999;**59**:1113

Heusler alloy. Scientific Report-UK. 2015;**5**:15143

2014;**616**:184

2005;**72**:094435

Materialia. 1999;**40**:523

Physics. 1998;**83**:7297

Applied Physics. 2002;**91**:8222

shape memory alloys. Materials Letters. 2003;**57**:4291

Materials Science and Engineering A. 2004;**370**:481

tron diffraction. Acta Materialia. **2002**;50:53

2001;**44**:475

transformation. Nature. 2006;**439**:957

30 Shape Memory Alloys - Fundamentals and Applications


[62] Wang X, Dapino MJ. Behavior of nimnga under dynamic magnetic fields considering magnetic diffusion and eddy current power loss. Proceedings of the ASME International Mechanical Engineering Congress and Exposition 2007, Vol. 10, Pts A and B: Mechanics of Solids and Structures; 2008. p 381

[48] Li Y, Jiang C, Liang T, Ma Y, Xu H. Martensitic transformation and magnetization of Ni-Fe-Ga ferromagnetic shape memory alloys. Scripta Materialia. 2003;**48**:1255

[49] Han ZD, Wang DH, Zhang CL, Xuan HC, Zhang JR, Gu BX, Du YW. The martensitic transformation and the magnetocaloric effect in Ni50−xMn38+xIn12 alloys. Solid State

[50] Liu Z, Wang H, Yu S, Dai X, Chen J, Wu G, Liu Y. Phase equilibrium of ferromagnetic

[51] Rolfs K, Chmielus M, Wimpory RC, Mecklenburg A, Müllner P, Schneider R. Double

[52] Guldbakke JM, Chmielus M, Rolfs K, Schneider R, Müllner P, Raatz A. Magnetic, mechanical and fatigue properties of a Ni45.4Mn29.1Ga21.6Fe3.9 single crystal. Scripta Mate-

[53] Gao L, Dong GF, Gao ZY, Cai W. Effect of Dy addition on mechanical and magnetic properties of Mn-rich Ni-Mn-Ga ferromagnetic shape memory alloys. Journal of Alloys

[54] Sozinov A, Likhachev AA, Lanska N, Ullakko K. Giant magnetic-field-induced strain in NiMnGa seven-layered martensitic phase. Applied Physics Letters. 2002;**80**:1746

[55] Cai PY, Feng SS, Chen WP, Xue SX, Li ZG, Zhou Y, Wang HB, Wang GP. Magnetic entropy change and magnetic-field-induced strain in polycrystalline Ni47Mn32Ga21 alloy.

[56] Guo SH, Zhang YH, Li JL, Qi Y, Quan BY, Wang XL. Magnetic-field-induced strains of

[57] Gaitzsch U, Potschke M, Roth S, Rellinghaus B, Schultz L. Mechanical training of polycrystalline 7M Ni50Mn30Ga20 magnetic shape memory alloy. Scripta Materialia.

[58] Gaitzsch U, Potschke M, Roth S, Rellinghaus B, Schultz L. A 1% magnetostrain in poly-

[59] Gaitzsch U, Romberg J, Potschke M, Roth S, Mullner P. Stable magnetic-field-induced strain above 1% in polycrystalline Ni-Mn-Ga. Scripta Materialia. 2011;**65**:679

[60] Potschke M, Weiss S, Gaitzsch U, Cong DY, Hurrich C, Roth S, Schultz L. Magnetically resettable 0.16% free strain in polycrystalline Ni-Mn-Ga plates. Scripta Materialia.

[61] Ham-Su R, Healey JP, Underhill RS, Farrell SP, Cheng LM, Hyatt C, Chen J, Gharghouri MA. Vibration damping material employing magnetic shape memory alloys. Aerospace Materials and Manufacturing: Development, Testing, and Life Cycle Issues - Honoring William Wallace. Canadian Institute of Mining, Metallurgy and Petroleum, Montreal,

bonded Ni-Mn-Ga melt-spun ribbons. Chinese Physics Letters. 2006;**23**:227

shape memory alloy Co39Ni33Al28. Scripta Materialia. 2006;**54**:1299

twinning in Ni-Mn-Ga-Co. Acta Materialia. 2010;**58**:2646

Acta Physica Sinica-Chinese Edition. 2011;**60**:623

crystalline 5M Ni-Mn-Ga. Acta Materialia. 2009;**57**:365

Communications. 2008;**146**:124

32 Shape Memory Alloys - Fundamentals and Applications

and Compounds. 2012;**520**:281

rialia. 2010;**62**:875

2007;**57**:493

2010;**63**:383

QC; 2004. p. 335


[88] Dong JD, Yan AR, Liu J. Microstructure and magnetocaloric properties of melt-extracted La-Fe-Si microwires. Journal of Magnetism and Magnetic Materials. 2014;**357**:73

[75] Nguyen HD, Zhang L, Ou ZQ, Bruck E. Magnetoelastic coupling and magnetocaloric

[76] Yibole H, Guillou F, Zhang L, van Dijk NH, Bruck E. Direct measurement of the magnetocaloric effect in MnFe(P,X) (X = As, Ge, Si) materials. Journal of Physics D: Applied

[77] Manekar M, Roy SB. Reproducible room temperature giant magnetocaloric effect in

[78] Krenke T, Duman E, Acet M, Wassermann EF, Moya X, Manosa L, Planes A. Inverse magnetocaloric effect in ferromagnetic Ni-Mn-Sn alloys. Nature Materials. **2005**;4:450

[79] Krenke T, Duman E, Acet M, Moya X, Manosa L, Planes A. Effect of Co and Fe on the inverse magnetocaloric properties of Ni-Mn-Sn. Journal of Applied Physics. 2007;

[80] Gao B, Hu FX, Shen J, Wang J, Sun JR, Shen BG. Field-induced structural transition and

[81] Takeuchi AY, Guimaraes CE, Passamani EC, Larica C. Enhancement of magnetocaloric properties near room temperature in Ga-doped Ni50Mn34.5In15.5 Heusler-type alloy.

[82] Zhang Y, Zheng Q, Xia W, Zhang J, Du J, Yan A. Enhanced large magnetic entropy change and adiabatic temperature change of Ni43Mn46Sn11 alloys by a rapid solidification

[84] Zhang X, Qian M, Miao S, Su R, Liu Y, Geng L, Sun J. Enhanced magnetic entropy change and working temperature interval in Ni-Mn-In-Co alloys. Journal of Alloys and

[85] Zhang Y, Zhang L, Zheng Q, Zheng X, Li M, Du J, Yan A. Enhanced magnetic refrigeration properties in Mn-rich Ni-Mn-Sn ribbons by optimal annealing. Scientific

[86] Dubenko I, Samanta T, Pathak AK, Kazakov A, Prudnikov V, Stadler S, Granovsky A, Zhukov A, Ali N. Magnetocaloric effect and multifunctional properties of Ni-Mn-based

[87] Quetz A, Koshkid'Ko YS, Titov I, Rodionov I, Pandey S, Aryal A, Ibarra-Gaytan PJ, Prudnikov V, Granovsky A, Dubenko I, Samanta T, Cwik J, Sanchez Llamazares JL, Stadler S, Lahderanta E, Ali N. Giant reversible inverse magnetocaloric effects in

Heusler alloys. Journal of Magnetism and Magnetic Materials. 2012;**324**:3530

Ni50Mn35In15 Heusler alloys. Journal of Alloys and Compounds. 2016;**683**:139

[83] Yan A, Mueller KH, Gutfleisch O. Magnetocaloric effect in LaFe11.8−xCo<sup>x</sup>

ribbons. Journal of Alloys and Compounds. 2008;**450**:18

Sn11 alloy. Journal of Magnetism and

Si1.2 melt-spun

effect in hexagonal Mn-Fe-P-Si compounds. Scripta Materialia. 2012;**67**:975

Fe-Rh. Journal of Physics D: Applied Physics. 2008;**41**:192004

the related magnetic entropy change in Ni43Mn43Co<sup>3</sup>

Magnetic Materials. 2009;**321**:2571

Journal of Applied Physics. 2012;**111**:3867

method. Scripta Materialia. 2015;**104**:41

Compounds. 2016;**656**:154

Reports-UK. 2015;**5**

Physics. 2014;**47**:75002

34 Shape Memory Alloys - Fundamentals and Applications

**102**:33903


#### **Chapter 2**

Provisional chapter

## **Shape Memory Wires in** *R***<sup>3</sup>**

Shape Memory Wires in R3

Shinya Okabe, Takashi Suzuki and Shuji Yoshikawa

Additional information is available at the end of the chapter Shinya Okabe, Takashi Suzuki and Shuji Yoshikawa

http://dx.doi.org/10.5772/intechopen.69175 Additional information is available at the end of the chapter

#### Abstract

[103] Xin Y, Li Y, Chai L, Xu HB. Shape memory characteristics of dual-phase Ni-Mn-Ga based high temperature shape memory alloys. Scripta Materialia. 2007;**57**:599

[104] Glavatskyy I, Glavatska N, Soderberg O, Hannula SP, Hoffmann JU. Transformation temperatures and magnetoplasticity of Ni-Mn-Ga alloyed with Si, In, Co or Fe. Scripta

[105] Glavatskyy I, Glavatska N, Dobrinsky A, Hoffmann JU, Sorberg O, Hannula SP. Crystal structure and high-temperature magnetoplasticity in the new Ni-Mn-Ga-Cu magnetic

[106] Ma YQ, Yang SY, Wang CP, Liu XJ. Tensile characteristics and shape memory effect of

[107] Soto D, Hernandez FA, Flores-Zuniga H, Moya X, Manosa L, Planes A, Aksoy S, Acet M, Krenke T. Phase diagram of Fe-doped Ni-Mn-Ga ferromagnetic shape-memory

[108] Soto-Parra DE, Vives E, Gonzalez-Alonso D, Manosa L, Planes A, Romero R, Matutes-Aquino JA, Ochoa-Gamboa RA, Flores-Zuniga H. Stress- and magnetic field-induced entropy changes in Fe-doped Ni-Mn-Ga shape-memory alloys. Applied Physics Letters.

[110] Wilson SA, Jourdain RPJ, Zhang Q, Dorey RA, Bowen CR, Willander M, Wahab QU, Willander M, Al-hilli SM, Nur O, Quandt E, Johansson C, Pagounis E, Kohl M, Matovic J, Samel B, van der Wijngaart W, Jager EWH, Carlsson D, Djinovic Z, Wegener M, Moldovan C, Iosub R, Abad E, Wendlandt M, Rusu C, Persson K. New materials for micro-scale sensors and actuators. Materials Science and Engineering: R: Reports.

[111] Boise State Micro-Pump Aids Neurological Research. 2014. https://news.boisestate.edu/ update/2014/09/24/micro-pump-developed-boise-state-aids-neurological-research-

[112] Franco V, Blazquez JS, Ingale B, Conde A. The magnetocaloric effect and magnetic refrigeration near room temperature: Materials and models. In: Clarke DR, editor. Annual Review of Materials Research, vol. 42. Palo Alto: Annual Reviews; 2012. p. 305

[113] Krautz M, Skokov K, Gottschall T, Teixeira CS, Waske A, Liu J, Schultz L, Gutfleisch O. Systematic investigation of Mn substituted La(Fe,Si)(13) alloys and their hydrides for room-temperature magnetocaloric application. Journal of Alloys and Compounds.

[109] Magnetic shape memory actuators. http://www.themsmnet.net/devices.html

Ga19 high-temperature shape memory alloy. Scripta Materialia. 2008;**58**:918

shape memory alloys. Scripta Materialia. 2007;**56**:565

Materialia. 2006;54:1891

36 Shape Memory Alloys - Fundamentals and Applications

alloys. Physical Review B. 2008;**77**:1

Ni56Mn21Co<sup>4</sup>

2010;**96**

2007;**56**:1

england/

2014;**598**:27

We propose a new model describing the dynamics of wire made of shape memory alloys, by combining an elastic curve theory and the Ginzburg-Landau theory. The wire is assumed to be a closed curve and is not to be stretched with deformation. The derived system of nonlinear partial differential equations consists of a thermoelastic system and a geometric evolution equation under the inextensible condition. We also show that the system has dual variation structure as well as a straight material case. The structure implies stability of infinitesimally stable stationary state in the Lyapunov sense.

DOI: 10.5772/intechopen.69175

Keywords: shape memory alloys, elastic curve, thermoelastic system, nonlinear partial differential equations, Ginzburg-Landau theory, phase transition, stability, dual variation principle

#### 1. Introduction

Shape memory effect arises from the phase transition of lattice structure. Although there are many models for shape memory alloys, one of the classical model is proposed by Falk, which we call the Falk model. Falk applied the Ginzburg-Landau theory for phase transition to shape memory alloys by regarding shear strain E as an order parameter (see e.g., [1]). That is, the Helmholz free energy density proposed by Falk is given by

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

$$\begin{aligned} \tilde{f}(\epsilon, \theta, \partial\_x^2 u) &:= \frac{1}{2} |\partial\_x^2 u|^2 + f(\epsilon, \theta) + f\_0(\theta) \\ &= \frac{1}{2} |\partial\_x^2 u|^2 + \left(\frac{1}{6} |\epsilon|^6 - \frac{1}{4} |\epsilon|^4 + \frac{1}{2} (\theta - \theta\_\epsilon) |\epsilon|^2\right) + f\_0(\theta) \end{aligned} \tag{1}$$

where u and θ are displacement and absolute temperature, respectively, and a positive constant <sup>θ</sup><sup>c</sup> denotes the critical temperature of the phase transition. We call the first term <sup>j</sup>∂<sup>2</sup> <sup>x</sup> uj 2 2 curvature energy density, the second term f(E, θ) nonlinear elastic energy density and the third term f0(θ) thermal energy density. In other words, Falk represented the phase transition by using the form of the nonlinear elastic energy density. We also remark that for simplicity, all physical constants without the critical temperature are normalized by unity. The Falk model was proposed for straight materials. In the model, the material is built up by a stack of layers parallel to the so-called habit plane (see [2]) and assumed that the displacement u in that direction to depend only on a coordinate x perpendicular to the habit plane, that is, the variable x runs in the stacking direction. Then, the material conserves its volume.

From the point of the small deformation theory, we may use a linearized approximation relation E = ∂xu. Moreover, we take f<sup>0</sup> as the following typical form:

$$f\_0(\theta) := \theta - \theta \log \theta.$$

Then following a standard procedure of derivation of thermoelastic system (see e.g., [3]), we can derive the system of nonlinear partial differential equations called the Falk model:

$$\begin{cases} \partial\_t^2 u \, \, \left. \partial\_x^4 u \right| = \partial\_x \{ \left( \partial\_x u \right)^5 - \left( \partial\_x u \right)^3 + (\theta - \theta\_c) \partial\_x u \}, \\ \partial\_t \theta - \partial\_x^2 \theta = \theta \partial\_x u \, \, \partial\_t \partial\_x u. \end{cases} \tag{2}$$

Here, unknowns are displacement u and absolute temperature θ, and ∂<sup>x</sup> and ∂<sup>t</sup> represent partial differential operator with respect to x and t, respectively. The model is well known as one of classical models describing shape memory alloys. For the other models, we refer Fremond [4], Fremond-Miyazaki [5] and reference therein. The Falk model (Eq. (2)) has been studied actively in the mathematical literature. In the isothermal case, well-posedness, stability of solitary-wave solution, existence of steady state, travelling wave solution and invariant measure have been investigated by Fang-Grillakis [6], Falk-Laedke-Spatschek [7], Friedman-Sprekels [8], Garcke [9] and Tsutsumi-Yoshikawa [10], respectively. For the full system (Eq. (1)), the well-posedness results are found in, for example, [11–13] and so on, and numerical results are found in Hoffmann-Zou [14], Niezgodka-Sprekels [15] by finite element method and in Matus-Melnik-Wang-Rybak [16] and Yoshikawa [17, 18] by the finite difference method. In particular, in Ref. [19], the stability of steady state in the Lyapunov sense was shown. More precisely, the stationary state of Eq. (2) is expressed as a nonlocal nonlinear elliptic problem. If there exists a linearized stable critical point for the functional corresponding to the elliptic problem, then for each neighbourhood U of the equilibrium, we can find a neighbourhood W of the equilibrium such that the solution of Eq. (2) with the initial data in W stays in U for any time. The proof can be shown by the dual variation principle which appears in most of the models in non-equilibrium statistical thermodynamics (see [20]).

The existence of several non-trivial steady states for low-temperature phase and low-energy case is proved in Ref. [21]. The numerical simulation given in Ref. [17] exactly indicates the properties mentioned above. The dual variation structure appears also in a multi-dimensional case [22]; however, well-posedness of the multi-dimensional model corresponding to Eq. (2) is still open in large initial data case due to the propagation of singularity. That is one of our motivations of this problem.

We mention mathematical studies on the motion of curves governed by geometric evolution equations. One of the typical objects is curve-shortening flow derived as an L<sup>2</sup> gradient flow for the length functional of curve γ:

$$L(\mathcal{Y}) := \int\_{\mathcal{Y}} ds$$

where s denotes the arc length parameter of γ. By Gage [23], Gage and Hamilton [24] and Grayson [25], it is well known that the curve-shortening flow shrinks simple closed curves to a point in a finite time. Since the curve-shortening flow can be regarded as a one-dimensional case of mean curvature flow for surfaces, the flow is applicable to various mathematical analysis. For example, the curve-shortening flow plays an important role in studies on phase transition. We also mention the curve-straightening flow which has been attracted a great interest and studied actively in mathematical literature. The flow is derived as an L<sup>2</sup> gradient flow for the elastic energy

$$K(\gamma) := \frac{1}{2} \int\_{\gamma} \kappa^2 \, ds$$

where κ denotes the scalar curvature of γ. It is well known that the flow is applicable to studies on elastic curve inspired by Bernoulli and Euler. Indeed, the curve-straightening flow under the length constraint L(γ) � C converges to a classical elastic curve so-called elastica. There is also an interest in the study on motion of curves governed by the L<sup>2</sup> -gradient flow for E under the inextensible condition. Under the condition, the length constraint L(γ) � C is also satisfied for the condition means that the curve does not stretch. As we will state in Section 2.2, the constraint is imposed on each point of curves. Thus, a standard Lagrange multiplier theory does not work. Therefore, we have to make use of geometric properties of curves governed by the flow.

The purpose of this chapter is to derive a mathematical model describing thermoelastic deformation of shape memory wire in R<sup>3</sup> . In particular, we regard the wire as a closed space curve satisfying the inextensible condition. From the physical point of view, it may be unnatural that the wire does not stretch. However, the contribution of this chapter is to adopt a geometric analysis into a classical thermoelastic theory with phase transition inspired by Falk.

#### 2. Setting and derivation of equations

<sup>~</sup><sup>f</sup> <sup>ð</sup>E, <sup>θ</sup>, <sup>∂</sup><sup>2</sup>

38 Shape Memory Alloys - Fundamentals and Applications

<sup>x</sup>u<sup>Þ</sup> :<sup>¼</sup> <sup>1</sup> 2 j∂2 xuj

¼ 1 2 j∂2 xuj 2 þ 1 6 jEj <sup>6</sup> � <sup>1</sup> 4 jEj 4 þ 1 2

<sup>2</sup> <sup>þ</sup> <sup>f</sup>ðE, <sup>θ</sup>Þ þ <sup>f</sup> <sup>0</sup>ðθ<sup>Þ</sup>

where u and θ are displacement and absolute temperature, respectively, and a positive constant <sup>θ</sup><sup>c</sup> denotes the critical temperature of the phase transition. We call the first term <sup>j</sup>∂<sup>2</sup>

curvature energy density, the second term f(E, θ) nonlinear elastic energy density and the third term f0(θ) thermal energy density. In other words, Falk represented the phase transition by using the form of the nonlinear elastic energy density. We also remark that for simplicity, all physical constants without the critical temperature are normalized by unity. The Falk model was proposed for straight materials. In the model, the material is built up by a stack of layers parallel to the so-called habit plane (see [2]) and assumed that the displacement u in that direction to depend only on a coordinate x perpendicular to the habit plane, that is, the

From the point of the small deformation theory, we may use a linearized approximation

f <sup>0</sup>ðθÞ :¼ θ � θlogθ:

Then following a standard procedure of derivation of thermoelastic system (see e.g., [3]), we

Here, unknowns are displacement u and absolute temperature θ, and ∂<sup>x</sup> and ∂<sup>t</sup> represent partial differential operator with respect to x and t, respectively. The model is well known as one of classical models describing shape memory alloys. For the other models, we refer Fremond [4], Fremond-Miyazaki [5] and reference therein. The Falk model (Eq. (2)) has been studied actively in the mathematical literature. In the isothermal case, well-posedness, stability of solitary-wave solution, existence of steady state, travelling wave solution and invariant measure have been investigated by Fang-Grillakis [6], Falk-Laedke-Spatschek [7], Friedman-Sprekels [8], Garcke [9] and Tsutsumi-Yoshikawa [10], respectively. For the full system (Eq. (1)), the well-posedness results are found in, for example, [11–13] and so on, and numerical results are found in Hoffmann-Zou [14], Niezgodka-Sprekels [15] by finite element method and in Matus-Melnik-Wang-Rybak [16] and Yoshikawa [17, 18] by the finite difference method. In particular, in Ref. [19], the stability of steady state in the Lyapunov sense was shown. More precisely, the stationary state of Eq. (2) is expressed as a nonlocal nonlinear elliptic problem. If there exists a linearized stable critical point for the functional corresponding to the elliptic problem, then for each neighbourhood U of the equilibrium, we can find a neighbourhood W of the equilibrium such that the solution of Eq. (2) with the initial data in W stays in U for any time. The proof can be shown by the dual variation principle which appears in most of the

<sup>5</sup> � ð∂xu<sup>Þ</sup>

<sup>x</sup>θ ¼ θ∂xu ∂t∂xu:

<sup>3</sup> þ ð<sup>θ</sup> � <sup>θ</sup>cÞ∂xug,

can derive the system of nonlinear partial differential equations called the Falk model:

<sup>x</sup>u ¼ ∂xfð∂xuÞ

<sup>∂</sup>t<sup>θ</sup> � <sup>∂</sup><sup>2</sup>

variable x runs in the stacking direction. Then, the material conserves its volume.

relation E = ∂xu. Moreover, we take f<sup>0</sup> as the following typical form:

models in non-equilibrium statistical thermodynamics (see [20]).

∂2 <sup>t</sup> <sup>u</sup> <sup>þ</sup> <sup>∂</sup><sup>4</sup>

(

� �

ðθ � θcÞjEj

2

þ f <sup>0</sup>ðθÞ

ð1Þ

<sup>x</sup> uj 2 2

ð2Þ

We denote the closed curve representing shape of wire by Γ = {γ(ξ): ξ ∈ Ξ}, where the variable ξ is an arbitrary parameter not necessarily the arc length parameter. Let us define a displacement vector from a point ξ in an original shape Γ<sup>0</sup> = {γ<sup>0</sup> (ξ): ξ ∈ Ξ} by u(ξ) (see Figure 1); namely, it holds that

$$\gamma(\xi) = \gamma^0(\xi) + \mathfrak{u}(\xi)$$

for


Throughout this chapter, we denote by L the length of Γ<sup>0</sup> , and hence, the length of Γ is also L from the non-stretching assumption. To apply the idea by Falk, we need to determine the form of strain and free energy (Eq. (1)) suitable for this setting.

Figure 1. Original shape Γ<sup>0</sup> and deformed curve Γ.

#### 2.1. Definition of strain

We first consider the strain. Let γ<sup>0</sup> (ξ) be a space closed curve, where ξ is a parameter (not necessary to be the arc length parameter). For γ<sup>0</sup> (ξ), we define the displacement vector by u(ξ), and we denote <sup>γ</sup>ðξÞ ¼ <sup>γ</sup><sup>0</sup>ðξÞ þ <sup>u</sup>ðξÞ. Since the relation "strain <sup>≈</sup> line element" holds, let us first pursue line element between γ<sup>0</sup> (ξ) and γ (ξ). From the direct calculation, we have

$$\begin{split} \left| \gamma'(\xi) \right|^2 - \left| \gamma^{0'}(\xi) \right|^2 &= \left\{ \gamma^{0'}(\xi) + \boldsymbol{\mu}'(\xi) \right\} \cdot \left\{ \gamma^{0'}(\xi) + \boldsymbol{\mu}'(\xi) \right\} - \boldsymbol{\gamma}^{0'}(\xi) \cdot \boldsymbol{\gamma}^{0'}(\xi) \\ &= 2\boldsymbol{\gamma}^{0'}(\xi) \cdot \boldsymbol{\mu}'(\xi) + \left| \boldsymbol{\mu}'(\xi) \right|^2. \end{split}$$

Here, if we assume a smallness of deformation, then we may assume

$$|\gamma^{0'}(\xi) \cdot \mathfrak{u}'(\xi)| \gg |\mathfrak{u}'(\xi)|^2.$$

From now on, we regard the strain as

$$
\gamma^{0'} (\xi) \cdot \mu'(\xi) .
$$

#### 2.2. Definition of energy functional

<sup>γ</sup>ðξÞ ¼ <sup>γ</sup><sup>0</sup>

: displacement vector.

from the non-stretching assumption. To apply the idea by Falk, we need to determine the form

necessary to be the arc length parameter). For γ<sup>0</sup> (ξ), we define the displacement vector by u(ξ), and we denote <sup>γ</sup>ðξÞ ¼ <sup>γ</sup><sup>0</sup>ðξÞ þ <sup>u</sup>ðξÞ. Since the relation "strain <sup>≈</sup> line element" holds, let us first

ðξÞg � fγ<sup>0</sup><sup>0</sup>

ðξÞj ≫ ju<sup>0</sup>

ðξÞ:

ðξÞþju<sup>0</sup>

ðξÞ þ u<sup>0</sup>

ðξÞj<sup>2</sup> :

ðξÞj<sup>2</sup> :

pursue line element between γ<sup>0</sup> (ξ) and γ (ξ). From the direct calculation, we have

ðξÞ þ u<sup>0</sup>

ðξÞ � u<sup>0</sup>

γ00 ðξÞ � u<sup>0</sup>

ðξÞ � u<sup>0</sup>

<sup>¼</sup> <sup>2</sup>γ<sup>0</sup><sup>0</sup>

jγ00

Here, if we assume a smallness of deformation, then we may assume

Throughout this chapter, we denote by L the length of Γ<sup>0</sup>

of strain and free energy (Eq. (1)) suitable for this setting.

for

• γðξÞ :¼

• <sup>γ</sup><sup>0</sup>ðξ<sup>Þ</sup> :<sup>¼</sup>

• uðξÞ :¼

 γ0 <sup>1</sup>ðξÞ, <sup>γ</sup><sup>0</sup>

2.1. Definition of strain

jγ0

We first consider the strain. Let γ<sup>0</sup>

Figure 1. Original shape Γ<sup>0</sup> and deformed curve Γ.

ðξÞj<sup>2</sup> � jγ<sup>0</sup><sup>0</sup>

From now on, we regard the strain as

ðξÞj<sup>2</sup> ¼ fγ<sup>0</sup><sup>0</sup>

γ1ðξÞ, γ2ðξÞ, γ3ðξÞ

40 Shape Memory Alloys - Fundamentals and Applications

u1ðξÞ, u2ðξÞ, u3ðξÞ

<sup>2</sup>ðξÞ, <sup>γ</sup><sup>0</sup> <sup>3</sup>ðξÞ  ðξÞ þ uðξÞ

: vector representing the shape,

: vector representing the original shape,

, and hence, the length of Γ is also L

(ξ) be a space closed curve, where ξ is a parameter (not

<sup>ð</sup>ξÞg � <sup>γ</sup><sup>0</sup><sup>0</sup>

<sup>ð</sup>ξÞ � <sup>γ</sup><sup>0</sup><sup>0</sup>

ðξÞ

Let γ<sup>0</sup> (ξ) be an initial closed curve and γ(ξ, t) denote a family of closed curves starting from γ0 (ξ). Recalling that the arc length parameter s(ξ, t)of γ(ξ, t) is given by

$$ds = \begin{vmatrix} \partial \boldsymbol{\gamma} \\ \partial \boldsymbol{\xi} \end{vmatrix} d\boldsymbol{\xi},$$

we can write kinetic energy M(γ) of γ as

$$M(\boldsymbol{\gamma}) := \mathcal{G} \left| \frac{\partial \boldsymbol{\gamma}}{\partial t} \right|^2 \left| \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \right| d\boldsymbol{\xi}.$$

In a similar manner, thermal energy is defined by

$$F\_0(\boldsymbol{\gamma}) := \left. \oint f\_0(\boldsymbol{\xi}) \right| \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \Big| d\boldsymbol{\xi},$$

and the curvature energy is expressed as

$$K(\boldsymbol{\gamma}) := \oint \kappa^2 \left| \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \right| d\boldsymbol{\xi} \dots$$

Observe that the scalar curvature κ is written as

$$\kappa = \left\{ \left| \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \right| \frac{\partial^2 \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}^2} - \left( \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \cdot \frac{\partial^2 \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}^2} \right) \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \right\} \left| \frac{\partial \boldsymbol{\gamma}}{\partial \boldsymbol{\xi}} \right|^{-3}.\right.$$

Lastly, the nonlinear elastic energy density is given by

$$f(\eth\_{\xi} \gamma, \theta; \eth\_{\xi} \gamma^{0}) \ := \frac{1}{6} \left( \frac{\partial \gamma^{0}}{\partial \xi} \cdot \frac{\partial u}{\partial \xi} \right)^{6} - \frac{1}{4} \left( \frac{\partial \gamma^{0}}{\partial \xi} \cdot \frac{\partial u}{\partial \xi} \right)^{4} + \frac{1}{2} (\theta - \theta\_{\varepsilon}) \left( \frac{\partial \gamma^{0}}{\partial \xi} \cdot \frac{\partial u}{\partial \xi} \right)^{2}$$

and then the nonlinear elastic energy is written as

$$F(\eth\_{\xi}\gamma, \theta; \eth\_{\xi}\gamma^{0}) := \oint f(\eth\_{\xi}\gamma, \theta; \eth\_{\xi}\gamma^{0}) \Big| \frac{\partial \gamma}{\partial \xi} \Big| d\xi$$

where <sup>u</sup> <sup>=</sup> <sup>γ</sup> � <sup>γ</sup><sup>0</sup> . Thus, we obtain the Helmholtz energy for our setting as

$$H(\boldsymbol{\gamma}, \boldsymbol{\theta}, \boldsymbol{\gamma}^0) := M(\boldsymbol{\gamma}) + K(\boldsymbol{\gamma}) + F(\partial\_{\boldsymbol{\xi}} \boldsymbol{\gamma}, \boldsymbol{\theta}; \partial\_{\boldsymbol{\xi}} \boldsymbol{\gamma}^0) + F\_0(\boldsymbol{\theta}).$$

From now on, let <sup>s</sup><sup>∈</sup> <sup>R</sup>=L<sup>Z</sup> <sup>¼</sup>: <sup>S</sup><sup>1</sup> <sup>L</sup> be the arc length parameter of the initial closed curve <sup>γ</sup><sup>0</sup> <sup>=</sup> <sup>γ</sup><sup>0</sup> (s). It follows from the property of arc length parameter that |γ0<sup>0</sup> (s)| � 1. In a similar fashion to the above equation, γ(s, t) means the closed curve deformed along evolution from γ<sup>0</sup> (s). Moreover, in what follows, we assume that γ(s, t) satisfies

$$|\partial\_s \gamma(s, t)| \equiv 1 \tag{3}$$

which means "s is arc length parameter of γ not only the initial time but also every time t". From the assumption, we can rewrite M, F<sup>0</sup> and F shortly as

$$M(\gamma) = \int\_0^L |\, \partial\_t \gamma'| \,^2 ds,$$

$$F\_0(\theta) = \int\_0^L f\_0(\theta) ds,$$

$$F(\partial\_s \, \gamma, \theta; \partial\_s \, \gamma^0) = \int\_0^L f(\partial\_s \gamma', \theta; \partial\_s \gamma^0) \, ds.$$

Moreover, since <sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>2</sup> <sup>s</sup>γ ¼ 0, from Eq. (3), it holds that

$$K(\boldsymbol{\gamma}) = \int\_0^L |\,\partial\_s^2 \boldsymbol{\gamma}|^2 ds.$$

Therefore, the Helmholtz free energy density is denoted by

$$H(t) := \frac{1}{2}||\partial\_t \gamma(\cdot, t)||\_{L^2(S\_\perp^1)}^2 + \frac{1}{2}||\partial\_s^2 \gamma(\cdot, t)||\_{L^2(S\_\perp^1)}^2 + F\Big(\partial\_s \gamma(\cdot, t), \,\theta(\cdot, t); \partial\_s \gamma^0(\cdot)\Big) + F\_0\Big(\theta(\cdot, t)\Big).$$

We will explain that for the free energy under some assumptions, the following system of nonlinear partial differential equations is derived:

$$\begin{cases} & \partial\_t^2 \mathcal{V} + \partial\_s^4 \mathcal{V} + \partial\_s f\_{,\mathsf{d}\_\mathcal{V}\boldsymbol{\gamma}} (\mathsf{\dot{\partial}}\_s \boldsymbol{\gamma}, \ \boldsymbol{\partial}\_s \mathsf{\dot{\partial}}\_s \boldsymbol{\gamma}^0) - \mathsf{\dot{\partial}}\_s \left\{ (\mathsf{v} - 2 \, |\, \mathsf{\dot{\partial}}\_s^2 \boldsymbol{\gamma} \right\}^2 \, \mathsf{\dot{\partial}}\_s \boldsymbol{\gamma} \right\} = 0, \\\\ & - \partial\_s^2 \boldsymbol{v} + |\mathsf{\dot{\partial}}\_s^2 \boldsymbol{\gamma}|^2 \boldsymbol{v} = 2 \left| \partial\_s^2 \boldsymbol{\gamma} \right|^4 - |\mathsf{\dot{\partial}}\_s^3 \boldsymbol{\gamma}|^2 + |\mathsf{\dot{\partial}}\_s \mathsf{\dot{\partial}}\_l \boldsymbol{\gamma}|^2 + \mathsf{\dot{\partial}}\_s^2 f\_{,\mathsf{d}\_\mathcal{V}} (\mathsf{\dot{\partial}}\_s \boldsymbol{\gamma}, \ \boldsymbol{\partial}; \mathsf{\dot{\partial}}\_s \boldsymbol{\gamma}^0) \cdot \mathsf{\dot{\partial}}\_s \boldsymbol{\gamma}, \\\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (4) \\\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (10) \end{cases}$$

where

$$\begin{split} f \lrcorner\_{\clubsuit \boldsymbol{\lambda} \boldsymbol{\gamma}} (\clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \urcorner \boldsymbol{\theta} ; \clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \urcorner \boldsymbol{\delta}^{0}) &= \left( \frac{\partial f}{\partial \boldsymbol{\delta} \boldsymbol{\gamma}\_{1}} \urcorner \frac{\partial f}{\partial \boldsymbol{\delta} \boldsymbol{\gamma}\_{2}} \urcorner \frac{\partial f}{\partial \boldsymbol{\delta} \boldsymbol{\gamma}\_{3}} \right) \\ &= \{ (\clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \boldsymbol{\gamma}^{0} \cdot \clubsuit \boldsymbol{\omega} \urcorner)^{5} - (\clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \boldsymbol{\gamma}^{0} \cdot \clubsuit \boldsymbol{\omega} \urcorner)^{3} + (\boldsymbol{\theta} - \boldsymbol{\theta}\_{c}) \clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \urcorner \clubsuit \boldsymbol{\delta}\_{\boldsymbol{s}} \urcorner \} \boldsymbol{\delta}\_{\boldsymbol{s}} \urcorner \} .\end{split}$$

#### 2.3. Equation of motion

By using the Hamilton principle, we derive an equation of the motion of γ. Namely, we will derive the Euler-Lagrange equation for the functional:

$$\tilde{H}(\boldsymbol{\gamma},\ \boldsymbol{\theta};\boldsymbol{\gamma}^{0}) = \int\_{t\_{\mathrm{2}}}^{t\_{\mathrm{1}}} \left\{ \frac{1}{2} ||\boldsymbol{\partial}\_{t}\boldsymbol{\eta}(\cdot,t)||\_{\mathrm{L}^{2}(\bar{\mathbf{s}}\_{\mathrm{1}}^{\mathrm{d}})}^{2} + \frac{1}{2} ||\boldsymbol{\partial}\_{t}^{2}\boldsymbol{\eta}(\cdot,t)||\_{\mathrm{L}^{2}(\bar{\mathbf{s}}\_{\mathrm{1}}^{\mathrm{d}})}^{2} - F\left(\boldsymbol{\partial}\_{t}\boldsymbol{\eta}(\cdot,t),\ \boldsymbol{\theta}(\cdot,t);\boldsymbol{\partial}\_{t}\boldsymbol{\eta}^{0}(\cdot)\right) - F\_{\boldsymbol{0}}\left(\boldsymbol{\theta}(\cdot,t)\right) \right\} dt. \tag{5}$$

Let us denote the variation of γ by

$$\gamma(\mathbf{s}, t; \theta) := \gamma(\mathbf{s}, t) + \varepsilon \,\,\boldsymbol{\varrho}(\mathbf{s}, t)$$

where ε is a sufficiently small positive parameter and ϕ is sufficiently smooth and satisfies ϕ(s, t1) = ϕ(s, t2) = 0. Moreover, from the assumption Eq. (3), it is also necessary to hold that

$$\left. \frac{d}{d\varepsilon} |\partial\_s \boldsymbol{\gamma}(\mathbf{s}, t; \boldsymbol{\varepsilon})| \right|\_{\boldsymbol{\varepsilon}=\mathbf{0}} = \mathbf{0}.$$

Since

j∂sγðs, tÞj � 1 ð3Þ

which means "s is arc length parameter of γ not only the initial time but also every time t".

ðL 0 j ∂tγ j

> ðL 0

> > ðL 0

ðL 0 j ∂2 sγj 2 ds:

<sup>2</sup> ds,

<sup>f</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>

Þ ds:

<sup>∂</sup>sγð�, tÞ, <sup>θ</sup>ð�, tÞ; <sup>∂</sup>sγ<sup>0</sup>

sγj 2

<sup>ð</sup>∂t∂s<sup>γ</sup> � <sup>∂</sup>sγ<sup>0</sup><sup>Þ</sup>

<sup>∂</sup>sγð�, tÞ, <sup>θ</sup>ð�, tÞ; <sup>∂</sup>sγ<sup>0</sup>

<sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

�

ð�Þ � þ F<sup>0</sup> � θð�, tÞ � :

Þ ∂sγg ¼ 0,

<sup>s</sup> <sup>f</sup> ,∂s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>Þ � <sup>∂</sup>sγ,

<sup>3</sup> þ ð<sup>θ</sup> � <sup>θ</sup>cÞ∂sγ<sup>0</sup> � <sup>∂</sup>su <sup>g</sup>∂sγ<sup>0</sup>:

ð�Þ � � F<sup>0</sup> � θð�, tÞ ð4Þ

dt: ð5Þ

f <sup>0</sup>ðθÞds,

MðγÞ ¼

F0ðθÞ ¼

<sup>F</sup>ð∂<sup>s</sup> <sup>γ</sup>, <sup>θ</sup>; <sup>∂</sup><sup>s</sup> <sup>γ</sup><sup>0</sup>Þ ¼

<sup>s</sup>γ ¼ 0, from Eq. (3), it holds that

sγð�, tÞjj<sup>2</sup>

<sup>L</sup>2ðS<sup>1</sup> <sup>L</sup><sup>Þ</sup> <sup>þ</sup> <sup>F</sup> �

<sup>s</sup><sup>γ</sup> <sup>þ</sup> <sup>∂</sup>sf , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>Þ � <sup>∂</sup><sup>s</sup> fð<sup>v</sup> � <sup>2</sup> <sup>j</sup>∂<sup>2</sup>

We will explain that for the free energy under some assumptions, the following system of

<sup>2</sup> þ j∂s∂tγ<sup>j</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> –γ<sup>0</sup><sup>Þ</sup>

<sup>5</sup> � ð <sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>su <sup>Þ</sup>

By using the Hamilton principle, we derive an equation of the motion of γ. Namely, we will

� ��

KðγÞ ¼

From the assumption, we can rewrite M, F<sup>0</sup> and F shortly as

42 Shape Memory Alloys - Fundamentals and Applications

Therefore, the Helmholtz free energy density is denoted by

<sup>L</sup>2ðS<sup>1</sup> <sup>L</sup><sup>Þ</sup> þ 1 2 jj∂2

nonlinear partial differential equations is derived:

sγj 2 <sup>v</sup> <sup>¼</sup> <sup>2</sup> <sup>j</sup>∂<sup>2</sup>

<sup>∂</sup>t<sup>θ</sup> � <sup>∂</sup><sup>2</sup>

∂sγ<sup>1</sup>

derive the Euler-Lagrange equation for the functional:

<sup>L</sup>2ðS<sup>1</sup> <sup>L</sup><sup>Þ</sup> þ 1 2 jj∂2 sγð�, tÞjj<sup>2</sup> <sup>L</sup>2ðS<sup>1</sup> <sup>L</sup><sup>Þ</sup> � <sup>F</sup> �

jj∂tγð�, tÞjj<sup>2</sup>

¼ fð <sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>su <sup>Þ</sup>

sγj <sup>4</sup> � j∂<sup>3</sup> sγj

, <sup>∂</sup><sup>f</sup> ∂sγ<sup>2</sup>

� �

, <sup>∂</sup><sup>f</sup> ∂sγ<sup>3</sup>

<sup>s</sup>θ ¼ θ �

Moreover, since <sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>H</sup>ðt<sup>Þ</sup> :<sup>¼</sup> <sup>1</sup> 2

> 8 >>>><

> >>>>:

2.3. Equation of motion

Þ ¼ ðt1 t2 1 2

<sup>H</sup><sup>~</sup> <sup>ð</sup>γ, <sup>θ</sup>; <sup>γ</sup><sup>0</sup>

where

jj∂tγð�, tÞjj<sup>2</sup>

∂2 <sup>t</sup> <sup>γ</sup> <sup>þ</sup> <sup>∂</sup><sup>4</sup>

<sup>s</sup> <sup>v</sup> þ j∂<sup>2</sup>

<sup>f</sup> ,∂s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup><sup>s</sup> <sup>γ</sup><sup>0</sup>Þ ¼ <sup>∂</sup><sup>f</sup>

� <sup>∂</sup><sup>2</sup>

$$\left. \frac{d}{d\varepsilon} |\eth\_s \gamma(s, t; \theta)| \right|\_{\varepsilon=0} = \eth\_s \gamma(s, t) \cdot \eth\_s \rho(s, t)$$

ϕ has to satisfy

$$
\partial\_s \gamma(\mathbf{s}, t) \cdot \partial\_s \varphi(\mathbf{s}, t) = 0
$$

for any s∈ S<sup>1</sup> <sup>L</sup> and t > 0. Calculating the first variation of the energy functional, we have

$$\left. \frac{d}{d\varepsilon} \tilde{H}(\boldsymbol{\gamma}, \boldsymbol{\theta}; \boldsymbol{\gamma}^0) \right|\_{\varepsilon=0} = \left. \int\_{t\_1}^{t\_2} \{ \langle \mathbf{\dot{\partial}}\_l \boldsymbol{\gamma}, \mathbf{\dot{\partial}}\_l \boldsymbol{\varphi} \rangle - \langle \mathbf{\dot{\partial}}\_s^2 \boldsymbol{\gamma}, \mathbf{\dot{\partial}}\_s^2 \boldsymbol{\varphi} \rangle - \langle f\_{, \mathbf{\dot{\partial}}\_l \boldsymbol{\gamma}}(\mathbf{\dot{\partial}}\_s \boldsymbol{\gamma}, \boldsymbol{\theta}; \mathbf{\dot{\partial}}\_s \boldsymbol{\gamma}^0), \mathbf{\dot{\partial}}\_s \boldsymbol{\varphi} \rangle \} dt \dots$$

From the integral by parts, the right-hand side is rewritten as follows:

$$-\int\_{t\_1}^{t\_2} \langle \eth\_t^2 \gamma + \eth\_s^4 \gamma - \eth\_s f\_{~, \eth\_s \gamma} (\eth\_s \gamma, \theta; \eth\_s \gamma^0), \neq \rangle dt. \tag{6}$$

Then the integral Eq. (6) is equal to 0 for any ϕ satisfying ϕðs, t1Þ ¼ ϕðs, t2Þ ¼ 0 and ∂sγ � ∂sϕ � 0. For the purpose, we define

$$V := \{ \boldsymbol{\varrho} \mid \partial\_s \boldsymbol{\jmath} \cdot \partial\_s \boldsymbol{\varrho} \equiv 0 \}.$$

The orthogonal complement V<sup>⊥</sup> of the space V with respect to L<sup>2</sup> ðS1 <sup>L</sup>Þ inner product is given by

$$V^{\perp} = \{ \partial\_{\mathbf{s}} (w \partial\_{\mathbf{s}} \gamma) w = w(\mathbf{s}, t) \text{ is a scalar function} \}. \tag{7}$$

Here, we remark that in the case where γ is a curve embedded in three-dimensional space (not a planar curve), ∂<sup>2</sup> <sup>s</sup><sup>γ</sup> 6¼ <sup>0</sup> has to be satisfied for every <sup>ð</sup>s, tÞ<sup>∈</sup> <sup>S</sup><sup>1</sup> <sup>L</sup> � Rþ. In the end of this section, we will show the reason why V<sup>⊥</sup> is given as above. Consequently, if for the direction ∂2 <sup>t</sup> <sup>γ</sup> <sup>þ</sup> <sup>∂</sup><sup>4</sup> <sup>s</sup><sup>γ</sup> � <sup>∂</sup>sf , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>Þ, there exists a scalar function <sup>w</sup> <sup>=</sup> <sup>w</sup>(s, t) such that

$$
\partial\_t^2 \gamma + \partial\_s^4 \gamma - \partial\_s f\_{,\mathfrak{d}\_\delta \gamma} (\mathfrak{d}\_s \gamma, \Theta; \mathfrak{d}\_s \gamma^0) = \mathfrak{d}\_s (w \mathfrak{d}\_s \gamma) \tag{8}
$$

then Eq. (6) is equal to 0.

Next, we derive the equation for w. From the assumption Eq. (3), we see that

$$0 = \partial\_t^2 |\partial\_s \boldsymbol{\gamma}|^2 = 2\partial\_s \partial\_t^2 \boldsymbol{\gamma} \cdot \partial\_s \boldsymbol{\gamma} + 2|\partial\_s \partial\_t \boldsymbol{\gamma}|^2$$

It follows from Eq. (8) that

$$\left\{-\partial\_s^5 \gamma + \partial\_s^2 f\_{,\mathfrak{d}\gamma}(\partial\_s \gamma, \theta; \mathfrak{d}\_s \gamma^0) + \partial\_s^2 (w \partial\_s \gamma) \right\} \cdot \mathfrak{d}\_s \gamma = -|\partial\_s \partial\_t \gamma|^2. \tag{9}$$

Differentiating Eq. (3), we obtain

$$\begin{aligned} \partial\_s \boldsymbol{\gamma} \cdot \partial\_s^2 \boldsymbol{\gamma} &= 0, \\ \partial\_s \boldsymbol{\gamma} \cdot \partial\_s^3 \boldsymbol{\gamma} &= - \left| \partial\_s^2 \boldsymbol{\gamma} \right|^2, \\ \partial\_s \boldsymbol{\gamma} \cdot \partial\_s^4 \boldsymbol{\gamma} &= -\frac{3}{2} \partial\_s (\left| \partial\_s^2 \boldsymbol{\gamma} \right|^2), \\ \partial\_s \boldsymbol{\gamma} \cdot \partial\_s^5 \boldsymbol{\gamma} &= -2 \partial\_s^2 (\left| \partial\_s^2 \boldsymbol{\gamma} \right|^2) + \left| \partial\_s^3 \boldsymbol{\gamma} \right|^2. \end{aligned}$$

By the relations, we will rewrite Eq. (9). It follows from the direct calculation that

$$
\partial\_s^2 \left( \mathfrak{w} \partial\_s \gamma \right) \cdot \partial\_s \gamma = \partial\_s^2 w \, - \, w |\partial\_s^2 \gamma|^2 \,.
$$

Since from the definition

$$\begin{aligned} &= \left\{ f\_{\sf d, \sf d, \sf s} (\eth\_{\sf s} \chi, \eth\_{\sf s} \lozenge \eth\_{\sf s} \lozenge) \right. \\ &= \left\{ \left( \eth\_{\sf s} \chi^{0} \cdot \eth\_{\sf s} (\chi - \sf y^{0}) \right)^{5} - \left( \eth\_{\sf s} \chi^{0} \cdot \eth\_{\sf s} (\chi - \sf y^{0}) \right)^{3} + (\theta - \theta\_{\sf c}) \left( \eth\_{\sf s} \chi^{0} \cdot \eth\_{\sf s} (\chi - \sf y^{0}) \right) \right\} \eth\_{\sf s} \chi^{0} . \end{aligned}$$

we also obtain

∂2 <sup>s</sup> <sup>f</sup> , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>Þ � <sup>∂</sup>s<sup>γ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup> s <sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup> 5 � <sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup> 3 þ ðθ � θcÞ <sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup> ∂sγ<sup>0</sup> � ∂sγ:

Therefore, substituting these into Eq. (9), we find

$$\begin{split} 0 &= 2\partial\_s^2(|\partial\_s^2\boldsymbol{\gamma}|^2) - |\partial\_s^3\boldsymbol{\gamma}|^2 + \partial\_s^2\boldsymbol{w} - \boldsymbol{w}|\partial\_s^2\boldsymbol{\gamma}|^2 + |\partial\_s\partial\_t\boldsymbol{\gamma}|^2 + \partial\_s^2\Big[\left\{\left(\partial\_s\boldsymbol{\gamma}^0 \cdot \partial\_s(\boldsymbol{\gamma}-\boldsymbol{\gamma}^0)\right)^5\right\} \\ &- \left(\partial\_s\boldsymbol{\gamma}^0 \cdot \partial\_s\left(\boldsymbol{\gamma}-\boldsymbol{\gamma}^0\right)\right)^3 + (\boldsymbol{\Theta}-\boldsymbol{\Theta}\_c)\Big(\partial\_s\ \boldsymbol{\gamma}^0 \cdot \partial\_s\left(\boldsymbol{\gamma}-\boldsymbol{\gamma}^0\right)\right)\Bigg\}\partial\_s\boldsymbol{\gamma}^0 \Big] \cdot \boldsymbol{\partial}\_s\boldsymbol{\gamma}. \end{split}$$

Here setting the new unknown <sup>v</sup> by <sup>v</sup> :<sup>¼</sup> <sup>w</sup> <sup>þ</sup> <sup>2</sup>j∂<sup>2</sup> sγj 2 , we can rewrite the equation as follows:

$$\begin{split}-\partial\_s^2 \boldsymbol{\upsilon} + |\partial\_s^2 \boldsymbol{\gamma}|^2 \boldsymbol{\upsilon} &= \mathsf{2} \left| \partial\_s^2 \boldsymbol{\gamma} \right|^4 - |\partial\_s^3 \boldsymbol{\gamma}|^2 + |\partial\_s \partial\_t \boldsymbol{\gamma}|^2 \\ + \mathsf{d}\_s^2 \left[ \left\{ \left( \partial\_s \boldsymbol{\gamma}^0 \cdot \partial\_s (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0) \right)^5 - \left( \partial\_s \boldsymbol{\gamma}^0 \cdot \partial\_s (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0) \right)^3 + (\boldsymbol{\theta} - \boldsymbol{\theta}\_c) \left( \partial\_s \boldsymbol{\gamma}^0 \cdot \partial\_s (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0) \right) \right\} \partial\_s \boldsymbol{\gamma}^0 \right] \cdot \mathsf{d}\_s \boldsymbol{\gamma} \cdot \boldsymbol{\Gamma} \end{split}$$

Consequently, under given temperature θ, the equation of motion is given by

$$\begin{cases} & \partial\_t^2 \gamma + \partial\_s^4 \gamma - \partial\_{\sf s} f\_{,\sf a\_\sf \chi} (\eth\_\sf \chi, \theta; \clubsuit \eth\_\sf \chi^0) - \eth\_\sf \{(\upsilon - 2|\partial\_s^2 \gamma|^2) \eth\_\sf \chi\} = 0, \\ & - \partial\_s^2 \upsilon + |\partial\_s^2 \gamma|^2 \upsilon = \left. 2|\partial\_s^2 \gamma|^4 - |\partial\_s^3 \gamma|^2 + |\partial\_s \eth\_\sf \chi|^2 - \partial\_s^2 f\_{,\sf \partial\_s \chi} (\eth\_\sf \chi, \theta; \clubsuit \partial\_s \chi^0) \cdot \eth\_\sf \chi. \end{cases}$$

At the rest of this section, we prove that the orthogonal complement of V is given by Eq. (7).

Lemma 1. Let <sup>γ</sup>ðs, t<sup>Þ</sup> be a smooth curve in <sup>R</sup><sup>3</sup> and s∈S<sup>1</sup> <sup>L</sup> be an arc length parameter of γ for any t. Suppose that ∂<sup>2</sup> <sup>s</sup><sup>γ</sup> 6¼ <sup>0</sup> holds for all <sup>ð</sup>s, tÞ∈S<sup>1</sup> <sup>L</sup> � <sup>R</sup><sup>þ</sup> then the orthogonal complement V<sup>⊥</sup> of the space <sup>V</sup> ¼ fϕj∂s<sup>γ</sup> � <sup>∂</sup>s<sup>ϕ</sup> � <sup>0</sup><sup>g</sup> with respect to L<sup>2</sup> ðS1 <sup>L</sup>Þ inner product is represented by

$$V^\perp = \{ \eth\_\circ(w \eth\_\circ \gamma) w = w(\square, t) \text{ is a scalar function} \}.$$

Proof. Observe that ∂sγ, ∂<sup>2</sup> <sup>s</sup><sup>γ</sup> and the outer product <sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>2</sup> <sup>s</sup>γ are orthogonal each other. Under the assumption ∂<sup>2</sup> <sup>s</sup>γ 6¼ 0, a coordinate system defined on γ consists of the vectors. Then arbitrary vector η = η(s, t) can be represented as

$$
\eta(\mathbf{s},t) = \eta\_1(\mathbf{s},t)\partial\_{\mathbf{s}}\gamma(\mathbf{s},t) + \eta\_2(\mathbf{s},t)\,\partial\_{\mathbf{s}}^2\gamma(\mathbf{s},t) + \eta\_3(\mathbf{s},t)\,\partial\_{\mathbf{s}}\gamma(\mathbf{s},t) \times \partial\_{\mathbf{s}}^2\gamma(\mathbf{s},t).
$$

If we assume additionally η ∈ V, then we obtain

$$\begin{split} 0 &= \partial\_t \eta \cdot \partial\_s \eta \\ &= \left\{ \partial\_t \eta\_1 \partial\_s \gamma + \eta\_1 \partial\_s^2 \gamma + \partial\_s \eta\_2 \partial\_s^2 \gamma + \eta\_2 \partial\_s^3 \gamma + \partial\_s \eta\_3 \partial\_s \gamma \times \partial\_s^2 \gamma + \eta\_3 \left( \partial\_s^2 \gamma \times \partial\_s^2 \gamma + \partial\_s \gamma \times \partial\_s^3 \gamma \right) \right\} \cdot \partial\_t \gamma \\ &= \partial\_s \eta\_1 + \eta\_2 \partial\_s \gamma \cdot \partial\_s^3 \gamma \\ &= \left\| \partial\_t \eta\_1 - \eta\_2 \right\| \partial\_s^2 \gamma \right\|^2 \end{split}$$

that is

∂2 <sup>t</sup> <sup>γ</sup> <sup>þ</sup> <sup>∂</sup><sup>4</sup>

then Eq. (6) is equal to 0.

It follows from Eq. (8) that

Since from the definition

¼ 

we also obtain

∂2

<sup>¼</sup> <sup>∂</sup><sup>2</sup> s 

<sup>f</sup> , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup><sup>Þ</sup>

<sup>s</sup> <sup>f</sup> , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>Þ � <sup>∂</sup>s<sup>γ</sup>

<sup>0</sup> <sup>¼</sup> <sup>2</sup>∂<sup>2</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>s</sup> ðj∂<sup>2</sup> sγj 2 Þ�j∂<sup>3</sup> sγj <sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

� 

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

f�∂<sup>5</sup>

44 Shape Memory Alloys - Fundamentals and Applications

Differentiating Eq. (3), we obtain

<sup>s</sup><sup>γ</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

<sup>s</sup><sup>γ</sup> � <sup>∂</sup>sf , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>

<sup>2</sup> <sup>¼</sup> <sup>2</sup>∂s∂<sup>2</sup>

Þ þ <sup>∂</sup><sup>2</sup>

sγj 2 ,

> <sup>s</sup> ðj∂<sup>2</sup> sγj 2 Þþ j∂<sup>3</sup> sγj 2 :

> > <sup>s</sup><sup>w</sup> � <sup>w</sup>j∂<sup>2</sup>

3

3

<sup>2</sup> þ j∂s∂tγ<sup>j</sup>

sγj 2 :

þ ðθ � θcÞ

þ ðθ � θcÞ

<sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup> s 

<sup>∂</sup><sup>s</sup> <sup>γ</sup><sup>0</sup> � <sup>∂</sup><sup>s</sup> <sup>ð</sup><sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

 ∂sγ<sup>0</sup> � ∂sγ:

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup> <sup>5</sup>

∂sγ<sup>0</sup>

∂sγ<sup>0</sup>

� ∂sγ:

<sup>t</sup> γ � ∂sγ þ 2j∂s∂tγj

2

<sup>s</sup> ðw∂sγÞg � ∂sγ ¼ �j∂s∂tγj

Next, we derive the equation for w. From the assumption Eq. (3), we see that

<sup>t</sup> j∂sγj

<sup>s</sup> <sup>f</sup> , <sup>∂</sup>s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>

<sup>s</sup>γ ¼ 0,

<sup>s</sup><sup>γ</sup> ¼ �j∂<sup>2</sup>

<sup>s</sup><sup>γ</sup> ¼ � <sup>3</sup> 2 <sup>∂</sup>sðj∂<sup>2</sup> sγj 2 Þ,

<sup>s</sup><sup>γ</sup> ¼ � <sup>2</sup>∂<sup>2</sup>

By the relations, we will rewrite Eq. (9). It follows from the direct calculation that

<sup>s</sup> <sup>ð</sup>w∂sγÞ � <sup>∂</sup>s<sup>γ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>s</sup><sup>w</sup> � <sup>w</sup>j∂<sup>2</sup>

þ ðθ � θcÞ

3

sγj

<sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>3</sup>

<sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>4</sup>

<sup>∂</sup>s<sup>γ</sup> � <sup>∂</sup><sup>5</sup>

∂2

5 � 

5 � 

Therefore, substituting these into Eq. (9), we find

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup><sup>s</sup> <sup>ð</sup><sup>γ</sup> � <sup>γ</sup><sup>0</sup><sup>Þ</sup>

<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup>

Þ ¼ ∂sðw∂sγÞ ð8Þ

2

: ð9Þ

$$
\partial\_s \eta\_1 = |\partial\_s^2 \gamma|^2 \eta\_2. \tag{10}
$$

In other words, an element of V consists of η1, η<sup>2</sup> satisfying Eq. (10) and arbitrary η3. However, we remark that we cannot take η<sup>2</sup> freely. Indeed, in order to verify L periodicity of η1, η2, we have to show the following condition:

$$\int\_{0}^{L} |\partial\_{s}^{2}\boldsymbol{\gamma}|^{2} \eta\_{2} d\boldsymbol{s} = 0. \tag{11}$$

If <sup>ζ</sup>ðs, tÞ ¼ <sup>ζ</sup>1ðs, tÞ∂sγðs, tÞ þ <sup>ζ</sup>2ðs, tÞ∂<sup>2</sup> <sup>s</sup>γðs, tÞ þ <sup>ζ</sup>3ðs, tÞ∂sγðs, tÞ � <sup>∂</sup><sup>2</sup> <sup>s</sup>γðs, tÞsatisfies 〈ζ, η〉 = 0, then we see that η<sup>1</sup> and η<sup>2</sup> satisfy Eqs. (10) and (11) and any η<sup>3</sup> satisfies

$$\int\_{0}^{L} \{\zeta\_1 \eta\_1 + \zeta\_2 \eta\_2 |\partial\_s^2 \gamma|^2 + \zeta\_3 \eta\_3 (\partial\_s \gamma \times \partial\_s^2 \gamma)^2\} ds = 0. \tag{12}$$

In particular, if we assume η<sup>2</sup> � 0 and η<sup>3</sup> � 0, then we infer from Eq. (10) that η<sup>1</sup> � C holds true. Then, we deduce from Eq. (12) that

$$\int\_{0}^{L} \zeta\_1 ds = 0.$$

Now we define

$$
\varphi(\mathbf{s},t) = \zeta\_1(0,t) + \int\_0^s \zeta\_1(\mathbf{s},t)d\mathbf{s}.
$$

Then, ϕ has the period L and satisfies ∂sϕ ¼ ζ1. Substituting it into Eq. (12) and using Eq. (10), we have

$$\begin{split} 0 &= \int\_{0}^{L} \{\eth\_{s}\wp \: \eta\_{1} + \zeta\_{2}\eta\_{2} |\eth\_{s}^{2}\wp|^{2} + \zeta\_{3}\eta\_{3} (\eth\_{s}\wp \times \eth\_{s}^{2}\wp)^{2} \} ds \\ &= \int\_{0}^{L} \{-\wp \eth\_{s}\eta\_{1} + \zeta\_{2}\eta\_{2} |\eth\_{s}^{2}\wp|^{2} + \zeta\_{3}\eta\_{3} (\eth\_{s}\wp \times \eth\_{s}^{2}\wp)^{2} \} ds \\ &= \int\_{0}^{L} \{-\wp \: |\eth\_{s}^{2}\wp|^{2}\eta\_{2} + \zeta\_{2}\eta\_{2} |\eth\_{s}^{2}\wp|^{2} + \zeta\_{3}\eta\_{3} (\eth\_{s}\wp \times \eth\_{s}^{2}\wp)^{2} \} ds \\ &= \int\_{0}^{L} \{(-\wp + \zeta\_{2})\eta\_{2} |\eth\_{s}^{2}\wp|^{2} + \zeta\_{3}\eta\_{3} (\eth\_{s}\wp \times \eth\_{s}^{2}\wp)^{2} \} ds. \end{split}$$

Recalling Eq. (11), we see that the vector-valued function (η2, <sup>η</sup>3) is orthogonal with ðj∂<sup>2</sup> sγj 2 , 0Þ in the sense of L<sup>2</sup> inner product. Therefore, there exists some function μ = μ(t) depending only on t such that

$$\left( \left( -\varphi + \zeta\_2 \right) |\partial\_s^2 \mathcal{\gamma}|^2 , \left( \partial\_s \mathcal{\gamma} \times \partial\_s^2 \mathcal{\gamma} \right)^2 \zeta\_3 \right) = \mu (|\partial\_s^2 \mathcal{\gamma}|^2 , 0)^2$$

that is,

$$-\varphi + \zeta\_2 = \mu,\ \zeta\_3 \equiv 0.\tag{13}$$

Setting

$$
\mu + \varphi(\mathbf{s}, t) = w(\mathbf{s}, t)\_{\prime}
$$

the function w(s, t) is the L periodic function and satisfies ∂sw ¼ ζ1. It follows from Eq. (13) that

$$
\zeta\_2(s, t) = w(s, t).
$$

Then ζ(s) is orthogonal with elements of V with respect to L<sup>2</sup> inner product. Thus, we get

$$
\zeta(\mathbf{s},t) = \partial\_t w(\mathbf{s},t)\partial\_s \gamma(\mathbf{s},t) + w(\mathbf{s},t)\partial\_s^2 \gamma(\mathbf{s},t) = \partial\_t \left(w(\mathbf{s},t)\partial\_s \gamma(\mathbf{s})\right),
$$

which completes the proof.

Q.E.D.

If <sup>ζ</sup>ðs, tÞ ¼ <sup>ζ</sup>1ðs, tÞ∂sγðs, tÞ þ <sup>ζ</sup>2ðs, tÞ∂<sup>2</sup>

46 Shape Memory Alloys - Fundamentals and Applications

Then, we deduce from Eq. (12) that

0 ¼ ðL 0

> ¼ ðL 0

> ¼ ðL 0

> ¼ ðL 0

> > �

Now we define

we have

on t such that

that is,

Setting

ðL 0

we see that η<sup>1</sup> and η<sup>2</sup> satisfy Eqs. (10) and (11) and any η<sup>3</sup> satisfies

<sup>f</sup>ζ1η<sup>1</sup> <sup>þ</sup> <sup>ζ</sup>2η2j∂<sup>2</sup>

sγj

ðL 0

ϕðs, tÞ ¼ ζ1ð0, tÞ þ

sγj

sγj

sγj

<sup>2</sup> <sup>þ</sup> <sup>ζ</sup>3η3ð∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>s</sup>γÞ 2 ζ3 �

<sup>η</sup><sup>2</sup> <sup>þ</sup> <sup>ζ</sup>2η2j∂<sup>2</sup>

sγj

Recalling Eq. (11), we see that the vector-valued function (η2, <sup>η</sup>3) is orthogonal with ðj∂<sup>2</sup>

in the sense of L<sup>2</sup> inner product. Therefore, there exists some function μ = μ(t) depending only

, <sup>ð</sup>∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

μ þ ϕðs, tÞ ¼ wðs, tÞ,

the function w(s, t) is the L periodic function and satisfies ∂sw ¼ ζ1. It follows from Eq. (13) that

<sup>f</sup>∂sϕ η<sup>1</sup> <sup>þ</sup> <sup>ζ</sup>2η2j∂<sup>2</sup>

f�ϕ∂sη<sup>1</sup> <sup>þ</sup> <sup>ζ</sup>2η2j∂<sup>2</sup>

fð�<sup>ϕ</sup> <sup>þ</sup> <sup>ζ</sup>2Þη2j∂<sup>2</sup>

f�<sup>ϕ</sup> <sup>þ</sup> <sup>ζ</sup>2g j∂<sup>2</sup>

sγj 2

f�<sup>ϕ</sup> <sup>j</sup>∂<sup>2</sup> sγj 2

<sup>s</sup>γðs, tÞ þ <sup>ζ</sup>3ðs, tÞ∂sγðs, tÞ � <sup>∂</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>ζ</sup>3η3ð∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

In particular, if we assume η<sup>2</sup> � 0 and η<sup>3</sup> � 0, then we infer from Eq. (10) that η<sup>1</sup> � C holds true.

ζ1ds ¼ 0:

Then, ϕ has the period L and satisfies ∂sϕ ¼ ζ1. Substituting it into Eq. (12) and using Eq. (10),

ðs 0

<sup>2</sup> <sup>þ</sup> <sup>ζ</sup>3η3ð∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>ζ</sup>3η3ð∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>ζ</sup>3η3ð∂s<sup>γ</sup> � <sup>∂</sup><sup>2</sup>

<sup>s</sup>γÞ 2 gds:

<sup>¼</sup> <sup>μ</sup>ðj∂<sup>2</sup> sγj 2 , 0Þ

�ϕ þ ζ<sup>2</sup> ¼ μ, ζ<sup>3</sup> � 0: ð13Þ

ζ1ðs, tÞds:

<sup>s</sup>γÞ 2 gds

> <sup>s</sup>γÞ 2 gds

> > <sup>s</sup>γÞ 2 gds

> > > sγj 2 , 0Þ

<sup>s</sup>γÞ 2 <sup>s</sup>γðs, tÞsatisfies 〈ζ, η〉 = 0, then

gds ¼ 0: ð12Þ

Remark 1. The assumption ∂<sup>2</sup> <sup>s</sup>γ 6¼ 0 in Lemma 1 means that the curvature is always non-zero. If ∂2 <sup>s</sup>γ ¼ 0 at some point, we cannot determine the tangential vector ∂sγ at the point uniquely. Therefore, we need the assumption in order to give a coordinate system at every point of Γ. On the other hand, in the case of a planar curve, we do not need the assumption. Indeed, by rotating the tangential vector, we can construct a coordinate system.

Remark 2. We mention the elastic flow with the inextensible condition (Eq. (3)), more precisely, L<sup>2</sup> gradient flow for K(γ) under the constraint (Eq. (3)). To the best of our knowledge, the problem was first considered by N. Koiso [26] for planar closed curves. With the aid of smoothing effect of the elastic energy E, the Cauchy problem on the elastic flow has a unique classical solution and the solution converges to an equilibrium state as <sup>t</sup> ! <sup>∞</sup> in the <sup>C</sup>∞ topology. The result can be extended to the following case: (i) L<sup>2</sup> gradient flow for E under the area-preserving condition and (C) [27] and (ii) L<sup>2</sup> gradient flow for Tadjbakhsh-Odeh energy functional under the constraint (C) [28]. Moreover, the result [26] was also extended to the case of space curves [29].

#### 2.4. Derivation of heat equation

In this subsection, we study the energy law. We confirm thermal energy conservation law (the first law of thermodynamics) and the increasing law of entropy (the second law of thermodynamics). To begin with, we consider the first law of thermodynamics. According to Ref. [30], thermal energy conservation law for thermoelastic system is given by

$$
\theta \partial \partial \mathcal{S} + \nabla \cdot \boldsymbol{q} = \boldsymbol{h} \tag{14}
$$

where S, q and h are entropy, thermal velocity and external heat, respectively. In our setting, thermal transfer occurs only on wire, and the wire does not expand. Then, we may regard ∇ � q as ∂sq as the same as one-dimensional case, where s is necessary to be arc length parameter. By the same reason, the Fourier law q = ∇θ is replaced by

$$
\mathfrak{q} = -\mathfrak{d}\_s \theta. \tag{15}
$$

The Helmholtz free energy density

$$\begin{aligned} \tilde{f} &= \tilde{f} \left( \partial\_s^2 \boldsymbol{\gamma} \,\,\,\,\partial\_s \boldsymbol{\gamma} \,\,\,\theta \,\, ; \boldsymbol{\gamma}^0 \right) \\ &= \frac{1}{2} |\partial\_s^2 \boldsymbol{\gamma}|^2 + f(\partial\_s \boldsymbol{\gamma} \,\,\,\theta \,\, ; \boldsymbol{\gamma}^0) + f\_0(\theta) \end{aligned}$$

and the entropy S are connected with the relation

$$\mathcal{S} = -\frac{\partial \bar{\mathcal{Y}}}{\partial \theta}.$$

Then, the conservation law, Eq. (14), is rewritten as follows:

$$-\left\|f\_0^{''}(\theta)\,\partial\_t\theta\,-\partial\_s^2\theta = \theta\left(\partial\_t\gamma^0\cdot\partial\_s(\gamma-\gamma^0)\right)(\partial\_t\partial\_s\gamma\cdot\partial\_s\gamma^0)+h.\tag{16}$$

We note that the Clausius-Duhem inequality holds automatically:

$$
\partial\_t S + \partial\_s \left( \frac{q}{\Theta} \right) \ge \frac{h}{\Theta} \quad \dots
$$

Indeed, we observe from Eqs. (14) and (15) that

$$\begin{aligned} \left(\partial\_t S + \partial\_s \left(\frac{q}{\theta}\right) - \frac{h - \partial\_s q}{\theta} + \partial\_s \left(\frac{q}{\theta}\right) \right) \\ &= \frac{h}{\theta} - \frac{q \partial\_s \Theta}{\theta^2} \\ &= \frac{h}{\theta} + |\frac{\partial\_s \Theta}{\theta}|^2 \succeq \frac{h}{\Theta} \dots \end{aligned}$$

The Clausius-Duhem inequality corresponds to the second law of thermodynamics. For more precise information of the inequality, we refer to, for example, 1.11 of chapter 4 in Ref. [2]. Here, we assume external heat source h = 0 and adopt the well-known form:

$$f\_0(\theta) = \theta - \theta \log \theta.$$

Then since f <sup>0</sup> 0 0ðθ޼�1=θ, Eq. (16) is reduced to

$$
\partial\_t \partial - \partial\_s^2 \theta = \Theta \Big(\eth\_s \gamma^0 \cdot \eth\_s (\gamma - \gamma^0)\Big) (\eth\_t \eth\_s \gamma \cdot \eth\_s \gamma^0).
$$

We thus obtain the system of equation as Eq. (4).

#### 3. Dual variation structure

Let us rewrite Eq. (4):

$$
\partial\_t^2 \gamma + \partial\_s^4 \gamma \; \, + \, \partial\_s f\_{,\partial\_t \gamma} (\partial\_s \gamma \; \; \; \theta \colon \partial\_s \gamma^0) - \partial\_s \{ (\upsilon - 2 \left| \partial\_s^2 \gamma \right|^2) \partial\_s \gamma \} = 0,\tag{17}
$$

$$-\left|\partial\_s^2 \boldsymbol{\nu} + |\partial\_s^2 \boldsymbol{\gamma}|^2 \boldsymbol{\nu} = 2|\partial\_s^2 \boldsymbol{\gamma}|^4 - |\partial\_s^3 \boldsymbol{\gamma}|^2 + |\partial\_s \partial\_t \boldsymbol{\gamma}|^2 + \partial\_s^2 f\_{,\mathbf{d}\_\circ \boldsymbol{\gamma}}(\partial\_s \boldsymbol{\gamma}, \boldsymbol{\theta}; \partial\_s \boldsymbol{\gamma}^0) \cdot \partial\_s \boldsymbol{\gamma}|\right| \tag{18}$$

$$
\partial\_t \partial\_t - \partial\_s^2 \theta = \theta \Big(\partial\_s \gamma^0 \cdot \partial\_s (\gamma - \gamma^0)\Big) (\partial\_t \partial\_s \gamma \cdot \partial\_s \gamma^0), \ (t, s) \in (0, T) \times S^1\_{\mathcal{L}}\tag{19}
$$

$$\gamma(0,\mathbf{s}) = \gamma\_0(\mathbf{s}),\ \mathbf{\partial}\gamma(0,\mathbf{s}) = \gamma\_{1'}\ \theta(0,\mathbf{s}) = \theta\_{0'}\ \mathbf{s} \in \mathbf{S}^1\_L. \tag{20}$$

In this section, we show that the problem in Eqs. (17)–(20) has also dual variation structure as well as the problem (Eq. (2)). The structure plays an important role to prove the dynamical stability of infinitesimally stable stationary state.

We assume that the system has sufficiently smooth solution (γ, θ) satisfying θ > 0. Then initial data also has to satisfy

$$|\eth\_{\sf s} \mathcal{V}\_0| = 1,\ \eth\_{\sf s} \mathcal{V}\_0 \cdot \eth\_{\sf s} \mathcal{V}\_1 = 0.$$

Setting

<sup>~</sup><sup>f</sup> <sup>¼</sup> <sup>~</sup><sup>f</sup> <sup>ð</sup>∂<sup>2</sup>

¼ 1 2 j∂2 <sup>s</sup>γ j

Then, the conservation law, Eq. (14), is rewritten as follows:

<sup>0</sup>ðθ<sup>Þ</sup> <sup>∂</sup>t<sup>θ</sup> � <sup>∂</sup><sup>2</sup>

q θ 

We note that the Clausius-Duhem inequality holds automatically:

<sup>s</sup>θ ¼ θ 

<sup>¼</sup> <sup>h</sup> � <sup>∂</sup>sq

¼ h <sup>θ</sup> � <sup>q</sup>∂s<sup>θ</sup> θ2

¼ h <sup>θ</sup> þ j <sup>∂</sup>s<sup>θ</sup> θ j 2 ≥ h θ :

ðθ޼�1=θ, Eq. (16) is reduced to

<sup>s</sup>θ ¼ θ 

<sup>s</sup><sup>γ</sup> <sup>þ</sup> <sup>∂</sup>sf ,∂s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>

<sup>∂</sup>t<sup>θ</sup> � <sup>∂</sup><sup>2</sup>

We thus obtain the system of equation as Eq. (4).

Here, we assume external heat source h = 0 and adopt the well-known form:

∂tS þ ∂<sup>s</sup>

<sup>θ</sup> <sup>þ</sup> <sup>∂</sup><sup>s</sup>

and the entropy S are connected with the relation

� θ f 00

48 Shape Memory Alloys - Fundamentals and Applications

Indeed, we observe from Eqs. (14) and (15) that

∂tS þ ∂<sup>s</sup>

Then since f <sup>0</sup>

0 0

3. Dual variation structure

∂2 <sup>t</sup> <sup>γ</sup> <sup>þ</sup> <sup>∂</sup><sup>4</sup>

<sup>s</sup> <sup>v</sup> þ j∂<sup>2</sup> sγj 2 <sup>v</sup> <sup>¼</sup> <sup>2</sup>j∂<sup>2</sup> sγj <sup>4</sup> � j∂<sup>3</sup> sγj

Let us rewrite Eq. (4):

� <sup>∂</sup><sup>2</sup>

<sup>s</sup>γ, ∂sγ, θ ; γ<sup>0</sup>

Þ

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup>

q θ ≥ h θ :

q θ 

The Clausius-Duhem inequality corresponds to the second law of thermodynamics. For more precise information of the inequality, we refer to, for example, 1.11 of chapter 4 in Ref. [2].

f <sup>0</sup>ðθÞ ¼ θ � θlogθ:

<sup>∂</sup>sγ<sup>0</sup> � <sup>∂</sup>sð<sup>γ</sup> � <sup>γ</sup><sup>0</sup>

Þ 

Þ � <sup>∂</sup>sfð<sup>v</sup> � <sup>2</sup> <sup>j</sup>∂<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>∂</sup><sup>2</sup>

<sup>2</sup> þ j∂s∂tγ<sup>j</sup>

<sup>ð</sup>∂t∂s<sup>γ</sup> � <sup>∂</sup>sγ<sup>0</sup>

sγj 2

<sup>s</sup> <sup>f</sup> ,∂s<sup>γ</sup>ð∂sγ, <sup>θ</sup>; <sup>∂</sup>sγ<sup>0</sup>

Þ∂sγg ¼ 0, ð17Þ

Þ � ∂sγ, ð18Þ

Þ:

Þ þ f <sup>0</sup>ðθÞ

Þ 

<sup>ð</sup>∂t∂s<sup>γ</sup> � <sup>∂</sup>sγ<sup>0</sup>

Þ þ h: ð16Þ

<sup>2</sup> <sup>þ</sup> <sup>f</sup>ð∂sγ, <sup>θ</sup> ; <sup>γ</sup><sup>0</sup>

<sup>S</sup> ¼ � <sup>∂</sup>~<sup>f</sup> ∂θ :

$$\begin{aligned} f\_1(\mathfrak{d}\_\mathfrak{d}\boldsymbol{\gamma}) &:= \frac{1}{2} \left( \frac{\partial \boldsymbol{\gamma}^0}{\partial \mathbf{s}} \cdot \frac{\partial (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0)}{\partial \mathbf{s}} \right)^2, \\ f\_2(\mathfrak{d}\_\mathbf{\bar{s}}\boldsymbol{\gamma}) &:= \frac{1}{6} \left( \frac{\partial \boldsymbol{\gamma}^0}{\partial \mathbf{s}} \cdot \frac{\partial (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0)}{\partial \mathbf{s}} \right)^6 - \frac{1}{4} \left( \frac{\partial \boldsymbol{\gamma}^0}{\partial \mathbf{s}} \cdot \frac{\partial (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0)}{\partial \mathbf{s}} \right)^4 - \frac{\theta\_c}{2} \left( \frac{\partial \boldsymbol{\gamma}^0}{\partial \mathbf{s}} \cdot \frac{\partial (\boldsymbol{\gamma} - \boldsymbol{\gamma}^0)}{\partial \mathbf{s}} \right)^2, \end{aligned}$$

we have the relation f ¼ θf <sup>1</sup> þ f <sup>2</sup>. Multiplying Eq. (17) by ∂tγ and integrating it with respect to s, we obtain

$$\frac{d}{dt}\left(\frac{1}{2}||\partial\_t\gamma||\_{L^2}^2 + \frac{1}{2}||\partial\_s^2\gamma||\_{L^2}^2\right) = -\langle f\_{,\partial\_t\gamma'},\partial\_t\partial\_{t'}\gamma\rangle = -\frac{d}{dt}\int\_0^L f\_2(\partial\_s\gamma)ds - \int\_0^L \theta\partial\_t f\_1(\partial\_s\gamma)ds.$$

Integrating Eq. (19), we find

$$\frac{d}{dt}\int\_0^L \theta ds = \int\_0^L \theta \partial\_t f\_1(\partial\_s \gamma) ds.$$

Then for the quantity

$$E(\boldsymbol{\gamma}, \partial\_t \boldsymbol{\gamma}, \boldsymbol{\theta}) := \frac{1}{2} ||\partial\_t \boldsymbol{\gamma}||\_{L^2}^2 + \frac{1}{2} ||\partial\_s^2 \boldsymbol{\gamma}||\_{L^2}^2 + \int\_0^L \boldsymbol{\theta} d\mathbf{s} + \int\_0^L f\_2(\partial\_t \boldsymbol{\gamma}) d\mathbf{s} \,\boldsymbol{\zeta}$$

it holds that

$$\frac{d}{dt}E(\boldsymbol{\gamma}, \partial\_t \boldsymbol{\gamma}, \boldsymbol{\theta}) = 0.$$

Moreover, for the quantity

$$W(\eth\_s \gamma, \theta) := \int\_0^L \{f\_1(\eth\_s \gamma) - \log \theta\} ds$$

we deduce from Eq. (19) that

$$\frac{d}{dt}W(\partial\_s \gamma, \theta) = \int\_0^L \frac{\partial\_s^2 \theta}{\theta} ds = -\int\_0^L |\frac{\partial\_s \theta}{\theta}|^2 ds \le 0.1$$

Finally, we confirm an important structure of the system (17)–(20). We denote the stationary state of θ by θ > 0, and the corresponding equilibrium by γ satisfy the following constraint:

$$b \equiv E(\gamma\_{0'} \; \gamma\_{1'} \; \; \; \theta\_0) = E(\gamma\_{\prime} \; \; \; 0, \; \overline{\theta}),$$

$$\partial\_s^4 \gamma = \eth\_s \{ \overline{\theta} f\_{1, \flat\_{\ast} \; \prime} + f\_{2, \flat\_{\ast} \; \prime} \}.$$

Eliminating θ by the relation

$$b = L\overline{\theta} + \frac{1}{2}||\partial\_s^2 \gamma||\_{L^2}^2 + \int\_0^L f\_2(\partial\_s \gamma) \, ds$$

the stationary state of this problem satisfies the following nonlinear nonlocal problem:

$$\mathfrak{d}\_s^4 \gamma = \mathfrak{d}\_s \left\{ \frac{1}{L} \left( b - \frac{1}{2} ||\partial\_s^2 \gamma||\_{L^2}^2 - \int\_0^L f\_2(\mathfrak{d}\_s \gamma) ds \right) f\_{1, \mathfrak{d}\_\uparrow \gamma}(\mathfrak{d}\_s \gamma) + f\_{2, \mathfrak{d}\_\uparrow \gamma}(\mathfrak{d}\_s \gamma) \right\}. \tag{21}$$

Eq. (21) is derived as the Euler-Lagrange equation of the functional

$$J\_b(y) := \frac{1}{L} \int\_0^L f\_1(y)ds - \log\left(b - \frac{1}{2}||\partial\_s y||\_{L^2}^2 - \int\_0^L f\_2(y)ds\right) + \log L$$

where <sup>y</sup> <sup>¼</sup> <sup>∂</sup>sγ<sup>∈</sup> <sup>H</sup><sup>2</sup> ðS1 L, S<sup>2</sup> Þ and

$$\mathcal{S}^2 := \{ \omega \in \mathbb{R}^3 | |\omega| = 1 \}.$$

We remark that the following relation between Jb and W holds true:

$$W(\eth\_s \gamma, \theta) \ge L J\_b(\eth\_s \gamma).$$

The relation is called semi-unfolding minimality. Thus, if (γ, θ) is a non-stationary state, <sup>b</sup> <sup>¼</sup> <sup>E</sup>ðγ0, <sup>γ</sup>1, <sup>θ</sup>0Þ, <sup>y</sup> <sup>¼</sup> <sup>∂</sup>s<sup>γ</sup> is a linearized stable critical point of Jb <sup>¼</sup> JbðyÞ, y<sup>∈</sup> <sup>H</sup><sup>1</sup> ðS1 L, S<sup>2</sup> Þ and θ > 0 is a constant satisfying Eðγ, 0, θÞ ¼ b, then it holds that for y ¼ ∂sγ

$$J\_b(\underline{y}) - J\_b(\overline{\underline{y}}) \le W(\partial\_s \underline{\gamma}\_{0'}, \; \Theta\_0) - W(\overline{\underline{y}}, \overline{\underline{\Theta}}) .$$

By this structure, we can infer that any infinitesimally stable stationary state is dynamically stable, that is, stable in the Lyapunov sense. A critical point <sup>y</sup> <sup>¼</sup> <sup>∂</sup>s<sup>γ</sup> of Jb for <sup>γ</sup><sup>∈</sup> <sup>H</sup><sup>2</sup> ðS1 L, S<sup>2</sup> Þ is infinitesimally stable if there exists ε<sup>0</sup> > 0 such that any ε<sup>1</sup> ∈ 0, <sup>ε</sup><sup>0</sup> 2 � � admits δ<sup>0</sup> > 0 such that if jj∂sðγ � γÞjjH<sup>1</sup> < ε<sup>0</sup> and Jbð∂sγÞ � Jbð∂sγÞ < δ<sup>0</sup> then

$$||\mathfrak{d}\_{\mathfrak{s}}(\mathcal{Y}-\overline{\mathcal{Y}})||\_{H^{4}} < \varepsilon\_{1}.$$

The definition of infinitesimally stable is obviously weaker than the one of well-known linearized stable which means that for a critical point y ¼ ∂sγ of Jb, the quadratic form

$$Q\_{\mathfrak{y}}(w, w) = \frac{d}{d\varepsilon^2} J\_b(\overline{y} + \varepsilon w)|\_{\varepsilon = 0}$$

is a positive definite for any w ∈ H<sup>2</sup> ðS1 L, S<sup>2</sup> Þ.

Theorem 1. Assume that θ > 0 is a constant and that γ is an infinitesimally stable critical point of Jb with constraint ∂sγ∈ H<sup>2</sup> ðS1 L, S<sup>2</sup> Þ. Then ðγ, θÞ is a dynamically stable in the sense that for any ε > 0, there exists δ > 0 such that if

$$E(\boldsymbol{\gamma}\_{0'}, \boldsymbol{\gamma}\_{1'}, \theta\_0) = |b\_\prime| |\partial\_t(\boldsymbol{\gamma}\_0 - \overline{\boldsymbol{\gamma}})|\_{\mathcal{H}^1} < \delta,\\ \left|\frac{1}{L}\right|\_0^L \log \theta\_0(\mathbf{s}) d\mathbf{s} - \log \overline{\theta}| < \delta \tag{22}$$

then

d

50 Shape Memory Alloys - Fundamentals and Applications

Eliminating θ by the relation

∂4 <sup>s</sup>γ ¼ ∂<sup>s</sup>

where <sup>y</sup> <sup>¼</sup> <sup>∂</sup>sγ<sup>∈</sup> <sup>H</sup><sup>2</sup>

1 <sup>L</sup> <sup>b</sup> � <sup>1</sup> 2 jj∂2 sγjj<sup>2</sup> <sup>L</sup><sup>2</sup> � ðL 0

Jbðy<sup>Þ</sup> :<sup>¼</sup> <sup>1</sup> L ðL 0

> ðS1 L, S<sup>2</sup> Þ and

dt <sup>W</sup>ð∂sγ, <sup>θ</sup>Þ ¼

∂4

b ¼ Lθ þ

Eq. (21) is derived as the Euler-Lagrange equation of the functional

We remark that the following relation between Jb and W holds true:

1 2 jj∂2 sγjj<sup>2</sup> <sup>L</sup><sup>2</sup> þ ðL 0

� �

<sup>f</sup> <sup>1</sup>ðyÞds � log <sup>b</sup> � <sup>1</sup>

<sup>S</sup><sup>2</sup> :¼ f<sup>ω</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup>

<sup>b</sup> <sup>¼</sup> <sup>E</sup>ðγ0, <sup>γ</sup>1, <sup>θ</sup>0Þ, <sup>y</sup> <sup>¼</sup> <sup>∂</sup>s<sup>γ</sup> is a linearized stable critical point of Jb <sup>¼</sup> JbðyÞ, y<sup>∈</sup> <sup>H</sup><sup>1</sup>

θ > 0 is a constant satisfying Eðγ, 0, θÞ ¼ b, then it holds that for y ¼ ∂sγ

infinitesimally stable if there exists ε<sup>0</sup> > 0 such that any ε<sup>1</sup> ∈ 0, <sup>ε</sup><sup>0</sup>

jj∂sðγ � γÞjjH<sup>1</sup> < ε<sup>0</sup> and Jbð∂sγÞ � Jbð∂sγÞ < δ<sup>0</sup> then

the stationary state of this problem satisfies the following nonlinear nonlocal problem:

f <sup>2</sup>ð∂sγÞds

2 jj∂syjj<sup>2</sup> <sup>L</sup><sup>2</sup> � ðL 0

Wð∂sγ, θÞ ≥ LJbð∂sγÞ:

The relation is called semi-unfolding minimality. Thus, if (γ, θ) is a non-stationary state,

JbðyÞ � JbðyÞ ≤ Wð∂sγ0, θ0Þ � Wðy, θÞ:

By this structure, we can infer that any infinitesimally stable stationary state is dynamically

stable, that is, stable in the Lyapunov sense. A critical point <sup>y</sup> <sup>¼</sup> <sup>∂</sup>s<sup>γ</sup> of Jb for <sup>γ</sup><sup>∈</sup> <sup>H</sup><sup>2</sup>

� �

jjωj ¼ 1g:

� �

ðL 0 ∂2 sθ <sup>θ</sup> ds ¼ �

Finally, we confirm an important structure of the system (17)–(20). We denote the stationary state of θ by θ > 0, and the corresponding equilibrium by γ satisfy the following constraint:

b � Eðγ0, γ1, θ0Þ ¼ Eðγ, 0, θÞ,

<sup>s</sup>γ ¼ ∂sfθf <sup>1</sup>,∂s<sup>γ</sup> þ f <sup>2</sup>, <sup>∂</sup>s<sup>γ</sup>g:

ðL 0 j ∂sθ θ j 2 ds ≤ 0:

f <sup>2</sup>ð∂sγÞ ds

f <sup>1</sup>, <sup>∂</sup>s<sup>γ</sup>ð∂sγÞ þ f <sup>2</sup>,∂s<sup>γ</sup>ð∂sγÞ

f <sup>2</sup>ðyÞds

2

þ logL

: ð21Þ

ðS1 L, S<sup>2</sup>

> ðS1 L, S<sup>2</sup> Þ is

� � admits δ<sup>0</sup> > 0 such that if

Þ and

$$\sup\_{t \ge 0} ||\eth\_\circ(\gamma(t) - \overline{\gamma})||\_{H^1} < \varepsilon,\\ \left| \frac{1}{L} \int\_0^L \log \theta(t, s) ds - \log \overline{\theta} \right| < \varepsilon.$$

Proof. We first show the semi-unfolding minimality. From the energy conservation law, we see that

$$b\_- = \frac{1}{2}||\partial\_t \gamma||\_{L^2}^2 + \frac{1}{2}||\partial\_s^2 \gamma||\_{L^2}^2 + \int\_0^L \theta ds + \int\_0^L f\_2(\partial\_s \gamma) ds.$$

It follows from the Jensen inequality that

$$\frac{1}{L} \int\_0^L \log \theta ds \le \log \left( \frac{1}{L} \int\_0^L \theta ds \right).$$

Then we have

$$\begin{aligned} &\mathcal{W}(\mathfrak{d}\_{\sf s}\boldsymbol{\gamma},\ \boldsymbol{\Theta}) \geq \int\_{0}^{L} f\_{1}(\mathfrak{d}\_{\sf s}\boldsymbol{\gamma}) \mathrm{d}s - \log\left(\frac{1}{L} \int\_{0}^{L} \boldsymbol{\Theta} \mathrm{d}s\right) \\ &\geq \int\_{0}^{L} f\_{1}(\mathfrak{d}\_{\sf s}\boldsymbol{\gamma}) \mathrm{d}s - L \log\frac{1}{L} \left(b - \frac{1}{2}||\mathfrak{d}\_{\sf s}\boldsymbol{y}||\_{\boldsymbol{L}^{2}}^{2} - \int\_{0}^{L} f\_{2}(\boldsymbol{y}) \mathrm{d}s\right) \\ &= L f\_{b}(\mathfrak{d}\_{\sf s}\boldsymbol{\gamma}). \end{aligned}$$

We have thus completed to show the semi-unfolding minimality. Recall that γ∈ H<sup>2</sup> ðS1 L, S<sup>2</sup> Þ is an infinitesimally stable critical point of Jbð∂sγÞ. Thus, we find <sup>ε</sup><sup>0</sup> > 0such that any <sup>ε</sup><sup>1</sup> <sup>∈</sup> <sup>ð</sup>0, <sup>ε</sup><sup>0</sup> <sup>2</sup> � admits δ<sup>0</sup> > 0 such that if jj∂sðγ � γÞjjH<sup>1</sup> < ε<sup>0</sup> and Jbð∂sγÞ � Jbð∂sγÞ < δ<sup>0</sup> then

$$||\mathfrak{d}\_{\mathfrak{s}}(\mathcal{Y}-\overline{\mathcal{Y}})||\_{H^{l}} < \varepsilon\_{1}.\tag{23}$$

From the above properties, it holds that

$$J\_b\left(\partial\_s \boldsymbol{\gamma}(t)\right) \le \frac{1}{L} \mathcal{W}\left(\partial\_s \boldsymbol{\gamma}(t), \ \boldsymbol{\Theta}(t)\right) \le \frac{1}{L} \mathcal{W}(\partial\_s \boldsymbol{\gamma}\_{0'}, \ \boldsymbol{\Theta}\_0). \tag{24}$$

Moreover, for the constant θ > 0, we obtain

$$\mathcal{W}(\partial\_s \overline{\gamma}, \ \overline{\theta}) = L \Big[ \frac{1}{L} \int\_0^L f\_1(\partial\_s \underline{\gamma}) ds \ - \log \frac{1}{L} \left( b - \frac{1}{2} ||\partial\_s \overline{y}||\_{L^2}^2 - \int\_0^L f\_2(\overline{y}) ds \right) \right] = L f\_b(\partial\_s \ \overline{\gamma}).$$

namely,

$$J\_b(\partial\_s \overline{\gamma}) = \frac{1}{L} \mathcal{W}(\partial\_s \overline{\gamma}, \overline{\theta}). \tag{25}$$

Given ε > 0, setting δ∈ 0, <sup>ε</sup><sup>0</sup> 2 � � and satisfying Eq. (22), we have

$$\begin{split} \frac{1}{L}|\mathcal{W}(\eth\_{\eth}\mathcal{V}\_{0^{\square}},\theta\_{0}) - \mathcal{W}(\eth\_{\eth}\overline{\mathcal{V}},\overline{\theta})| &\leq \frac{1}{L} \Big( \|f\_{1,\operatorname{\bf b}\underline{\mathcal{V}}}(\eth\_{\eth}\mathcal{V}\_{0})\|\_{L^{-}} + \|f\_{1,\operatorname{\bf b}\underline{\mathcal{V}}}(\eth\_{\eth}\overline{\mathcal{V}})\|\_{L^{-}} \Big) \|\eth\_{\eth}(\mathcal{V}\_{0} - \overline{\mathcal{V}})\|\_{L^{1}} \\ &+ |\frac{1}{L}\Big| \int\_{0}^{L} \log \theta\_{0} ds - \log \overline{\theta}| < \min(\delta\_{0}, \varepsilon). \end{split} \tag{26}$$

Therefore, it follows from Eq. (24) to Eq. (26) that

$$J\_b\left(\mathfrak{d}\_s\boldsymbol{\gamma}(t)\right) - J\_b(\mathfrak{d}\_s\overline{\boldsymbol{\gamma}}) = \frac{1}{L}\left(W(\mathfrak{d}\_s\boldsymbol{\gamma}\_{0'}\ \boldsymbol{\theta}\_0) - W(\mathfrak{d}\_s\overline{\boldsymbol{\gamma}}, \overline{\boldsymbol{\theta}})\right) < \delta\_0.$$

If jj∂sðγðtÞ � γÞjjH<sup>1</sup> ¼ δð ≤ ε0=2 < ε0Þ, then we apply Eq. (23) for ε<sup>1</sup> = δ, and hence

$$\|\partial\_s \left( \mathcal{V}(t) - \overline{\mathcal{V}} \right)\|\_{H^1} < \delta\_\prime$$

which is a contradiction. Thus, we have

$$\|\partial\_s \left( \mathcal{V}(t) - \overline{\mathcal{V}} \right)\|\_{H^1} \neq \delta.$$

Here, from <sup>γ</sup><sup>∈</sup> <sup>C</sup>ð½0, <sup>∞</sup>Þ; H<sup>2</sup> � and k∂sðγ<sup>0</sup> � γÞkH<sup>1</sup> < δ, it follows that

$$||\eth\_s(\gamma(t) - \overline{\gamma})||\_{\mathcal{H}^1} < \delta\tag{27}$$

for any t ≥ 0.

From the semi-unfolding minimality (Eqs. (24) and (25)) and the linearized stability of Jb, we observe that

$$\mathcal{W}(\eth\_{\sf s}\mathcal{V}, \theta) \ge Lf\_b(\eth\_{\sf s}\mathcal{V}) \ge Lf\_b(\eth\_{\sf s}\overline{\mathcal{V}}) = \mathcal{W}(\eth\_{\sf s}\overline{\mathcal{V}}, \theta).$$

Then, combining Eq. (26) with Eq. (27), we have

$$\begin{aligned} \left| \frac{1}{L} \int\_0^L \log \theta(t,s)ds - \log \overline{\theta} \right| &\leq \frac{1}{L} \left( W(\eth\_s \gamma, \theta) - W(\eth\_s \overline{\gamma}, \overline{\theta}) \right) + \frac{1}{L} \left| \int\_0^L \{f\_1(\eth\_s \gamma) - f\_1(\eth\_s \overline{\gamma})\} ds \right| \\ &\leq \frac{1}{L} \left( W(\eth\_s \gamma\_{0'}, \theta\_0) - W(\eth\_s \overline{\gamma}, \overline{\theta}) \right) + \mathbb{C} ||\eth\_s (\gamma - \overline{\gamma})||\_{H^1} \leq \varepsilon + \mathbb{C} \delta \leq 2\varepsilon \end{aligned}$$

where δ is small enough such that δ < ε / C. This completes the proof.

Q.E.D.

jj∂sðγ � γÞjjH<sup>1</sup> < ε1: ð23Þ

f <sup>2</sup>ðyÞds

<sup>L</sup> <sup>W</sup>ð∂sγ, <sup>θ</sup>Þ: <sup>ð</sup>25<sup>Þ</sup>

�

� < δ0:

jjH<sup>1</sup> < δ ð27Þ

k∂sðγ<sup>0</sup> � γÞkL<sup>1</sup>

ð26Þ

<sup>L</sup> <sup>W</sup>ð∂sγ0, <sup>θ</sup>0Þ: <sup>ð</sup>24<sup>Þ</sup>

¼ LJbð∂<sup>s</sup> γÞ,

∂sγðtÞ, θðtÞ

<sup>L</sup> <sup>b</sup> � <sup>1</sup> 2 jj∂syjj<sup>2</sup> <sup>L</sup><sup>2</sup> � ðL 0

� � � �

� ≤ 1

kf <sup>1</sup>, <sup>∂</sup>s<sup>γ</sup>ð∂sγ0ÞkL<sup>∞</sup> þ kf <sup>1</sup>, <sup>∂</sup>s<sup>γ</sup>ð∂sγÞkL<sup>∞</sup>

Wð∂sγ0, θ0Þ � Wð∂sγ, θÞ

logθ0ds � logθj < minðδ0, εÞ:

From the above properties, it holds that

52 Shape Memory Alloys - Fundamentals and Applications

Jb � ∂sγðtÞ � ≤ 1 L W �

Moreover, for the constant θ > 0, we obtain

L ðL 0

2

þj 1 L ðL 0

Therefore, it follows from Eq. (24) to Eq. (26) that

<sup>f</sup> <sup>1</sup>ð∂sγÞds � log <sup>1</sup>

1 L �

� Jbð∂sγÞ ¼ <sup>1</sup>

L �

γðtÞ � γ

γðtÞ � γ

γðtÞ � γ

�

�

and k∂sðγ<sup>0</sup> � γÞkH<sup>1</sup> < δ, it follows that

From the semi-unfolding minimality (Eqs. (24) and (25)) and the linearized stability of Jb, we

�

kH<sup>1</sup> < δ,

kH<sup>1</sup> 6¼ δ:

If jj∂sðγðtÞ � γÞjjH<sup>1</sup> ¼ δð ≤ ε0=2 < ε0Þ, then we apply Eq. (23) for ε<sup>1</sup> = δ, and hence

k∂s �

k∂s �

jj∂s �

Jbð∂sγÞ ¼ <sup>1</sup>

� � and satisfying Eq. (22), we have

<sup>W</sup>ð∂sγ, <sup>θ</sup>Þ ¼ <sup>L</sup> <sup>1</sup>

Given ε > 0, setting δ∈ 0, <sup>ε</sup><sup>0</sup>

<sup>L</sup> <sup>j</sup>Wð∂sγ0, <sup>θ</sup>0Þ � <sup>W</sup>ð∂sγ, <sup>θ</sup>Þj <sup>≤</sup>

Jb � ∂sγðtÞ �

which is a contradiction. Thus, we have

�

Here, from <sup>γ</sup><sup>∈</sup> <sup>C</sup>ð½0, <sup>∞</sup>Þ; H<sup>2</sup>

for any t ≥ 0.

observe that

namely,

1

Remark 3. Both the existence of solution for evolution equations (Eq. (4)) and non-trivial solutions for stationary problem (Eq. (21)) are open problems. In the straight material case (i.e. the problem (2)), smooth solution for Eq. (2) is assured in [11] (we also refer to chapter 5 in [2]). The existence results of non-trivial solution for stationary problem (Eq. (21)) in lowtemperature and low-energy cases can be found in [21, 31].

#### 4. Concluding remarks

In this chapter, we propose the new mathematical model describing the movement of wire made of shape memory alloys. The derived system of nonlinear partial differential equations is a thermoelastic system with phase transition and non-stretching constraint. The Falk model (Eq. (2)) represents the dynamics for crystal as a stack of layers, whose displacement is restricted to move only on one direction. On the other hand, our model describes the dynamics of wire. We emphasize that our model allows the displacement of each direction. Thus, our model may describe a more realistic motion of wire made of shape memory alloys. Moreover, it is also interesting to regard our model as a mathematical problem on elastic curve with heat conduction. To the best of our knowledge, there is no result considering such a mathematical problem.

We mention the mathematical contribution of the present chapter. We prove the dynamical stability of an infinitesimally stable stationary state by finding the dual variation structure in our model. This property shall be applicable, for example, to assure the strength of not only a wire in an original shape but also a deformed wire. Indeed, in the straight material case, namely in the Falk model (Eq. (2)), numerical simulation shows the stability in this sense (see [17]).

#### Acknowledgements

This work was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) and (C), Grant Numbers 26247013 and 16K05234.

#### Author details

Shinya Okabe<sup>1</sup> , Takashi Suzuki<sup>2</sup> \* and Shuji Yoshikawa3


#### References


[11] Sprekels J, Zheng S. Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Journal of Physics D. 1989;39:59–76

Author details

, Takashi Suzuki<sup>2</sup>

54 Shape Memory Alloys - Fundamentals and Applications

\*Address all correspondence to: suzuki@sigmath.es.osaka-u.ac.jp

3 Faculty of Science and Technology, Oita University, Oita, Japan

2 Graduate School of Engineering Science, Osaka University, Osaka, Japan

1 Mathematical Institute, Tohoku University, Sendai, Japan

121st ed. New York: Springer-Verlag; 1996

p. 480. DOI: 10.1007/978-3-662-04800-9

alloys. Physical Review B. 1987;36:3031–3041

alloys. Journal of Differential Equations. 1995;121:203–231

DOI: 10.1007/978-3-7091-4348-3

\* and Shuji Yoshikawa3

[1] Falk F. Elastic phase transitions and nonconvex energy functions. In: Pitman Research Notes in Mathematics Series. Vol 185. IRSEE. Harlow: Longman Science and Technology;

[2] Brokate M, Sprekels J. Hysteresis and phase transitions (Applied Mathematical Sciences).

[3] Jiang S, Racke R. Evolution Equations in Thermoelasticity. Monographs and Surveys in Pure and Applied Mathematics. 112nd ed. Boca Raton: Chapman & Hall/CRC; 2000 [4] Fremond M. Non-Smooth Thermomechanics. Berlin, Heidelberg: Springer-Verlag; 2002.

[5] Fremond M, Miyazaki S. Shape Memory Alloys. Wien: Springer-Verlag; 1996. p. 147.

[6] Fang Y-F, Grillakis MG. Existence and uniqueness for Boussinesq type equations on a

[7] Falk F, Laedke EW, Spatschek KH. Stability of solitary-wave pulses in shape-memory

[8] Friedman A, Sprekels J. Steady states of austenitic-martensitic domains in the Ginzburg-Landau theory of shape memory alloys. Continuum Mechanics and Thermodynamics

[9] Garcke H. Travelling wave solutions as dynamic phase transitions in shape memory

[10] Tsutsumi Y, Yoshikawa S. Invariant measures for the isothermal Falk model of shape memory alloys. GAKUTO International Series. Mathematical Sciences and Applications.

circle. Communications in Partial Differential Equations. 1996;21:1253–1277

Shinya Okabe<sup>1</sup>

References

1990. pp. 45–59

1990;2:199–213

2015;37:163–182


**Shape Memory Alloys: Applications**

[26] Koiso N. Motion of a springy circle wire. Science Reports—College of General Education,

[27] Okabe S. The motion of elastic planar closed curves under the area-preserving condition.

[28] Okabe S. The dynamics of elastic closed curves under uniform high pressure. Calculus of

[29] Koiso N. On motion of a curve towards elastica. In: Actes da la Table Ronde de Geometrie Differentielle, Semin Congr. Vol. 1, 1992; Luminy. Paris: Soc. Math. France; 1996. pp. 403–

[30] Lagnese J, Lions J-L. Modelling Analysis and Control of Thin Plates. Recherches en Mathematiques Appliquees (Research in Applied Mathematics). 6th ed. Paris, Masson; 1988

[31] Suzuki T, Tasaki S. Stationary solutions to a thermoelastic system on shape memory

Indiana University Mathematics Journal. 2007;56:1871–1912

Variations and Partial Differential Equations. 2008;33:493–521

Osaka University. 1991;40:15–19

56 Shape Memory Alloys - Fundamentals and Applications

materials. Nonlinearity. 2010;23:2023–2656

436

Provisional chapter

#### **Magnetocaloric Effects in Metamagnetic Shape Memory Alloys** Magnetocaloric Effects in Metamagnetic

DOI: 10.5772/intechopen.69116

Shape Memory Alloys

Takumi Kihara, Xiao Xu, Wataru Ito, Ryosuke Kainuma, Yoshiya Adachi, Takeshi Kanomata and Masashi Tokunaga Takumi Kihara, Xiao Xu, Wataru Ito, Ryosuke Kainuma, Yoshiya Adachi, Takeshi Kanomata

Additional information is available at the end of the chapter and Masashi Tokunaga

http://dx.doi.org/10.5772/intechopen.69116 Additional information is available at the end of the chapter

#### Abstract

Recently, metamagnetic shape memory alloys have attracted much attention as candidates for the rare-earth free magnetic refrigerants. These materials undergo the martensitic transformation (MT) at around room temperature accompanied by a significant entropy change. The application of the magnetic field at the low-temperature martensitic phase realizes the magnetic field-induced martensitic transformation (MFIMT). Through the MFIMT, the materials show an unconventional magnetocaloric effect (MCE), which is called inverse magnetocaloric effect (IMCE). In this chapter, the direct measurement system of MCE in pulsed-high-magnetic fields is introduced. With taking the advantage of the fast field-sweep rate of pulsed field, adiabatic measurements of MCE are carried out at various temperatures. Using this technique, the IMCEs of the metamagnetic shape memory alloys NiCoMnIn and NiCoMnGa are directly measured as adiabatic temperature changes in pulsed fields. From the experimental data of MCE for NiCoMnIn, the entropy of spin system in the austenite phase is estimated through a simple mean-field model. By the combination of MCE, magnetization and specific heat measurements, the electronic, lattice and magnetic contributions to the IMCE are individually evaluated. The result for NiCoMnIn demonstrates that lattice entropy plays the dominant role for IMCE in this material.

Keywords: shape memory alloys, Heusler alloys, magnetocaloric effects, specific heat, pulsed magnetic field

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

#### 1. Introduction

#### 1.1. Magnetocaloric effects and magnetic refrigeration

Magnetocaloric effect (MCE) is thermodynamically defined as a temperature change of magnetic material due to the variation of an externally applied magnetic field. This effect is a consequence of the field variation of the entropy of a material. For instance, the application of a magnetic field to a paramagnet in the adiabatic condition reduces the disorder of spins, which lowers the magnetic entropy in the material. Since the total entropy is conserved in an adiabatic condition, the lattice entropy (and the electronic entropy in case of metals) must increase by a comparable amount of the magnetic entropy decrease. Consequently, the temperature of the material increases. In the adiabatic demagnetization process, the magnetic entropy is restored from the lattice and electronic systems, and hence, the temperature of the material decreases. In the MCE measurement, the adiabatic temperature change or the isothermal entropy change as a function of applied field is the experimentally measurable quantity. Since the MCE is sensitive to the entropy change of the material at the magnetic phase transitions, it is useful to map out the magnetic phase diagram and evaluate the temperature and field variations of entropy by the combination with the specific heat measurement. The MCE is also applied to the magnetic refrigeration, which is realized by means of the cycle of adiabatic magnetization/demagnetization and heat exchange between the magnetic material and the surrounding. This technique has long been used to realize extremely low temperatures [1, 2].

In recent decades, the magnetic refrigeration based on the MCE has attracted much attention as an alternative technique to the vapor-compression cycle for the ambient temperatures, which has been triggered by the discovery of the giant MCE (GMCE) at around room temperatures [3–5]. The GMCE is first observed in Gd5Si2Ge2, which was reported by Pecharsky and Gschneidner [3]. This material undergoes the first-order magnetic and structural phase transition at 276 K with a giant entropy change. The maximum entropy change at the transition temperature in this material is about twice larger than that of pure Gd, which is considered as a benchmark of magnetocaloric material at around room temperature. The GMCE, in this material, appears as a result of the simultaneous changes in the magnetic and lattice entropy through the first-order phase transition (FOT). Hence, the total entropy change can exceed the limit of the magnetic entropy change caused by the spin ordering. For the simultaneous change in the magnetic and lattice entropy, the strong coupling between the spin and the lattice systems is needed. For this reason, the GMCE undergoes in many cases the first-order magneto-structural phase transition [3–12].

#### 1.2. MCEs in Ni-Mn-based Heusler alloys

Ni50Mn50xZx (Z = Ga, In, Sn and Sb) Heusler alloys have attracted a lot of attention because of the potential applications for novel rare-earth free magnetic refrigerants [13–17]. These alloys have a cubic (L21) Heusler structure with a space group of Fm3 ¯ m [15]. Some of the non-stoichiometric compositions undergo martensitic transformation (MT) from a high-temperature austenitic phase (A-phase: cubic) to a low-temperature martensitic phase of reduced symmetry (M-phase: tetragonal, orthorhombic, or monoclinic) at around room temperature (Ni50Mn25Ga25 is the only Ni-Mn-based Heusler alloy that shows the MT in the stoichiometric composition [18]). In these alloys, the MT temperatures (TMT) and the magnetic properties in each phase strongly depend on the Z species and its compositional ratio x [15]. The TMT increases with decreasing x. In the small x region, the ferromagnetic phase appears at the Curie temperatures in both the A phase (T<sup>A</sup> <sup>C</sup>) and the M phase (T<sup>M</sup> <sup>C</sup> ), respectively [15, 16, 19–21]. In addition, the partial substitution of Ni with Co enhances the exchange coupling between the spins in the alloys, which contributes to increase T<sup>A</sup> C and decrease T<sup>M</sup> <sup>C</sup> and TMT [14, 22]. Ni45Co5Mn36.7In13.3 undergoes MT accompanied with a magnetic phase transition from a ferromagnetic A phase to a paramagnetic M phase (monoclinic). The application of an external magnetic field to the M phase realizes magnetic-field-induced martensitic transformation (MFIMT). Kainuma et al. [14] reported, for the first time, the magneticfield-induced shape recovery by MFIMT in this material, which is called metamagnetic shape memory effect. Owing to the large magnetic-field-induced strain and the large output stress through MFIMT, Ni45Co5Mn36.7In13.3 is considered to be a potential magnetic actuator material [14].

In the Ni45Co5Mn36.7In13.3, the MFIMT occurs as it is accompanied by significant increase of entropy, whereas the applied magnetic field aligns the spins parallel to the field direction and reduces the magnetic entropy [14, 22]. This unconventional phenomenon is called the inverse magnetocaloric effect (IMCE). The IMCEs were also observed in other compositions Ni50Mn50�xZx (Z = Ga, In, Sn and Sb) [13–16, 19–23]. The IMCE indicates the significant positive change in entropy through the MFIMT that exceeds the negative contribution from the spin ordering. Therefore, the individual evaluations of the electronic, lattice and magnetic entropy changes are important to elucidate the origin of such positive entropy changes at the MFIMT. In addition, the direct measurement of MCE is also important because it enables us to evaluate the magnetic part of entropy as presented in Section 4. However, numerous studies have attempted to understand the IMCE in these materials through indirect methods such as magnetization and/or specific heat, which can only reveal the total entropy change [13–17, 19–23].

There are several studies investigating adiabatic temperature change of these Heusler alloys under magnetic fields. For instance, Liu et al. [17] carried out the direct measurement of IMCEs up to 1.9 T for NiMnInCo. In their experiments, the samples undergo the MFIMT from a pure M phase to a mixed phase because the magnetic field of 1.9 T is insufficient to complete the MFIMT. In that case, the quantitative interpretation of the MCE results is difficult. Therefore, the direct MCE measurements in the wide range of temperatures and magnetic fields are crucial to provide a greater understanding of this phenomenon.

#### 2. Thermodynamics of MCE

1. Introduction

extremely low temperatures [1, 2].

1.2. MCEs in Ni-Mn-based Heusler alloys

cubic (L21) Heusler structure with a space group of Fm3

1.1. Magnetocaloric effects and magnetic refrigeration

60 Shape Memory Alloys - Fundamentals and Applications

Magnetocaloric effect (MCE) is thermodynamically defined as a temperature change of magnetic material due to the variation of an externally applied magnetic field. This effect is a consequence of the field variation of the entropy of a material. For instance, the application of a magnetic field to a paramagnet in the adiabatic condition reduces the disorder of spins, which lowers the magnetic entropy in the material. Since the total entropy is conserved in an adiabatic condition, the lattice entropy (and the electronic entropy in case of metals) must increase by a comparable amount of the magnetic entropy decrease. Consequently, the temperature of the material increases. In the adiabatic demagnetization process, the magnetic entropy is restored from the lattice and electronic systems, and hence, the temperature of the material decreases. In the MCE measurement, the adiabatic temperature change or the isothermal entropy change as a function of applied field is the experimentally measurable quantity. Since the MCE is sensitive to the entropy change of the material at the magnetic phase transitions, it is useful to map out the magnetic phase diagram and evaluate the temperature and field variations of entropy by the combination with the specific heat measurement. The MCE is also applied to the magnetic refrigeration, which is realized by means of the cycle of adiabatic magnetization/demagnetization and heat exchange between the magnetic material and the surrounding. This technique has long been used to realize

In recent decades, the magnetic refrigeration based on the MCE has attracted much attention as an alternative technique to the vapor-compression cycle for the ambient temperatures, which has been triggered by the discovery of the giant MCE (GMCE) at around room temperatures [3–5]. The GMCE is first observed in Gd5Si2Ge2, which was reported by Pecharsky and Gschneidner [3]. This material undergoes the first-order magnetic and structural phase transition at 276 K with a giant entropy change. The maximum entropy change at the transition temperature in this material is about twice larger than that of pure Gd, which is considered as a benchmark of magnetocaloric material at around room temperature. The GMCE, in this material, appears as a result of the simultaneous changes in the magnetic and lattice entropy through the first-order phase transition (FOT). Hence, the total entropy change can exceed the limit of the magnetic entropy change caused by the spin ordering. For the simultaneous change in the magnetic and lattice entropy, the strong coupling between the spin and the lattice systems is needed. For this reason, the GMCE

undergoes in many cases the first-order magneto-structural phase transition [3–12].

Ni50Mn50xZx (Z = Ga, In, Sn and Sb) Heusler alloys have attracted a lot of attention because of the potential applications for novel rare-earth free magnetic refrigerants [13–17]. These alloys have a

compositions undergo martensitic transformation (MT) from a high-temperature austenitic phase (A-phase: cubic) to a low-temperature martensitic phase of reduced symmetry (M-phase: tetragonal, orthorhombic, or monoclinic) at around room temperature (Ni50Mn25Ga25 is the only Ni-Mn-based Heusler alloy that shows the MT in the stoichiometric composition [18]). In these alloys, the MT temperatures (TMT) and the magnetic properties in each phase strongly depend on

¯

m [15]. Some of the non-stoichiometric

In this section, a brief explanation of thermodynamics of the MCE is provided (refer to Ref. [24] for more details). Let us start with the total entropy of the system S(T, H, p). The total differential of S(T, H, p) can be written as:

$$dS = \left(\frac{\partial S}{\partial T}\right)\_{H,p} dT + \left(\frac{\partial S}{\partial H}\right)\_{T,p} dH + \left(\frac{\partial S}{\partial p}\right)\_{T,H} dp \tag{1}$$

Here, T, H and p denote temperature, external magnetic field and pressure, respectively. In the adiabatic and isobaric condition (dS = 0 and dp = 0), one obtains:

$$dT = -\left(\frac{\partial T}{\partial \mathcal{S}}\right)\_H \left(\frac{\partial \mathcal{S}}{\partial H}\right)\_T dH. \tag{2}$$

Since the heat capacity of a system at a constant magnetic field CH(T) is defined as CH(T) = T(∂S / ∂T)H, the isentropic temperature change due to the variation of the magnetic field from H<sup>1</sup> to H<sup>2</sup> can be expressed as:

$$
\Delta T\_{ad}(H\_1 \to H\_2) = -\int\_{H\_1}^{H\_2} \frac{T}{\mathbb{C}\_H(T)} \left(\frac{\partial \mathcal{S}}{\partial H}\right)\_H dH. \tag{3}
$$

On the other hand, the isothermal-isobaric entropy change is given by the well-known Maxwell relation:

$$
\left(\frac{\partial \mathcal{S}}{\partial H}\right)\_{T,p} = \left(\frac{\partial M}{\partial T}\right)\_{H,p} \tag{4}
$$

Here, M is the magnetization of a material. After integration, Eq. (4) gives

$$
\Delta \mathbb{S}\_T(H\_1 \to H\_2) = -\int\_{H\_1}^{H\_2} \left( \frac{\partial M(T, H)}{\partial T} \right)\_H dH. \tag{5}
$$

By combining Eqs. (3) and (4), the ΔTad can be expressed as:

$$
\Delta T\_{ad}(H\_1 \to H\_2) = -\int\_{H\_1}^{H\_2} \frac{T}{\mathbb{C}\_H(T)} \left(\frac{\partial M}{\partial T}\right)\_H dH. \tag{6}
$$

Using Eqs. (5) and (6), ΔTad and ΔST can indirectly be estimated from the magnetization and specific heat measurements. Therefore, many studies employed so far use this method to evaluate the magnetocaloric properties for the several materials [3–16].

When a material undergoes magnetic phase transitions, the MCEs at the phase boundary show different behaviour between first- and second-order phase transitions. In the case of the second-order phase transition, the entropy of a material continuously and reversibly changes, and hence, Eqs. (5) and (6) can safely be used. On the other hand, the entropy discontinuously changes through an FOT, and the heat capacity becomes infinite at the transition temperature. Since the Maxwell relation is relevant only when the entropy is a continuous function of temperature and magnetic field, the direct application of the Eqs. (5) and (6) for the FOT is not allowed. Recently, several works proposed some approaches to apply the Maxwell relations to the FOTs, but it is still in debate [25–31]. Therefore, the direct measurement of MCE (ΔTad) is crucial to gain a deep insight into the entropic behaviour through the FOTs.

#### 3. Direct MCE measurement system in pulsed-high-magnetic field

As described in the previous sections, the MCE measurements are of particular interest in both fundamental (investigation of the magnetic phase transitions) and applied physics (magnetic refrigeration). Therefore, a lot of experimental techniques of the MCE measurements have been developed so far [32–37]. Recently, Kihara et al. [36] developed a direct measurement system of MCE under pulsed high-magnetic fields up to 56 T. In this section, the details of their experimental technique are provided.

dT ¼ � <sup>∂</sup><sup>T</sup>

ΔTadðH<sup>1</sup> ! H2޼�

∂S ∂H � �

Here, M is the magnetization of a material. After integration, Eq. (4) gives

ΔSTðH<sup>1</sup> ! H2޼�

ΔTadðH<sup>1</sup> ! H2޼�

evaluate the magnetocaloric properties for the several materials [3–16].

By combining Eqs. (3) and (4), the ΔTad can be expressed as:

T,p

can be expressed as:

62 Shape Memory Alloys - Fundamentals and Applications

well relation:

∂S � �

H

Since the heat capacity of a system at a constant magnetic field CH(T) is defined as CH(T) = T(∂S / ∂T)H, the isentropic temperature change due to the variation of the magnetic field from H<sup>1</sup> to H<sup>2</sup>

> ð<sup>H</sup><sup>2</sup> H<sup>1</sup>

On the other hand, the isothermal-isobaric entropy change is given by the well-known Max-

<sup>¼</sup> <sup>∂</sup><sup>M</sup> ∂T � �

ð<sup>H</sup><sup>2</sup> H<sup>1</sup>

ð<sup>H</sup><sup>2</sup> H<sup>1</sup>

Using Eqs. (5) and (6), ΔTad and ΔST can indirectly be estimated from the magnetization and specific heat measurements. Therefore, many studies employed so far use this method to

When a material undergoes magnetic phase transitions, the MCEs at the phase boundary show different behaviour between first- and second-order phase transitions. In the case of the second-order phase transition, the entropy of a material continuously and reversibly changes, and hence, Eqs. (5) and (6) can safely be used. On the other hand, the entropy discontinuously changes through an FOT, and the heat capacity becomes infinite at the transition temperature. Since the Maxwell relation is relevant only when the entropy is a continuous function of temperature and magnetic field, the direct application of the Eqs. (5) and (6) for the FOT is not allowed. Recently, several works proposed some approaches to apply the Maxwell relations to the FOTs, but it is still in debate [25–31]. Therefore, the direct measurement of MCE

(ΔTad) is crucial to gain a deep insight into the entropic behaviour through the FOTs.

3. Direct MCE measurement system in pulsed-high-magnetic field

As described in the previous sections, the MCE measurements are of particular interest in both fundamental (investigation of the magnetic phase transitions) and applied physics (magnetic

T CHðTÞ

H, p

∂MðT,HÞ ∂T � �

∂S ∂H � �

> T CHðTÞ

∂S ∂H � �

H

H

H

∂M ∂T � �

T

dH: ð2Þ

dH: ð3Þ

dH: ð5Þ

dH: ð6Þ

ð4Þ

Figure 1(a) shows the schematic view of the electric circuit for the pulsed field generation system. The magnetic fields are generated in the solenoid magnet coil by discharging the energy stored in the capacitor bank [38]. Figure 1(b) shows the time variations of the pulsed fields generated in the nondestructive magnet installed at the Institute for Solid State Physics (ISSP), the University of Tokyo. The three curves correspond to the field profiles for charging voltages of 1, 5 and 8 kV to the capacitor bank, respectively. As shown in Figure 1(b), the maximum field depends on the charging voltage to the capacitor bank. The total duration of the pulsed field is about 36 ms. This fast sweep rate of the pulsed field (the maximum sweep rate is about 104 T/s) has an advantage to realize the adiabatic conditions.

Figures 2(a) and (b) show the schematic and the picture of the sample setup. The sample is shaped into the thin plate with the thickness of less than 0.1 mm to reduce the eddy current heating caused by the application of the pulsed field (in case of metallic samples). The mag-

Figure 1. (a) A schematic view of the electric circuit of the pulse magnet system and (b) Magnetic field profiles of the 56-T pulse magnet at the Institute for Solid State Physics, The University of Tokyo.

Figure 2. (a) A schematic drawing of the sample with the thermometer and (b) A picture of the sample (Ni41Co9Mn31.5Ga18.5) mounted on the MCE probe.

netic field is applied parallel to the sample plane to reduce the cross-section of the sample. In the thermometry under pulsed fields, the sample temperature changes rapidly as a function of the magnetic field. Therefore, commercial thermometers such as a Cernox® bare chip cannot be used because of the limitation of their thermal response time [35]. To detect the instantaneous change in the sample temperature in the pulsed field, a very small resistive thermometer is used, in which a patterned Au sensor (thickness: 100 nm) is deposited on a sapphire disk (thickness: 5 μm). This thermometer is mounted on top of the sample with a small amount of Apiezon® N grease as shown in Figure 2(b). The small heat capacity of this thermometer (typical value is about 200 μJ/K at 300 K) and the large thermal conductance of the sapphire disk realize the fast response to the sample temperature and enable us to measure the MCEs in the pulsed fields as demonstrated for the NiCoMnIn in the next section. The sample with the thermometer is fixed on the Pyrex® glass plates, which have low thermal conductance, by the small amount of glue in order to reduce the heat leak to the surrounding. The probe (assembly of the sample, the thermometer, the sample holder, the bath heater, etc.) is inserted in the thinwalled tube made of non-magnetic stainless steel as shown in Figure 3. The sample space (inside of the tube) is evacuated to reduce the heat exchange between the sample and the surrounding through the residual gas. The tube is immersed in liquid helium for low temperature measurements. For high temperature measurements, small amount of helium gas is introduced into the space indicated by "Liq. He" in Figure 3 as a heat exchange gas. During the measurement, the probe is cooled through the cold finger that is connected to the tube at the bottom of the probe. The sample temperature is regulated by the feedback operation of the bath heater (Figure 3). As shown in Figure 2(b), the four Au wires (diameter: 30 μm) are connected to the thermometer by the small amount of Ag paste. The resistance of the thermometer is measured by the ac method using numerical lock-in technique [35, 36].

Figure 3. A schematic drawing of the probe setup.

To calibrate the Au thermometer grown on the sapphire disk, the longitudinal magnetoresistances (MRs) are measured in the pulsed fields before the MCE measurements. To carry out the isothermal measurements, the sample is removed, and the sapphire disk with the thermometer is placed directly on the Pyrex® glass substrate. Moreover, 1 atm of helium gas is introduced into the sample space at room temperature. Figure 4 shows the MRs of the thermometer measured up to 56 T at the various temperatures (black curves). At 0 T, one can confirm the linear temperature dependence of the resistance of the metallic Au film. The slope is dR / dT = 8.97 mΩ/K. In the temperature region between 200 and 330 K, the Au film shows the very small MR as shown in Figure 4, and therefore, the linear temperature dependence of resistance can be seen in the whole field range up to 56 T. To convert the resistance to temperature, the MRs are fitted by the polynomial function of the temperature (T) and the magnetic field (H):

netic field is applied parallel to the sample plane to reduce the cross-section of the sample. In the thermometry under pulsed fields, the sample temperature changes rapidly as a function of the magnetic field. Therefore, commercial thermometers such as a Cernox® bare chip cannot be used because of the limitation of their thermal response time [35]. To detect the instantaneous change in the sample temperature in the pulsed field, a very small resistive thermometer is used, in which a patterned Au sensor (thickness: 100 nm) is deposited on a sapphire disk (thickness: 5 μm). This thermometer is mounted on top of the sample with a small amount of Apiezon® N grease as shown in Figure 2(b). The small heat capacity of this thermometer (typical value is about 200 μJ/K at 300 K) and the large thermal conductance of the sapphire disk realize the fast response to the sample temperature and enable us to measure the MCEs in the pulsed fields as demonstrated for the NiCoMnIn in the next section. The sample with the thermometer is fixed on the Pyrex® glass plates, which have low thermal conductance, by the small amount of glue in order to reduce the heat leak to the surrounding. The probe (assembly of the sample, the thermometer, the sample holder, the bath heater, etc.) is inserted in the thinwalled tube made of non-magnetic stainless steel as shown in Figure 3. The sample space (inside of the tube) is evacuated to reduce the heat exchange between the sample and the surrounding through the residual gas. The tube is immersed in liquid helium for low temperature measurements. For high temperature measurements, small amount of helium gas is introduced into the space indicated by "Liq. He" in Figure 3 as a heat exchange gas. During the measurement, the probe is cooled through the cold finger that is connected to the tube at the bottom of the probe. The sample temperature is regulated by the feedback operation of the bath heater (Figure 3). As shown in Figure 2(b), the four Au wires (diameter: 30 μm) are connected to the thermometer by the small amount of Ag paste. The resistance of the ther-

mometer is measured by the ac method using numerical lock-in technique [35, 36].

Figure 3. A schematic drawing of the probe setup.

64 Shape Memory Alloys - Fundamentals and Applications

$$R(T,H) = a\_0 + a\_1T + a\_2H + a\_3T^2 + a\_4TH + a\_5H^2 + an,\tag{7}$$

where a<sup>n</sup> for n = 0,1,2,… is the polynomial coefficient. The result of the fitting by this polynomial function for n = 0,1,2,…,35 is in good agreement with the data as shown by the rainbow surface in Figure 4. The deviation of the fitting function from the data points is within 4 mΩ over the entire range of temperatures and magnetic fields, which corresponds to the error of the temperature of 0.45 K. Hence, this Au film thermometer enables the accurate MCE measurements in the pulsed fields up to 56 T. The validity of this technique was demonstrated in the wide range of temperature through the measurements on Gd at around room temperature and on Gd3Ga5O12 at low temperatures [36].

Figure 4. Magnetoresistance of the Au film thermometer deposited on a sapphire disk measured in the pulsed field up to 56 T at the various temperatures (black curves). The rainbow surface is the fitting to the polynomial function discussed in the text.

#### 4. Magnetocaloric properties of metamagnetic shape memory alloys

#### 4.1. Experimental procedures

In this chapter, polycrystalline samples of Ni45Co5Mn50xInx (x = 13.3 and 13.5) and Ni41Co9 Mn31.5Ga18.5 were used for the MCE, magnetization and specific heat measurements. The samples of Ni45Co5Mn50xInx (x = 13.3 and 13.5) were prepared by the induction melting method. The ingots were annealed at 900C for 24 h under argon atmosphere. The samples of Ni41Co9Mn31.5Ga18.5 were prepared by the arc melting method. The ingots vacuum encapsulated in a quartz tube were annealed at 800C for 72 h and then quenched in cold water.

Magnetization and MCE measurements were performed in pulsed-high-magnetic fields. The pulsed fields are generated by the 56-T magnet at the ISSP. The heat capacity measurements in steady fields were carried out using a thermal-relaxation method in a physical property measurement system, QuantumDesign (PPMS).

#### 4.2. NiCoMnIn

#### 4.2.1. Magnetization measurements

As described in Section 1.2, the metamagnetic shape memory alloy Ni45Co5Mn36.7In13.3 undergoes the MT at around room temperature. Figures 5(a) and (b) show the isothermal magnetizations of the Ni45Co5Mn36.7In13.3 measured at 310 and 280 K, respectively. The sample is in the ferromagnetic A phase at 310 K, where the M-H curve shows the ferromagnetic behaviour as shown in Figure 5(a). On the other hand, as shown in Figure 5(b), the magnetization increases steeply at the MFIMT, when the magnetic field is applied to the paramagnetic M phase. Here, the transition fields are defined as Haf for the field increasing process and Hms for the field decreasing process in Figure 5(b), respectively. The M-H curves measured at the various temperatures are shown in Figure 6(a). One may note that the application of the pulsedhigh-magnetic field up to 25 T can complete the MFIMTs in the entire temperature range. From the M-H curve at 4.2 K, the saturation magnetization moment (Ms) in the ferromagnetic A phase is estimated as Ms = 1.68 μB/f.u. Here, μ<sup>B</sup> is the Bohr magneton. A formula unit (f.u.) is defined as Ni0.45Co0.05Mn0.367In0.133. The magnetic phase diagram of Ni45Co5Mn36.7In13.3 determined from

Figure 5. M H curves measured in the pulsed magnetic fields at (a) 310 and (b) 280 K for Ni45Co5Mn36.7In13.3 [39].

the M � H curves is shown in Figure 6(b). The M � H curves and the magnetic phase diagram for Ni45Co5Mn36.5In13.5 are also provided in Figures 7(a) and (b). Hereafter, we denote In13.3 for Ni45Co5Mn36.7In13.3 and In13.5 for Ni45Co5Mn36.5In13.5, respectively. The saturated magnetization in the ferromagnetic A phase for In13.5 is also estimated as Ms = 1.76 μB/f.u. As shown in Figures 6(b) and 7(b) and described in Section 1.2, the magnetic phase diagram of Ni45Co5 Mn50�xInx is sensitive to the compositional ratio x. However, the saturated magnetizations of the two compositions are similar.

4. Magnetocaloric properties of metamagnetic shape memory alloys

In this chapter, polycrystalline samples of Ni45Co5Mn50xInx (x = 13.3 and 13.5) and Ni41Co9 Mn31.5Ga18.5 were used for the MCE, magnetization and specific heat measurements. The samples of Ni45Co5Mn50xInx (x = 13.3 and 13.5) were prepared by the induction melting method. The ingots were annealed at 900C for 24 h under argon atmosphere. The samples of Ni41Co9Mn31.5Ga18.5 were prepared by the arc melting method. The ingots vacuum encapsulated in a quartz tube were annealed at 800C for 72 h and then quenched in cold water.

Magnetization and MCE measurements were performed in pulsed-high-magnetic fields. The pulsed fields are generated by the 56-T magnet at the ISSP. The heat capacity measurements in steady fields were carried out using a thermal-relaxation method in a physical property

As described in Section 1.2, the metamagnetic shape memory alloy Ni45Co5Mn36.7In13.3 undergoes the MT at around room temperature. Figures 5(a) and (b) show the isothermal magnetizations of the Ni45Co5Mn36.7In13.3 measured at 310 and 280 K, respectively. The sample is in the ferromagnetic A phase at 310 K, where the M-H curve shows the ferromagnetic behaviour as shown in Figure 5(a). On the other hand, as shown in Figure 5(b), the magnetization increases steeply at the MFIMT, when the magnetic field is applied to the paramagnetic M phase. Here, the transition fields are defined as Haf for the field increasing process and Hms for the field decreasing process in Figure 5(b), respectively. The M-H curves measured at the various temperatures are shown in Figure 6(a). One may note that the application of the pulsedhigh-magnetic field up to 25 T can complete the MFIMTs in the entire temperature range. From the M-H curve at 4.2 K, the saturation magnetization moment (Ms) in the ferromagnetic A phase is estimated as Ms = 1.68 μB/f.u. Here, μ<sup>B</sup> is the Bohr magneton. A formula unit (f.u.) is defined as Ni0.45Co0.05Mn0.367In0.133. The magnetic phase diagram of Ni45Co5Mn36.7In13.3 determined from

Figure 5. M H curves measured in the pulsed magnetic fields at (a) 310 and (b) 280 K for Ni45Co5Mn36.7In13.3 [39].

4.1. Experimental procedures

66 Shape Memory Alloys - Fundamentals and Applications

4.2. NiCoMnIn

4.2.1. Magnetization measurements

measurement system, QuantumDesign (PPMS).

The total entropy change through MFIMT (ΔStot) can be estimated from the well-known Clausius-Clapeyron equation:

$$
\mu\_0 \frac{dH\_0}{dT} = -\frac{\Delta S\_{\text{tot}}}{\Delta M}.\tag{8}
$$

Here, <sup>μ</sup><sup>0</sup> = 4 <sup>π</sup> � <sup>10</sup>�<sup>7</sup> H/m is the space permeability. The <sup>Δ</sup><sup>M</sup> is the difference of magnetization between the A and M phase. The H<sup>0</sup> = (Hms + Haf) / 2 is used for the transition field. The results are shown in Figure 8, where the ΔStot is defined as the entropy change from the M to the A phase. The ΔStot at the MT temperature at the zero field is estimated to be 26 J/kg K for In13.3 and 23 J/kg K for In13.5. These values are in good agreement with that obtained from the

Figure 6. (a) M � H curves measured in the pulsed magnetic fields at the various temperatures and (b) the magnetic phase diagram for Ni45Co5Mn36.7In13.3 (In13.3) [39].

Figure 7. (a) M � H curves measured in the pulsed magnetic fields at the various temperatures and (b) the magnetic phase diagram for Ni45Co5Mn36.5In13.5 (In13.5) [39].

Figure 8. Temperature dependencies of the entropy change at the MFIMT estimated from the magnetic phase diagrams for Ni45Co5Mn36.7In13.3 (In13.3) and Ni45Co5Mn36.5In13.5 (In13.5) [39].

specific heat measurements [39]. The ΔStot decreases with decreasing temperature in both In13.3 and In13.5 and reaches zero at around 100 K. This indicates that the entropies of A and M phase coincide with each other below 100 K. In other words, the driving force of the MT decreases and reaches to zero below 100 K, which increases the hysteresis of the M H curves at low temperatures. The MCEs can be calculated by the M H curves using Eq. (5), which are compared with the results of direct measurements in the next section.

#### 4.2.2. MCE measurements in the pulsed magnetic fields

The direct MCE measurement technique described in Section 3 was first applied to the In13.3 [39]. Figures 9(a) and (b) show the adiabatic changes of the sample temperature of In13.3 measured at 310 and 280 K. The typical mass of the sample for the MCE measurements is about 50 mg, which is more than 100 times larger than that of the thermometer. Therefore, we can neglect the heat capacity of the thermometer in the measurements. Since the sample is in the ferromagnetic A phase at 310 K, the conventional MCE (monotonic heating/cooling with increasing/decreasing magnetic field) due to the forced spin alignment by the applied field is observed. It is important to note that the reversible temperature change demonstrates the experimental validity: negligibly small heat exchange between the sample and the surroundings and fast response of the thermometer [36]. The result at 280 K is the MCE when the magnetic field is applied to the M phase [Figure 9(b)]. The steep cooling (IMCE) occurs at the MFIMT in the field-increasing process. In the successive field-decreasing process, the sample temperature heats up at MFIMT and approximately reaches to the initial temperature at the zero field as

Figure 9. Magnetic field variations of the sample temperature in the adiabatic condition for In13.3. (a) The magnetic field applied to the ferromagnetic A phase and (b) the magnetic field applied to the paramagnetic M phase [39].

expected for the adiabatic measurements. The difference between the initial and final temperatures is less than 0.5 K, which indicates that the heating caused by the hysteresis loss is negligibly smaller than the MCE. In addition, a gentle slope is observed in the field-induced A phase, which is due to the change in the entropy of the ferromagnetic spins. This conventional MCE can be calculated by using the following mean-field model:

$$M(T, H) = \text{Ng}\mu\_B \text{J}B\_l \left(\frac{\text{g}\mu\_B \text{J}\mu\_0 (H - \lambda M)}{k\_B T}\right). \tag{9}$$

Here, <sup>N</sup> = 6.02 � 1023 mol�<sup>1</sup> is Avogadro's number, <sup>g</sup> is the g factor, <sup>J</sup> is the spin quantum number, <sup>B</sup>J(x) is the Brillouin function, and <sup>k</sup><sup>B</sup> = 1.38 � <sup>10</sup>�<sup>23</sup> J/K is Boltzmann constant. The coefficient <sup>λ</sup> represents the magnetic interaction, which is defined as <sup>λ</sup> <sup>¼</sup> <sup>3</sup>kBTΘ=Ng<sup>2</sup>μ<sup>2</sup> <sup>B</sup>JðJ þ 1Þ by the Weiss temperature TΘ. Using the experimentally obtained values g<sup>J</sup> = 1.68 and T<sup>Θ</sup> = TC = 387 K, the magnetization of A phase can be calculated as a function of the temperature and magnetic field. Hence, the MCEs of A phase are calculated by this M(T, H) curve through Eq. (6). The results are plotted as the red-dashed lines in Figures 9(a) and (b) and are in reasonable agreement with the experimental data. On the other hand, the MCE in the paramagnetic M phase is very small and therefore, cannot be calculated by this model.

Figure 10 shows the temperature dependence of ΔTad (15T), which is indicated in Figure 9(b), the difference of the sample temperatures between 15 and 0 T as a function of the initial temperature. The ΔTad (15T) decreases linearly with increasing temperature and is in good agreement with the indirectly estimated values from the magnetization as shown in Figure 10.

#### 4.2.3. Specific heat measurements

specific heat measurements [39]. The ΔStot decreases with decreasing temperature in both In13.3 and In13.5 and reaches zero at around 100 K. This indicates that the entropies of A and M phase coincide with each other below 100 K. In other words, the driving force of the MT decreases and reaches to zero below 100 K, which increases the hysteresis of the M H curves at low temperatures. The MCEs can be calculated by the M H curves using Eq. (5), which are

Figure 8. Temperature dependencies of the entropy change at the MFIMT estimated from the magnetic phase diagrams

The direct MCE measurement technique described in Section 3 was first applied to the In13.3 [39]. Figures 9(a) and (b) show the adiabatic changes of the sample temperature of In13.3 measured at 310 and 280 K. The typical mass of the sample for the MCE measurements is about 50 mg, which is more than 100 times larger than that of the thermometer. Therefore, we can neglect the heat capacity of the thermometer in the measurements. Since the sample is in the ferromagnetic A phase at 310 K, the conventional MCE (monotonic heating/cooling with increasing/decreasing magnetic field) due to the forced spin alignment by the applied field is observed. It is important to note that the reversible temperature change demonstrates the experimental validity: negligibly small heat exchange between the sample and the surroundings and fast response of the thermometer [36]. The result at 280 K is the MCE when the magnetic field is applied to the M phase [Figure 9(b)]. The steep cooling (IMCE) occurs at the MFIMT in the field-increasing process. In the successive field-decreasing process, the sample temperature heats up at MFIMT and approximately reaches to the initial temperature at the zero field as

compared with the results of direct measurements in the next section.

4.2.2. MCE measurements in the pulsed magnetic fields

for Ni45Co5Mn36.7In13.3 (In13.3) and Ni45Co5Mn36.5In13.5 (In13.5) [39].

68 Shape Memory Alloys - Fundamentals and Applications

From the magnetization and MCE measurements, the total entropy change at MFIMT and the magnetic entropy in the field-induced A phase is estimated. In this section, the specific heat

Figure 10. The temperature dependence of ΔTad(15T) (red triangles), which is defined in Figure 9(b). The open squares are the MCEs calculated from isothermal magnetization curves using Eq. (5) [39].

measurements are performed to evaluate the electronic and lattice contributions to the total entropy. As shown in Figures 6(b) and 7(b), the samples remain in the A phase at the low temperatures when they are cooled down in the magnetic field above Hms. Therefore, the application of high magnetic fields enables us to measure the specific heat in both A and M phase for a sample. Figure 11 shows the results of specific heat measurements for In13.5 at low temperatures, which are plotted as C/T versus T<sup>2</sup> . The data of A phase (open circles) are measured after the field cooling at 12 T from room temperature. The data in both phases can be fitted by the function:

$$
\mathbb{C} = \gamma T + \beta T^3,\tag{10}
$$

where the first term represents the electronic contribution and the second term corresponds to the phonon contribution. The electronic contribution to the specific heat is characterized by Sommerfeld coefficient, γ, which is proportional to the density of states at Fermi level. The phonon contribution with the coefficient β provides us with the Debye temperature ΘD. The values [γ, ΘD] are estimated to be [γ<sup>A</sup> = 52 mJ/kg K2 (3.4 mJ/mol K<sup>2</sup> ), Θ<sup>A</sup> <sup>D</sup> ¼ 314 K] for A phase and [γ<sup>M</sup> = 48 mJ/kg K<sup>2</sup> (3.1 mJ/mol K<sup>2</sup> ), Θ<sup>M</sup> <sup>D</sup> ¼ 361 K] for M phase, respectively. The electronic contribution to the total entropy change ΔSele is estimated as:

$$
\Delta \mathcal{S}\_{\text{ele}}(T) = (\boldsymbol{\gamma}^{\mathcal{A}} - \boldsymbol{\gamma}^{\mathcal{M}})T.\tag{11}
$$

At 300 K, ΔSele becomes 1.2 J/kg K. This value corresponds to about 5% of the total entropy change at MT. Thus, the electronic contribution to the IMCE does not play dominant role.

Figure 11. Specific heat of In13.5 plotted as C/T versus T<sup>2</sup> [39].

measurements are performed to evaluate the electronic and lattice contributions to the total entropy. As shown in Figures 6(b) and 7(b), the samples remain in the A phase at the low temperatures when they are cooled down in the magnetic field above Hms. Therefore, the application of high magnetic fields enables us to measure the specific heat in both A and M phase for a sample. Figure 11 shows the results of specific heat measurements for In13.5 at low temperatures,

Figure 10. The temperature dependence of ΔTad(15T) (red triangles), which is defined in Figure 9(b). The open squares are

cooling at 12 T from room temperature. The data in both phases can be fitted by the function:

<sup>C</sup> <sup>¼</sup> <sup>γ</sup><sup>T</sup> <sup>þ</sup> <sup>β</sup>T<sup>3</sup>

where the first term represents the electronic contribution and the second term corresponds to the phonon contribution. The electronic contribution to the specific heat is characterized by Sommerfeld coefficient, γ, which is proportional to the density of states at Fermi level. The phonon contribution with the coefficient β provides us with the Debye temperature ΘD. The

At 300 K, ΔSele becomes 1.2 J/kg K. This value corresponds to about 5% of the total entropy change at MT. Thus, the electronic contribution to the IMCE does not play dominant role.

values [γ, ΘD] are estimated to be [γ<sup>A</sup> = 52 mJ/kg K2 (3.4 mJ/mol K<sup>2</sup>

contribution to the total entropy change ΔSele is estimated as:

the MCEs calculated from isothermal magnetization curves using Eq. (5) [39].

), Θ<sup>M</sup>

. The data of A phase (open circles) are measured after the field

; ð10Þ

<sup>D</sup> ¼ 314 K] for A phase

), Θ<sup>A</sup>

<sup>D</sup> ¼ 361 K] for M phase, respectively. The electronic

<sup>Δ</sup>SeleðTÞ¼ðγ<sup>A</sup> � <sup>γ</sup><sup>M</sup>ÞT: <sup>ð</sup>11<sup>Þ</sup>

which are plotted as C/T versus T<sup>2</sup>

70 Shape Memory Alloys - Fundamentals and Applications

and [γ<sup>M</sup> = 48 mJ/kg K<sup>2</sup> (3.1 mJ/mol K<sup>2</sup>

The specific heat of lattice system can be calculated by assuming the Debye model:

$$\mathbf{C}\_{\rm V} = 9 \text{N}k\_{\rm B} \left(\frac{T}{\Theta\_{\rm D}}\right)^{3} \int\_{0}^{\Theta\_{\rm D}/T} \frac{\mathbf{x}^{4} e^{\mathbf{x}}}{\left(e^{\mathbf{x}} - 1\right)^{2}} d\mathbf{x}.\tag{12}$$

Therefore, the lattice contribution to the total entropy change ΔSlat(T) due to the difference of the Debye temperature between A and M phase can be calculated. Here, the CV in Eq. (10) is the specific heat under the constant volume, while the measurements are carried out in the isobaric condition. Hence, one has to consider the difference of the specific heat between at the constant pressure and at the constant volume conditions, which can be calculated as:

$$
\mathbf{C}\_p - \mathbf{C}\_V = 9\alpha^2 \text{BVT}\_\prime \tag{13}
$$

where α is bulk thermal expansion coefficient, B is bulk elastic modulus, and V is molar volume. Using the values <sup>α</sup> = 6.67 � <sup>10</sup>�<sup>6</sup> <sup>K</sup>�<sup>1</sup> for Ni45Co5Mn37In13 [40] and <sup>B</sup> =140 GPa for Ni50Mn25In25 [41], one obtains Cp � CV = 2.3 J/kg K at 300 K, which is smaller than the 0.6% of the experimentally obtained C<sup>p</sup> at 300 K [39]. Therefore, the calculated data Clat can be compared with the C<sup>p</sup> for the entire temperature range. The ΔSlat is calculated as 51 J/kg K at 300 K, which is about twice larger than ΔStot value obtained from the phase diagram [Figure 7(b)]. This indicates that the lattice contribution plays the central role in the IMCE.

#### 4.2.4. Estimations of the electronic, lattice and magnetic entropy changes

Using Eqs. (11) and (12), ΔSele and ΔSlat can be calculated for the entire temperature range, which are plotted as the blue- and green-solid curves in Figure 12. The magnetic entropy change ΔSmag is calculated by subtracting ΔSele and ΔSlat from the total entropy change (ΔSmag = ΔStot � ΔSele �ΔSlat). The TMT dependence of ΔSmag estimated from the experimental data of ΔStot below 300 K is plotted as the red-dashed curve in Figure 12. The ΔSmag is estimated as ΔSmag = �29 J/kg K (1.9 J/mol K) at 300 K. As mentioned in Section 4.2.2, the magnetic entropy in A phase (S<sup>A</sup> mag) can be calculated by using the mean-field model. Using the values T<sup>C</sup> = 383 K and J = 0.88 for In13.5, the S<sup>A</sup> mag is calculated as 79 J/kg K (5.1 J/mol K) at 300 K. Therefore, the magnetic entropy of the M phase can be estimated as S<sup>M</sup> mag ¼ 108 J/kg K (7.0 J/mol K). If one assumes a random arrangement of spins in the paramagnetic M phase, the magnetic moment of the M phase gJ<sup>M</sup> ¼ 1:32 is obtained from the relation: S<sup>M</sup> mag ¼ NkBlnð2J<sup>M</sup> þ 1Þ. This magnetic moment is smaller than that of the A phase by 25%. This result indicates that the magnetic contribution to the total entropy change is composed of not only the negative change due to the spin ordering but also the positive change due to the change in the magnetic moment.

Figure 12. Calculated entropy changes through the MT as functions of the transition temperature (ΔSele, ΔSlat, ΔSmag, and ΔStot versus TMT) [39]. The solid and open squares are the experimentally obtained ΔStot [22, 42]. The open triangles are the ΔStot estimated by the M � H curves shown in Figure 7(a).

By assuming the constant value of S<sup>M</sup> mag ¼ 108 J/kg K, the temperature dependence of the magnetic entropy change can be calculated as ΔS<sup>0</sup> magðTÞ ¼ <sup>S</sup><sup>A</sup> magðTÞ � <sup>S</sup><sup>M</sup> mag. The result is plotted as a red-solid curve in Figure 12. The calculated ΔStot is also plotted as the black-dashed curve in Figure 12, which coincides with the experimentally obtained data for Ni50�yCoyMn50�xInx for the various x and y values above 300 K. Above TC, the ΔStot shows the small temperature dependence. In this temperature region, both A and M phase are paramagnetic, where the contribution of the spin ordering can be neglected. Hence, the ΔStot was determined by the difference of the lattice entropy and the difference of the magnetic moment. Below TC, the A phase is ferromagnetic, and therefore, the contribution of the spin alignment increases with decreasing TMT. This effect dominates the TMT dependence of the ΔStot. This behaviour of the ΔStot in the vicinity of the T<sup>C</sup> can be confirmed in the x dependencies of the Ni50�yCoyMn50�xInx [22, 43].

Below 300 K, the calculated ΔStot no longer reproduces the experimental results. This discrepancy can be attributed to the improper assumption of the constant S<sup>M</sup> mag. To obtain the deeper insight into the magnetic contribution to the IMCE at lower temperatures, the detailed information about the magnetic structure of M phase is necessary.

#### 4.3. NiCoMnGa

4.2.4. Estimations of the electronic, lattice and magnetic entropy changes

positive change due to the change in the magnetic moment.

the ΔStot estimated by the M � H curves shown in Figure 7(a).

the S<sup>A</sup>

phase can be estimated as S<sup>M</sup>

72 Shape Memory Alloys - Fundamentals and Applications

obtained from the relation: S<sup>M</sup>

Using Eqs. (11) and (12), ΔSele and ΔSlat can be calculated for the entire temperature range, which are plotted as the blue- and green-solid curves in Figure 12. The magnetic entropy change ΔSmag is calculated by subtracting ΔSele and ΔSlat from the total entropy change (ΔSmag = ΔStot � ΔSele �ΔSlat). The TMT dependence of ΔSmag estimated from the experimental data of ΔStot below 300 K is plotted as the red-dashed curve in Figure 12. The ΔSmag is estimated as ΔSmag = �29 J/kg K (1.9 J/mol K) at 300 K. As mentioned in Section 4.2.2, the magnetic entropy in A phase (S<sup>A</sup>

be calculated by using the mean-field model. Using the values T<sup>C</sup> = 383 K and J = 0.88 for In13.5,

of spins in the paramagnetic M phase, the magnetic moment of the M phase gJ<sup>M</sup> ¼ 1:32 is

of the A phase by 25%. This result indicates that the magnetic contribution to the total entropy change is composed of not only the negative change due to the spin ordering but also the

Figure 12. Calculated entropy changes through the MT as functions of the transition temperature (ΔSele, ΔSlat, ΔSmag, and ΔStot versus TMT) [39]. The solid and open squares are the experimentally obtained ΔStot [22, 42]. The open triangles are

mag is calculated as 79 J/kg K (5.1 J/mol K) at 300 K. Therefore, the magnetic entropy of the M

mag ¼ 108 J/kg K (7.0 J/mol K). If one assumes a random arrangement

mag ¼ NkBlnð2J<sup>M</sup> þ 1Þ. This magnetic moment is smaller than that

mag) can

In the NiCoMnIn, the negative entropy change due to the spin ordering through the MFIMT suppresses the ΔStot and dominates the TMT dependence of the ΔStot, as shown in Figure 12. Therefore, the large IMCE due to the lattice entropy change and due to the change of magnetic moment is expected to occur when a material undergoes MT from the ferromagnetic/paramagnetic M phase to the ferromagnetic/paramagnetic A phase.

Recently, Kihara et al. carried out the magnetization and MCE measurements for Ni41Co9Mn31.5 Ga18.5. Figure 13 shows the isothermal M � H curve measured in the pulsed magnetic field at 260 K. At this temperature, the sample is in ferromagnetic M phase at zero-field. As shown in Figure 13, the sample undergoes the MFIMT from the ferromagnetic M phase to the ferromagnetic A phase. In this case, it can be predicted that the negative entropy change due to the spin ordering is smaller than that when the magnetic field is applied to the paramagnetic M phase. Hence, the large and TMT independent IMCE is expected to be observed.

Figure 14 shows the result of the direct MCE measurement at 257.5 K. The sample temperature increases with increasing magnetic field below 16 T in both field increasing (the M phase) and decreasing (the field induced A phase) processes. This temperature increase can be considered as the conventional MCEs due to the forced spin alignment by the magnetic field. In the field increasing process, the sample temperature steeply decreases above 16 T, which can be considered as the IMCE through the MFIMT. However, this temperature decrease is observed up to 36 T, where the MFIMT seems to be completed above 30 T according to the magnetization measurement (Figure 13). In the successive field decreasing process, the sample temperature increases reversibly and starts decreasing at around 25 T, where there is no anomaly in the M � H

Figure 13. M H curve measured in the pulsed magnetic field at 260 K for Ni41Co9Mn31.5Ga18.5.

Figure 14. Magnetic field variation of the sample temperature in the adiabatic condition for Ni41Co9Mn31.5Ga18.5 measured at the initial temperature of 257.5 K.

curve. The temperature difference between 0 and 36 T is ΔTad (36T) = 3.9 K, which is indicated in Figure 14. Contrary to the expectation, the large IMCE is not observed. The origin of this unexpected MCE in the Ni41Co9Mn31.5Ga18.5 is still unclear.

#### 5. Conclusion

In this chapter, the direct measurement system of magnetocaloric effect (MCE) under pulsedhigh-magnetic field is introduced. Using this technique, the inverse magnetocaloric effects (IMCE) of NiCoMnIn and NiCoMnGa were measured as adiabatic temperature changes in pulsed fields. The electronic, lattice and magnetic contributions to the total entropy change through the magnetic field induced martensitic transformation (MFIMT) are individually evaluated by the combination of the results of MCE, specific heat and magnetization measurements. Through the analysis for NiCoMnIn, the validity of the present experiments for the metamagnetic shape memory alloys is demonstrated.

In conclusion, the origins of the IMCE of NiCoMnIn are as follows. The electronic contribution to the IMCE is negligibly small. On the other hand, the significant change in entropy of the lattice system plays a dominant role in the IMCE. The magnetic contribution involves the two different contributions competing with each other: the increase of magnetic moment (positive entropy change) and the spin ordering (negative entropy change). Moreover, the magnetic contribution determines the composition dependence of the total entropy change (ΔStot) at the martensitic transformation (MT), when the MT temperature (TMT) and Curie temperature in the austenite phase (TC) are close to each other. However, the model presented in this chapter cannot reproduce the composition dependence of the ΔStot when the TMT and T<sup>C</sup> are far away from each other. To clarify the origin of the IMCE for the entire temperature range, the detailed experiments about the magnetic structure of the martensitic phase are important.

The magnetization and MCE measurements are carried out at 260 K for Ni41Co9Mn31.5Ga18.5. In the MCE measurement, a complex field dependence of the sample temperature is observed. To clarify the origin of the MCE in the Ni41Co9Mn31.5Ga18.5, the MCE measurements in the wide range of temperatures and the instantaneous observation of the structural change through the MFIMT are highly desirable.

#### Acknowledgements

Figure 14. Magnetic field variation of the sample temperature in the adiabatic condition for Ni41Co9Mn31.5Ga18.5 mea-

Figure 13. M H curve measured in the pulsed magnetic field at 260 K for Ni41Co9Mn31.5Ga18.5.

sured at the initial temperature of 257.5 K.

74 Shape Memory Alloys - Fundamentals and Applications

The authors thank Dr A. Miyake for helping with the magnetization measurements under the pulsed magnetic fields at the Institute for Solid State Physics, the University of Tokyo. This chapter was partly supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan, through Grant-in-Aid for Scientific Researches (Grant Nos. 23340096 and 16K17734).

#### Author details

Takumi Kihara<sup>1</sup> \*, Xiao Xu2 , Wataru Ito3 , Ryosuke Kainuma<sup>2</sup> , Yoshiya Adachi4 , Takeshi Kanomata5 and Masashi Tokunaga<sup>6</sup>

\*Address all correspondence to: t\_kihara@imr.tohoku.ac.jp

1 Institute for Materials Research, Tohoku University, Sendai, Japan

2 Department of Materials Science, Tohoku University, Sendai, Japan


5 Research Institute for Engineering and Technology, Tohoku Gakuin University, Tagajo, Japan

6 The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan

#### References


[10] Wada H, Tanabe Y. Giant magnetocaloric effect of MnAs1-xSbx. Applied Physics Letters. 2001;58:3657

Author details

Takumi Kihara<sup>1</sup>

Japan

References

1926;81:1154

Chemical Society. 1927;49(8):1870

materials. Reports on Progress in Physics. 2005;68:1479

material Gd5(Si2Ge2). Physical Review Letters. 2000;84:4617

Tb5Si2Ge2. Physical Review Letters. 2004;93:137201

Review Letters. 1997;78:4494

\*, Xiao Xu2

76 Shape Memory Alloys - Fundamentals and Applications

Takeshi Kanomata5 and Masashi Tokunaga<sup>6</sup>

, Wataru Ito3

1 Institute for Materials Research, Tohoku University, Sendai, Japan 2 Department of Materials Science, Tohoku University, Sendai, Japan

\*Address all correspondence to: t\_kihara@imr.tohoku.ac.jp

, Ryosuke Kainuma<sup>2</sup>

3 Materials Science and Engineering, Sendai National College of Technology, Natori, Japan

[1] Debye P. Some observations on magnetization at low temperature. Annals of Physics.

[2] Giauque WF. A thermodynamic treatment of certain magnetic effects. A proposed method of producing temperatures considerably below 1 degree absolute. Journal of the American

[3] Pecharsky VK, Gschneidner KA Jr. Giant magnetocaloric effect in Gd5(Si2Ge2). Physical

[4] Brück E. Developments in magnetocaloric refrigeration. Journal of Physics D. 2005;38:R381

[5] Gschneidner KA Jr, Pecharsky VK, Tsokol AO. Recent developments in magnetocaloric

[6] Choe W, Pecharsky VK, Pecharsky AO, Gschneidner KA Jr, Young VG Jr, Miller GJ. Making and breaking covalent bonds across the magnetic transition in the giant magnetocaloric

[7] Morellon L, Arnold Z, Magen C, Ritter C, Prokhnenko O, Skorokhod Y, Algarabel PA, Ibarra MR, Kamarad J. Pressure enhancement of the giant magnetocaloric effect in

[8] Tegus O, Bruck E, Buschow KHJ, de Boer FR. Transition-metal-based magnetic refriger-

[9] Fujieda S, Fujita A, Fukamichi K. Large magnetocaloric effect in La(FexSi1-x)13 itinerant-

ants for room temperature applications. Nature (London). 2002;415:150

electron metamagnetic compounds. Applied Physics Letters. 2002;81:1276

4 Graduate School of Science and Engineering, Yamagata University, Yonezawa, Japan 5 Research Institute for Engineering and Technology, Tohoku Gakuin University, Tagajo,

6 The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan

, Yoshiya Adachi4

,


[40] Li Z, Jing C, Zhang HL, Yu DH, Chen L, Kang BJ, Cao SX, Zhang JC. A large and reproducible metamagnetic shape memory effect in polycrystalline Ni45Co5Mn37In13 Heusler alloy. Journal of Applied Physics. 2010;108:113908-1-4

[26] Amaral JS, Amaral VS. On estimating the magnetocaloric effect from magnetization measurements. Journal of Magnetism and Magnetic Materials. 2010;322:1552–1557 [27] Kataoka M, Kanomata T, Umetsu RY, Kainuma R. Normal and inverse magnetocaloric effects in the ferromagnetic shape memory alloys. Journal of Magnetism and Magnetic

[28] Balli M, Fruchart D, Gignoux D, Zach R. The "colossal" magnetocaloric effect in Mn1 xFexAs: What are we really measuring. Applied Physics Letters. 2009;95:072509-1-3 [29] Cui W, Liu W, Zhang Z. The origin of large overestimation of the magnetic entropy changes calculated directly by Maxwell relation. Applied Physics Letters. 2010;96:222509-1-3 [30] Tocado L, Palacios E, Burriel R. Entropy determinations and magnetocaloric parameters in systems with first-order transitions: Study of MnAs. Journal of Applied Physics.

[31] Liu GJ, Sun R, Shen J, Gao B, Zhang HW, Hu FX, Shen BG. Determination of the entropy changes in the compounds with a first-order magnetic transition. Applied Physics Let-

[32] Kuz'min MD, Tishin AM. Magnetocaloric effect. Part 1: An introduction to various

[33] Levitin RZ, Snegirev VV, Kopylov AV, Lagutin AS, Gerber A. Magnetic method of magnetocaloric effect determination in high pulsed magnetic fields. Journal of Magne-

[34] Dan'kov SY, Tishin AM, Pecharsky VK, Gschneidner KA. Experimental device for studying the magnetocaloric effect in pulse magnetic fields. Review of Scientific Instruments.

[35] Kohama Y, Marcenat C, Klein T, Jaime M. AC measurement of heat capacity and magnetocaloric effect for pulsed magnetic fields. Review of Scientific Instruments. 2010;81:104902-

[36] Kihara T, Kohama Y, Hashimoto Y, Katsumoto S, Tokunaga M. Adiabatic measurements of magneto-caloric effects in pulsed high magnetic fields up to 55 T. Review of Scientific

[37] Kohama Y, Hashimoto Y, Katsumoto S, Tokunaga M, Kindo K. Heat-pulse measurements of specific heat in 36 ms pulsed magnetic fields. Measurement Science and Technology.

[39] Kihara T, Xu X, Ito W, Kainuma R, Tokunaga M. Direct measurements of inverse magnetocaloric effects in metamagnetic shape-memory alloy NiCoMnIn. Physical Review

[38] Kindo K. 100 T magnet developed in Osaka. Physica B. 2001;294–295:585–590

aspects of theory and practice. Cryogenics. 1992;32(6):545–557

tism and Magnetic Materials. 1997;170:223–227

Materials. 2014;361:34–43

78 Shape Memory Alloys - Fundamentals and Applications

2009;105:093918-1-3

ters. 2007;90:032507-1-3

1997;68(6):2432–2437

2013;24:115005

B. 2014;90:214409-1-6

Instruments. 2013;84:074901-1-7

1-13


**Provisional chapter**

#### **Applications of Nanostructural NiTi Alloys for Medical Devices Devices**

**Applications of Nanostructural NiTi Alloys for Medical** 

DOI: 10.5772/intechopen.69238

Elena O. Nasakina, Mikhail A. Sevostyanov, Alexander S. Baikin, Alexey V. Seryogin, Sergey V. Konushkin, Konstantin V. Sergienko, Alexander V. Leonov and Alexey G. Kolmakov Alexander S. Baikin, Alexey V. Seryogin, Sergey V. Konushkin, Konstantin V. Sergienko, Alexander V. Leonov and Alexey G. Kolmakov Additional information is available at the end of the chapter

Elena O. Nasakina, Mikhail A. Sevostyanov,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69238

#### **Abstract**

New nanostructural shape memory alloy (55.91 wt% of Ni and 44.03 wt% of Ti) for the production of minimally invasive implantation medical devices (stents) was tested for corrosion resistance under static conditions by dipping it into solutions with various acidities (pH from 1.68 to 9.18) for 2 years, for static mechanical properties and for biocompatibility. The material for investigations was 280‐μm wires before and after thermal treatment at 450°C for 15 min in air and surface mechanical treatment. The characteristic image and size of grains were determined using the transmission electron microscope (TEM), and the phase composition; surface morphology; and the layer‐by‐ layer composition were investigated using an X‐ray diffractometer; a scanning electron microscope (SEM); and an Auger spectrometer. The nickel release from the investigated nanostructural nitinol is less in comparison with data for microstructural nitinol in a solution of any acidity. Dissolution in the alkali medium is absent. A significant retar‐ dation of the nickel ion release (and insignificant concentration as a whole) and the absence of titanium ion release in the weakly acidic and neutral solutions with pol‐ ished samples are observed. A simultaneous 7–11% increase in strength and plasticity in comparison with microstructural nitinol was attained. Toxicity of samples has not been revealed.

**Keywords:** nitinol, biocompatibility, corrosion resistance, shape memory effect, superelasticity

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **1. Introduction**

Nitinol (NiTi) alloys possessing a shape memory effect (SME) and mechanical characteristics similar to the behavior of living tissues have been already used for years as the material for production of medical devices, including implants, for example stents [1–6].

SME promotes the production of the least traumatic self‐expanding stents, which are implanted into the human body to relieve obstruction or constriction of the respiratory, digestive, excre‐ tory, and cardiovascular systems and to restore the normal circulation of physiological flows without the need for additional devices except the catheter carrier [6]. But nitinol contains nickel (including on its surface) which is toxic for organisms [7–9]. Different authors give completely different durations (up to constant) and magnitudes (from fractions to hundreds of mg/L) of the nickel ion release from microstructural nitinol into the medium [8–16]. And also, a different level of nitinol biocompatibility and of its electro‐chemical corrosion characteristics is noted [2, 8–10, 17, 18]. Nitinol properties are generally determined by the structure and the composition, which depend on the production process and treatments [2, 8–9, 11, 17, 19–22].

As is well known, formation of nanostructures imparts special, controlled characteristics to materials [23, 24]. Nitinol, as titanium, is a self‐passivated material, that is, it forms a complex surface oxide layer, which protects the material from corrosion, and also is biocompatible in itself [2, 10–12]. This oxide layer also surrounds each grain. Therefore, it is possible to assume that increase in a volume fraction of such grain boundaries in alloy at its nanostructuring will positively affect its corrosion resistance and biocompatibility. On the other hand, the high density of intergranular surface defects could lead to a poor corrosion performance since cor‐ rosion attack typically initiates at surface heterogeneities [25].

As described in Ref. [25], one of the most efficient methods of fabrication of bulk nanocrys‐ talline materials is the severe plastic deformation (SPD) leading to the breakdown of coarse grains into nanosized (with the size less than 100 nm) grains. The main SPD techniques are equal channel angular extrusion (ECAP) including thermomechanical treatment, high pres‐ sure torsion (HPT), hydrostatic extrusion, and others.

It is noticed in literature that nanostructure has an ambiguous effect on the nitinol proper‐ ties and this is little studied. Investigations showed that in HCl nitinol in the nanosized state (grain diameter of 10 nm) reveals a significantly lower corrosion resistance than in the micro‐ structural state because of an increase in the boundary length and in the amount of defects of the grain structure; and nitinol in NaCl, on the contrary, is more passive (consequently corrosion resistant) [26]. In the study [25, 27–29], there is no difference in corrosion processes between micro‐ and nanostructured nitinol. But in case of nanostructured titanium, corrosion resistance reduction was observed [30] though in all works positive influence on mechani‐ cal properties of materials was noted. In Refs. [31–33], large recoverable strains, resistance against cyclic plasticity mechanisms, plasticity, and good fatigue response were achieved by grain refinement by means of ECAP. HPT produced an ultrafine‐grained structure with grain sizes in the range from 5 to 100 nm, which leads to a very high strength, good ductility, high recovery stress, and to a maximum reverse strain, significantly improving cyclic endur‐ ance and pseudoelasticity [34]. Additional electropulse current treatment of coarse‐grained and ultrafine‐grained NiTi (grain diameter more than 100 nm) results in the formation of structures with a grain size less than 80 and 100 nm, correspondingly (was more effective for microstructural material), with increased deformability and mechanical properties [35]. At the same time, according to the computational model and the experimental data in Ref. [36], it was shown that when the grain size is smaller than some critical value (around 50–80 nm in all directions), the martensitic phase transformation is totally suppressed even though the material severely deforms; whereas in another experiment, nanostructure does not prevent phase transformations that are influenced only by aging regime and cooling rate [37]. Authors of Ref. [38] noted that the cold‐drawn nitinol contains 50 nm thick grains and possesses quite high tensile strength, hardness, and fatigue life, the maxima of which are reached after heat treatments at 450°C within 15 min; and heat treatments at above 450°C induced recrystalliza‐ tion and grain and precipitate growth. It is shown that the nanostructural morphology of the nitinol and titanium surfaces at the tissue‐implant interface positively affects biocompatibility *in vitro* and *in vivo* and enhances functional activity of desirable cell cultures and enzymes without any fibrous tissue intervention. However, not all nanomorphologies on implants imparted a similar biological response [39, 40].

**1. Introduction**

82 Shape Memory Alloys - Fundamentals and Applications

Nitinol (NiTi) alloys possessing a shape memory effect (SME) and mechanical characteristics similar to the behavior of living tissues have been already used for years as the material for

SME promotes the production of the least traumatic self‐expanding stents, which are implanted into the human body to relieve obstruction or constriction of the respiratory, digestive, excre‐ tory, and cardiovascular systems and to restore the normal circulation of physiological flows without the need for additional devices except the catheter carrier [6]. But nitinol contains nickel (including on its surface) which is toxic for organisms [7–9]. Different authors give completely different durations (up to constant) and magnitudes (from fractions to hundreds of mg/L) of the nickel ion release from microstructural nitinol into the medium [8–16]. And also, a different level of nitinol biocompatibility and of its electro‐chemical corrosion characteristics is noted [2, 8–10, 17, 18]. Nitinol properties are generally determined by the structure and the composition,

As is well known, formation of nanostructures imparts special, controlled characteristics to materials [23, 24]. Nitinol, as titanium, is a self‐passivated material, that is, it forms a complex surface oxide layer, which protects the material from corrosion, and also is biocompatible in itself [2, 10–12]. This oxide layer also surrounds each grain. Therefore, it is possible to assume that increase in a volume fraction of such grain boundaries in alloy at its nanostructuring will positively affect its corrosion resistance and biocompatibility. On the other hand, the high density of intergranular surface defects could lead to a poor corrosion performance since cor‐

As described in Ref. [25], one of the most efficient methods of fabrication of bulk nanocrys‐ talline materials is the severe plastic deformation (SPD) leading to the breakdown of coarse grains into nanosized (with the size less than 100 nm) grains. The main SPD techniques are equal channel angular extrusion (ECAP) including thermomechanical treatment, high pres‐

It is noticed in literature that nanostructure has an ambiguous effect on the nitinol proper‐ ties and this is little studied. Investigations showed that in HCl nitinol in the nanosized state (grain diameter of 10 nm) reveals a significantly lower corrosion resistance than in the micro‐ structural state because of an increase in the boundary length and in the amount of defects of the grain structure; and nitinol in NaCl, on the contrary, is more passive (consequently corrosion resistant) [26]. In the study [25, 27–29], there is no difference in corrosion processes between micro‐ and nanostructured nitinol. But in case of nanostructured titanium, corrosion resistance reduction was observed [30] though in all works positive influence on mechani‐ cal properties of materials was noted. In Refs. [31–33], large recoverable strains, resistance against cyclic plasticity mechanisms, plasticity, and good fatigue response were achieved by grain refinement by means of ECAP. HPT produced an ultrafine‐grained structure with grain sizes in the range from 5 to 100 nm, which leads to a very high strength, good ductility, high recovery stress, and to a maximum reverse strain, significantly improving cyclic endur‐ ance and pseudoelasticity [34]. Additional electropulse current treatment of coarse‐grained

production of medical devices, including implants, for example stents [1–6].

which depend on the production process and treatments [2, 8–9, 11, 17, 19–22].

rosion attack typically initiates at surface heterogeneities [25].

sure torsion (HPT), hydrostatic extrusion, and others.

Thus, it is still of interest how nanostructuring can influence operational characteristics of nitinol as a medical (implantation) material. The aim of this chapter is to investigate the com‐ position, structure, and properties of polycrystalline nitinol with nanograins.

### **2. Obtaining and investigating nanostructural NiTi alloy for medical application**

Nanostructured wires with a diameter of 280 μm made of nitinol (50.9 wt% of Ni) were obtained from the Institution of Russian Academy of Sciences, A.A. Baikov Institute of Metallurgy and Material Science, before processing and after mechanical and/or thermal treatments were investigated.

To obtain the material, we used the modernized complex plastic deformation technology developed in the IMET RAS. The corresponding charge was re‐melted several times in a vac‐ uum furnace in argon. Ingots were transformed by rolling and rotary forging at a temperature of 750–1000°C into bars with a diameter of up to 4 mm, from which the wire was obtained by step‐wise hot drawing through a synthetic diamond die and by intermediate thermal treat‐ ment for stabilization of the material structure and removal of mechanical stress.

The mechanical treatment (by abrasive paper from 180 to 1000 grit and finally by GOI (State Optical Institute) paste to a mirror surface) of the wire surface was carried out by an abrasive cloth for the removal of flat indentations and defects in the form of dimples after wire draw‐ ing. The decrease in the diameter was to 10 μm in comparison with the original. The thermal treatment of nitinol that allows one to vary static properties and cyclic loadings in operating conditions with a wide range of deformations is extremely important for stabilization of the properties, constraining (shaping) the samples, and successful application of the product. The optimal conditions for the thermal treatment were chosen from the previous researches as *T* = 450°C and an annealing time of 15 min (what has coincided with [38]), and carried out in a LOIP LF 7/13\_G2 muffler (Russia) equipped with a programmed controller.

To investigate the wire microstructure, the preliminary etching of the surface was carried out in a mixture of 1 mL HF + 2 mL HNO<sup>3</sup> + 47 mL H<sup>2</sup> O composition for 2–3 min, upon ter‐ mination of which the sample was washed several times in distilled water and dried in air. Investigations were carried out on an Axiovert 40 MAT optical metallographic microscope (Carl Zeiss, Germany) with digital image processing. The characteristic image and size of grains were determined with the use of a TECNAI 12 transmission electron microscope (TEM) (FEI COMPANY, USA), and samples were prepared by means of a GATAN 691 ion‐etching device (Gatan Inc., USA). To determine the phase composition, we used an Ultima IV X‐ray dif‐ fractometer (Rigaku Co., Japan) in Cu Kα—radiation on the basis of Bragg‐Brentano method. Phase analysis was prepared in the PDXL program complex using the ICDD database. The surface morphology and the layer‐by‐layer composition were investigated on a scanning elec‐ tion microscope (SEM) VEGA II SBU with the module INCA Energy for energy‐dispersive analysis (TESCAN, Czech Republic) and on a JAMP‐9500F Auger spectrometer (JEOL Co., Japan) in combination with ion etching at argon bombardment under an angle of 30°.

The bright‐field TEM image in **Figure 1a** shows that nitinol grains resemble nanofibers with a cross‐section size from 30 to 70 nm, and the longitudinal section size of fibers is several microns. Thus, grains are extended along the wire axis. Rings are seen in the microdiffraction pattern (**Figure 1b**), which point to polycrystalline structure of the sample. The calculation and analysis of the interplanar spacing distances from the electron diffraction patterns show that they correspond to the B2 phase.

Also, according to the energy‐dispersive and X‐ray structural analyses (**Figure 2a**), the material volume is represented by the base of the B2 phase of TiNi and inclusions of Ti2 Ni

**Figure 1.** Transmission electron microscopy data: (a) bright‐field TEM image and (b) microdiffraction pattern.

*T* = 450°C and an annealing time of 15 min (what has coincided with [38]), and carried out in

To investigate the wire microstructure, the preliminary etching of the surface was carried

mination of which the sample was washed several times in distilled water and dried in air. Investigations were carried out on an Axiovert 40 MAT optical metallographic microscope (Carl Zeiss, Germany) with digital image processing. The characteristic image and size of grains were determined with the use of a TECNAI 12 transmission electron microscope (TEM) (FEI COMPANY, USA), and samples were prepared by means of a GATAN 691 ion‐etching device (Gatan Inc., USA). To determine the phase composition, we used an Ultima IV X‐ray dif‐ fractometer (Rigaku Co., Japan) in Cu Kα—radiation on the basis of Bragg‐Brentano method. Phase analysis was prepared in the PDXL program complex using the ICDD database. The surface morphology and the layer‐by‐layer composition were investigated on a scanning elec‐ tion microscope (SEM) VEGA II SBU with the module INCA Energy for energy‐dispersive analysis (TESCAN, Czech Republic) and on a JAMP‐9500F Auger spectrometer (JEOL Co.,

Japan) in combination with ion etching at argon bombardment under an angle of 30°.

The bright‐field TEM image in **Figure 1a** shows that nitinol grains resemble nanofibers with a cross‐section size from 30 to 70 nm, and the longitudinal section size of fibers is several microns. Thus, grains are extended along the wire axis. Rings are seen in the microdiffraction pattern (**Figure 1b**), which point to polycrystalline structure of the sample. The calculation and analysis of the interplanar spacing distances from the electron diffraction patterns show

Also, according to the energy‐dispersive and X‐ray structural analyses (**Figure 2a**), the material volume is represented by the base of the B2 phase of TiNi and inclusions of Ti2

**Figure 1.** Transmission electron microscopy data: (a) bright‐field TEM image and (b) microdiffraction pattern.

+ 47 mL H<sup>2</sup>

O composition for 2–3 min, upon ter‐

Ni

a LOIP LF 7/13\_G2 muffler (Russia) equipped with a programmed controller.

out in a mixture of 1 mL HF + 2 mL HNO<sup>3</sup>

84 Shape Memory Alloys - Fundamentals and Applications

that they correspond to the B2 phase.

**Figure 2.** Nitinol structure data: (a) X‐ray diffraction patterns and (b) microstructure analysis.

intermetallides. The character of X‐ray patterns indicates that the composition is unchanged after thermal treatment. The microstructure analysis (**Figure 2b**) verifies that the base of the alloy investigated is represented by the B2 phase, and the material contains Ti2 Ni intermetal‐ lides, and the sizes and the volume fraction of which are unchanged after annealing.

According to the SEM images (**Figure 3a**), the wire surface before treatment is heteroge‐ neous and covered by spots, dark areas alternating with bright ones and the high rough‐ ness is clearly expressed. After annealing (**Figure 3b**), the surface externally is similar to the initial one, whereas after polishing (**Figure 3c**) practically all defects and the roughness are smoothed, spots are absent, and only traces of treatment are visible. Surface microdefects are grooves with a depth and width of less than 1 μm, and after polishing and annealing, the wire surface is the most‐smoothed and uniform (**Figure 3d**).

**Figure 3.** SEM image of the wire surface: (a) before treatment, (b) after annealing, (c) after polishing, and (d) after polishing and annealing.

The compositions of bright and dark spots are different (**Figure 4a**): a high content of titanium oxide is revealed in bright ones and carbon, in dark ones. Both layers attain 3 μm in thickness and are not placed over each other. Such a thick surface layer, as was believed, was a result of the long intermediate thermal treatment during the wire production [8, 9, 41]. Most likely, therefore, annealing for 15 min does not have an effect on features of changing the com‐ position. Carbon is present on the wire surface, probably, due to graphite‐containing lubri‐ cant (used during wire drawing), which remains on the wire surface and then sticks during annealing. The same effect (impurities on the surface after several cycles of treatment due to lubricant) was observed in Ref. [11]. The composition of the polished surface is homogeneous (**Figure 4b**): the entire wire is covered by an oxide layer less than 20 nm in thickness, which, according to the literature [41], must positively affect corrosion resistance under dynamic conditions because thin oxide layers show greater flexibility and may adapt to loads applied to the material. The minus is the presence of nickel in the surface layer, though in an insig‐ nificant amount. The subsequent annealing promotes the formation of an oxide‐nitride layer up to 80–150 nm in depth, to 50–60 nm free from nickel that could affect corrosion resistance of material positively (**Figure 4c**), that is also connected with thermal treatment which leads to the formation of mixed surface layer consisting of titanium and nickel oxides and to the growth of the oxide layer thickness [8, 9, 41].

intermetallides. The character of X‐ray patterns indicates that the composition is unchanged after thermal treatment. The microstructure analysis (**Figure 2b**) verifies that the base of the

According to the SEM images (**Figure 3a**), the wire surface before treatment is heteroge‐ neous and covered by spots, dark areas alternating with bright ones and the high rough‐ ness is clearly expressed. After annealing (**Figure 3b**), the surface externally is similar to the initial one, whereas after polishing (**Figure 3c**) practically all defects and the roughness are smoothed, spots are absent, and only traces of treatment are visible. Surface microdefects are grooves with a depth and width of less than 1 μm, and after polishing and annealing, the wire

**Figure 3.** SEM image of the wire surface: (a) before treatment, (b) after annealing, (c) after polishing, and (d) after

polishing and annealing.

Ni intermetal‐

alloy investigated is represented by the B2 phase, and the material contains Ti2

surface is the most‐smoothed and uniform (**Figure 3d**).

86 Shape Memory Alloys - Fundamentals and Applications

lides, and the sizes and the volume fraction of which are unchanged after annealing.

The mechanical properties of samples with a working part length of 45 mm were determined under the conditions of static stretching on an Instron 3382 (Instron, USA) universal testing machine with a loading speed of 2 mm/min. The base diameter was used in the calculation of strength properties. Three to five samples were tested per one experimental point. The conventional yield strength *σ*0.2, the ultimate strength *σ*u, and the relative elongation *δ* were determined (**Table 1**).

Results of the mechanical stretching tests showed positive influence of annealing on strength properties of alloy in both structural states, and strength properties of a nanostructural nitinol exceeded strength properties of microstructural nitinol by 1.3–1.5 times. Nitinol wire polish‐ ing after drawing promotes an increase in the static properties of a nanostructural alloy of 18%. Relative elongations of all samples were of 51–53%.

Heat treatment significantly increases cyclic durability of nanostructural material (**Figure 5**). Its durability in the initial state is insignificant, and all fatigue curves lie in the low‐cyclic area (below *N* = 6 × 103 cycles). After annealing, durability sharply increases both in the field of low‐cycle and multicycle fatigue. The durability limit after annealing on the basis of 4 × 106 cycles is 400 MPa.

The wire disintegrates with the formation of a neck (**Figure 6**). The fracture surface is aligned almost perpendicularly to the stretching axis. It consists of a set of unequally sized self‐similar pores (cups) of the viscous break. The destruction starts from the most serious defects on the sample surface in the zone of the neck where the main crosscut crack, to which the formed pores are merged, is formed. Heat treatment practically does not influence the nature of destruction.

Micro‐Vickers hardness measurements determined at loading 1–2 N by the WOLPERT GROUP 401/402 device—MVD (WILSON Instruments, USA) equipped with a light microscope.

**Figure 4.** Wire surface structure according to Auger spectroscopy prior to the immersion testing: (a) before treatment and after annealing (dark and bright spots in **Figure 3a** and **b**), (b) after polishing, and (c) after polishing and annealing.


**Table 1.** Effect of annealing on nitinol mechanical properties.

**Figure 4.** Wire surface structure according to Auger spectroscopy prior to the immersion testing: (a) before treatment and after annealing (dark and bright spots in **Figure 3a** and **b**), (b) after polishing, and (c) after polishing and annealing.

88 Shape Memory Alloys - Fundamentals and Applications

**Figure 5.** Fatigue curves of a nanostructural nitinol before treatment (1) and after annealing at 450°C, 15 min (2).

Microhardness of nanostructural material is higher by ≈ 38% in comparison with a microstruc‐ tural analog (332 ± 3 and 240 ± 3 HV, respectively). It can be explained by the increase in volume fraction of grain boundaries and also by the presence of titanium oxides on these borders, the hardness of which is higher than the basic hardness. As a result of surface polishing, a formation of a thin uniform oxide layer with a small share of impurity takes place, and micro‐ hardness increases by 17.5% (from 332 ± 3 to 390 ± 4 HV). The subsequent annealing within 15 min at 450°C promotes the thickening of the oxide layer and an increase in the share of titanium oxides in it that leads to additional increase in microhardness by 2.3% (to 399 ± 3 HV).

The corrosion resistance of the material was determined under static conditions by immersion into solutions with various acidities because pH in the human body changes from 1 to 9 (for example, 1.05 at the duodenum; 1.53–1.67 is the norm of gastric juice; 3.8–4 in the near‐surface area of the bowels; 7.34–7.43 is the norm of blood; 8.5–9 in the bowels; etc.). We used a neutral 0.9 wt% solu‐ tion of sodium chloride, artificial plasma, and four standard buffer solutions (to reproduce acidic and alkaline media at the given level) [8, 9, 42], which are listed in **Table 2**, and for comparison, we also used a 0.03‐M solution of hydrochloric acid. The standard buffer solutions were prepared from corresponding standard trimetric substances (fixanals) made by Merk (USA).

Wire samples (1—TiNi before treatment, 2—TiNi after annealing, 3—TiNi after polishing, and 4—TiNi after polishing and annealing), with a weight of 32.6 mg each (separately from

**Figure 6.** Disintegration behavior of the nitinol wire at static stretching.


**Table 2.** pH and composition of the solutions used for immersion tests.

each other) were placed into flasks of heat‐resistant laboratory glass (acidic and neutral media) or polypropylene (alkali medium) with 100 mL of the selected solution. The flasks were hermetically closed by a lapped cover and aged in a dark place. Sampling from flasks for analysis was done after a selected period (10, 25, 45, 60, 75, 236 or 287, 704, or 754 days). The initial buffer solutions were used as reference solutions. Analysis was carried out by an ULTIMA 2 sequential atomic emission spectrometry (HORIBA Jobin Yvon, Japan) for using atomic emission spectrometry (AES) with inductively coupled plasma (ICP) for direct simul‐ taneous determination of titanium and nickel in solutions.

After immersion, the surface morphology and layer‐by‐layer composition were also investigated.

The results of measuring the ion release into solutions are represented in **Figures 7** and **8**. Insignificant corrosion is observed in acidic and neutral media, and the nickel concentration is less than the average magnitudes cited in the literature [8, 9, 11–13, 43]; however, the titanium content is revealed in solutions. There are no results about all samples in the alkaline environ‐ ment and artificial plasma, and also about TiNi‐3 and TiNi‐4 samples in solutions with acidity 3.56–6.31 since, in these cases, the release of elements was zero or below a limit of detection for the overall time of the investigation.

Comparison of the treatment effect on the corrosion resistance of samples (**Figure 7**) makes it possible to conclude that the samples undergo the most corrosion after annealing, and the mechanical treatment, as was expected, strongly increases the corrosion resistance of nitinol.

According to the literature, the thermal treatment at a temperature from 400 to 1000°C, which is required for stabilization of the mechanical properties, always results in the significant wors‐ ening of the corrosion resistance due to the formation of nonuniform surface layers consisting of titanium and nickel oxides [21, 22, 41]. The undesirable effect of the thermal treatment on the corrosion resistance can also be explained by the occurrence of tempering and recrystallization of the outer cold‐hardened (strengthening) layer on the surface of untreated nitinol, which forms during its production. At the same time, the surface treatment, which facilitates the for‐ mation of the most perfect and homogeneous passive film, increases the corrosion resistance.

And in case of TiNi‐4 and TiNi‐3 samples on an initial stage of researches regularity has the return nature. It is possible to assume that in the beginning thicker oxide surface layer of a sample after polishing and annealing acts as the better barrier against diffusion of nickel ions in solution;

**Figure 7.** Nickel concentration in solution with pH 1.68 as a function of sample treatment and immersion time.

each other) were placed into flasks of heat‐resistant laboratory glass (acidic and neutral media) or polypropylene (alkali medium) with 100 mL of the selected solution. The flasks were hermetically closed by a lapped cover and aged in a dark place. Sampling from flasks for analysis was done after a selected period (10, 25, 45, 60, 75, 236 or 287, 704, or 754 days). The initial buffer solutions were used as reference solutions. Analysis was carried out by an ULTIMA 2 sequential atomic emission spectrometry (HORIBA Jobin Yvon, Japan) for using atomic emission spectrometry (AES) with inductively coupled plasma (ICP) for direct simul‐

С4 О8 × 2Н<sup>2</sup>

Н5 О6

> Н5 О4 К, 0.05 М

В4 О7 × 10Н<sup>2</sup>

(0.22 mМ), CaCl<sup>2</sup>

Н12О<sup>6</sup>

О, 0.05 М

О, 0.05 М

(2.5 mМ), MgSO<sup>4</sup>

(26.3 mМ), K<sup>2</sup>

HPO<sup>4</sup>

·7H2

(0.9 mМ), KCl

SO<sup>4</sup>

O (0.82 mМ), Na<sup>2</sup>

К, 0.025 М

(5.55 mM) [8–9]

taneous determination of titanium and nickel in solutions.

**Figure 6.** Disintegration behavior of the nitinol wire at static stretching.

1 1.68 Potassium tetraoxalate КН<sup>3</sup>

2 3.56 Acid potassium tartrate С<sup>4</sup>

3 4.01 Acid potassium phthalate С<sup>8</sup>

4 6.31 Sodium chloride NаСl, 0.9 wt% 5 9.18 Acid sodium tetraborate Nа<sup>2</sup>

**Table 2.** pH and composition of the solutions used for immersion tests.

6 7.36 Artificial plasma: NaCl (92.3 mМ), NaHCO<sup>3</sup>

(2.7 mМ), NaH<sup>2</sup>

(1.48 mМ), D‐glucose С<sup>6</sup>

PO<sup>4</sup>

**No рН Сomposition**

90 Shape Memory Alloys - Fundamentals and Applications

**Figure 8.** Nickel concentration in investigated solutions with various acidities with nitinol sample without treatment as a function of sampling time. The marked curves correspond to solution pH: 1—1.68, 2—3.56, 3—4.01, and 4—6.31.

however, after a long period of immersion, its nonuniformity does not allow to slow down leach‐ ing of elements as effectively as in case of a more uniform passive film which is initially received when polishing. The ratio of metal concentration in solutions with the annealed and the nonan‐ nealed samples at existence and lack of polishing treatment approximately coincides.

The surface examination after holding in solutions also reveals the hardest corrosive attack after annealing and the lowest after polishing. However, the wire diameter changes dispro‐ portionately to the ion concentration in the solution. For example, the diameters of untreated and annealed wires are almost equal after holding in the media with pH 1.68 (**Table 3**); that is, the surface fails nonuniformly, and it is corroded by single deep holes and pores.

In previous researches [42], a release of ions both of nickel and of titanium was observed; in the most acidic environment, the Ni released is about 1.5–2 times more (however, both concentra‐ tions are small, and the titanium concentration is not toxic for an organism), at pH 3.56 nickel concentration was 2–4 times more, and in other solutions titanium was not revealed which is consistent with literary data about its consumption on a passive film formation [8, 9, 42].

The ion concentrations of both nickel and titanium in solutions increase with time; however, this increase has a different character depending on the medium (**Figures 7** and **8**). During short‐term tests [42], we believed that the slope of time‐dependent concentration curves decreased gradually in all media, flattening out because of the termination of metal etching from the surface. It was so in general, however, the long‐term investigation showed small


**Table 3.** Diameters of each type of NiTi wire sample before and after immersion for 2 years in solutions with various pH values.

fluctuations in the slope of the concentration curve toward increase and reduction of metal release in acid media. The last in comparison with published data can be related to sequential processes of the destruction and renewal of the protective oxide film (de‐ and repassivation) on defect areas [12, 44, 45].

however, after a long period of immersion, its nonuniformity does not allow to slow down leach‐ ing of elements as effectively as in case of a more uniform passive film which is initially received when polishing. The ratio of metal concentration in solutions with the annealed and the nonan‐

**Figure 8.** Nickel concentration in investigated solutions with various acidities with nitinol sample without treatment as a function of sampling time. The marked curves correspond to solution pH: 1—1.68, 2—3.56, 3—4.01, and 4—6.31.

The surface examination after holding in solutions also reveals the hardest corrosive attack after annealing and the lowest after polishing. However, the wire diameter changes dispro‐ portionately to the ion concentration in the solution. For example, the diameters of untreated and annealed wires are almost equal after holding in the media with pH 1.68 (**Table 3**); that is,

In previous researches [42], a release of ions both of nickel and of titanium was observed; in the most acidic environment, the Ni released is about 1.5–2 times more (however, both concentra‐ tions are small, and the titanium concentration is not toxic for an organism), at pH 3.56 nickel concentration was 2–4 times more, and in other solutions titanium was not revealed which is consistent with literary data about its consumption on a passive film formation [8, 9, 42].

The ion concentrations of both nickel and titanium in solutions increase with time; however, this increase has a different character depending on the medium (**Figures 7** and **8**). During short‐term tests [42], we believed that the slope of time‐dependent concentration curves decreased gradually in all media, flattening out because of the termination of metal etching from the surface. It was so in general, however, the long‐term investigation showed small

nealed samples at existence and lack of polishing treatment approximately coincides.

92 Shape Memory Alloys - Fundamentals and Applications

the surface fails nonuniformly, and it is corroded by single deep holes and pores.

The greater ion release from samples of a single type is observed in the most acidic medium (**Figure 8**), which abruptly drops with increasing pH and again increases in the NaCl solu‐ tion. This corresponds to expectations. On the one hand, the metal ion release increases with increasing acidity, which in the theory corresponds to an increase in the corrosiveness of the medium. On the other hand, a high metal yield (it is greater than that for solution with acidity of 4.01 and 3.56 in the initial period and for pH 4.01 after 600 days) is revealed in the physi‐ ological saline, being a fairly concentrated source of chlorine ions, which are related to corro‐ sion activator ions, having a depassivation and pitting corrosion effect, having a metal affinity greater than oxygen, and displacing the latter from its compounds with metals [46, 47].

Visually, the most damage occurs also at pH 1.68 after holding for 2 years (**Figure 9a** and **Table 3**). According to the location of pitting, the corrosive attack occurs coaxially to the direction of defects in production during drawing. Small traces of corrosion are observed at the surface at an acidity of 3.56. Defects of the metallic surface are no longer revealed at pH 4.01; however, depositions of organic nature are clearly expressed, which cover the wire with great strata and fibers but do not cover it completely (**Figure 9b**). Wires in the neutral solutions are covered by a smooth homogeneous surface layer (**Figure 9c**); and traces of pitting are absent, which supposedly is a result of repassivation of the damaged surface. Changes in the surface and diameter (**Table 3**) are not observed after holding in the alkaline medium. Because the used buffer solutions were not physiological media and served only for formation of the required pH, their effect on the alloy was compared with the effect of hydrochloric acid [42]. The wire surface after long holding in HCl solution (**Figure 9d**) is more like the surface after holding in the physiological saline than in the acidic buffer, and the change in diameter (**Table 3**) is significantly less than that in a solution with pH 1.68. At the same time the concentrations of solved nickel for a short time in solutions of hydrochloric acid with pH 1.56 (the acidity changed to 2 after 2 years of immersion) and the buffer with pH 1.68 were close and that of titanium differed by more than a factor of two (after 10 days nickel concentration in HCl solution was 1.94 mg/l, titanium concentration was 0.515 mg/l). This gives occasion to suppose that the behavior mechanism in the two chloride‐containing media is similar, etching of metals from the surface slows down at the initial similarity of magnitudes of solved nickel at long‐term holding in the acid, and the surface is covered by a protective film.

**Figure 9.** SEM image of the untreated NiTi sample surface after 2 years of immersion in various solutions: (a) buffer at pH 1.68 (Solution 1; **Table 2**); (b) buffer at pH 4.01 (Solution 3, **Table 2**); (c) 0.9 wt% NaCl (Solution 4, **Table 2**); and (d) HCl (pH 1.56).

In the case with solutions of salts of organic acids (pH 1.68–4.01), the dissolution of nickel and titanium is explained because both elements react with them [42]. In addition, regions of selective dissolution of metals were revealed, titanium in acidic media and nickel in acidic and neutral media [48]. However, nickel is subjected to corrosion damage in oxidizing media, and titanium is considered as satisfactorily stable (has a pitting corrosion potential almost two orders of magnitude higher than that for nickel in NaCl solution) [46, 47]. In addition, the investigation of the corrosion mechanism of microstructural nitinol in chlorine‐containing solutions showed that the corrosive surface demonstrated a low content of nickel, and the result of the process was the Ni ion release into solution and the formation of titanium oxide in the damaged region because titanium remained at the surface and reacted with dissolved oxygen [12]. The region between volume NiTi and surface TiO<sup>2</sup> contained layers of Ni3 Ti and Ni4 Ti and clusters of pure nickel [1, 41, 11], the formation of which was caused also by the formation of the titanium oxide layer: nickel atoms are released from the Ni‐Ti interatomic bond almost at room temperature when the thin oxide layer spontaneously forms at the niti‐ nol surface, and nickel may be released into the solution. However, published data of the titanium ion release at the corrosion of microstructural nitinol were not found. Therefore, the presence of titanium ions in the NaCl solution turned out to be unexpected. Here, we suggest that this is related to the nanostructure of nitinol: nanograins failed during the "washout" of nickel from them and released a greater amount of titanium into the solution at the begin‐ ning [42]. Though titanium is not considered to be toxic for humans even in amounts much greater than those obtained here and its concentration in the solution is much less than the nickel content, it can be noted, however, that the nanostructural properties doubly affect the corrosion resistance of nitinol.

hydrochloric acid [42]. The wire surface after long holding in HCl solution (**Figure 9d**) is more like the surface after holding in the physiological saline than in the acidic buffer, and the change in diameter (**Table 3**) is significantly less than that in a solution with pH 1.68. At the same time the concentrations of solved nickel for a short time in solutions of hydrochloric acid with pH 1.56 (the acidity changed to 2 after 2 years of immersion) and the buffer with pH 1.68 were close and that of titanium differed by more than a factor of two (after 10 days nickel concentration in HCl solution was 1.94 mg/l, titanium concentration was 0.515 mg/l). This gives occasion to suppose that the behavior mechanism in the two chloride‐containing media is similar, etching of metals from the surface slows down at the initial similarity of magnitudes of solved nickel at long‐term holding in the acid, and the surface is covered by

**Figure 9.** SEM image of the untreated NiTi sample surface after 2 years of immersion in various solutions: (a) buffer at pH 1.68 (Solution 1; **Table 2**); (b) buffer at pH 4.01 (Solution 3, **Table 2**); (c) 0.9 wt% NaCl (Solution 4, **Table 2**); and (d)

a protective film.

94 Shape Memory Alloys - Fundamentals and Applications

HCl (pH 1.56).

We observed that the significant retardation of the nickel ion release (and insignificant con‐ centration as a whole) and the absence of titanium ion release form nanostructural nitinol samples in weakly acidic and neutral solutions after mechanical polishing. Traces of pitting corrosion on the surface of polished samples are visible only after 2 years of holding in the most acidic medium (**Figure 10**); wires held in the remaining media look intact equally. This leads to a conclusion of the occurrence of a strong and homogeneous protective surface layer

**Figure 10.** SEM image of the polished wire surface after holding in (a) solution with pH 1.68 and (b) remaining media.

of titanium oxide, which serves as a barrier to the release of nickel into the medium, and the high corrosion resistance of investigated nanostructural nitinol.

According to the literature, with increasing holding period of samples in the chlorine‐contain‐ ing solution, the thickness of the oxide layer increases [12, 14]. The investigation of untreated nitinol wires after 4 months of holding in the salt solution showed that their breakdown potential of the passive film changed from 200 to 800 mV, which the authors related to a decrease in the Ni content on the surface [17].

In this chapter, the surface composition of TiNi wires is practically invariable in the case with untreated and annealed samples after holding in solutions. Dark carbon‐containing spots with insignificant nickel content and bright areas of titanium oxide with nickel inclusions, as before, alternate with each other, and the total analysis over an area of 100 × 100 μm2 demon‐ strate the invariably high carbon content at a depth greater than 1 μm.

The greatest depth of the surface oxide layer, approximately 25 nm, is observed for polished wires after holding in neutral solutions (**Table 4**). At the same time, the titanium distribution in the elementary and bound states after the NaCl medium was determined, and elementary titanium was not revealed to a depth of 17 nm (**Figure 11**). The nickel contents at the surface


**Table 4.** Surface composition of polished NiTi samples after immersion in various solutions for 2 years.

**Figure 11.** Composition of the polished NiTi wire surface after immersion for 2 years in 0.9 wt% NaCl (Solution 4, Table 2) as determined by Auger spectroscopy.

itself are shown in Table 4, in the NaCl and artificial plasma comes to a plateau at a depth of ≈20 nm, whereas in the remaining media, at about 10 nm. The oxygen concentration in the surface layer in all samples traverses the maximum at ≈60–70 wt% at a depth of 2.5–7 nm. At the same time, the carbon content on these dependences drops abruptly from a fairly high one at a depth to 2.5 nm, which can be related to the presence of outer mechanical contaminations at the surface. Release of metal ions into the solution with pH 9.18 was not observed com‐ pletely, and this explains the identity of compositions before and after immersion.

of titanium oxide, which serves as a barrier to the release of nickel into the medium, and the

According to the literature, with increasing holding period of samples in the chlorine‐contain‐ ing solution, the thickness of the oxide layer increases [12, 14]. The investigation of untreated nitinol wires after 4 months of holding in the salt solution showed that their breakdown potential of the passive film changed from 200 to 800 mV, which the authors related to a

In this chapter, the surface composition of TiNi wires is practically invariable in the case with untreated and annealed samples after holding in solutions. Dark carbon‐containing spots with insignificant nickel content and bright areas of titanium oxide with nickel inclusions, as

The greatest depth of the surface oxide layer, approximately 25 nm, is observed for polished wires after holding in neutral solutions (**Table 4**). At the same time, the titanium distribution in the elementary and bound states after the NaCl medium was determined, and elementary titanium was not revealed to a depth of 17 nm (**Figure 11**). The nickel contents at the surface

**Solution рН 1.68 3.56 4.01 6.31/7.36 9.18** Oxide layer thickness/nm 8–13 13–17 15–20 23–28 ≈10 Nickel surface concentration/wt% 8 5 3 0 3

**Figure 11.** Composition of the polished NiTi wire surface after immersion for 2 years in 0.9 wt% NaCl (Solution 4, Table 2)

**Table 4.** Surface composition of polished NiTi samples after immersion in various solutions for 2 years.

demon‐

before, alternate with each other, and the total analysis over an area of 100 × 100 μm2

strate the invariably high carbon content at a depth greater than 1 μm.

high corrosion resistance of investigated nanostructural nitinol.

decrease in the Ni content on the surface [17].

96 Shape Memory Alloys - Fundamentals and Applications

as determined by Auger spectroscopy.

Thus, the mechanically polished surface held in the chlorine‐containing medium really looks protected from corrosion, including under dynamic conditions, even after subsequent anneal‐ ing. However, the contact of nickel with the surrounding physiological environment never‐ theless is possible that it demands to consider an individual susceptibility of patients.

The biocompatibility of the nanostructured nitinol was measured *in vitro* using standard test systems: the cultures of myofibroblasts from human peripheral vessels and human bone mar‐ row mesenchymal stromal cells (MSC) were used as standard cell models. The myofibroblasts were isolated from cut peripheral veins [49] and grown in the DMEM medium (Biolot, Russia) with the addition of 10% fetal calf serum (Gibco, United States), 40 μg/mL of gentamicin at 37°C, and 5% carbon dioxide in a CO<sup>2</sup> incubator (Binder, Germany). The MSC (Biolot, Russia) were grown in the alpha‐MEM medium (Sigma, United States) under the same conditions.

Samples of materials were placed into the wells of a 6‐well plate (Greiner, Germany). Then cells were inoculated on the sample surface (5 × 103 cells per cm2 ) and cultured for 5 days. To determine the numbers of vital and dead cells, the cells growing on the material surface were stained with fluorescent dyes—acridine orange (Sigma, USA, 1 μg/mL) and propidium iodide (Sigma, USA, 1 μg/mL). Acridine orange stains both vital and dead cells, while prop‐ idium iodide stains only dead cells. After that, the samples were incubated for 10 min at 37°C [50]. Then the samples were examined under a DM 6000 fluorescence microscope (Leica, Germany). For the assay, at least 500 cells on the sample surface were counted. In the case of myofibroblasts and MSC, the percentages of vital cells for NiTi were 91 ± 3 and 95 ± 1, respec‐ tively. Thus, the material samples used in the study did not have a short‐term toxic effect on the cells that overgrew its surfaces *de novo*.

The mitotic activity of the cells was assessed considering the mitotic index of the cells in the logarithmic growth phase (the 3rd day after inoculation). The number of mitotic cells was determined by fluorescence microscopy using the vital staining with the Hoechst 33342 fluorescent dye (Sigma, USA). The mitotic cells were identified based on the distribution of chromatin inherent in the prophase, metaphase, anaphase, and telophase. At least 500 cells on the sample surface were counted for the assay. The calculated MI value [51] for the cells grow‐ ing on the NiTi surface was 3.1% for the myofibroblast culture and 1.8% for the MSC culture.

After 5 days of culturing, the morphological analysis of the myofibroblasts and MSC on the surface of material showed that myofibroblasts occupy only ≈75% of the NiTi surface acces‐ sible for the growth, while MSC occupy ≈50% of the accessible NiTi surface and so no mono‐ layer is formed for either myofibroblasts or MSC.

This means that toxicity of samples was not revealed, but the cell reaction was not the best possible.

Thus it is shown that the use of nanostructured nitinol, obtained and investigated in this work, in medicine as a material for noninvasive implants is prospective compared with microstructured material, but formation of a corrosion‐resistant and biocompatible surface layer in future still seems desirable.

Investigated nanostructural material is currently used to manufacture the noninvasive medi‐ cal implants—stents—successfully applied to restore respiratory and urinary systems and esophageal and intestinal patency (**Figure 12**).

In connection with the presence of increased corrosion resistance, strength and plasticity, shape memory effect, controlled by the composition and heat treatment phase transformations, this

**Figure 12.** Patency restoration process: (a) stent applied to restore esophageal and intestinal patency, (b) example of the restoration of patency in the case of large intestine cancer via the implantation of a stent (operation and photo of the Blokhin Russian Cancer Research Center, Russian Academy of Medical Sciences).

new nanostructural material can also be used to create other medical products and instru‐ ments, as well as can be used in other areas that require the use of smart materials.

#### **3. Conclusions**

Thus it is shown that the use of nanostructured nitinol, obtained and investigated in this work, in medicine as a material for noninvasive implants is prospective compared with microstructured material, but formation of a corrosion‐resistant and biocompatible surface

Investigated nanostructural material is currently used to manufacture the noninvasive medi‐ cal implants—stents—successfully applied to restore respiratory and urinary systems and

In connection with the presence of increased corrosion resistance, strength and plasticity, shape memory effect, controlled by the composition and heat treatment phase transformations, this

**Figure 12.** Patency restoration process: (a) stent applied to restore esophageal and intestinal patency, (b) example of the restoration of patency in the case of large intestine cancer via the implantation of a stent (operation and photo of the

Blokhin Russian Cancer Research Center, Russian Academy of Medical Sciences).

layer in future still seems desirable.

98 Shape Memory Alloys - Fundamentals and Applications

esophageal and intestinal patency (**Figure 12**).

New nanostructural NiTi alloy that can be used in medical applications was obtained and investigated.

The material base is the B2 phase in the form of nanofibers with a diameter of 30–70 nm, which are elongated along the wire axis. Inclusions of Ti<sup>2</sup> Ni intermetallides are also observed, the distribution and the size of which are not affected by annealing at 450°C for 15 min.

The nickel release from investigated nanostructural nitinol is less in comparison with data for microstructural nitinol in a solution of any acidity. Although earlier in the literature, there were no similar results. Dissolution in the alkali medium is absent.

Mechanical treatment increases nitinol corrosion resistance, and thermal treatment decreases it, and the nanostructure possibly serves as the cause of the titanium ion presence in the solu‐ tion of any acidity with unpolished samples. A significant retardation of the nickel ion release (and insignificant concentration as a whole) and the absence of titanium ion release in the weakly acidic and neutral solutions with polished samples are observed.

The mechanical treatment of the surface results in the formation of a thin layer of titanium oxide, decreases the roughness and number of defects but does not completely remove nickel from the surface. The nickel‐free surface with the 25‐nm titanium oxide protective layer is attained by holding in a neutral 0.9 wt% solution of sodium chloride. Such a layer must be the barrier to nickel release into the medium even under dynamic conditions.

A simultaneous 7–11% increase in strength and plasticity in comparison with microstructural nitinol was attained. The presence of the shape memory effect and the absence of an uncon‐ trolled change in the temperature of phase transformations in connection with nanostructur‐ ing were noted.

Toxicity of samples has not been revealed. But no cell monolayer is formed on the surface of nanostructural alloy after 5 days of culturing of the myofibroblasts and mesenchymal stromal cells.

Nanostructured nitinol usage in medicine as a material for noninvasive implants is prospec‐ tive compared with microstructured material but formation of corrosion‐resistant and bio‐ compatible surface layer still seems desirable.

#### **Acknowledgements**

This work was supported by the RFBR № 15‐33‐70006 «mol\_a\_mos» and 14‐29‐10208 «ofi\_m». The authors wish to thank Golberg MA, Dyomin K Yu, Mikhailova AB, Volchenkova VA, and Kargin Yu F for their help in sample analysis.

#### **Author details**

Elena O. Nasakina\*, Mikhail A. Sevostyanov, Alexander S. Baikin, Alexey V. Seryogin, Sergey V. Konushkin, Konstantin V. Sergienko, Alexander V. Leonov and Alexey G. Kolmakov

\*Address all correspondence to: nacakina@mail.ru

A.A. Baikov Institute of Metallurgy and Material, Science, Russian Academy of Sciences, Moscow, Russia

#### **References**


[9] Nasakina EO, Sevostyanov MA, Goncharenko BA, Leonova YO, Kolmakov AG, Zabolotny VT. Metody issledovaniya I povysheniya korrozionnoy stoykosty medit‐ synskogo splava s effektom pamiaty formy NiTi. Sposoby izmeneniya korrozionnoy stoykosti nitinola [Methods of corrosion resistance research and increase of medical alloy with shape memory effect NiTi. Ways of a nitinol corrosion resistance change]. Perspektivnie materialy [Advanced materials]. 2014;**9**:19‐33

**Author details**

Moscow, Russia

**References**

\*Address all correspondence to: nacakina@mail.ru

100 Shape Memory Alloys - Fundamentals and Applications

1996;**6**:267‐289. DOI: 10.3233/BME‐1996‐6405

Tomsk: MITs; 2006. p. 296

DOI: 10.1134/S003602951105017X

1537. DOI: 10.1161/STROKEAHA.106.477711

148. DOI:10.1016/j.biomaterials.2008.09.011

Technologies. 2000;**9**:81‐88

s00330‐003‐2022‐5

Elena O. Nasakina\*, Mikhail A. Sevostyanov, Alexander S. Baikin, Alexey V. Seryogin, Sergey V. Konushkin, Konstantin V. Sergienko, Alexander V. Leonov and Alexey G. Kolmakov

A.A. Baikov Institute of Metallurgy and Material, Science, Russian Academy of Sciences,

[1] Shabalovskaya S. On the nature of the biocompatibility and medical applications of NiTi shape memory and superelastic alloys. Bio‐Medical Materials and Engineering.

[2] Gyunter VO, Khodorenko VN, Yasenchuk YF, Chekalkin TL. Nikelid titana. Meditsinskii material novogo pokoleniya (Titanium Nickelide. Medical Material of New Generation).

[3] Dotter CT, Buschmann PAC, McKinney MK, Rosch J. Transluminal expandable niti‐ nol coil stent grafting: Preliminary report. Radiology. 1983;**147**:259‐260. DOI: 10.1007/

[4] Zabolotnyi VT, Belousov OK, Palii NA, Goncharenko BA, Armaderova EA, Sevost'yanov MA. Materials science of the production, treatment and properties of titanium nickelide for application in endovascular surgery. Russian Metallurgy (Metally). 2011;**3**:437‐448.

[5] Stoeckel D. Nitinol medical devices and implants. Minimally Invasive Therapy & Allied

[6] Bose A, Hartmann M, Henkes HA. Novel. Self‐expanding nitinol stent in medically refractory intracranial atherosclerotic stenosis: Wingspan study. Stroke. 2007;**38**:1531‐

[7] Xiaoying Lu, Xiang Bao, Yan Huang, Yinghua Qu, Huiqin Lu, Zuhong Lu. Mechanisms of cytotoxicity of nickel ions based on gene expression profiles. Biomaterials. 2009;**30**:141‐

[8] Nasakina EO, Sevostyanov MA, Goncharenko BA, Leonova YO, Kolmakov AG, Zabolotny VT. Metody issledovaniya I povysheniya korrozionnoy stoykosty medit‐ synskogo splava s effektom pamiaty formy NiTi. Issledovanie korrosionnoy stoykosty I biosovmestimosty nitinola [Methods of corrosion resistance research and increase of medical alloy with shape memory effect NiTi. Research of corrosion resistance and bio‐ compatibility of nitinol]. Perspektivnie materialy [Advanced materials]. 2014;**7**:37‐49


[31] Aslan Ahadi, Qingping Sun. Effects of grain size on the rate‐dependent thermomechani‐ cal responses of nanostructured superelastic NiTi. Acta Materialia. 2014;**76**:186‐197. DOI: 10.1016/j.actamat.2014.05.007

[20] Her‐Hsiung Huang. Variation in corrosion resistance of nickel‐titanium wires from dif‐

[21] Trepanier C, Tabizian M, Yahia LH, Bilodeau L, Piron DL. Effect of modification of oxide layer on NiTi stent corrosion resistance. Journal of Biomedical Materials Research.

[22] Trepanier C, Fino J, Zhu L, Pelton AR. Corrosion resistance of oxidized nitinol. In: Proceedings of the International Conference on Shape Memory and Superelastic Technologies. SMST‐2003; 5‐8 May 2003; California. California: SMST Soc; 2004. pp. 267‐276

[23] Farrukh MA, editor. Functionalized Nanomaterials. Rijeka: InTech; 2016. pp. 172. DOI:

[24] Roldugin VI, Fedotov MA, Folmanis GE, Kovalenko LV, Tananaev IG. Formation of aqueous colloidal solutions of selenium and silicon by laser ablation. Doklady Physical

[25] Leon Mishnaevsky Jr., Levashov E, Ruslan Z, Valiev, Javier Segurado, Ilchat Sabirov, Nariman Enikeev, Sergey Prokoshkin, Andrey V Solov'yov, Andrey Korotitskiy, Elazar Gutmanas, Irene Gotman, Eugen Rabkin, Sergey Psakh'e, Luděk Dluhoš, Marc Seefeldt, Alexey Smolin. Nanostructured titanium‐based materials for medical implants: Modeling and development. Materials Science and Engineering R. 2014;**81**:1‐19. DOI:

[26] Amirhanova NA, Valiev RZ, Adasheva SL, Prokof'ev EA. Issledovanie korrozionnyh i jelektrohimicheskih svojstv splavov na osnove nikelida titana v krupnozernistom i ul'tramelkozernistom sostojanijah [Investigation of corrosion and electrochemical prop‐ erties of alloys based on nickel titanium in coarse and ultrafine states]. Journal of the Ufa

[27] Korotin DM, Bartkowski S, Kurmaev EZ, Cholakh SO, Müller M, Neumann M, Gunderov D, Valiev RZ, Cholakh SO. Arsenic contamination of coarse‐grained and nanostruc‐ tured nitinol surfaces induced by chemical treatment in hydrofluoric acid. Journal of Biomedical Materials Research Part B: Applied Biomaterials. 2012;**100B**(7):1812‐1816.

[28] Zheng CY, Nie FL, Zheng YF, Cheng Y, Wei SC, Valiev RZ. Enhanced in vitro biocompat‐ ibility of ultrafine‐grained titanium with hierarchical porous surface. Applied Surface

[29] Zheng CY, Nie FL, Zheng YF, Cheng Y, Wei CS, Liqun Ruan, Valiev RZ. Enhanced cor‐ rosion resistance and cellular behavior of ultrafine‐grained biomedical NiTi alloy with

[30] Zhang XN. Biomechanical and biocorrosion properties of nanostructured titanium. Advanced Materials Research. 2007;**29-30**:51‐54. DOI: 10.4028/www.scientific.net/AMR.

sol‐gel coating. Applied Surface Science. 2011;**257**(13):5913‐5918.

Science. 2011;**257**(13):5634‐5640. DOI: 10.1016/j.apsusc.2011.01.062

ferent manufacturers. Angle Orthodontist. 2005;**75**:661‐665

Chemistry. 2015;**463**:161‐164. DOI: 10.1134/S0012501615070064

State Aviation Technical University. 2006;**7**(1):143‐146

1998;43:433‐440

102 Shape Memory Alloys - Fundamentals and Applications

10.5772/63186

10.1016/j.mser.2014.04.002

DOI: 10.1002/jbm.b.32748

a novel SrO‐SiO<sup>2</sup>

29‐30.51

‐TiO<sup>2</sup>

DOI: 10.1016/j.apsusc.2011.02.006


### **Development of Faster SMA Actuators**

Atta Muhammad Nizamani, Jawaid Daudpoto and

Muhammad Ali Nizamani

[43] Tomic S, Rudolf R, Brun ko M, Anžel I, Savic V, Olic M. Response of monocytederived dendritic cells to rapidly solidified nickeltitanium ribbons with shape memory proper‐

[44] Venugopalan R, Trepanier C. Assessing the corrosion behavior of nitinol for minimally‐ invasive device design. Minimally Invasive Therapy & Allied Technologies. 2000;**9**:67‐74

[45] Sun EX, Fine S, Nowak WB. Electrochemical behavior of nitinol alloy in ringer's solution. Journal of Materials Science: Materials in Medicine. 2002;**13**:959‐964. DOI:

[46] Zhuk NP. Kurs korrozii i zashhity metallov [The course of corrosion and protection of

[47] Ulig GG, Revi GU. Korrozija i bor'ba s nej. Vvedenie v korrozionnuju nauku i tehniku

[48] Shcherbakov AI, Kasatkina IV, Zalavutdinov RK. Ustojchivost' splava «pamjati» TiNi k selektivnoj korrozii [Resistance of "Memory" alloy TiNi to selective corrosion]. Corro‐

[49] Akatov VC, Fadeeva IC, Chekanov AV, Solov'yev VV. Rol' kletok retsipiyenta v mekha‐ nizme patologicheskoy kal'tsifikatsii transplantatov klapanov serdtsa i sosudov [Role of the recipient cells in the mechanism of transplants pathological calcification of heart

[50] Garmash SA, Smirnova VS, Karp OE, Usacheva AM, Berezhnov AV, Ivanov VE, Chernikov AV, Bruskov VI, Gudkov SV. Pro‐oxidative, genotoxic and cytotoxic prop‐ erties of uranyl ions. Journal of Environmental Radioactivity. 2014;**127**:163‐170. DOI:

[51] Permyakov SE, Knyazeva EL, Khasanova LM, Fadeev RS, Zhadan AP, Roche‐Hakansson H, Håkansson AP, Akatov VS, Permyakov EA. Oleic acid is a key cytotoxic component

of HAMLET‐like complexes. Biological Chemistry. 2012;**393**:85‐92

[Corrosion and its prevetion]. Leningrad: Chemestry; 1989

ties. European Cells & Materials. 2012;**23**:58‐81. DOI: 10.22203/eCM.v023a05

10.1023/A:1019812729884

104 Shape Memory Alloys - Fundamentals and Applications

metals]. Moscow: Metallurgy; 1976

10.1016/j.jenvrad.2012.12.009

sion: Materials and Protection. 2007;**12**:14‐16

valves and blood vessels]. Biofizika. 2010;**55:**937‐942

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69868

#### **Abstract**

Large cycle time, resulted from slow cooling, is the core hindrance to the wide spread applications of shape memory alloys (SMAs) as actuators. This chapter discusses a novel cooling technique to decrease the cycle time of SMAs. Under this technique, the SMA actuator of 0.15 mm diameter was run through a grease-filled Polytetrafluoroethylene (PTFE) tube of 0.5 mm outside diameter. Later, same tests were repeated with oil filled PTFE tube. The test results conducted in ambient air were used as standard for comparison. The actuation current in ambient air was set at 210, 310 and 410 mA. While testing with heat sink, i.e. grease and oil, the SMA was heated with 210, 310, 410, 500, 615 and 720 mA currents for 1 and 2 seconds, whereas the SMA was heated for 1 second only with 810 mA current. It was found that the grease cooling reduced the cooling time up to 30% and oil cooling by 20%, as compared to the ambient air-cooling time. However, the grease-cooled actuators had shown less strain, and their response was non-linear at many instances. Heat loss to the sinks resulted to more power consumption than that in ambient air cooling for equivalent amount of strain.

**Keywords:** SMA actuators, long cycle time, SMA cooling, Teflon tubing

#### **1. Introduction**

The need for miniaturization and lighter systems has resulted in the development of smart actuators, which are compact in size and lighter in weight. The shape memory alloys (SMAs), also called the smart alloys, were discovered by Arne Ölander in 1932 [1]. Various properties of SMAs, like high work output as compared to other conventional actuators, silent, clean and spark-free operation, design simplicity and easy miniaturization, have attracted researchers and engineers to use them as actuators in several applications [2].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The operation of SMA actuators is considered to be simple. SMA actuators in many applications are in the form of a thin wire and their diameters ranging from 0.025 to 0.51 mm [3]. The motion created in SMAs is due to a molecular rearrangement in their crystalline structure when phase transformation takes place, as shown in **Figure 1**.

**Figure 1.** Phase transformation in SMAs [4].

The phase transformation takes place at certain temperatures called phase transformation temperature. The phase transformation temperature is determined by the composition and heat treatment methods applied to the SMAs. Low temperature phase is known as martensite, whereas, heating results in transformation to austenite phase. SMAs are relatively soft in Martensite phase and possess smaller value of Young's modulus, whereas they are hard in Austenite phase with higher value of Young's modulus.

#### **2. SMA design challenges**

The design challenges in SMA-based systems are to succeed over the limitations incorporated with SMAs. Some of the SMA limitations are relatively small usable strain, low operational frequency, low controllability, low accuracy and low-energy efficiency. However, low operational frequency resulted from large cycle time is widely reported in literature as core hindrance to wide spread applications of the SMAs [5–7]. The operational frequency of SMAs is given by Eq. (1) (working frequency of SMAs).

$$f\_w = \frac{1}{t\_h + t\_c} \tag{1}$$

Where

The operation of SMA actuators is considered to be simple. SMA actuators in many applications are in the form of a thin wire and their diameters ranging from 0.025 to 0.51 mm [3]. The motion created in SMAs is due to a molecular rearrangement in their crystalline structure

The phase transformation takes place at certain temperatures called phase transformation temperature. The phase transformation temperature is determined by the composition and heat treatment methods applied to the SMAs. Low temperature phase is known as martensite, whereas, heating results in transformation to austenite phase. SMAs are relatively soft in Martensite phase and possess smaller value of Young's modulus, whereas they are hard in

The design challenges in SMA-based systems are to succeed over the limitations incorporated with SMAs. Some of the SMA limitations are relatively small usable strain, low operational frequency, low controllability, low accuracy and low-energy efficiency. However, low operational frequency resulted from large cycle time is widely reported in literature as core hindrance to wide spread applications of the SMAs [5–7]. The operational frequency of SMAs is

> <sup>w</sup> <sup>=</sup> \_\_\_\_ <sup>1</sup> *t* <sup>h</sup> + *t* c

(1)

when phase transformation takes place, as shown in **Figure 1**.

106 Shape Memory Alloys - Fundamentals and Applications

Austenite phase with higher value of Young's modulus.

given by Eq. (1) (working frequency of SMAs).

*f*

**2. SMA design challenges**

**Figure 1.** Phase transformation in SMAs [4].


The cycle time in SMAs is defined as the total time required by the SMAs to contract and expand, hence completing its full cycle. Cycle time is the algebraic sum of heating time (i.e. contraction time) and cooling time (i.e. expansion time). The heating time of the SMAs can be easily reduced by increasing the magnitude of actuation current, whereas the cooling rate is redistricted by the rate of heat transfer rate to the environment surrounding the SMA specimen. The actuator response time also depends upon the size and shape of the actuators. The SMA actuators with smaller diameter cool faster as compared to those having larger diameters. However, this will affect the loading capacity of SMA actuators. **Figure 2** is developed by using the technical data provided by Dynalloy, Inc., the SMA manufacturer [3]. It shows the relationship between the loading capacity of SMA actuators and cooling time required by SMA actuators of various diameters in ambient air at room temperature. However, the time required to restore final 0.5% strain is not considered.

**Figure 2.** Relationship between wire diameter, cooling time and safe load for SMA wire actuators.

From **Figure 2**, it can be observed that the cooling time required by the smaller diameter SMAs is relatively small. However, the smaller diameter SMAs can carry little load as compared to those carried by SMAs having larger diameter. Therefore, the need is to develop the faster cooling mechanisms for SMAs.

Large cycle time is core hindrance to the SMA applications in several areas, including automobiles, robotics, biomedical, etc.

#### **3. Practical applications of faster SMA actuators**

Around 200 actuation tasks are performed in a family car [8]. The actuators required to operate several functions in a car are categorized as (a) low-power actuators for comfort and aesthetic aspects, (b) actuators having high power to operate various control mechanisms of vehicles and (c) the actuators with high frequency to control engine performance [9]. The SMA actuators are being used in first category of actuators, as applied by Mercedes, BMW, General Motors (GM), Hyundai, Ford, Porsche and Volkswagen (VW) to actuate lumbar support in the car seats for passenger comfort [10]. The SMAs are applicable in second category of actuators; however, these are still to be studied for this category, whereas due to low working frequency, resulting from large cooling time, the SMA actuators are either not suitable or less suitable to be applied against third category of actuators.

The SMAs due to their ability to produce linear motion, high power to weight ratio, light weight and compact size are widely being used in several areas of robotics. Some common SMA applications in robotics are robotic grippers [11], human organ orthotics like a foot ankle orthotics [12], rehabilitation robots like a wearable elbow exoskeleton [13] and bio-inspired fish-like robots like i-tuna and dragon fly robots [14, 15]. However, large cooling time is reported to be the core limitation of SMAs in the field of robotics [8, 12, 16, 17].

The biocompatibility of SMAs has made them feasible to be applied in several applications related to biomedical sciences. SMAs are being used in neurology, cardiology and interventional radiology in different ways [18]. SMA-actuated surgical needles and catheters, artificial heart, artificial head and drug-delivery valves are some of the most prominent SMA applications in area of biomedical [19–22]. However, low working frequency resulted from large cooling time is reported to be the serious issue [19, 22, 23].

The SMAs due to attractive morphing properties and ability to withstand dynamic loads are used in several aerospace applications. Some of the noticeable aerospace applications of SMAs are in wing morphing, for example, the DARPA project for development of smart wings [24], SAMPSON project to enhance SMA applications in aircraft engine nozzles at inlets to obtain flying benefits according to flight conditions [25] and reconfigurable engine nozzle fan chevron by Boeing [26]. SMAs are equally applicable in various space research applications, for example, folding and unfolding mechanisms for solar panels on Hubble telescope [27], actuation of valves and apertures on board of Rosetta mission (2004) and Pathfinder–Sojourner Mission (1997) [28]. Their applications can still be enhanced to various actuation applications in spacecrafts, either manned or unmanned [29]. However, the limited working frequency of SMAs resulting from large cooling time is core issue in many aerospace engineering applications [30, 31].

#### **4. Literature review**

From **Figure 2**, it can be observed that the cooling time required by the smaller diameter SMAs is relatively small. However, the smaller diameter SMAs can carry little load as compared to those carried by SMAs having larger diameter. Therefore, the need is to develop the

Large cycle time is core hindrance to the SMA applications in several areas, including auto-

Around 200 actuation tasks are performed in a family car [8]. The actuators required to operate several functions in a car are categorized as (a) low-power actuators for comfort and aesthetic aspects, (b) actuators having high power to operate various control mechanisms of vehicles and (c) the actuators with high frequency to control engine performance [9]. The SMA actuators are being used in first category of actuators, as applied by Mercedes, BMW, General Motors (GM), Hyundai, Ford, Porsche and Volkswagen (VW) to actuate lumbar support in the car seats for passenger comfort [10]. The SMAs are applicable in second category of actuators; however, these are still to be studied for this category, whereas due to low working frequency, resulting from large cooling time, the SMA actuators are either not suitable or less

The SMAs due to their ability to produce linear motion, high power to weight ratio, light weight and compact size are widely being used in several areas of robotics. Some common SMA applications in robotics are robotic grippers [11], human organ orthotics like a foot ankle orthotics [12], rehabilitation robots like a wearable elbow exoskeleton [13] and bio-inspired fish-like robots like i-tuna and dragon fly robots [14, 15]. However, large cooling time is

The biocompatibility of SMAs has made them feasible to be applied in several applications related to biomedical sciences. SMAs are being used in neurology, cardiology and interventional radiology in different ways [18]. SMA-actuated surgical needles and catheters, artificial heart, artificial head and drug-delivery valves are some of the most prominent SMA applications in area of biomedical [19–22]. However, low working frequency resulted from large

The SMAs due to attractive morphing properties and ability to withstand dynamic loads are used in several aerospace applications. Some of the noticeable aerospace applications of SMAs are in wing morphing, for example, the DARPA project for development of smart wings [24], SAMPSON project to enhance SMA applications in aircraft engine nozzles at inlets to obtain flying benefits according to flight conditions [25] and reconfigurable engine nozzle fan chevron by Boeing [26]. SMAs are equally applicable in various space research applications, for example, folding and unfolding mechanisms for solar panels on Hubble telescope [27], actuation of valves and apertures on board of Rosetta mission (2004) and Pathfinder–Sojourner Mission (1997) [28]. Their applications can still be enhanced to various actuation applications in spacecrafts, either manned or unmanned [29]. However, the limited working frequency of SMAs resulting from

large cooling time is core issue in many aerospace engineering applications [30, 31].

reported to be the core limitation of SMAs in the field of robotics [8, 12, 16, 17].

faster cooling mechanisms for SMAs.

108 Shape Memory Alloys - Fundamentals and Applications

**3. Practical applications of faster SMA actuators**

suitable to be applied against third category of actuators.

cooling time is reported to be the serious issue [19, 22, 23].

mobiles, robotics, biomedical, etc.

Several studies have been conducted to reduce the cooling time required by SMAs to regain martensite phase; however, no any unique method has been proposed. In the following passages, a review of such studies is presented.

Loh et al. [32] used silicone grease as heat sink. The authors filled the grease in a metal tube of outside diameter 0.8 mm and inserted 600 mm long and 0.3 mm diameter SMA actuator in the tube.

In reported method, the heat dissipation from SMA to sink was achieved at 5°C per second. The SMA actuator was heated by electrical current supply of 2 A, with a duty ratio of 0.4. The martensitic phase transformation also depends upon the mechanical load connected to the wire; however, in the reported work, the SMA actuators are made to lift up 3 kg mass. The system had following listed limitations:


In another study, Loh et al. [33], while developing the SMA-actuated prosthetic hand, used stainless steel (SS) tubes to enhance the heat transfer rate from SMAs to the environment. The authors ran 2 SMA wires, each 500 mm long and 0.3 mm diameter, through the SS tubes. The SMA actuators were actuated with 50–70 V via pulse width module (PWM). In the reported method, the actuation voltage ranging from 50 to 70 V is unsafe for many applications and uneconomic as well.

Taylor and Au [17] developed an SMA-actuated prosthetic hand. The authors applied the forced air-cooling technique for rapid heat transfer from the SMA wire actuators to the surrounding environment using a small fan. The forced air cooling with a fan produced satisfactory results; however, this method is not applicable in the miniature applications where space is a major constraint.

Cheng and Desai [34] applied water circulation through the SMA actuator arrangement for cooling purpose. The SMA spring actuator was enclosed inside a silicone tube of inside diameter 1.98 mm. The reported method resulted in 0.33 Hz frequency. However, in the presented research, the working frequency of 0.33 Hz was achieved in ambient air cooling when the SMA actuator was heated with 410 mA actuation current. The water circulation, as applied in reported research, is likely to increase the system's complexity as a pump will be required to circulate the water through the actuator, also special sealing arrangements are necessary to avoid water leakage out of actuator. Therefore, the reported system will be bulky and not feasible in miniature applications.

Pathak et al. [35], in order to examine the performance of various cooling media, conducted a comparative study. The authors conducted tests on various SMA actuator samples whose diameters were ranging from 0.1524 to 0.508 mm and each SMA actuator sample was 177.8 mm long. The authors tested the SMA actuator samples in still air, forced air convection, mineral oil, thermal grease and water as cooling media. In still air, the authors found that SMA wire diameter had major impact on the cooling time. It was observed that the values for coefficient of heat transfer (*h*) for 0.1524 mm diameter SMA wire was 153 W m−2 K−1, whereas value of *h* for 0.508 mm diameter SMA wire was 68 W m−2 K−1, which is 44% smaller than that of 0.1524 mm diameter SMA wire. In forced convection in air, the authors set the air flow rate at 625 ft min−1 using a fan. It was noticed that the value of *h* for 0.1524 mm diameter SMA wire was nearly 650 W m−2 K−1, whereas that of 0.508 mm diameter SMA wire was nearly 400 W m−2 K−1 which is 40% smaller than *h* values for 0.1524 mm diameter SMA wire. Since the air was enforced using a small fan which is not feasible for miniature applications, for example, in minimal invasive surgery, a separate controller will be required to vary the fan speed in order to adjust the speed of airflow in the SMA actuator, at the same time, this will lead to more power consumption to operate a fan and a possible controller. Later, the authors tested the SMA actuators jacketed around with mineral oil. The value of *h* for 0.1524 mm diameter SMA wire was found to be 1000 W m−2 K−1, whereas the *h* value for 0.508 mm diameter SMA wire was nearly 510 W m−2 K−1. This shows a reduction of 49% in the *h* values for 0.508 mm diameter SMA wire as compared to that of 0.1524 mm diameter SMA wire. As compared to forced air convection, the *h* value for 0.1524 mm diameter SMA wire was improved by 350 W m−2 K−1, whereas the *h* value for 0.508 mm diameter SMA wire was improved by 110 W m−2 K−1. However, when compared to still air cooling, the *h* value increased by 6.5 times for 0.1524 mm diameter SMA wire and 8.9 times for 0.508 mm diameter SMA wire. Although this is a significant improvement in the values of coefficient of heat transfer, the oil cooling is incorporated with certain limitations, which include sealing complexity to avoid seepage of oil, also in certain conditions, the viscosity of oil may provide the resistance to SMA motion. The authors later characterized the SMA actuator in thermal grease. It was found that the value of *h* for 0.508 mm diameter SMA wire was 55% lower as compared to *h* value of 0.1524 mm diameter SMA wire when thermal grease was used as heat sink. The value of *h* was significantly higher in thermal grease as compared to the *h* values in mineral oil and air. For example, the value of *h* for 0.1524 mm diameter SMA wire was 4.3 times higher as compared to that in mineral oil and 28 times higher than that in still air for same diameter SMA wire. The cooling with distilled water produced better results and cooled off the SMA actuators in least time as compared to all other cooling media. The water quenching increased the value of *h* by 1.3 times for 0.1524 mm diameter SMA wire and 1.6 times for 0.508 mm diameter SMA wire as compared to *h* values in thermal grease. However, water cooling has certain costs. However, the major limitation is the boiling temperature of water, as the temperature will exceed 100°C, the water will start boiling; therefore, it is only applicable when the transformation temperature is lower, for example, in Flexinol® 70°C. For water, some special sealing arrangements are necessary. Also, the water circulation without a pump is difficult.

Tadesse et al. [36] conducted a series of tests on Flexinol® and Biometal® SMA actuators manufactured by Dynolloy, Inc and Toki Corporation, respectively. Each sample had 0.1 mm diameter and 100 mm long. Considering the actuation current limitations, the authors actuated the Flexinol® SMA actuator with 180 mA and Biometal® wire with 200 mA actuation current.

The authors tested the SMA samples in various cooling media including forced air at 0.3 m s−1 using a small computer fan, forced air cooling at 4.6 m s−1 using a compressor, thermal grease, solid heat sink and water quenching. However, the cooling time in ambient conditions was used as standard to compare the results of other cooling media.

The SMA actuators cooled off in 1.1 second while let to cool in ambient air, whereas forced air cooling at 0.3 m s−1 improved the cooling time to 0.7 second. The high-speed air circulation cooled off the SMA actuators in about 0.3 second. Since the high-speed air circulation was achieved using a compressor, it is not practically applicable in miniature and weight conscious SMA systems. Solid heat sink was used in the form of aluminium tubes which were brought in contact with SMA actuators up on the power cut off to cool the SMA elements. The solid heat sink cooled the SMA actuators in 0.41 second. Solid heat sinking as proposed by the authors is not applicable in miniature applications, also it will require a close eye over the power shut down to bring the sink in contact with SMA actuator as soon as power supply is cut off. Water quenching cooled the SMA actuators in 0.2 second, which is significant reduction in the cooling time of SMAs as compared to ambient air cooling. However, the water circulation without a compressor is difficult, whereas water spray with a syringe is not practical in many applications including prosthesis, robotics and minimal invasive surgical applications. The thermal grease cooling was applied by filling it in a copper tube along with SMA actuators. The SMA actuator failed to contract while cooling with thermal grease. The author claimed that the reason for the failure in contraction was the fast dissipation of heat from the SMA actuator, which retained insufficient heat to cause a phase transformation in the actuator. However, in the proposed research, the SMA element was strained up to 4% when actuated with 810 mA actuation current, 410 mA actuation current also generated sufficient strain in the SMA actuator.

Russell and Gorbet [37] developed mobile heat sink with a two-wired differential type configuration. The authors used a 350 mm long SMA wire actuator of 0.3 mm diameter. The mechanism was arranged in such a way that the midpoint of the SMA actuator was anchored to a 6 mm diameter shaft, hence leaving 150 mm long SMA actuator on either sides of shaft. Furthermore, the heat sink, in a strip form, was attached to the shaft at the centre. As the wire contracted upon heating, it caused the shaft to rotate which simultaneously rotated the heat sink, attached to the shaft, towards the hot portion of the SMA wire, whose current supply was just disconnected. The reported SMA cooling mechanism is complex and is not applicable in conditions when smaller lengths of SMA actuator are used.


A comprehensive literature review is given in **Table 1**.

oil, thermal grease and water as cooling media. In still air, the authors found that SMA wire diameter had major impact on the cooling time. It was observed that the values for coefficient of heat transfer (*h*) for 0.1524 mm diameter SMA wire was 153 W m−2 K−1, whereas value of *h* for 0.508 mm diameter SMA wire was 68 W m−2 K−1, which is 44% smaller than that of 0.1524 mm diameter SMA wire. In forced convection in air, the authors set the air flow rate at 625 ft min−1 using a fan. It was noticed that the value of *h* for 0.1524 mm diameter SMA wire was nearly 650 W m−2 K−1, whereas that of 0.508 mm diameter SMA wire was nearly 400 W m−2 K−1 which is 40% smaller than *h* values for 0.1524 mm diameter SMA wire. Since the air was enforced using a small fan which is not feasible for miniature applications, for example, in minimal invasive surgery, a separate controller will be required to vary the fan speed in order to adjust the speed of airflow in the SMA actuator, at the same time, this will lead to more power consumption to operate a fan and a possible controller. Later, the authors tested the SMA actuators jacketed around with mineral oil. The value of *h* for 0.1524 mm diameter SMA wire was found to be 1000 W m−2 K−1, whereas the *h* value for 0.508 mm diameter SMA wire was nearly 510 W m−2 K−1. This shows a reduction of 49% in the *h* values for 0.508 mm diameter SMA wire as compared to that of 0.1524 mm diameter SMA wire. As compared to forced air convection, the *h* value for 0.1524 mm diameter SMA wire was improved by 350 W m−2 K−1, whereas the *h* value for 0.508 mm diameter SMA wire was improved by 110 W m−2 K−1. However, when compared to still air cooling, the *h* value increased by 6.5 times for 0.1524 mm diameter SMA wire and 8.9 times for 0.508 mm diameter SMA wire. Although this is a significant improvement in the values of coefficient of heat transfer, the oil cooling is incorporated with certain limitations, which include sealing complexity to avoid seepage of oil, also in certain conditions, the viscosity of oil may provide the resistance to SMA motion. The authors later characterized the SMA actuator in thermal grease. It was found that the value of *h* for 0.508 mm diameter SMA wire was 55% lower as compared to *h* value of 0.1524 mm diameter SMA wire when thermal grease was used as heat sink. The value of *h* was significantly higher in thermal grease as compared to the *h* values in mineral oil and air. For example, the value of *h* for 0.1524 mm diameter SMA wire was 4.3 times higher as compared to that in mineral oil and 28 times higher than that in still air for same diameter SMA wire. The cooling with distilled water produced better results and cooled off the SMA actuators in least time as compared to all other cooling media. The water quenching increased the value of *h* by 1.3 times for 0.1524 mm diameter SMA wire and 1.6 times for 0.508 mm diameter SMA wire as compared to *h* values in thermal grease. However, water cooling has certain costs. However, the major limitation is the boiling temperature of water, as the temperature will exceed 100°C, the water will start boiling; therefore, it is only applicable when the transformation temperature is lower, for example, in Flexinol® 70°C. For water, some special sealing arrangements are necessary. Also, the water circulation without a pump is difficult. Tadesse et al. [36] conducted a series of tests on Flexinol® and Biometal® SMA actuators manufactured by Dynolloy, Inc and Toki Corporation, respectively. Each sample had 0.1 mm diameter and 100 mm long. Considering the actuation current limitations, the authors actuated the Flexinol® SMA actuator with 180 mA and Biometal® wire with 200 mA actuation current. The authors tested the SMA samples in various cooling media including forced air at 0.3 m s−1 using a small computer fan, forced air cooling at 4.6 m s−1 using a compressor, thermal grease, solid heat sink and water quenching. However, the cooling time in ambient conditions was

110 Shape Memory Alloys - Fundamentals and Applications

used as standard to compare the results of other cooling media.


**Table 1.** Review of various cooling media.

### **5. Materials and methods**

#### **5.1. Experimental setup**

**Sr. No. Author Cooling Media Remarks**

112 Shape Memory Alloys - Fundamentals and Applications

6 Tadesse et al. [36] Forced air using

7 Russell and Gorbet [37] Mobile heat sink with

**Table 1.** Review of various cooling media.

4 Cheng and Desai [34] Water circulation Water circulation produced the satisfactory

5 Pathak et al. [35] Still air The cooling time increases with increase in

computer fan

Forced air using compressors

two-wire differential type configuration

results. However, the boiling temperature of water is core hindrance, also the water being a low viscous fluid will require special sealing

fan, which is not practical in miniature applications, like minimal invasive surgery.

improvement in cooling time, as it was reduced up to 48% as compared ambient air cooling. However, in certain applications, oil

serious issue as reported by [6, 32].

to avoid leakage is core problem.

be required to control air speed.

applications of SMAs. Thermal grease The SMA actuator failed to contract due to

grease.

Solid heat sink Aluminium tubes, used as solid heat sink,

control algorithms. Water Cooling with water produced best results

55%. However, heat loss during actuation is a

amongst all other coolants. However, sealing

Airflow using a fan, which is not practical in miniature applications, like minimal invasive surgery. Also, special control algorithms may

The use of compressor is not applicable in miniature, biomedical weight conscious

rapid heat loss to the coolant i.e. thermal

were brought in contact with SMA actuator as power was cutoff. This will require a close look on power supply so that the heat sink may be brought in contact as soon as the power is cut off, or this will require complex

amongst all other coolants. However, sealing to avoid leakage is core problem and boiling temperature of water are core hindrances.

The reported mechanism is complex and is not applicable with short SMA wires.

sealing may be a problem.

Thermal grease Thermal grease reduced the cooling time by

Water Cooling with water produced best results

arrangement.

Forced air convection Forced airflow can be achieved using a

Mineral oil Mineral oil produced a significant

SMA wire diameter.

A special purpose setup was developed to test the SMA actuators for various parameters. The experimental setup comprised of a laser displacement sensor, a load cell, a k-type thermocouple, a current transducer, an NI Elvis prototyping board, my RIO devices and power supply units to actuate the SMA actuator and to power the sensors. The experimental setup is shown in **Figure 3**.

**Figure 3.** Experimental setup.

#### *5.1.1. Sensors*

The special-purpose experimental setup consisted of different sensors, including a laser displacement sensor to determine the contraction (i.e. strain) in the SMA actuators, a load sensor to determine the force exerted by SMA in lifting up the dead weight, a k-type thermocouple to measure the temperature of actuator and a current transducer to determine the current flow across the SMA actuator.

#### *5.1.1.1. Laser displacement sensor*

These are the non-contact sensors used to determine the displacement or deformation in case of SMA actuators. Due to high accuracy, these sensors are preferred over the traditionally used displacement sensors, like proximity sensors. In the presented research, the LK-G157 laser displacement sensor made by Keyence was used to detect any deformation, i.e. strain in the SMA actuators. This laser displacement sensor (LK-G157) is capable to measure the deformation when the object is within the range of 150 mm; however, it can measure a maximum of 40 mm displacement. It is provided with a red semiconductor laser which emits light of wavelength 655 nm.

#### *5.1.1.2. Load cell*

The tension produced in the SMA actuator while lifting dead weight was determined using a load sensor. Transducers Kit load cell model mdb 2.5 lb, having capacity to measure 2.5 lb, was used in this research.

#### *5.1.1.3. Thermocouple*

A close look at the temperature of the SMA actuator is necessary in order to avoid over heating of the SMA actuator, which can cause a permanent deformation. A fine gauge k-type thermocouple along with a MAX7785 amplifier was used to measure the temperature of the SMA actuators. Since the SMA wire diameter was very fine (i.e. 0.15 mm), the contact loss between SMA and the thermocouple was a serious issue, which is addressed by Ref. [38].

#### *5.1.1.4. Current transducer*

The strain in the SMA actuator is proportional to the amount of actuation current. An ACS 712 current transducer having analogue output was used to keep a close look on the amount of current being flowing through the SMA actuator.

#### *5.1.2. Actuators*

The presented work aims at developing the faster SMA actuators. A series of experiments were conducted on a high-temperature (90°C transformation temperature) Flexinol® SMA actuator having 0.15 mm diameter. The length of SMA actuator while testing in ambient air was 80 mm; however, the length of SMA actuator in heat sink was 90 mm. The Flexinol® SMA actuators can withstand a maximum strain of 8%, whereas 4% strain is considered to be safe for cyclic operations.

#### *5.1.3. Tubing*

In this research, the SMA actuator was run through the Polytetrafluoroethylene (PTFE) tube along with the coolant. The care was taken that SMA actuator was completely jacketed around with the coolant. PTFE tube, due to its favourable mechanical and chemical properties, was preferred over other tubing. Various chemical characteristics that make PTFE material superior are its resistance to corrosive reagents, non-solubility, long-term weatherability, non-adhesiveness and non-flammability, whereas its mechanical properties include stability at high temperatures, flexibility at low temperature and low coefficient of friction [39].

#### *5.1.4. Coolants*

In order to achieve faster cooling, the PTFE tube was initially filled with high-temperature synthetic base grease (NLGI-3). Later, the tests were conducted with the PTFE tube filled with oil (Shell Helix Hx3 20W-50 Mobil Oil).

#### **5.2. Methods**

displacement sensor made by Keyence was used to detect any deformation, i.e. strain in the SMA actuators. This laser displacement sensor (LK-G157) is capable to measure the deformation when the object is within the range of 150 mm; however, it can measure a maximum of 40 mm displacement. It is provided with a red semiconductor laser which emits light of wavelength 655 nm.

The tension produced in the SMA actuator while lifting dead weight was determined using a load sensor. Transducers Kit load cell model mdb 2.5 lb, having capacity to measure 2.5 lb,

A close look at the temperature of the SMA actuator is necessary in order to avoid over heating of the SMA actuator, which can cause a permanent deformation. A fine gauge k-type thermocouple along with a MAX7785 amplifier was used to measure the temperature of the SMA actuators. Since the SMA wire diameter was very fine (i.e. 0.15 mm), the contact loss between

The strain in the SMA actuator is proportional to the amount of actuation current. An ACS 712 current transducer having analogue output was used to keep a close look on the amount

The presented work aims at developing the faster SMA actuators. A series of experiments were conducted on a high-temperature (90°C transformation temperature) Flexinol® SMA actuator having 0.15 mm diameter. The length of SMA actuator while testing in ambient air was 80 mm; however, the length of SMA actuator in heat sink was 90 mm. The Flexinol® SMA actuators can withstand a maximum strain of 8%, whereas 4% strain is considered to be safe for cyclic operations.

In this research, the SMA actuator was run through the Polytetrafluoroethylene (PTFE) tube along with the coolant. The care was taken that SMA actuator was completely jacketed around with the coolant. PTFE tube, due to its favourable mechanical and chemical properties, was preferred over other tubing. Various chemical characteristics that make PTFE material superior are its resistance to corrosive reagents, non-solubility, long-term weatherability, non-adhesiveness and non-flammability, whereas its mechanical properties include stability at high temperatures, flexibility at low temperature and low coefficient of friction [39].

In order to achieve faster cooling, the PTFE tube was initially filled with high-temperature synthetic base grease (NLGI-3). Later, the tests were conducted with the PTFE tube filled with

SMA and the thermocouple was a serious issue, which is addressed by Ref. [38].

*5.1.1.2. Load cell*

was used in this research.

114 Shape Memory Alloys - Fundamentals and Applications

*5.1.1.4. Current transducer*

*5.1.2. Actuators*

*5.1.3. Tubing*

*5.1.4. Coolants*

oil (Shell Helix Hx3 20W-50 Mobil Oil).

of current being flowing through the SMA actuator.

*5.1.1.3. Thermocouple*

The SMA actuator was actuated using Joule heating method, with different level of actuation current in ambient air, grease and oil. In Joule heating, the SMA element is heated taking advantage of the resistance offered by it to the passage of electrical current. When the current passes through the SMA element, power losses are produced which in result heat the SMA actuator. This method of heating can be described by Eq. (2) (heating power).

$$P = I^\circ R \tag{2}$$

Here,

*P* = Electrical power to heat the SMA actuator,

*I* = Current across actuator, measured in Ampere (A) and

*R* = Resistance offered by SMA actuator to the flow of current, measured in Ohm (Ω).

For experiments in ambient air, bare SMA actuator was used. Therefore, close look on the actuation current was necessary to avoid over heating of the actuator. The maximum current in ambient air tests was set as per given specification of the SMA actuator manufacturer. However, the SMA actuator was run through the PTFE tube while testing in grease and oil. PTFE material is flexible at low temperatures and is stable at high temperatures, also it offers less friction. However, during preparation for test, the sealing of the coolant, i.e. grease or oil inside the PTFE tube was a serious issue. This issue was later overcome by inserting a smaller diameter PTFE tube in the bottoms of the main PTFE tube, carrying the coolant and SMA actuator. Since grease and oil worked as heat sink, SMA actuator was tested at higher actuation currents with negligible chances of overheating. **Figure 4** shows the schematic diagram of the SMA actuator in PTFE tube along with the coolant.

**Figure 4.** Proposed heat sink.

The SMA actuator was made to lift up a dead weight of 180 g, which induced 100 MPa stress in the SMA actuator. The data acquisition rate was set at 5 samples per second, which was sufficient to thoroughly examine the condition of SMA actuator during actuation and after the current supply was cutoff.

The thermal conductivity values of grease and oil are higher than the air (i.e. λOIL, GREASE > λAIR) as given in **Table 2**; therefore, the proposed heat sink will result in faster SMA cooling as compared to ambient air cooling.

Other factor affecting the cooling time is the surface area. Since the surface area of SMA is smaller than the surface area of PTFE tube (i.e. *A*SMA < *A*PTFE), the heat from the SMA will spread to the greater surface area of PTFE tube through the thermal conductive oil/grease, as shown in **Figure 5**.


**Table 2.** Thermal conductivity of different material.

**Figure 5.** Area and heat dissipation model of (a) SMA with sink and (b) bare SMA.

#### **6. Experiments and results**

The SMA actuator was tested in ambient air, grease and oil, and time required by these cooling media was analysed. However, the time required to restore last 0.5% strain is not included, as it is not considered by the manufacturer. To assure the SMA properties, fresh specimen was used in each test. The strain induced in the SMA actuator is calculated by using Eq. (3) (strain induced in SMA actuator).

$$\text{\textbullet Strain} = \frac{\Delta l}{l} \text{\textbullet100} \tag{3}$$

Here,

The SMA actuator was made to lift up a dead weight of 180 g, which induced 100 MPa stress in the SMA actuator. The data acquisition rate was set at 5 samples per second, which was sufficient to thoroughly examine the condition of SMA actuator during actuation and after the

The thermal conductivity values of grease and oil are higher than the air (i.e. λOIL, GREASE > λAIR) as given in **Table 2**; therefore, the proposed heat sink will result in faster SMA cooling as compared

Other factor affecting the cooling time is the surface area. Since the surface area of SMA is smaller than the surface area of PTFE tube (i.e. *A*SMA < *A*PTFE), the heat from the SMA will spread to the greater surface area of PTFE tube through the thermal conductive oil/grease, as

18 [3] 0.25 [40] 0.79 [41] 0.145 [40] 0.0257 [40]

Material SMA Teflon tubing Thermal grease Thermal oil Air

The SMA actuator was tested in ambient air, grease and oil, and time required by these cooling media was analysed. However, the time required to restore last 0.5% strain is not included, as it is not considered by the manufacturer. To assure the SMA properties, fresh specimen was

current supply was cutoff.

116 Shape Memory Alloys - Fundamentals and Applications

to ambient air cooling.

shown in **Figure 5**.

**Table 2.** Thermal conductivity of different material.

Thermal conductivity (λ) in W m−1 K−1

**6. Experiments and results**

**Figure 5.** Area and heat dissipation model of (a) SMA with sink and (b) bare SMA.

∆*l* = Change in length of SMA actuator due to heating, taken in mm

*l* = Pre-deformed length of SMA actuator, taken in mm

#### **6.1. Tests in ambient air**

An 80 mm long bare SMA actuator specimen was tested in ambient air at 210, 310 and 410 mA actuation currents. The time for a cycle was set 10 seconds, of them 1 second for heating and remaining 9 seconds for relaxing of actuator. It was assured that SMA actuator should return to its original condition before the next cycle started. The response of SMA actuator in the ambient air actuation is given in **Figure 6**.

**Figure 6.** SMA actuator response in ambient air.

The negative sign on vertical axis in **Figure 6** shows the contraction in SMA actuator. It was found that heat energy produced at 210 mA heating current was too small to transform the SMA from martensite to austenite, hence resulting in negligible strain. When SMA actuator was actuated with 310 mA actuation current in ambient air, it could only be strained by 0.311%. This amount of strain is too small to be considered for any practical applications. The reason for small strain was the insufficient heat to cause a crystal rearrangement in the crystalline structure of the SMA. However, when SMA actuator was heated with 410 mA actuation current, a strain of 3.68%, i.e. 3 mm of contraction in SMA actuator was observed. This amount of strain is sufficient for many applications including minimal invasive surgery and other robotic applications. Upon the cutoff of power supply, the wire relaxed in 2 seconds. In such circumstances, the SMA actuator will have the working frequency of 0.333 Hz.

#### **6.2. Tests in grease**

For tests in grease, 90 mm long SMA actuator specimen was passed through an 80 mm long grease-filled PTFE tube. Being surrounded with grease, the chances of overheating were minimum; therefore, SMA actuator was actuated for 2 seconds as well. The SMA actuator sample was actuated with 210, 310, 410, 500, 615 and 720 mA for 1 and 2 seconds, whereas 810 mA actuation current was supplied for 1 second only.

#### *6.2.1. One second actuation in grease*

The SMA actuator specimen in grease-filled PTFE tube was actuated for 1 second at 210, 310, 410, 500, 615, 720 and 810 mA actuation currents. The time for a cycle was set 10 seconds, of them 1 second for heating and remaining 9 seconds for relaxing of actuator. It was assured that SMA could fully recover its initial form before the next cycle starts. **Figure 7** shows the response of SMA against 1 second actuation in grease.

Negative sign on the vertical axis in **Figure 7** shows contraction in the SMA actuator against the actuation.

**Figure 7.** SMA actuator response against 1 second actuation in grease.

It can be observed that no strain was induced in the SMA actuator at 210 mA actuation current. The heat produced during actuation at 210 mA actuation was too low to cause a phase transformation in the actuator. At 310 mA heating current, the SMA actuator was strained by 0.0167%, which is too small for any application of SMA as actuator. At 410 mA heating current, due to rapid loss of heat to the coolant, i.e. grease, during heating, little amount of strain was induced in the SMA actuator, as compared to that produced in ambient air cooling, when heated with same level of actuation current. The actuation current of 410 mA could strain the grease-cooled actuator by 0.167%, which is only 4.54% of the strain produced in the ambient air-cooled SMA actuator at same actuation current. As discussed in the previous portions that due to higher thermal conductivity of grease than that of air, the heat loss rate to the grease was higher. Other reason for the rapid heat loss was the greater surface area of grease-cooled actuator, as it also included PTFE tube. The 500 and 615 mA actuation current resulted to 0.3635 and 0.7921% strain, respectively. Dynolloy Inc, in their technical data sheet for Flexinol SMA actuators, does not consider the time required to restore last 0.5% strain [3]. Therefore, in the grease-cooled SMA actuators, the cooling time for smaller strains is not considered. The SMA actuator deformed up to 3.047 mm when heated with 720 mA actuation current, this will induce a strain of 3.4%. This amount of strain is enough for SMAs to be applied as actuators in several applications like robotics, systems for drug delivery, catheters, etc. The SMA actuator cooled in 1.2 seconds. Sufficient quantity of heat was induced in the SMA to cause a phase transformation, when actuated with 810 mA actuation current. The heating was too fast for grease to absorb during actuation, due to which significant amount of heat was available inside the system to heat the SMA wire and cause a phase transformation. The 810 mA actuation current expanded the SMA actuator by 3.257 mm, resulting to 3.62% strain. However, the wire cooled off in 1.4 seconds, which is 30% improvement in cooling time as compared to that in ambient air. It should be noted that SMA actuator was strained by 3.68% when actuated with 410 mA actuation current in ambient air and recovered in 2 seconds.

#### *6.2.2. Two seconds actuation in grease*

robotic applications. Upon the cutoff of power supply, the wire relaxed in 2 seconds. In such

For tests in grease, 90 mm long SMA actuator specimen was passed through an 80 mm long grease-filled PTFE tube. Being surrounded with grease, the chances of overheating were minimum; therefore, SMA actuator was actuated for 2 seconds as well. The SMA actuator sample was actuated with 210, 310, 410, 500, 615 and 720 mA for 1 and 2 seconds, whereas 810 mA

The SMA actuator specimen in grease-filled PTFE tube was actuated for 1 second at 210, 310, 410, 500, 615, 720 and 810 mA actuation currents. The time for a cycle was set 10 seconds, of them 1 second for heating and remaining 9 seconds for relaxing of actuator. It was assured that SMA could fully recover its initial form before the next cycle starts. **Figure 7** shows the

Negative sign on the vertical axis in **Figure 7** shows contraction in the SMA actuator against

It can be observed that no strain was induced in the SMA actuator at 210 mA actuation current. The heat produced during actuation at 210 mA actuation was too low to cause a phase transformation in the actuator. At 310 mA heating current, the SMA actuator was strained by 0.0167%, which is too small for any application of SMA as actuator. At 410 mA heating current, due to rapid loss of heat to the coolant, i.e. grease, during heating, little amount of strain was induced in

circumstances, the SMA actuator will have the working frequency of 0.333 Hz.

**6.2. Tests in grease**

the actuation.

actuation current was supplied for 1 second only.

response of SMA against 1 second actuation in grease.

**Figure 7.** SMA actuator response against 1 second actuation in grease.

*6.2.1. One second actuation in grease*

118 Shape Memory Alloys - Fundamentals and Applications

The grease-cooled SMA actuators were actuated for 2 seconds with 210, 310, 410, 500, 615 and 720 mA actuation currents. These tests will help understand the effect of actuation time on the cooling efficiency of a heat sink. In these tests, the time for a cycle was set 10 seconds, of them 2 seconds for heating and remaining 8 seconds for relaxing of SMA actuator, in order to be assured that SMA fully recovers its pre-deformed shape and form before the next cycle starts. **Figure 8** shows the response of grease-cooled SMA actuator when actuated for 2 seconds at different actuation currents.

The negative sign on vertical axis represents contraction in SMA actuator, which is caused by heating.

It was observed that 210 mA actuation could not result in any deformation in SMA actuator, whereas a negligible amount of strain, about 0.04%, was observed in the SMA actuator at 310 mA actuation current. The heat energy produced by 210 and 310 mA actuation currents was too low to cause any phase transformation in the actuators.

When heated with 410 and 500 mA actuation currents, the SMA wire was strained by 0.35 and 0.6%, respectively. Because of rapid heat loss to the grease during heating, insufficient quantity of heat remained inside the SMA actuator to cause a complete phase transformation from martensite to the austenite. Whereas heating with 615 mA actuation current was sufficient to cause a significant amount of phase transformation in the SMA actuator. This could strain the actuator up to 3.2%, which is around four times higher than the strain produced by same level of actuation current when supplied for 1 second. The SMA wire actuator cooled off in 2.5 seconds, which is too high. The SMA actuator

**Figure 8.** SMA actuator response against 2 seconds actuation in grease.

contracted up to 3.1 mm when heated with 720 mA actuation current supplied for 2 seconds. This resulted in 3.44% strain in SMA actuator, which is nearly equivalent to the strain produced with similar amount of actuation current when supplied for 1 second. However, the SMA wire cooled off in 6 seconds, which is five times greater than that required in 1 second actuation at same level of actuation current. This increase in cooling time shows that the efficiency of grease is affected with increasing heating time, especially at high actuation currents.

#### **6.3. Tests in oil**

To find out the effect of oil on SMA actuator cooling time, a 90 mm long SMA actuator specimen was run through an 80 mm long PTFE tube. The SMA actuator was heated with 210, 310, 410, 500, 615 and 720 mA current supplied for 1 and 2 seconds, whereas 810 mA actuation current was supplied for 1 second only.

#### *6.3.1. One second actuation in oil*

The SMA actuator specimen in oil-filled PTFE tube was actuated for 1 second at 210, 310, 410, 500, 615, 720 and 810 mA actuation currents. The time for a cycle was set 10 seconds, of them 1 second for heating and remaining 9 seconds for relaxing of SMA actuator. It was assured that SMA could fully recover its initial form before the next cycle starts. **Figure 9** shows the response of SMA against 1 second actuation in oil.

It was observed that 210 mA did not result in any strain in the SMA actuator; however, negligible strain in SMA actuator was observed at 310 and 410 mA actuation currents due to

**Figure 9.** SMA actuator response against 1 second actuation in oil.

significant amount of heat loss to the oil during actuation. At 500 and 615 mA actuations, the SMA actuators strained by only 0.3472 and 0.6076%, respectively, due to insufficient amount of heat to cause phase transformation.

The 720 mA actuation current could produce sufficient heat to cause the phase transformation in the oil-covered SMA actuator. The 720 mA strained the SMA actuator up to 3%, producing 2.685 mm stroke length, which is sufficient for many SMA applications including catheters and latches and micro robots. The SMA actuator relaxed in 1 second hence capable to give 0.5 Hz frequency. While actuated with 810 mA current supply, the SMA was deformed by 3.5 mm, inducing 3.9%, which is 0.28% more than that produced with same actuation current in grease-cooled actuator. However, the cooling time required for SMA actuator was found to be 1.6 seconds.

#### *6.3.2. Two seconds actuation in oil*

contracted up to 3.1 mm when heated with 720 mA actuation current supplied for 2 seconds. This resulted in 3.44% strain in SMA actuator, which is nearly equivalent to the strain produced with similar amount of actuation current when supplied for 1 second. However, the SMA wire cooled off in 6 seconds, which is five times greater than that required in 1 second actuation at same level of actuation current. This increase in cooling time shows that the efficiency of grease is affected with increasing heating time, especially at high actuation currents.

To find out the effect of oil on SMA actuator cooling time, a 90 mm long SMA actuator specimen was run through an 80 mm long PTFE tube. The SMA actuator was heated with 210, 310, 410, 500, 615 and 720 mA current supplied for 1 and 2 seconds, whereas 810 mA actuation

The SMA actuator specimen in oil-filled PTFE tube was actuated for 1 second at 210, 310, 410, 500, 615, 720 and 810 mA actuation currents. The time for a cycle was set 10 seconds, of them 1 second for heating and remaining 9 seconds for relaxing of SMA actuator. It was assured that SMA could fully recover its initial form before the next cycle starts. **Figure 9** shows the

It was observed that 210 mA did not result in any strain in the SMA actuator; however, negligible strain in SMA actuator was observed at 310 and 410 mA actuation currents due to

**6.3. Tests in oil**

current was supplied for 1 second only.

response of SMA against 1 second actuation in oil.

**Figure 8.** SMA actuator response against 2 seconds actuation in grease.

120 Shape Memory Alloys - Fundamentals and Applications

*6.3.1. One second actuation in oil*

The oil-cooled SMA actuators were actuated for 2 seconds with 210, 310, 410, 500, 615 and 720 mA actuation currents. SMA actuator heating for 2 seconds at various actuation currents will help understand the SMA behaviour against higher actuation times. These tests will also help understand the effect of actuation time on the cooling efficiency of a heat sink. In these set of tests, the time for a cycle was set 10 seconds, of them 2 seconds for heating and remaining 8 seconds for relaxing of SMA actuator. **Figure 10** shows the response of oil-cooled SMA actuator when actuated for 2 seconds at different actuation currents.

The negative sign on vertical axis in **Figure 10** denotes the shrinking in SMA actuator produced by heating.

**Figure 10.** SMA actuator response against 2 seconds actuation in oil.

At 210 mA actuation current, no deformation in the SMA actuator was observed hence no strain in the SMA was induced, whereas negligible strain equal to 0.027% was observed at 310 mA actuation current. When actuated with 410 and 500 mA actuation currents, little strain equal to 0.4 and 0.56%, respectively, was observed. The observed quantity of strain is too small to for any real-time application of SMAs as actuators. However, 615 mA actuation current resulted in 3.5% strain in SMA, which is adequate for a number of real-world applications. The SMA specimen relaxed in 2 seconds, hence capable to give 0.33 Hz frequency. The SMA actuator contracted up to 3.213 mm, hence inducing 3.57% strain in the actuator, when heated with 720 mA actuation current. The actuator cooled off in 4.4 seconds. However, 410 mA actuation current in ambient air produced same amount of strain in the SMA actuator, whereas the wire had cooled off in 2 seconds, which is only 45.45% of cooling time required by SMA actuator when heated with 720 mA actuation current for 2 seconds in oil. The reason for the increase in cooling time is that the rejection of considerable amount of heat to the oil, which increased the temperature of oil, hence reducing the heat flow rate from actuator to the oil. It is general consideration that the heat transfer rate is proportional to difference of temperature between the two mediums.

#### **7. Conclusions**

From the results, it is derived that at same level of actuation current, grease- and oil-cooled SMA actuators underwent smaller strain as compared to the strain produced in ambient air cooling. The maximum heating current in ambient air was 410 mA, which strained the SMA actuator by 3.68%; however, the wire relaxed in 2 seconds, whereas an equivalent amount of strain in grease-cooled actuators was produced at 720 and 810 mA heating currents, and SMA actuators relaxed in 1.2 and 1.4 seconds, respectively, which is 40 and 30% improvement in cooling time of SMA as compared to that in ambient air. When actuated for 2 seconds in grease at 720 mA current supply, the actuator strained by 3.4% but recovered in 6 seconds, which is five times higher than the cooling time observed against 1 second actuation at 720 mA current in grease. Similarly, oil-cooled actuators produced considerable strain against 720 and 810 mA actuation current when heated for 1 second. The cooling times in this case were 1 and 1.6 seconds, respectively. Like grease-cooled actuators, the oil-cooled actuators required long cooling time against 2 seconds actuations, as the temperature of coolants, i.e. grease or oil was too high to efficiently absorb the heat. However, in case of ambient air cooling, close watch over the temperature of SMA is compulsory to prevent overheating which is likely to cause a permanent deformation in the actuator. However, in grease and oil cooling, the heat was rapidly being rejected to the coolant (i.e. grease or oil); therefore, likelihood of overheating of the SMA actuator is less. From the results and discussions sections, it can be observed that the oil-cooled actuator produced linear response, whereas the response of grease-cooled SMA actuator fluctuated at various points. The oil-cooled actuators produced more stroke length as compared to grease-cooled actuators and cooled off in less time. Therefore, for more strain, less cooling time, linear behaviour and safe actuation, the oil cooling is suggested over grease and ambient air cooling.

The presented research also reveals that increasing the actuation time also results in increased cooling time with no significant effect on the stroke length (i.e. strain). Therefore, it is further concluded that rather than increasing the heating duration with smaller magnitude currents, the magnitude of the heating current should be increased in order to avoid the heating of the sink.

#### **Author details**

At 210 mA actuation current, no deformation in the SMA actuator was observed hence no strain in the SMA was induced, whereas negligible strain equal to 0.027% was observed at 310 mA actuation current. When actuated with 410 and 500 mA actuation currents, little strain equal to 0.4 and 0.56%, respectively, was observed. The observed quantity of strain is too small to for any real-time application of SMAs as actuators. However, 615 mA actuation current resulted in 3.5% strain in SMA, which is adequate for a number of real-world applications. The SMA specimen relaxed in 2 seconds, hence capable to give 0.33 Hz frequency. The SMA actuator contracted up to 3.213 mm, hence inducing 3.57% strain in the actuator, when heated with 720 mA actuation current. The actuator cooled off in 4.4 seconds. However, 410 mA actuation current in ambient air produced same amount of strain in the SMA actuator, whereas the wire had cooled off in 2 seconds, which is only 45.45% of cooling time required by SMA actuator when heated with 720 mA actuation current for 2 seconds in oil. The reason for the increase in cooling time is that the rejection of considerable amount of heat to the oil, which increased the temperature of oil, hence reducing the heat flow rate from actuator to the oil. It is general consideration that the heat transfer rate is proportional to difference of temperature

**Figure 10.** SMA actuator response against 2 seconds actuation in oil.

122 Shape Memory Alloys - Fundamentals and Applications

From the results, it is derived that at same level of actuation current, grease- and oil-cooled SMA actuators underwent smaller strain as compared to the strain produced in ambient air cooling. The maximum heating current in ambient air was 410 mA, which strained the SMA

between the two mediums.

**7. Conclusions**

Atta Muhammad Nizamani<sup>1</sup> \*, Jawaid Daudpoto<sup>1</sup> and Muhammad Ali Nizamani2


#### **References**


[17] Taylor F, Au C. Forced air cooling of shape memory alloy actuators for a prosthetic hand. Journal of Computing and Information Science in Engineering (2016):16(4), 041004 (Nov 07, 2016) (5 pages) Paper No: JCISE-16-1051; doi:10.1115/1.4033233, Received January 29, 2016; Revised March 16, 2016 https://computingengineering.asmedigitalcollection.asme. org/article.aspx?articleID=2511361

[3] Dynalloy I. Technical Characteristics of Flexinol Actuator Wires. Dyanalloy Inc. Tustin,

[4] Daudpoto, Jawaid, Ali Asghar Memon, and Imtiaz Hussain. "Actuation Characteristics of 0.15 Mm Diameter Flexinol® and Biometal® Wire Actuators for Robotic Applications." Mehran University of Research Journal of Engineering and Tecnology 32 (2013). Print. [5] Lara-Quintanilla A, Bersee HE. A study on the contraction and cooling times of actively cooled shape memory alloy wires. Journal of Intelligent Material Systems and Structures

[6] Nizamani, AM, J. Daudpoto, and MA Soomro. "Improving Cycle Time of Ni-Ti Shape Memory Alloy Actuators Using a Novel Cooling Technique." Sindh University Research

[7] Ikuta K. Micro/miniature shape memory alloy actuator. In: Proceedings, 1990 IEEE International Conference on Robotics and Automation, 1990; 13-18 May 1990; Cincinnati,

[8] Jani JM, Leary M, Subic A, Gibson MA. A review of shape memory alloy research, appli-

[9] Butera F, Coda A, Vergani G, SpA SG. Shape memory actuators for automotive applica-

[10] Jani JM, Leary M, Subic A. Shape memory alloys in automotive applications. Applied

[11] Zhong Z, Yeong C. Development of a gripper using SMA wire. Sensors and Actuators A:

[12] Esfahani ET. Developing an Active Ankle foot Orthosis based on Shape Memory Alloys.

[13] Copaci D, Flores A, Rueda F, Alguacil I, Blanco D, Moreno L. Wearable elbow exoskeleton actuated with shape memory alloy. In: Converging Clinical and Engineering Research on Neurorehabilitation II. Springer; 2017. pp. 477-481. Proceedings of the 3rd International Conference on NeuroRehabilitation (ICNR2016), October 18-21, 2016,

Segovia, Spain. https://link.springer.com/chapter/10.1007/978-3-319-46669-9\_79

A novel robotic fish design. Bioinspiration & Biomimetics. 2011;**6**:045005

28--30, 2013. pp. 90-100. Springer, ISBN: 978-3-662-43645-5

[14] Rossi C, Colorado J, Coral W, Barrientos A. Bending continuous structures with SMAs:

[15] Gaissert N, Mugrauer R, Mugrauer G, Jebens A, Jebens K, Knubben EM. Inventing a micro aerial vehicle inspired by the mechanics of dragonfly flight. 14th International Conference Towards Autonomous Robotic Systems,TAROS 2013, Oxford, UK, August

[16] Coral Cuellar W, Rossi C, ColoradoMontaño J, Barrientos Cruz A. SMA-BasedMuscleLike Actuation in Biologically Inspired Robots: A State of the Art Review. Intech; 2012

CA: 2011. www.dynalloy.com/pdfs/TCF1140.pdf

Journal-SURJ (Science Series) 48.4D (2016), p 161-164

Mechanics & Materials. 2014;**663**:248-253

The University of Toledo, Ohio, USA; 2007

Physical. 2006;**126**:375-381

Ohio, United States of America. IEEE; 1990. pp. 2156-2161

cations and opportunities. Materials & Design. 2014;**56**:1078-1113

tions. Nanotec IT newsletter. Roma: AIRI/Nanotec IT; 2007. pp. 12-16

27.3 (2016): 403-17

124 Shape Memory Alloys - Fundamentals and Applications


Actuator. In the proceedings of the 14th International Conference on New Actuators, Messe Bremen WFB Wirtschaftsförderung Bremen GmbH, Bremen, Germany. 742 p, ISBN 978-3-933339-22-5 . Proceedings available online at, http://www.actuator.de/userfiles/File/ACTUATOR\_14/ACTUATOR\_2014\_ordform.pdf


Actuator. In the proceedings of the 14th International Conference on New Actuators, Messe Bremen WFB Wirtschaftsförderung Bremen GmbH, Bremen, Germany. 742 p, ISBN 978-3-933339-22-5 . Proceedings available online at, http://www.actuator.de/user-

[29] Olivier J. Godard ; Magdalini Z. Lagoudas and Dimitris C. Lagoudas "Design of space systems using shape memory alloys", Proc. SPIE 5056, Smart Structures and Materials 2003: Smart Structures and Integrated Systems, 545 (August 5, 2003) Volume 5056, San

[30] Hartl DJ, Lagoudas DC. Aerospace applications of shape memory alloys. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. 2007;

[31] Sonawane NS, Thangavelautham J. Precision pointing of antennas in space using arrays of shape memory alloy based linear actuators. arXiv preprint arXiv:1701.07561, 2017 [32] Loh CS, Yokoi H, Arai T. Natural heat-sinking control method for high-speed actuation

[33] Loh CS, Yokoi H, Arai T. New shape memory alloy actuator: Design and application in the prosthetic hand. In: IEEE-EMBS 2005. 27th Annual International Conference of the Engineering in Medicine and Biology Society, 2005; 17-18 January 2006; Shanghai,

[34] Cheng SS, Desai JP. Towards high frequency actuation of SMA spring forthe neurosurgical robot-MINIR-II. In: 2015 IEEE International Conference on Robotics and Automation.

[35] Pathak A, Brei D, Luntz J. Transformation strain based method for characterization of convective heat transfer from shape memory alloy wires. Smart Materials and Structures.

[36] Tadesse Y, Thayer N, Priya S. Tailoring the response time of shape memory alloy wires through active cooling and pre-stress. Journal of Intelligent Material Systems and

[37] Russell RA, Gorbet RB. Improving the response of SMA actuators. In: Proceedings, 1995 IEEE International Conference on Robotics and Automation, 1995; 21-27 May 1995;

[38] Soomro, MA, J. Daudpoto, and AM Nizamani. "Design, Development and Validation of an Experimental Test Bed for SMA Actuators." Sindh University Research Journal-SURJ

[39] Teflon P, Resin PF. Properties Handbook. Washington: DuPont Fluoroproducts; 1996

[40] Box ET. Thermal Conductivity of Materials and Gases [Internet]. Available from: http:// www.engineeringtoolbox.com/thermal-conductivity-d\_429.html [Accessed: 25 April 2016]

[41] Thermalloy. Thermal Greases [Internet]. Available from: htt://www.aavid.com/product-

(ICRA); 26-30 May 2015; Seattle, Washington, USA. IEEE; 2015. pp. 2580-2585

of the SMA. International Journal of Advanced Robotic Systems. 2006;**3**:42

files/File/ACTUATOR\_14/ACTUATOR\_2014\_ordform.pdf

Diego, California, USA. http://dx.doi.org/10.1117/12.483469

**221**:535-552

126 Shape Memory Alloys - Fundamentals and Applications

2010;**19**:035005

Structures. 2010;**21**:19-40

Nagoya, Japan. IEEE; 1995. pp. 2299-2304

group/interface/greases [Accessed: 25 April 2016]

(Science Series) 48.4D (2016). p 39-44

China. IEEE; 2006. pp. 6900-6903

## *Edited by Farzad Ebrahim*

This book is a result of contributions of experts from international scientific community working in different aspects of shape memory alloys (SMAs) and reports on the state-of-the-art research and development findings on this topic through original and innovative research studies. Through its five chapters, the reader will have access to works related to ferromagnetic SMAs, while it introduces some specific applications like development of faster SMA actuators and application of nanostructural SMAs in medical devices. The book contains up-to-date publications of leading experts, and the edition is intended to furnish valuable recent information to the professionals involved in shape memory alloys analysis and applications. The text is addressed not only to researchers but also to professional engineers, students, and other experts in a variety of disciplines, both academic and industrial, seeking to gain a better understanding of what has been done in the field recently and what kind of open problems are in this area.

Photo by Robert St-Pierre / iStock

Shape Memory Alloys - Fundamentals and Applications

Shape Memory Alloys

Fundamentals and Applications

*Edited by Farzad Ebrahim*