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Shape Memory Alloys

edited by

Corneliu Cismasiu

SCIYO

Shape Memory Alloys

Edited by Corneliu Cismasiu

Published by Sciyo Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2010 Sciyo

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Shape Memory Alloys, Edited by Corneliu Cismasiu p. cm. ISBN 978-953-307-106-

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Preface VII

Molecular Dynamics Simulation of Shape-memory Behavior 1

Takuya Uehara

Thermo-mechanical behaviour of NiTi at impact 17

Zurbitu, J.; Kustov, S.; Zabaleta, A.; Cesari, E. and Aurrekoetxea, J.

Bending Deformation and Fatigue Properties of

Precision-Casting TiNi Shape Memory Alloy Brain Spatula 41

Hisaaki Tobushi, Kazuhiro Kitamura, Yukiharu Yoshimi and Kousuke Date

Hysteresis behaviour and modeling of SMA actuators 61

Hongyan Luo, Yanjian Liao, Eric Abel, Zhigang Wang and Xia Liu

Experimental Study of a Shape Memory Alloy

Actuation System for a Novel Prosthetic Hand 81

Konstantinos Andrianesis, Yannis Koveos,

George Nikolakopoulos and Anthony Tzes

Active Bending Catheter and Endoscope

Using Shape Memory Alloy Actuators 107

Yoichi Haga, Takashi Mineta, Wataru Makishi,

Tadao Matsunaga and Masayoshi Esashi

Numerical simulation of a semi-active vibration control

device based on superelastic shape memory alloy wires 127

Corneliu Cismaşiu and Filipe P. Amarante dos Santos

Seismic Vibration Control of Structures

Using Superelastic Shape Memory Alloys 155

Hongnan Li and Hui Qian

Joining of shape memory alloys 183

Odd M. Akselsen

Contents

The Shape Memory Alloys (SMAs) represent a unique material class exhibiting peculiar properties like the shape memory effect, the superelasticity associated with damping capabilities, high corrosion and extraordinary fatigue resistance. Due to their potential use in an expanding variety of technological applications, an increasing interest in the study of the SMAs has been recorded in the research community during the previous decades.

This book includes fundamental issues related to the SMAs thermo-mechanical properties, constitutive models and numerical simulation, medical and civil engineering applications and aspects related to the processing of these materials, and aims to provide readers with the following:

• It presents an incremental form of a constitutive model for shape memory alloys. When compared to experimental tests, it proves to perform well, especially when the stress drops during tension processes.
• It describes single-crystal and multi-grained molecular models that are used in the dynamic simulation of the shape memory behaviour.
• It explains and characterizes the temperature memory effect in TiNi and CU-based alloys including wires, slabs and films by electronic resistance, elongation and DSC methods.
• It analyses the thermo-mechanical behaviour of superelastic NiTi wires from low to impact strain rates, including the evolution of the phase transformation fronts.
• It presents an experimental testing programme aimed to characterize the bending deformation and the fatigue properties of precision-casted TiNi SMA used for instruments in surgery operations.
• It introduces a computational model based on the theory of hysteresis operator, able to accurately characterize the non-linear behaviour of SMA actuators and well suited for real- time control applications.
• It describes the development and testing of an SMA-based, highly sophisticated, lightweight prosthetic hand used for multifunctional upper-limb restoration.
• It presents the design of an active catheter and endoscope based on shape memory alloy actuators, expected to allow low cost endoscopic procedures.
• It exemplifies the use of superelastic shape memory alloys for the seismic vibration control of civil engineering structures, considering both passive and semi-active devices.
• It discusses some aspects related to the processing of the shape memory alloys and presents special techniques that provide bonds without severe loss of the initial SMA properties.

Preface

VIII

With its distinguished team of international contributors, Shape Memory Alloys is an essential reference for students, materials scientists and engineers interested in understanding the complex behaviour exhibited by the SMAs, their properties and potential for industrial applications.

Lisbon, July 2010

Editor

Corneliu Cismasiu Centro de Investigação em Estruturas e Construção - UNIC Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2829-516 Caparica Portugal

2 Shape Memory Alloys

Fig. 1. Schematic illustration of deformation and shape recovery of a SMA.

reproduced using the EAM potential as reported by Rubini and Ballone (Rubini & Ballone,

1993. and Farkas et al. (Farkas et al., 1995). Uehara et al. then utilized the EAM potential to demonstrate the shape-memory behavior of Ni-Al alloy in terms of a small single crystal (Uehara et al., 2001; Uehara & Tamai, 2004, 2005, 2006), the size dependency (Uehara et al., 2006), and the polycrystalline model (Uehara et al., 2008, 2009). Ozgen and Adiguzel also investigated the shape-memory behavior of Ni-Al alloy using a Lennard-Jones (LJ) potential (Ozgen & Adiguzel, 2003, 2004). In addition, for Ni-Ti alloy, martensitic transformation was simulated by Sato et al. (Sato et al., 2004) and Ackland et al. (Ackland et al., 2008). It was also reported by Kastner (Kastner, 2003, 2006) that the shape-memory effect can be represented even by a two-dimensional model with a general LJ potential on the basis of thermodynami- cal discussion on the effect of temperature on the phase transformation. For a more practical purpose, Park et al. demonstrated shape-memory and pseudoelastic behavior during uniaxial loading of an fcc silver nanowire, and discussed the effect of the initial defects and mechanism of twin-boundary propagation (Park et al., 2005, 2006). In this chapter, atomistic behavior and a stress–strain diagram obtained by molecular dy- namics simulation are presented, following our simulation of Ni-Al alloy. A summary of the molecular dynamics method as well as EAM potential is given in Sec. 2. The simulation con- ditions are explained in Sec. 3. Simulation results obtained using the single-crystal model and polycrystal model are presented in Sec. 4 and 5, respectively, and concluding remarks are given in Sec. 6.

2. Molecular Dynamics Method

2.1 Fundamental equations Employing the molecular dynamics (MD) method, the position and velocity of all atoms con- sidered are traced by numerically solving Newton’s equation of motion. Various physical and mechanical properties as well as dynamic behavior on the atomistic or crystal-structure scale are then obtained using a statistical procedure.

Molecular Dynamics Simulation of Shape-memory Behavior 3

The fundamental equation of MD method is Newton’s equation of motion for all atoms con- sidered in the system: r ¨i = f i /m (^) i, (1)

where r i and m (^) i are the position vector and mass of the i-th atom, respectively, and f i is the force acting on the i-th atom, which is represented as

f i = − ∂ Φ/ ∂ r i, (2)

with the potential energy Φ of the system considered. This equation is solved numerically. Verlet’s scheme, which is often used in MD simulations, is utilized:

r i (t + ∆t) = r i (t) + v i (t)∆t + F i (t)∆t^2 /( 2 m (^) i ) (3) v i (t + ∆t) = v i (t) + ( F i (t + ∆t) + F i (t))∆t/( 2 m (^) i ), (4)

where (t) represents the value at time t, and ∆t is the time increment. Temperature is expressed as

T =

2 K

3 Nk (^) b

3 Nk (^) b

N ∑ i

m (^) i v^2 i , (5)

where K is the total kinetic energy, k (^) b is the Boltzmann constant, and the notation <> rep- resents the time average.√ Temperature is controlled by scaling the velocity with the factor T/T 0. Pressure is defined using the virial theorem, and it can be controlled by adjusting the length of the axes, which is referred to as the scaling method. The stress tensor is defined and controlled employing the so-called Parrinello-Rahman (PR) method (Parrinello & Rahman, 1980,1981).

2.2 EAM potential Various interatomic energy functions have been proposed and are classified as empirical, semi-empirical, and first-principle potentials. The precision is highest for first-principle po- tentials, although only a small number of atoms are considered owing to the computational cost. This study employs the EAM potential, which was developed by Daw, Baskes (Daw & Baskes, 1984) and others, and the precision for metals is relatively fine. The potential function is written as

Φ = (^) ∑ i

F( ρ i ) +

2 ∑ i^ ∑ j =i

φ ij(r (^) ij). (6)

Here, Φ is the total potential energy in the system considered, the first term on the right-hand side is a many-body term as a function of the local electron density ρ i around the i-th atom, and the second term is a two-body term that expresses a repulsive force at close range. The electron density ρ i is assumed to be (Clementi & Roetti, 1974)

ρ i = (^) ∑ j =i

ρ ˜(r (^) ij) = (^) ∑ j =i

{N s^ ρ ˜s(r (^) ij) + N d^ ρ ˜d(r (^) ij)}, (7)

where

ρ ˜s(r (^) ij) = ρ ˜d(r (^) ij) = | (^) ∑ I

C (^) I R (^) I |^2 /4 π , (8)

Molecular Dynamics Simulation of Shape-memory Behavior 5

potential. The lattice points of B2 structure are assigned to Ni and Al atoms alternately to make Ni-50%Al alloy. An Al atom is then randomly chosen and replaced by a Ni atom. This procedure is repeated until the designated Ni concentration is reached. Using this model, MD simulations are carried out at a constant temperature under zero pressure with a periodic boundary condition. Figure 2 presents the initial configuration of Ni and Al atoms and snapshots after a 20000- step calculation for (a) 60%Ni and (b) 68% Ni at 10 K. The crystal structure of the 68% model has apparently different, while no change is observed for the 60% model. The transformed phase is regarded as martensite. The stable structure is found to be B2 or martensite for all Ni concentrations and temperatures, as summarized in Fig. 3. Martensite phase is obtained at low temperature for the high-Ni alloy, while B2 is stable at high temperature in the low-Ni range. Since both structures are obtained in the 64%-70% range, the 68% Ni concentration is used in the following simulations.

Fig. 2. Initial configuration of atoms and stable structures for (a) 60% Ni and (b) 68% Ni at 10 K.

3.2 Load and temperature condition Figure 4 represents the external load and temperature profile divided into four stages: loading (I), unloading (II), heating (III) and cooling (IV). The time increment is set as ∆t=1.5 fs, and the computation comprises 40000 steps. A shear load is imposed in the loading stage by leaning the edge that runs in the y direction in the x direction at a constant rate. The normal stress component is kept to zero by adjusting the three edge lengths. When the shear strain γ xy reaches 0.33, the stress is released employing the Parrinello-Rahman (PR) method. This stage corresponds to the unloading. The temperature is kept constant at T=10 K through these

6 Shape Memory Alloys

Fig. 3. Phase diagram representing stable structure.

stages, while it is raised to 1000 K and decreased to 10 K in Stages III and IV. Each component of the stress is controlled to zero through heat treatment employing the PR method.

4. Single-Crystal Model

4.1 Deformation and phase transformation In this section, the results obtained using a single-crystal model with 68% Ni are presented following the work of Uehara et al. (Uehara et al., 2006a). The initial phase is set as martensite

Fig. 4. Temperature and loading profile.

8 Shape Memory Alloys

stress drops corresponds to the instant that a variant layer changes orientation, and this con- tinues until all layers have the same orientation. The elastic recovery is clearly shown in this figure, and the shape recovery is expressed by the curve returning to the origin. As a result, the S-S curve draws a hysteresis loop, although the loading curve obtained experimentally or macroscopically is much smoother. Nevertheless, if neglecting the zigzag profile in loading, we conclude that the shape recovery is represented.

4.3 Size dependency To show the size dependency, extensive simulations are carried out using larger models (Ue- hara et al., 2006b). The S-S curve obtained for a larger model is shown by the dashed line in Fig. 7(a), where the result for a smaller model identical to that in Fig. 6 is also drawn for com- parison. The S-S curves have similar tendencies in the sense that there are repeated gradual stress rises and abrupt drops, although the number of peaks and drops is greater for the larger model; i.e., the curve as a whole is smoother for the larger model. The gradients of the two curves in the loading stage are identical, indicating that the elastic modulus is independent of the model size. Random numbers are used in the preparation of the 68% Ni alloy, and to set the initial velocity of the atoms. Therefore, the results may be affected by the randomness. Figure 7(b) shows the S-S curves obtained in six trials, each of which had the same model size and conditions but different random-number set. The curves do not coincide completely, but mostly show identical tendencies.

Fig. 6. Stress–strain relation during loading, unloading, heating, and cooling.

Molecular Dynamics Simulation of Shape-memory Behavior 9

Fig. 7. Stress–strain curves during loading and unloading for larger models.

5. Polycrystalline Model

5.1 Model Rather large discrepancy in the S-S curve, especially in the loading process, between the MD results and experimental observations is considered to be due to the MD model being highly simplified. To investigate to what degree the model affects the S-S curve, Uehara et al. carried out MD simulations using multi-grain models (Uehara et al., 2008, 2009), and the results are presented in this section. Two models with different grain shapes and distributions are shown in Fig. 8. Model A consists of two square and two octagonal grains, while Model B has four hexagonal grains. Both models guarantee continuity under periodic boundary conditions. The initial con- figuration is set as martensite phase, while the orientations of the variants are specific to each grain. The model size is around 40× 40 ×5 in units of the lattice constant (i.e., about 16.6nm×16.6nm×1.5nm) in the x, y, and z directions, respectively, and the total number of atoms is about 31000. The conditions for shear loading and heat treatment are the same as those in the previous section.

Molecular Dynamics Simulation of Shape-memory Behavior 11

sudden drop in stress is due to the simultaneous change in the orientation of a specific layer through the model in a single crystal. In the multi-grain model, however, the motion of the layer is interfered by the grain boundary, and the orientation change does not pass thorough the model. Therefore, there is no sudden drop in stress, and other layers begin to deform. This occurs continuously, resulting in the smoothing of the stress variation. The subsequent elastic recovery in the unloading process — macroscopic shape recovery due to B2 transformation in the heating process — and regeneration of the original martensite phases are similar to what occurs for the single-crystal model, and a hysteresis loop forms. As a result, we conclude that the shape-memory behavior is successfully simulated, and the S-S curves using a multi-grain model approach the experimental curves.

5.3 Results for Model B Figures 11 and 12 show the variation in the configuration of atoms and the S-S curves in the loading stage obtained using Model B. The snapshots in Figs. 11(a)–(d) correspond to the initial configuration, after relaxation, under loading, and at the end of loading, respectively. In Fig. 12, the S-S curves obtained in six cases, which differ in terms of the variant orientation in each grain, are plotted in a single diagram. Similarly to what is seen in Fig. 9, there is some relaxation around the grain boundaries, and B2 and M2 domains appear as shown in Fig. 11(b). Here again, since the overall state of M phase is maintained, the loading process is continued using this model. The orientation of the martensite varies as the shear load is imposed, and the M2 domain grows. Each variant has a particular orientation within the grain, and finally, M2 occupies almost all grains except at grain boundaries. Note that in some cases, there are multiple orientations in a grain, which is also considered to be the cause of the smoothing of the S-S curve. In Fig. 12, the overall ten- dency is common for all trials, while the plateau stress is classified into two groups. Detailed discussion is provided in the literature (Uehara et al., 2009).

Fig. 9. Variation of the configuration of atoms for Model A: (a) initial state, (b) after relaxation, (c) under loading, (d) after loading, (e) after unloading, (f) after heating, and (g) after cooling.

12 Shape Memory Alloys

Fig. 10. Stress–strain relation during loading, unloading, heating, and cooling for Model A.

6. Concluding Remarks

Deformation and shape-recovery processes are simulated employing the molecular dynamics method. There is shape memory even in a simple model of a single crystal, and a hystere- sis loop for the stress–strain curve is obtained. Deformation of martensite phase progresses through layer-by-layer change in the variant orientation, and the propagation results in a zigzag shape of the S-S curve. The shape of the loading curve drastically changes when using a multi-grain model, and the S-S curve approaches the experimentally observed curve, which is obtained on a macroscopic scale. It is revealed in this study that the smoothing of the curve is due to the existence of grain boundaries and variation in the crystal orientation. As future work, extensive simulations are required for detailed discussion on the role of the grain boundary and anisotropic tendency. Other effects and mechanisms based on defects, dislocations, sliding, and twinning, which are especially important in SMAs, should be con- sidered. The size dependency is also expected to be investigated in depth in terms of the above-mentioned mechanism, since it is one of the major remaining problems in solid me- chanics (Yamakov et al., 2002; Shimokawa et al., 2005). For quantitative evaluation, the preci- sion of the interatomic potential should be definitive and a three-dimensional model used.

Fig. 11. Variation of the configuration of atoms for Model B: (a) initial state, (b) after relaxation, (c) under loading, and (d) after loading.