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Thermomechanical couplings in shape memory alloy

1

materials

2

D.Soldatosa, S. P. Triantafylloub,V. P. Panoskaltsisa 3

aDepartment of Civil Engineering, Demokritos University of Thrace, University Campus 4

Xanthi-Kimmeria, 67100 Xanthi, GREECE 5

bDepartment of Civil Engineering, University of Nottingham, University Park, NottighamNG7 6

2RD, UK 7

8

AbstractIn this work we address several theoretical and computational issues which are related 9

to the thermomechanical modeling of shape memory alloy materials. More specifically, in this 10

paper we revisit a non-isothermal version of the theory of large deformation generalized 11

plasticity which is suitable for describing the multiple and complex mechanisms occurring in 12

these materials during phase transformations. We also discuss the computational 13

implementation of a generalized plasticity based constitutive model and we demonstrate the 14

ability of the theory in simulating the basic patterns of the experimentally observed behavior by 15

a set of representative numerical examples. 16

17

Keywords Shape memory alloys - Shape memory effect - Pseudoelasticity- Generalized 18

plasticity - Invariance - balance energy equation - Thermomechanical state equations 19

Isothermal split - Thermomechanical couplings 20

21 22 23 24 4

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1 Introduction 25

26

Shape Memory Alloys (SMAs) are an intriguing class of metal alloys with the ability to 27

undergo severe deformations and then recover their original shape. This can occur either 28

under the action of a thermomechanical cycle, with the corresponding response termed 29

as shape memory effect or a stress cycle within some appropriate temperature limits 30

(pseudoelasticity). Two are the fundamental mechanisms underlying this recovery. The 31

first one is a diffusionless transformation between the high ordered austenite phase 32

(parent phase) and the less ordered martensite (product) phase. The second evolves 33

through the reorientation (detwinning) of the martensite variants. These transformations 34

are termed as martensitic and may be met also in other metallic materials such as carbon 35

steels and invar alloys. 36

Due to these properties SMAs are being increasingly used in several innovating 37

applications which are met at almost all engineering fields. Thus, there is a pressing need 38

for simulation tools that can accurately describe their experimentally observed behavior, 39

especially under complex states of stress and temperature. 40

For the past three decades there has been substantial activity to model martensitic 41

transformations in shape memory alloys within a fully coupled thermomechanical 42

framework. This approach relies on the use of the so-called non-equilibrium (or 43

irreversible) thermodynamics. Within this approach, among others, Muller [28], Raniecki 44

et al. [43], Huo and Muller [15],Raniecki and Lexcellent [44], Leclercq and Lexcellent

45

[18], Boyd and Lagoudas [8], Lagoudas et al. [17], Peyroux et al. [40], Raniecki and 46

Lexcellent [45], Mü ller and Bruhns [29], Zio kowski [57] Christ and Reese [9], 47

(3)

Thamburaja [52], Morin et al. [27], Yu et al. [55] have proposed models based on the use 48

of a set of thermomechanical equations describing thekinematics of the martensitic 49

transformations. The constitutive equations are developed in a non-linear manner on the 50

basis of a free energy driving force and the laws of thermodynamics. 51

An alternative approach is the employment of plastic flow theories. Such an approach 52

is thermodynamically consistent and may furnish a concrete micromechanical 53

justification - see, e.g., the ideas exposed in the book by Smallman and Bishop ([50, pp. 54

278-280]); see also the concise discussion given in Panoskaltsis et al. [36].On the basis 55

of this idea, Anand and Gurtin [2], by following the equilibrium theory of austenite -56

martensite phase transitions of Ball and James [6], proposed a three-dimensional crystal 57

model which was able to reproduce the pseudoelastic response of SMAs under isothermal 58

and non-isothermal conditions. Related is the thermomechanical model by Lu and Weng 59

[20] - see also Yin and Weng [54] - where a set of explicit constitutive equations which 60

provide a direct link between the applied stress and the evolution of the product phase 61

during martensitic transformations, and between the stress and the overall strain of the 62

transforming system, is discussed. 63

Nevertheless, modeling a polycrystalline body remains a challenging task. Even in the 64

single crystal there exist 192 transformation systems (see Ball and James [6]; Anand and 65

Gurtin [2]; Yin and Weng [54]) and accordingly the number of active transformation 66

systems can be immense. Thus, a macroscopic approach within the context of plasticity 67

theories seems also attractive. Moreover, the macroscopic approach offers several 68

computational advantages since, as it is noted by Thamburaja [52], the numerical 69

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implementation of macroscopic models is easier than that of crystal models, while the 70

numerical simulations involving macroscopic models are computationally more efficient. 71

An interesting approach within the context of macroscopic theories of plasticity is the 72

one suggested by Lubliner and Auricchio [22] - see also the related work by Panoskaltsis 73

[32]; Panoskaltsis et al. [33 - 39] who developed a three-dimensional thermomechanical 74

constitutive model, based on non-isothermal generalized plasticity theory (Lubliner [21]). 75

Generalized plasticity is a general theory of rate-independent inelastic behavior which is 76

physically motivated by loading-unloading irreversibility and is mathematically founded 77

on set theory and topology. This general mathematical foundation provides the theory 78

- -connected elastic domains,

79

which is exactly the challenge in modeling SMAs. 80

The basic objective of this work is to revisit the previous work by Lubliner and Auricchio 81

[22] and Panoskaltsis et al. [36] - see also [33, 34] and to provide a general 82

thermomechanical framework, which in turn may constitute a basis for the derivation of 83

constitutive models for SMAs. Further to the aforementioned endeavors, the present 84

approach establishes the theory in a covariant setting and utilizes the modern invariance 85

(symmetry) principles for the derivation of the thermomechanical state equations. On the 86

computational side, novel aspects include: (1) The derivation of a (local) time integration 87

algorithm within the context of an isothermal operator split and (2) the numerical 88

simulation of non-conventional patterns of material response, where phase transformations 89

may be retarded or even inhibited due to self-heating/cooling effects. 90

This paper is organized as follows: In Section 2, we revisit the general multi-surface 91

formulation of non-isothermal generalized plasticity developed in Panoskaltsis et al. [33] 92

(5)

and we extend it in a covariant setting; we also render the theory fully covariant - see, e.g., 93

Marsden and Hughes [23, pp. 202-203]- upon studying the invariance properties of the 94

local balance of energy under general spatial transformations. In Section 3, as an 95

application we present a material model; this model constitutes a straight forward extension 96

to the non-isothermal regime of a model which has been recently discussed by the authors 97

in [36]. The extension is based on some basic results underlying the thermomechanical 98

response of an SMA material developed in Raniecki et al. [43], Raniecki and Lexcellent 99

[44] and Muller and Bruhns [29]. Finally, in Section 4 we discuss the computational 100

aspects which are related to the numerical implementation of the model and we present a 101

set of representative numerical examples. 102

103 104

2 Constitutive theory 105

106

2.1 Review of the basic equations

107 108

Similar to our previous work [36], a homogeneous body is considered, undergoing phase 109

transformations and occupying a region in the ambient space S R3, with points X 110

labeled by X X1, 2,X3 . The region is identified by the body material (reference)

111

configuration. A motion of the body within the ambient spaceSis defined accordingly as 112

the time dependent mappingx: 113

: S, t ( , )t

x x x x X (1)

114 4

(6)

which maps the pointsX of the material configuration onto the points x of the spatial 115

(current) configuration. Then the deformation gradient is defined as the tangent map of 116

(1), i.e. 117

, ,

t

T x X

F x

X 118

and the material (Green-St. Venant) strain tensor is defined as: 119

1( ), 2

E C I

120

where C is the right Cauchy-Green deformation tensor defined as C F FT and I is the

121

unit rank-2 tensor. Following [36], we assume that the basic kinematic assumption is based 122

on an additive decomposition of the strain tensor E into elastic Ee and inelastic 123

(transformation induced) ETrparts, i.e. 124

, e Tr

E E E (2)

125

where ETrrepresents inelastic deformation induced by generation, growth and annihilation 126

of the austenitic - martensitic fine structure (see, e.g., [6]) and defines an inelastically 127

deformed (intermediate) configuration andEe represents elastic deformation due to 128

stretching and rotation of the crystal lattice. 129

Since we deal with an internal variable theory, it is assumed that the local 130

thermomechanical state in a body - see, e.g., [21, 22] - is determined uniquely by the 131

couple (G,Q) whereG- belonging to a space G - stands for the vector of the controllable 132

state variables and Q -belonging to a space Q - stands for the vector of the internal 133

variables. According to the ideas presented in the review paper of Naghdi [30] the present 134

work is based on a referential (material) approach within a strain-space formulation. 135

(7)

Accordingly, G may be identified by the couple (E,T), where T is the (absolute) 136

temperature. In view of the additive decomposition (2), the internal variable vector may 137

be assumed to be composed by the transformation strain tensor ETr and an additional 138

internal variable vector . 139

The mathematical foundations of generalized plasticity - see Lubliner [21] - rely 140

crucially on a shift of emphasis from the yield surface concept to that of the elastic range. 141

This is defined at any material state as the region in the strain-temperature space 142

comprising the values of G's that can be attained elastically - i.e. with no change in the 143

internal variables (ETr, )Z - from the current strain-temperature point. The boundary of 144

this set may be defined as a loading surface (see further [21]). In turn a material state may 145

be defined as elastic if it is an interior point of its elastic range and inelastic if it is a 146

boundary point of its elastic range. It should be added that the notion of process is 147

introduced implicitly here. In a recent paper Panoskaltsis et al. [34] - see also [33] - argued 148

that for a material undergoing phase transformations the loading surface may be assumed 149

to be defined by a set ofnsmooth surfaces which are defined by expressions of the form 150

( ,T, , ) 0, =1,2,..., .

F E ETr n

151

Each of these surfaces is associated with a particular transformation mechanism - denoted 152

here symbolically by - which may be active at the current state. It is further assumed 153

that each equation F ( ,T,E ETr, ) 0 defines independent (non-redundant) active 154

surfaces at the current value of (E,T) and that the elastic range is a convex set. Then, on 155

the basis of the defining property of an inelastic state and the irreversibility of an inelastic 156

process from such a state it can be shown (see [21]; see also [33, 34] for the case of SMAs) 157

that the rate equations underlying the evolution of the internal variables may be stated as 158

(8)

=1

=1

( ,T, , ) ( ,T, , ) , ( ,T, , ) ( ,T, , ) n

n

H L

H L

Tr Tr Tr

Tr Tr

E E E Z L E E Z

Z E E Z M E E Z

(3) 159

where <·> stands for the Macaulay bracket which is defined as 160

if 0 ,

0 if 0

x x

x

x

161

and the H 's stand for scalar functions which enforce the defining property of an inelastic 162

state. Accordingly, the values of H 'smust be positive at any inelastic state and zero at 163

any elastic one. Finally, L and M represent non-vanishing functions, which are 164

associated with the properties of the phase transformation connected with the part of the 165

loading surface defined by F 0, while the L 's stand for the non-isothermal loading 166

rates which are defined to be 167

: T.

T

F F

L E

E

168

From Eqs. (3), one can deduce directly the loading-unloading criteria for the proposed 169

formulation, which may be systematically formulated as [34], in terms of the sets 170

adm

J { {1,2,..., }/n H >0} 171

and 172

act adm

J { J /L >0}, 173

as follows 174

(9)

adm

adm act

adm

If J : elastic state.

If J and J :

i. If 0 for all J : elastic unloading, ii. If 0

L L

<

adm

adm act

for at least one J : neutral loading,

If J and J : inelastic loading.

(4) 175

An equivalent assessment of the governing equations in the spatial configuration can be 176

done on the basis of a push-forward operation (see, e.g., Marsden and Hughes [23, pp.67-177

68]; Stumpf and Hoppe [51]; Holzapfel [14, pp. 82-84]) to the basic equations. For 178

instance, by performing a push-forward operation onto Eq. (2) the latter can be written in 179

the form 180

, e Tr

e e e

181

where e is the spatial (Almansi) strain tensor, defined as the push-forward of E, that is 182

1, T

e F EF and ,ee eTrare the corresponding elastic and transformation induced parts. In 183

a similar manner the rate equations for the evolution of the internal variables in the spatial 184

configuration read 185

=1

=1

L ( ,T, , , ) ( ,T, , , ) ,

L ( ,T, , , ) ( ,T, , , ) ,

n

n

h l

h l

V Tr Tr Tr

V Tr Tr

e e e z F l e e z F

z e e z F m e e z F

(5) 186

where zstands for the push-forward of the internal variable vector, and L ( )V stands for 187

the Lie derivative (see further [23, pp.93-104]; [51]; [14, pp. 106-108]), defined as the 188

convected derivative relative to the spatial configuration. Finally, the h 's stand for the 189

expression of the (scalar invariant) functions H in terms of the spatial variables 190

( ,T,e eTr, )z and the deformation gradient F, l and m stand for the push-forward of the 191

(10)

functions L and M respectively and the l 's stand for the (scalar invariant) loading 192

rates which in the spatial configuration are given as 193

:L T.

T

f f

l Ve

e 194

In this equation f is the expression for the loading surface associated with the index in 195

terms of the spatial variables. The (spatial) loading-unloading criteria follow naturally from 196

Eqs. (5) as 197

adm

adm act

adm

If j : elastic state.

If j and j :

i. If 0 for all j : elastic unloading, ii. If

l <

adm

adm act

0 for at least one j : neutral loading,

If j and j : inelastic loading.

l

198

where the sets jadm and jact are now defined in terms of the spatial variables as 199

adm

j { {1,2,...,n}/h >0} and jact { J /adm l >0}. 200

201 202

2.2 Covariant constitutive theory

203 204

In the classical literature of the thermomechanical modeling of SMAs it is common to use 205

approaches which are based on the second law of thermodynamics for the derivation of the 206

thermomechanical state equations. An alternative formulation may be established on the 207

basis of an invariance (symmetry) principle (see, e.g., Marsden and Hughes [23, pp. 154-208

176, 199-204, 275-288]; Yavari et al. [53]; see also the philosophical reflections given in 209

Earman [10] and the recent account by Ganghoffer [12]). Such an approach is based on the 210

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exploitation of the invariance properties of a quantity underlying the response of a 211

dynamical system under the action of some group of transformations. For instance, 212

Marsden and Hughes in [23, pp. 199-204], derived the classical stress-deformation and 213

entropy-temperature relations for an elastic material by postulating the invariance of the 214

local form of the referential energy balance equation under the superposition of a group of 215

spatial transformations. 216

The basic objective of this Section is to revisit the approach of Marsden and Hughes in 217

[23] and introduce it within a shape memory alloy behavior setting. In particular, the 218

derivation of both the stress tensor and the specific entropy from the Helmholtz free energy 219

is demonstrated, when the local form of the material balance of energy equation is

220

invariantunder superposition of a special group of transformations. This group consists of

221

arbitrary spatial diffeomorphisms, that is transformations of the ambient spacewhich may

222

change the spatial strain tensor( ).e It is noted that in order to change the local

223

thermomechanical state such a group of transformations is not enough since this will 224

change the mechanical state, but not the thermomechanical one. Therefore, in addition we 225

need also a transformation of the temperature, that is a diffeomorphism of R+(see the

226

footnote in p. 202 in Marsden and Hughes [23]). The simplest case for such a 227

diffeomorphism is atemperature rescaling. 228

It is noted that, unlike the original approach by Marsden and Hughes [23] where the 229

ambient space is considered to be a Riemannian manifold, within the present approach this 230

space is the (rigid) Euclidean space. In this case the basic axioms of Marsden and Hughes 231

[23, pp. 202-203], for the material which obeys the rate Eqs. (3) (or equivalently Eqs. (5)) 232

in the course of phase transformations, can be stated as follows: 233

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Axiom 1:At the material pointXand a given thermomechanical process Gthere exists a 234

scalar function E of the state variables (E E( ,T,E ETr, ))Z such as the energy balance 235

equation holds, that is 236

: ,

refE DIVH S E refR

237

where ref is the mass density in the material configuration, S is the second Piola-Kirchhoff 238

stress tensor,His the heat flux vector andRis the heat supply per unit mass. By introducing 239

the Helmholtz free energy function , which is obtained by the usual Legendre 240

transformation E NT, where Nis the specific entropy (N N( ,T,E ETr, ))Z , the 241

local form of the energy balance can be written in the form 242

( T T) : ,

ref N N DIVH S E refR (6)

243

where DIV( ) stands for the divergence operator in the material description. 244

Axiom 2: We denote by g and q the spaces of the control variables in the spatial

245

configuration - that is the spaces G andQ, configuration - and by 246

the set of the C scalar fields in the spatial configuration, and we assume the existenceS 247

of a map : ( , , ,S g q R ) such that for any diffeomorphism ( , ):( ,S R ) ( ,S R ), 248

( , ,T,x e e zTr, )= ( , , T, Tr( , T), ( , T), ). 249

Axiom 3:For curves t:S S and t( )x R , assume thatxt t xt, T = Tt t t satisfy

250

the balance of energy, that is 251

( T T ) : .

ref N N DIVH S E ref R (7) 252

(13)

where it has been further assumed that ref, , N and theF 's are transformed as scalars, 253

the heat flux vector is transformed as Ht t t* t 254

entropy production Rt TtNt ,is transformed asRt TtNt t(Rt TtNt). 255

In this case the internal variables etr and z, under the application of the spatial 256

diffeomorphism and the temperature rescaling,do not follow their own mode of evolution

257

since are always related to the spatial strain tensor eand the temperature Tby Eqs. (5).

258

Moreover, and more importantly it is noted that for both xtand xt t xt, the balance of 259

energy equation is written at the same material point X. Accordingly, the transformed 260

values of the strain and temperature rates in the primed system- see also [53] - will be given 261

as 262

,

T T T .

T

t

t

t t t t

t t t

E E E

(8) 263

The invariance properties of the balance of energy equation are exploited as in Marsden 264

and Hughes [23] by evaluating Eq. (7) at time t0, when t to (identity),

t t

t 0

w and

265

1, t t0

t t

u t

0

whereuis the velocity of at t0. 266

The time derivative of the transformed Helmholtz free energy in this case reads 267

0 0 0 0

: : T : : .

T

t t0 t t t t Tr t t t t Tr

E E Z

E E Z (9)

268

By means of Eqs. (8) the time derivatives

0

t t E and

0

T

t t are found to be 269

(14)

0

0

,

T T+ T. t t

t t u

E E w E

(10) 270

The time derivatives

0 t t Tr E and 0 t t

Z are evaluated by means of the transformation 271

formula of the loading rates, which in view of Eq. (10) reads 272

0 ( : T T ) 0=

= : : ( ) T+ ( T).

T T

t t t t

F F

L

F F F F

u

E E

E w E

E E

273

Accordingly, the rate

0

t t Tr

E is evaluated as 274 0 0 n =1 n n =1 =1 n =1

( ,T , , )

( ,T, , ) ( ,T, , )[ : ( ) ( T)]

T

( ,T, , )[ : ( ) ( T)].

T t t

t t

H L

F F

H L H u

F F

H u

Tr Tr

Tr Tr

Tr Tr

E L E E Z

L E E Z L E E Z w E

E

E L E E Z w E

E 275

In a similar manner for

0

t t

Z we derive 276

0

n

=1

( , T, , )[( : ( ) ( T)], T t t

F F

H Tr u

Z Z M E E Z w E

E 277

so that the transformed Helmholtz free energy reads 278

n

=1 n

=1

: ( ) : ( T)

T

: ( ,T, , )[ : ( ) ( T)]+

T

: ( ,T, , )[ : ( ) ( T)].

T

t t u

F F H u F F H u 0 Tr Tr Tr w E E

L E E Z w E

E E

M E E Z w E

(15)

' : : : ( ), t t0

S E S E S w E (12)

281

In light of Eqs. (9), (11) and (12) the transformed balance of energy equation (at t t0) 282

can be written as 283

n

=1 n

=1

{ : ( ) : ( T)

T

: ( ,T, , )[ : ( ) ( T)]+

T

: ( ,T, , )[ : ( ) ( T)]}+ T+ ( T)+

T

: : ( ) ( T ).

ref ref ref ref u F F H u F F

H u N N u DIV

R N Tr Tr Tr w E E

L E E Z w E

E E

M E E Z w E H

Z E

S E S w E 284

(13) 285

By subtracting Eq. (6) from Eq. (13) and by involving the transformation formulae for the 286

heat flux vector and the apparent heat supply (recall Axiom 3) we can derive the following 287 identity 288 n n =1 =1 n n =1 =1

: ( ) ( T)+

T

[ : ( ,T, , ) : ( ,T, , ) ]( )

[ : ( ,T, , ) : ( ,T, , ) ]( T)+

T T

( T) : ( ) 0

ref ref ref ref ref u F F H H F F

H H u

N u Tr Tr Tr Tr Tr Tr w E E

L E E Z M E E Z w E

E E Z E

L E E Z M E E Z

E Z

S w E ,

289 or equivalently 290 n =1 n =1 n =1 n =1

{ [ : ( ,T, , )

: ( ,T, , ) ] }: ( )}

[ : ( ,T, , )

T T

: ( ,T, , ) ]( T)=0.

T ref ref F H F H F H F

H N u

Tr Tr Tr Tr Tr Tr

L E E Z

E E E

M E E Z S w E

Z E

L E E Z

E

M E E Z

(16)

from which by noting that w anducan be arbitrarily specified, we can derive 292

n

=1 n

=1

n

=1 n

=1

[ : ( ,T, , )

: ( ,T, , ) ],

[ : ( ,T, , )

T T

: ( ,T, , ) ].

T ref

F H

F H

F

N H

F H

Tr Tr

Tr

Tr Tr

Tr

S L E E Z

E E E

M E E Z

Z E

L E E Z

E

M E E Z

Z

(15) 293

i.e., unlike the classical elastic case discussed in Marsden and Hughes [23, pp. 202-203], 294

for the material undergoing phase transformations, the covariance of the local form of the 295

energy balance, does not yield the standard thermomechanical relations

296

ref , N T, S

E (16)

297

unless a further assumption is made, namely that there exists a spatial diffeomorphism

298

( , ) which results in an unloading process from an inelastic state (i.e. a process with 299

adm

J and Jact ) which is quasi-reversible (see Fosdick and Serrinin [11]). This 300

means that in such a process the inelastic (transformation) workWin, defined as 301

n

=1 n

=1 n

=1 n

=1

[ : ( ,T, , )

: ( ,T, , ) ]: ( )

[ : ( ,T, , )

T

: ( ,T, , ) ](uT)

T ref

F

W H

F H

F H

F H

in Tr

Tr

Tr

Tr Tr

Tr

L E E Z

E E

M E E Z w E

Z E

L E E Z

E

M E E Z

Z 302

(17)

that is, the work performed by the internal variables during the action of ( , )vanishes.

303

Then, in this case the standard thermomechanical relations follow directly from Eq. (14) 304

for Win 0. It is noted that the expressions 305

n

=1 n

=1

n

=1 n

=1

[ : ( ,T, , )

: ( ,T, , ) ]: ( ),

[ : ( ,T, , )

T

: ( ,T, , ) ]( T),

T mech ref

thermal

F

W H

F H

F

W H

F

H u

in Tr

Tr

Tr t

in Tr

Tr

Tr t

L E E Z

E E

M E E Z w E

Z E

L E E Z

E

M E E Z

Z 306

stand for the mechanical and thermal work produced by the superposed spatial 307

diffeomorphism and the temperature rescaling, respectively. 308

Note that the result derived herein is in absolute accordance with the one derived on the 309

basis of the second law of thermodynamics by Lubliner and Auricchio in [22]. More 310

information on this point is provided by Panoskaltsis in [32]. 311

312

REMARK 1 The concept of the covariant energy balance has been also exploited by 313

Panoskaltsis in [32], but unlike the present case where we consider the covariance of the 314

referential balance of energy equation, Panoskaltsis considers covariance of the spatial

315

energy balance. Contrary to the present formulation, Panoskaltsis in [32] derives the

stress-316

strain relations in a spatial setting in terms of the Cauchy stress tensor and the Almansi 317

straine. More specifically, Panoskaltsis in [32] derives the standard stress-strain relations 318

, e 319

(18)

where and stand for the mass density and the Helmholtz free energy in the spatial 320

configuration. 321

322

REMARK 2Eqs. (15) constitute thecovariance conditionsfor the energy balance equation 323

(6) i.e. the necessary conditions, so that this equation is invariantunder the superposition 324

of arbitrary diffeomorphisms acting on the Euclidean space, DiffS, which include also a 325

temperature rescaling R . Moreover, it can be proved that the conditions (15) are also 326

sufficient. This meansthat if we do not consider the assumption related to the existence of

327

quasi-reversible processes, the invariance (symmetry) group of the balance of energy

328

equation is 329

n

=1 n

=1

n

=1 n

=1

{( , ) /

[ : ( ,T, , )

: ( ,T, , ) ] and

[ : ( ,T, , )

T T

: ( ,T, , ) ]}.

T ref

G DiffS R

F H

F H

F

N H

F H

Tr Tr

Tr

Tr Tr

Tr

S L E E Z

E E E

M E E Z

Z E

L E E Z

E

M E E Z

Z 330

In this case it can be proved - see Panoskaltsis and Soldatos in [35] - that the material

331

response is elastic (non-dissipative). Upon theconsideration of elastic-inelastic response,

332

the covariance group G is restricted tothe group

333

ref

{( , ) / , , 0 and 0},

T mech hermal

G DiffS R S N Win Win

E 334

In this sense, the present approach and its basic conclusions are consistent with the modern 335

approach to symmetries in physics, as emphasized for instance by Earman [10]: 336

(19)

The symmetries are in the laws of the phenomena, not in the phenomena

337

themselves

338

The phenomena break the symmetries of laws.

339 340

REMARK 3An alternative approach to the concept of invariance could be provided by noting 341

the natural connection which exists between conservation laws and the symmetries of the 342

(dynamical) system in question. In particular, if the Euler-Lagrange equations of the system 343

are satisfied and the Lagrangian is invariant under the action of some group of 344

345

corresponding conserved quantities. Such an approach has been favored, among others by 346

Rahuadjet al.[41, 42] and Romero [46] (see also [12, 53]). 347

348 349

2.3 The temperature evolution equation

350 351

As a final step we derive a general equation for the temperature evolution which occurs in 352

the course of phase transformations. This is done on the basis of the energy balance 353

equation (recall Eq. (6)). In this case, the time derivative of the Helmholtz free energy 354

yields 355

ref( : : Tr + TT)+ ref NT+ ref NT+DIV ref R+ : , Tr

E E Z H S E

E E Z (17)

356

which in turn upon substitution of the thermomechanical state Eqs. (16) yields 357

ref( ETr :ETr Z : )+Z refNT+DIVH refR. (18) 358

(20)

The time derivative of the entropy density is determined by the second of Eqs. (16) as 359

2 2 2 2

2

: : : T,

T T T T

N Tr

Tr

E E Z

E E Z (19)

360

which upon defining the specific heatcat constant deformation and internal variables as 361

2

2 T, T

c (20)

362

and upon substitution of Eqs. (19) and (20), reads 363

2 2 2

ref 1

T ( : : )+( : : : )T+(R ),

T T T

c Tr Tr DIV

Tr Tr

E Z E E Z H

E Z E E Z

364

which constitutes the temperature evolution equation in a non-isothermal process. This 365

equation upon defining theelastic contribution to heatingas 366

2 2 2

2 2

T( : : ) T[ : ( )]

T T T

T[ : ( )] T : ,

( ) T T

Q Tr

e Tr

Tr

Tr e

Tr e

E E E E

E E E

E E E

E E E

367

and theinelastic (transformation)one as 368

2

( : : )+T : ,

T

QTr Tr

Tr

E Z Z

E Z Z

369

takes the following remarkably simple form (see Rosakis et al. in [47]) 370

ref 1

T +(R ).

c Qe QTr DIVH (21)

371

which has the obvious advantage of decoupling the elastic and inelastic contributions to 372

material heating and is well suited for computational use. 373

374

REMARK 4Upon defining, the inelastic dissipationDas 375

(21)

( : : ),

D Tr

Tr

E Z

E Z

376

and the elastic-inelastic structural heatingHas 377

T ( : ),

T

H S E D

378

the temperature evolution equation takes the alternative form 379

ref 1

T ( ) ( R).

c D H DIVH (22)

380

In this equationH is associated to the non-dissipative (latent) elastic-plastic changes due 381

to thermal phenomena. This term as it will be clear in the foregoing see Section 4.3.1 -382

plays a very important role in thermo-mechanically coupled problems in SMAs. 383

384

REMARK 5 An equivalent equation for the temperature evolution equation can be also 385

derived in the spatial configuration by either working in a similar manner in terms of the 386

spatial variables or by means of a push-forward operation to Eq. (22). The resulting 387

expression in is 388

1

T ( ) ( ),

c d h divh r (23)

389

where 390

( : : ), T ( : ).

T

L h L d

d v Tr v

Tr

e z e

e z (23)

391

are the expressions for the plastic dissipationd and the structural heating structural h

392

nfiguration. Further, in Eq. (23), ( ,T,e eTr, )z is the 393

Helmholtz free energy in terms of the spatial variables, while , div( ),handr,stand 394

for the mass density, the divergence operator, the heat flux vector and the heat supply 395

(22)

in the spatial configuration. Such a form of the temperature evolution equation has been 396

favored by Simo and Miehe [48] and may be implemented in cases where a spatial 397

formulation of a material model is simpler than the material one (see e.g. Panoskaltsis 398

et al. in [39]). 399

400 401

3 A model problem 402

403

In the preceding Sections, the proposed formulation is presented largely in an abstract 404

manner by leaving the number and the nature of the internal variables, underlying the phase 405

transformations, unspecified. To clarify the application of generalized plasticity within a 406

thermomechanical modelling setting for phase transformations, a material model is 407

presented in this Section. 408

Without loss of generality, we confine our attention to phase transformations between the 409

austenite and a single (favorably) oriented martensite variant. The internal variable vector 410

Z, as it is common with this class of the models for SMAs (see, e.g., [43, 8, 22, 29, 33, 38, 411

52]) is assumed to be composed by a single scalar internal variable - say -the phase 412

fraction of martensite within the continuum. The (forward)austenite to martensite 413

transformation will be denoted symbolically as the (M) transformation, while the (reverse) 414

martensite to austenite transformation will be denoted as the (A) one. 415

In view of the additive decomposition of the strain tensor (2), the Helmholtz free energy 416

can be additively decomposed in a part e which corresponds to elastic and thermal 417

expansion behavior and an inelastic (due to phase transformations) part , as follows 418

(23)

= ( ( ),T, )+ ( ,T).

ref ref e E ETr ref Tr (24) 419

It is emphasized that this is not the conventional decomposition of the free energy function 420

performed within the classical inelastic theories (e.g. plasticity, viscoelasticity, 421

viscoplasticity), since the elastic part edepends on the internal variable . In this sense 422

the decomposition (24) resembles the decompositions employed within the 423

thermomechanical treatment of damage (see [33]). The elastic part of the Helmholtz free 424

energy, under the valid assumption that elastic thermal effects are negligible in comparison 425

to transformation induced thermal effects, may be assumed to be given by the expression 426

of the stored energy function of a St. Venant-Kirchhoff material (see, e.g., [23, pp. 223, 427

225]; [13, pp. 250-251]), that is 428

2 2

( )

( ( ),T) ( ( )) { [ ( )} ( ) [( ( )] ,

2 tr tr

e E ETr e E ETr E ETr E ETr

429

where and are the Lame parameters ( 0, >0), which are defined in terms of the 430

standard elastic constantsE, as 431

, .

(1 )(1 2 ) 2(1 )

E E

432

These are assumed to be dependent on the martensite fraction of the SMA, according to 433

the standard law of mixtures 434

( ) A ( M A), ( ) A ( M A), 435

where A, A are the Lame parameters when the material is fully austenite, and M, M 436

are these when the material is fully martensite. For the transformation part of the Helmholtz 437

free energy, by following Raniecki et al. in [42] - see also [44], [29] - we consider an 438

expression of the form 439

(24)

(1 ) A + M AM ,

Tr Chem Chem

440

where AChem and M

Chem are the chemical energies of the austenite and martensite phases 441

respectively, and AM results from the interaction between these phases. For these 442

energies we assume the following expressions (see also [29]) 443

0 0 0

0

0 0 0

0

0 0

T ( T ) [(T T ) Tln ]

T

T ( T ) [(T T ) Tln ],

T Z(1 Z)( Ts ),

u s c

u s c

u

A A A

Chem

M M M

Chem

AM 444

where T0 is the reference temperature and u0A, 0

s A , 0

u M, 0

s M, 0

u and s0 are the thermal 445

parameters of the model. 446

Then in light of the first of Eqs. (16) the second Piola-Kirchhoff stress tensor is found to 447

be 448

( ) 2 ( ),

tr Tr Tr

S E E 1 E E (25)

449

where1is the unit rank-2 tensor, and the dependence of the involved quantities on has 450

been dropped for convenience. 451

As in Panoskaltsis et al. [36], the loading surfaces are assumed to be given in the stress-452

space as a two parameter family of von-Mises type surfaces, that is 453

F( ,T) =S DEVS CT R= 0, (26)

454

where stands for the Euclidean norm, DEV( )stands for the deviatoric part of the stress 455

tensor in the reference configuration and C R are the family parameters. On substituting 456

from Eq. (25) into equation (26) the equivalent expression for the loading surfaces in strain-457

space may be derived as 458

(25)

( , ,T) =2 ( ) T = 0.

F E ETr DEV E ETr C R

459

For the rate equation for the evolution of the transformation strain we assume a normality 460

rule in the strain-space which is given as 461

3

2 = ,

2

F

Tr L

E

E (27)

462

where L is a material constant, which is defined as the maximum inelastic strain (see, e.g., 463

[7,22]), which is attained in the case of one-dimensional unloading in simple tension when 464

the material is fully martensite. 465

In order to close the model, as in our previous work in [36] - see also Auricchio et al. [5] 466

-, we consider a linear expression for the evolution of , which within the present 467

formulation can be expressed as 468

1

,

2 (1 ) 2

F F M T F F

L L

F F T M F F F F

Mf Ms d Af As

M A

Mf Ms d Mf L Af As Af L

469

(28) 470

where 471

2

( , ,T) =2 ( ) (T ),

3

FMf E ETr DEV E ETr CM Mf

472

2

( , ,T) =2 ( ) (T ),

3

FMs E ETr DEV E ETr CM Ms

473

2

( , ,T) =2 ( ) (T ),

3

FAs E ETr DEV E ETr CA As

474

2

( , ,T) =2 ( ) (T ),

3

FAf E ETr DEV E ETr CA Af

475

: T,

T

F F

LM LA E

E 476

(26)

in whichCM, CA, M Mf, s, Asand Af are (standard) material parameters which can be 477

determined by means of the well-known (see, e.g., [22]) critical stress-temperature diagram 478

for the SMAs transformations; the geometrical interpretation of Eqs. 479

0, 0, 0

FMf FMs FAs and FAf 0 can be determined also by means of the 480

aforementioned critical diagram (see also [22]). Finally, Md is a critical temperature value 481

above which the austenite is stable and the forward ( )M transformation cannot be 482

activated (see McKelvey and Ritchie in [24]). 483

The formulation is supplemented by a constitutive law for the heat flux vector (see Eqs. 484

(21), (22)) w [48, 29])

485

as per 486

T,

kGRAD

H (29)

487

where GRAD( ) is the gradient operator and k is the material conductivity. 488

489 490 491 492 493 494 495

4 Numerical simulations 496

497

In this Section we implement numerically the material model introduced in Section 3andwe 498

(27)

simulating several patterns of the extremely complex behavior of SMAs under non-500

isothermal conditions. 501

The numerical implementation of the proposed model can be performed by means of the 502

so-calledisothermal split, suggested in the pioneering work of Simo and Miehe [48]. The 503

basic idea is to solve the governing equations of the coupled thermomechanical problem 504

(equations of motion, constitutive equations, energy balance equation and appropriate 505

boundary conditions) by performinga natural split into a non-linear inelastic problem with

506

frozen thermal variables (step 1), followed by a heat conduction problem at fixed

507

configuration (step 2). These two steps are coupled via the elastic ( )Qe and the inelastic

508

(QTr) contributions to heating. The inelastic problem (step 1), can be pursued by means 509

of a predictor-corrector algorithm. Nevertheless, since the theory of plasticity employed 510

hereinis not a conventional one, the proposed algorithmic scheme - see [33, 38] - differs 511

vastly from the standard return mapping algorithm employed within the context of classical 512

plasticity theories. 513

514

4.1 Basic computational aspects

515 516

The basic point for the numerical implementation of the model relies crucially on realizing 517

that when the deformation gradient Fand the heat fluxHare known at the material point 518

X, the rate equations for the evolution of the internal variables (Eqs. (27) and (28)), the 519

balance of energy equation (21) and the stress-strain relations (25), form,at the local level, 520

a system of four equations in the four unknownsS, T ETr and . Thus the implementation 521

(28)

problem is just reduced to the problem of solving numerically the aforementioned system. 522

The details of the solution procedure follow. 523

Let J [0, ]T be the time interval of interest. It is assumed that at timetn J, the 524

configuration of the body of interest n xn( ), i.e. xn {xn x X Xn( ) / }, along 525

with the state variables 526

n n n n n n {E T ETr , ,H S, }, 527

are known. 528

Assume a time increment t, which drives 529

the time to tn+1, 530

the body configuration to 531

n+1 { n+1 n+1( ) / },

x x x X X

532

where 533

n+1( ) n( ) ( ( )),n x X x X u x X 534

anduis the given incremental displacement field, 535

the temperature to T .n+1 536

Then the algorithmic problem at hand is to update the internal variable vector ETr , and 537

the stress tensor Sto the time step tn+1 in a manner consistent with the (time continuous) 538

Eqs. (27), (28), (21) and (25). Note that the heat flux vector Hn+1is a function of the 539

temperature at time tn+1 s law (29). The solution of this problem is 540

pursued by a two-step algorithm - see further [48]-, which comprises an isothermal elastic-541

inelastic problem followed by a heat conduction problem as follows 542

(29)

Step 1: Isothermal elastic-inelastic problem

544

As a first step we assume an isothermal problem (Tn+1 T ) defined at the configurationn 545

n+1.

x The material strain tensor En+1 is determined by means of the corresponding 546

deformation gradient Fn+1 and the right Cauchy-Green tensor Cn+1 as 547

n+1

n+1 n+1 n+1 n+1 n+1 n+1 n+1 1

( ) , , ( ).

2 T

x

F X F C F F E C I

X 548

The application of the backward-Euler difference scheme for fixed temperature Tn+1 leads 549

to the following problem of evolution 550

n+1 n+1 n n+1 n

n+1 3

2 ( ) = ( )

2

F

Tr Tr L

E E

E (30)

551

n+1 n+1 n+1 n+1

n+1 n+1

n+1 n+1 n+1 n+1 n+1

n+1 n+1 n+1

n+1 n+1 n+1 n+1 n+1

T 1

2 (1 )

T

2 n

F F M

L F

F F M

F F

L F

F F

Mf Ms d

M

M L

Mf Ms d

Af As

A

Af L

Af As

(31) 552

553

where the F's stand for the time discrete expressions of the loading surfaces in terms of 554

the basic variables; for instance FMsn+1 reads 555

n+1 n+1 n+1 n+1 n+1 n+1 n+1 2

( , ,T ) =2 ( ) (T ),

3

FMs E ETr DEV E ETr CM Ms

556

while the L's stand for the isothermal loading rates, e.g., 557

n+1 n+1 n+1

n+1

: .

F

L M

M E

E 558

Note that within the context of the present strain-temperature space formulation the stress 559

tensor Sn+1, does not appear in Eqs. (30) and (31), so that the (isothermal) problem is 560

(30)

reduced to solving these equations for the unknowns n+1 n+1

. Tr

E

This problem can be solved 561

by means of a three-step predictor-corrector algorithm. The computational details 562

underlying the solution of the isothermal problem can be found in our previous work in 563

Panoskaltsis et al. [36] (see Section 4.1). 564

This provides theinitial updateof the internal variables, i.e. 565

n n+1

n n+1 n+1 n

1n n+1

, for T =T .

Tr Tr

E E

566

567

Step 2: Non-isothermal inelastic problem at fixed configuration

568

At this step, the total configuration, mediated herein by Fn+1- or equivalently by En+1 -569

remains fixed, while the solution of the isothermal problem is considered as an initial 570

condition, that is the known data at the beginning of this step are the elements of the set 571

n+1 n+1 n n {E T ETr , }, 572

while the application of the backward Euler scheme yields the following algorithm 573

n+1 n+1 n n+1 n

n+1 3

2 ( ) = ( )

2

F

Tr Tr L

E E

E (32)

574

n+1 n+1 n+1

n+1 n n+1

n+1 n+1 n+1 n+1

n+1 n+1 n+1

n+1 n+1 n+1 n+1 n+1

1

2 (1 )

2

F F

L

F F F

F F

L

F F F

Mf Ms

M

Mf Ms M L

Af As

A

Af As Af L

(33) 575

n+1 n+1 n+1 n+1 n+1 ref

1

(T T ) ,

c Qe QTr DIVH (34)

576

where 577

(31)

2

n+1 n+1 n+1 n n+1 n n+1

T [ :[( ) ( )],

T

Qe E E ETr ETr

E 578

2

n+1 n+1 n n+1 n n+1 n+1 n

n+1 n+1

n+1

( : ( ) ( )+T ( ),

T

QTr Tr Tr

Tr

E E

E 579

the F's stand for the time discrete expressions of the loading surfaces at time strep tn+1

580

and the L's stand now for thenon-isothermal loadingrates, e.g. 581

n+1 n+1

n+1 n+1 n n+1 n+1

n+1 n+1

: ( )+ (T T ).

T

F F

L M M

M E E E

582

The solution of this algorithmic problem yields the values of the basic variables

n+1

n+1

n+1

T ,

Tr

E 583

so that the second Piola-Kirchhoff stress tensor S can be determined by means of the 584

thermomechanical state equation 585

n+1 tr( n+1 Trn+1) 2 ( n+1 Trn+1),

S E E 1 E E (35)

586

Note that Eqs. (31) and (33) can be reduced further, depending on whether the (M) or 587

the (A) transformation is active (see [33, 36] for further details). 588

589

REMARK 6 The proposed isothermal split has the disadvantage of not being

590

unconditionally stable. An alternative numerical treatment relies on the isentropic split

-591

see Armero and Simo [4]; see also Agelet de Saracibar et al. [1] - where unlike the present

592

case, the coupled problem is divided intoan isentropic mechanical phase in which the total

593

entropy is held constant, followed by athermal phase at a fixed configuration, which leads

594

to an unconditionally stable algorithm. In the present case and since we deal with

595

homogeneous problems, where the coupling between the governing equations is relatively

596 4

(32)

small, the stability problem is not an issue; as a result, the isothermal split seems to be

597

adequate.

598 599

4.2 Numerical simulations

600

4.2.1 Simple shear

601

The first problem we study is a standard one within the context of finite inelasticity and 602

is that of simple shear which is defined as 603

1 1 2, 2 2, 3 3,

x X X x X x X

604

where is the shearing parameter. In order to show the computational versatility enjoyed 605

by the model we work with three set of parameters. The first set is that reported in Boyd 606

and Lagoudas [7] for a generic SMA, that is 607

M A

f s f

M A L

s

13,000 MPa, 30,000 MPa, =0.3, M 5 C, M 23 C, A 29 C, A 51 C, C 11.3 MPa/ C, C 4.5 MPa/ C, =0.0635.

E E

o o o o

o o

608

The second set of parameters is that reported in Auricchio et al. [5], for a commercial NiTi, 609

i.e., 610

M A

f s f

M A L

s

24,600 MPa, 31,000 MPa, =0.33, M 250

C 10.50 MPa/

E E

o o o o

o o

611

while the third set of parameters is that given in Speicher et al. [49] for a Ni Ti50.8 49.2 SMA, 612

that is 613

M A

f s f

L

s

38, 200 MPa, 48,500 MPa, =0.42, M 218.5

=0.0475,

E E

o o o o

614 4

(33)

For this SMA, the remaining parameters CM and CA are considered to be equal to those 615

of the alloy discussed by Boyd and Lagoudas in [7]. 616

The parameters of the non-isothermal part of the model are set equalfor allSMAs to those 617

used in Mullerand Bruhns [29], that is: 618

3 2 6 o o

ref t

* * o

0 0 0 0

o

0 0

6.45 10 k/m mm , 8.8 10 1/ K, c 837.36 J/kg K

16800.0 J/kg, 64.50 J/kg K,

4264.5 J/kg, 11.5 J/kg K.

u u u s s s

u s

A M A M

619

Within this simulation we examine the ability of the model in predicting pseudoelastic 620

phenomena under non-isothermal conditions. For this purpose, an adiabatic testis 621

considered. We further assume that due to the dynamic rates resulting in adiabatic response, 622

heat exchanges due to conduction, convection and radiation can be neglected in comparison 623

to the material heating/cooling induced by the inelastic (transformation) contribution to the 624

heating (QTr), a fact which leads to thermomechanical processes that can be considered as 625

homogeneous (see Rosakis et al. [47]). Accordingly, within this simulation the temperature 626

evolution equation reads 627

2

T ( : )+T .

T

c Tr

Tr E E 628

Our purpose in this example is to discuss a complete stress-induced transformation cycle 629

at a temperature T0 60 C A ,f where the SMA material exhibits pseudoelastic response. 630

The stress-deformation curves for this finite shear problem are shown in Figs. 1, 2 and 3 631

while the corresponding temperature-deformation curves are depicted in Fig. 4. Consistent 632

with the adiabatic response of an SMA material at temperature T0 Af (see, e.g., Grabe 633

and Bruhns [13]), the model predicts heat generation during the forward (M) transformation 634

(34)

and heat absorption during the reverse (A) transformation. Moreover, we note that at the 635

end of the forward (M) the model predicts that the temperature increase is the same for the 636

three alloys in question. This response is consistent with the experimentally observed one 637

(see, e.g., Peyroux et al. in [39]), and relies on the fact that, unlike the case of metals, in 638

SMAs the dissipated mechanical work ( : Tr ), Tr

E

E remains very small

639

compared to the latent heat 2 T

T due to phase changes; note that within the context of 640

present problem the latent heat is dominated by the non-isothermal part of the model which 641

has been considered the same for the three alloys. 642

A comparison between the corresponding isothermal and adiabatic responses for the 643

alloy discussed by Auricchio et al. in [5] is illustrated in Figs. 5 and 6. By referring to these 644

results we note that, under adiabatic conditions the forward (M) transformation occurs at 645

higher levels of stress, than the isothermal one. This fact has its origins to the temperature 646

increase which tries to stabilize the austenite and inhibit the transformation. As a result, a 647

higher level of stress is required to induce the forward (M) transformation. Moreover, by 648

referring to Fig. 6 we note that in general temperature changes retard both the forward (M) 649

and the reverse (A) transformations. In the latter case, this result has to be attributed to the 650

temperature decrease during the reverse transformation which now tries to stabilize the 651

martensite phase. 652

653

4.2.2 Uniaxial tension: validation of the model against actual experimental data

654 4

(35)

In order to verify further the ability of the proposed framework for simulating isothermal

655

(quasi-static) and adiabatic (dynamic) responses we consider a uniaxial tension problem.

656

This problem is defined as 657

1 (1 ) ,1 2 (1 ) 2, 3 3

x X x X x X ,

658

where 1+ are the straining parameters (principal stretches) along the axial and 659

660

the corresponding principal normal strains 11 and 22. Our purpose here is to compare the

661

predictions of the proposed model with the experimental results reported by Auricchio et

662

al. in [5]; see also Fig. 3 in [26]. The basic (isothermal) parameters are those used in the

663

simple shear problem, while the remaining parameters underlying the dynamic response

664

are set equal to

665

* *

L

0 0

0.038, 5800.0 J/kg, 64.50 J/kgK, 265 J/kg, 10.0 J/kgK.

u s

u s

666

As in the simple shear problem the material is subjected to a stress (loading-unloading)

667

cycle a temperature o

0 f

T 295 K( A ). The results are shown in Fig. 7. By referring to

668

these results, we observe that the proposed model can capture adequately both the change

669

in slope and the change of the hysteresis loop both in the case of quasi-static ( = 1000 sec,

670

where = is time of a loading-unloading cycle) and dynamic loading ( = 1 sec - see also

671

Fig. 3 in [26]). We also observe that the simple linear expression for the evolution of the

672

material martensite fraction used herein cannot capture precisely the shape of the

673

transformation branches. Nevertheless, this does not consist a serious drawback, since this

674

expression - recall section 2.1 - can be replaced adequately by a more sophisticated (e.g.

675 4

(36)

polynomial, exponential, hyperbolic) one; further details on the selection of the evolution

676

function for a two-shape memory alloy can be found in [39].

677

As a further illustration, in Figs 8 and 9, we show the effect of the (basic) thermal

678

parameters of the model in the stress-deformation curve. It is observed that, while the effect

679

of u0 is practically negligible, the thermal parameter u* plays an eminent role in the 680

predicted behavior upon controlling the slope of the stress-deformation curve.

681 682

4.2.3 Shape memory effect

683

As a next simulation we will evaluate the ability of the model in predicting the shape 684

memory effect. For this purpose, we revisit the uniaxial tension test. For this problem the 685

basic material parameters are set equal to those given in [5], while the material stiffness is 686

assumed to be constant (E EA 30,000Mpa). 687

The isothermal stress-strain curves for three different material temperatures 688

0 f

(T 55 C=A ,o

f 0 s

A >T 40 C A ,o >

s 0

A >T 28 Co

) are shown in Fig. 10. By referring 689

to this figure for T =A ,0 f the ability of the model in predicting pseudoelastic phenomena 690

under isothermal conditions is verified; note that since the temperature has been set exactly 691

equal to A ,f the reverse transformation ends at zero stress. The isothermal tests for 692

f 0 s

A >T >A and As >T0 are conducted in order to show the ability of the model in 693

predicting the shape memory effect. In the first of them, upon loading the (M) 694

transformation is activated. Upon unloading and since the temperature is less than the 695

temperature required for the complete reverse transformation at zero stress, the two phases 696

coexist and permanent deformations appear. However, as it will be clear in the subsequent, 697

(37)

the temperature has been set at a value less than the austenite start temperature at zero 699

stress, at the end of the stress cycle the material is completely in the martensite phase. This 700

results in the appearance of large permanent deformations, which are manifested by the 701

maximum inelastic strain L. Nevertheless, like the previous test this deformation may be 702

eliminated upon heating. 703

For the material heating problem, we assume thermal boundary conditions corresponding 704

toconvective heat exchange between the specimen and the surrounding mediumon the free

705

faces (with areaA) of the specimen; each face is assumed to have a unit area. In this case 706

the normal heat flux Hu, e.g., [48]) as

707

0 (T T ),

hA

u

H 708

where h stands for the convection coefficient which is chosen to be 709

3 o

20 . 10 N/mm K

h , and T is the surrounding medium temperature. By assuming 710

that the size of the material tested is small, the contribution to the material heating due to 711

the heat conduction can be neglected; a similar assumption has been also made by 712

Auricchio et al. [5]. As a result, in the absence of phase transformations the material 713

behaves as a rigid body hear convector; the corresponding temperature evolution 714

equation reads 715

ref 1

T .

c QTr Hu

716

The results of this test are illustrated in Fig. 11 where the strain along X1 axis is plotted 717

versus the surrounding medium temperature. 718

719

4.2.4 Thermally induced martensitic transformations at zero stress

720 4

Figure

Figure 1: Finite Shear: Shear stressS12 vs. shear strain.
Figure 3: Finite Shear: Normal stressS22vs. shear strain.
Figure 4: Finite Shear: Temperature T vs. shear strain.
Figure 6: Finite Shear: Comparison of isothermal and adiabatic responses.Martensite fractionvs
+7

References

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