Shape memory behaviour: modelling within continuum thermomechanics

Shape memory behaviour:modelling within continuum thermomechanics

Dirk Helm *, Peter Haupt

Institute of Mechanics, University of Kassel, M€oonchebergstr. 7, D-34109 Kassel, Germany

Received 9 May 2002; received in revised form 9 October 2002

Abstract

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is

proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory

effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as

the transition range between pseudoelasticity and pseudoplasticity.

The material model is based on a free energy function and evolution equations for internal variables. By means of the

free energy function, the energy storage during thermomechanical processes is described. Evolution equations for in-

ternal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative

material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced

into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies

the Clausius–Duhem inequality.

Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress

processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for

uniaxial processes and for non-proportional loadings as well.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Shape memory alloys; Thermomechanical modelling; Phase transitions; Thermoviscoplasticity

1. Introduction

Shape memory alloys exhibit a thermomechanical behaviour which cannot be observed in other materials

(see e.g. Funakubo, 1987; Otsuka and Wayman, 1998): pseudoelasticity, pseudoplasticity as well as one-and two-way shape memory effects. These effects are based on two basic deformation mechanisms oc-

curring in the microstructure, namely the stress- and temperature-induced martensitic (diffusionless) phase

transitions and the orientation of the martensite twins. Both deformation mechanisms are connected

with temperature changes and dissipation. However, the thermomechanical coupling effects during the

International Journal of Solids and Structures 40 (2003) 827–849

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* Corresponding author.

E-mail addresses: [email protected] (D. Helm), [email protected] (P. Haupt).

URLs: http://www.ifm.maschinenbau.uni-kassel.de/�helm, http://www.ifm.maschinenbau.uni-kassel.de/�haupt.

0020-7683/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0020-7683 (02 )00621-2

mail to: [email protected]

http://www.ifm.maschinenbau.uni-kassel.de/~helm, http://www.ifm.maschinenbau.uni-kassel.de/~haupt



martensitic phase transitions are much more pronounced in comparison to the case of the orientation of the

martensite twins (see e.g. Shaw and Kyriakides, 1995; Helm and Haupt, 2001).

Due to the interesting material properties of shape memory alloys, the development of exceptional new

products is possible. To support the engineering applications, a number of material models were developed.Some of these (see e.g. Bertram, 1982; Graesser and Cozzarelli, 1994) are not formulated in the context of

thermomechanics, although the material behaviour is strongly temperature-dependent. Other theories are

confined to the one-dimensional material behaviour (see e.g. Falk, 1983; Liang and Rogers, 1990), or they

are limited to represent pseudoelasticity only (see e.g. Raniecki et al., 1992; Auricchio and Taylor, 1997).

Mostly, rate-dependent effects are disregarded (see e.g. Boyd and Lagoudas, 1996).

In the present paper, a thermomechanically consistent material model is proposed to represent the three-

dimensional material behaviour of shape memory alloys. This includes the one-way shape memory effect,

the two-way shape memory effect due to external loads as well as the pseudoelastic and the pseudoplasticmaterial behaviour. The transition range between pseudoelasticity and pseudoplasticity is also represented.

Furthermore, the constitutive model is able to depict viscous characteristics of the material. As it is outlined

in Section 2, the model is based on a free energy function depending on the temperature and internal

variables. In order to model the energy storage during the martensitic phase transition, the free energy

function is formulated according to a mixture theory. For the description of the history-dependence,

evolution equations for the internal variables are set up. One internal variable is the fraction of martensite;

further internal variables are the inelastic strain tensor and a strain-like variable describing internal stress

fields (residual stresses). The viscous behaviour of shape memory materials, which is established experi-mentally (see Helm and Haupt (2001) for NiTi alloys) during the martensitic phase transition and the

orientation of the martensite twins, is modelled by means of an inelastic multiplier of Perzyna-type (cf.

Perzyna, 1963). Of course, it is difficult to observe the viscous phenomena in the case of pseudoelasticity

because the effects of viscosity interact with the thermomechanical coupling phenomena. However, in the

case of pseudoplasticity distinct rate-dependent effects like stress relaxation as well as creep occur. Thermo-

mechanical coupling phenomena play only a minor role. In this case, phenomena of viscosity can be

separately identified.

The proposed model describes the material behaviour of shape memory alloys under the assumption ofsmall deformations. In addition, the elastic part of the material behaviour is assumed to be isotropic. This

corresponds to a polycrystalline material structure with a large number of grains. As a consequence, effects

of texture cannot be represented within the present constitutive theory.

Shape memory alloys show strong thermomechanical coupling phenomena. Consequently, it is necessary

that the material model complies with the 2nd law of thermodynamic. Here, we prove the thermome-

chanical consistency in the sense of the Clausius–Duhem inequality, which is a special formulation of the

2nd Law of thermodynamics. In Section 3 some numerical solutions of the constitutive model in the context

of isothermal and non-isothermal strain and stress processes are presented. The numerical simulationsdemonstrate that the developed theory is able to represent the exceptional behaviour of these so-called

smart materials.

2. Phenomenological material model

In this section the constitutive model is formulated in the framework of phenomenological continuum

thermomechanics. First, the basic structure of the material model is developed on the basis of a free energy

function and observing the Clausius–Duhem inequality. Thereafter, the free energy and the evolution

equations for the internal variables are set up in detail. Finally, the thermomechanical consistency of the

complete material model is proven, and the thermomechanical coupling phenomena are discussed by meansof the heat conduction equation.

828 D. Helm, P. Haupt / International Journal of Solids and Structures 40 (2003) 827–849

2.1. Basic structure of the material model

The material model to be proposed is based on two assumptions. First, the linearised Green strain tensor

E (i.e. only small deformations are considered) is decomposed additively into an elastic part Ee and aninelastic part Ed:

E ¼ Ee þ Ed: ð1Þ

Here, Ed denotes the state of inelastic strain as a consequence of the stress-induced martensitic phase

transitions as well as due to the orientation and reorientation of martensite.

Furthermore, the inelastic part Ed of the linearised Green strain tensor is decomposed n times:

Ed ¼ Yei þ Ydi ; i ¼ 1; . . . ; n: ð2Þ

The parts Yei are used to model the energy storage due to internal stress fields. On the other hand, the partsYdi are introduced to model dissipation effects which occur during the production of internal stresses. The

several decompositions of the inelastic strain tensor make it possible to describe the material behaviour of

shape memory alloys in more detail. The free energy function is utilised to account for the energy storage in

a material due to thermomechanical loads. As a result of the physical understanding of the deformation

mechanisms in combination with experimental results (cf. Shaw and Kyriakides, 1995; Rogueda et al., 1996;

Lim and McDowell, 1999; Helm and Haupt, 2001), we use a free energy function of the form

w ¼ wwðh;Ee; z;Ye1 ; . . . ;YenÞ ¼ wweðh;Ee; zÞ þ wwsðh;Ye1 ; . . . ;YenÞ; ð3Þ

depending on the absolute temperature h, the elastic part of the linearised Green strain tensor Ee, the

fraction of martensite z and the internal variables Yei . According to Eq. (3) the free energy is decomposed

into two parts. The first part we represents the energy storage in consequence of elastic deformations,

temperature variations and martensite fraction. The second part ws represents the energy storage due to

internal stress fields. Furthermore, this part of the free energy can be used to describe other storage

mechanisms, e.g. energy changes due to interfacial effects or dislocations.

In the field of continuum mechanics one usually applies the 2nd Law of thermodynamics in the form ofthe Clausius–Duhem inequality,

hc ¼ � _ww � _hhg þ 1

qT � _EE� 1

qhq � gradh P 0; ð4Þ

postulating the positiveness of the entropy production c (see e.g. Haupt (2002) for details). Here, g is the

entropy, q the mass density, T the stress tensor and q the heat flux vector. At this point we remark that thereis some criticism on the Clausius–Duhem inequality as a general formulation of the 2nd Law of thermo-

dynamics; therefore, several other irreversibility constraints have been proposed in the literature (see Hutter

(1977), Jou et al. (1996), and M€uuller and Ruggeri (1998) for more details). In our case, it is sufficient to

apply the Clausius–Duhem inequality.

In most cases, it is expedient to divide the Clausius–Duhem inequality (4) into the internal dissipation

inequality,

d ¼ � _ww � _hhg þ 1

qT � _EEP 0; ð5Þ

and the heat conduction inequality,

� 1

qhq � gradh P 0: ð6Þ

D. Helm, P. Haupt / International Journal of Solids and Structures 40 (2003) 827–849 829

The heat conduction inequality (6) is of minor interest for the following; its validity is guaranteed if we

choose the Fourier-model, q ¼ �kgradh, with k > 0.

With these preliminary considerations the basic structure of the material model is explained. The in-

troduction of the free energy (3) into the internal dissipation inequality (5) in combination with the de-compositions (1) and (2) leads to

d ¼ � owoh

�þ g

�_hh þ 1

qT

�� owe

oEe

�� _EEe þ

Xn

i¼1

ows

oYei

� _YYdi �owe

oz_zzþ 1

qT

"�

Xn

i¼1

qows

oYei

#� _EEd P 0: ð7Þ

Following standard arguments, this inequality implies potential relations for the stress tensor and the

entropy,

T ¼ qowe

oEe

and g ¼ � owoh

; ð8Þ

and motivates the definition of an internal stress field (residual stress),

Xe ¼Xn

i¼1

Xei ¼Xn

i¼1

qows

oYei

: ð9Þ

To describe the changes in the microstructure, the decomposition of the martensite fraction into two

parts,

z ¼ zTIM þ zSIM; ð10Þis advantageous. zTIM represents the temperature-induced martensite (twinned martensite) and zSIM the

stress-induced martensite (detwinned martensite). The fraction of martensite is always between 0 and 1,

z 2 ½0; 1 (z ¼ 1 () pure martensite; z ¼ 0 () pure austenite). The present amount of the temperature-induced martensite does not dependent on the state of strain. In contrast to the temperature-induced

martensite the stress-induced martensite depends on the current strain. Experimental observations (cf.

Rogueda et al., 1996; Fu et al., 1992) show that a correlation between the present strain and the fraction of

martensite (zSIM) exists. This dependence is a result of shearing processes occurring in the microstructure

during thermomechanical loadings. However, both deformation mechanisms (orientation of the martensite

twins as well as the martensitic phase transition) are finished if a certain deformation is reached. In the

present model we describe the coupling between the inelastic strain and the fraction of stress-induced

martensite through the relation (cf. Levitas, 1998; Souza et al., 1998; Juh�aasz et al., 2000)

zSIM ¼ kEdkffiffi32

qcd

() _zzSIM ¼ Ed � _EEdffiffi32

qcdkEdk

; ð11Þ

where cd is a material parameter. Case distinctions are introduced later to satisfy the constraint condition

zSIM 2 ½0; 1; zSIM is equal to one if only stress-induced martensite is present. Of course, this is an idealizationof the constitutive theory: a shape memory alloy cannot consist of pure stress-induced martensite due to

various imperfections in the microstructure. The strategy according to Eq. (11) corresponds to the defi-

nition of a limit function, dependent on kEdk, which was already introduced in Bertram (1982).

As a consequence of the Eqs. (8)–(11), the internal dissipation inequality (7) reduces to a residual dis-

sipation inequality,

qd ¼Xn

i¼1

Xei � _YYdi � qowe

oz_zzTIM þ T

264 � Xe �

qffiffi32

qcd

owe

ozEd

kEdk

375 � _EEd P 0: ð12Þ


This inequality allows us to define two basic quantities of the model: the internal variable Xh,

Xh ¼qffiffi32

qcd

owe

oz

� �Ed

kEdk; ð13Þ

and the internal variable X,

X ¼ Xe þ Xh; ð14Þ

which is the sum of Xe and Xh. In the definition of Xh, the MacCauley-bracket,

hxi ¼ x for x > 0;0 for x6 0;

�ð15Þ

is introduced in order to describe the continuous transition between pseudoplasticity and pseudoelasticity.

The internal variable Xh with the property

okXhkoh

¼ qffiffi32

qcd

o

ohowe

oz

� �ð16Þ

leads to the effect that the constitutive theory describes the experimentally observed temperature-depen-

dence of the stress, necessary to initiate the martensitic phase transitions (cf. Fig. 5). The definitions (13)

and (14) lead to a more compact form of the residual dissipation inequality (12):

qd ¼Xn

i¼1

Xei � _YYdi � qowe

oz_zzTIM þ ½T� X � _EEd P 0: ð17Þ

In order to guarantee the positiveness of the internal dissipation d and furthermore the deviatoric evolution

of the inelastic strains, the following proportionality relations are suggested for the evolution of the internal

variables:

_EEd � ½T� XD; _YYdi � XDei; _zzTIM � � owe

oz: ð18Þ

Here, the deviatoric evolution of the internal variables Ed and Ydi is assumed. Consequently, the experi-

mentally observed small volume changes during the inelastic deformations (cf. Funakubo, 1987) are ne-

glected in the present theory. Proportionality relations like (18) are the most simple way to construct

evolution equations implying a strictly non-negative entropy production for all thermomechanical pro-

cesses. There is a considerable degree of freedom for an appropriate choice of the proportionality factors

in order to model the complex process-dependence of the material response. In addition to an adequate

description of the shape memory behaviour (Section 3.1) the proportionality relations contain furtherphysical meaning: The proportionality _EEd � ½T� XD ¼ ½T� Xe � XhD is suitable to represent the uniaxial

as well as the multiaxial evolution of the inelastic strains as a consequence of the internal variable Xh (cf.

Eq. (14)). The driving forces for the evolution of the internal variables Ydi are the internal stress fields XDei.

Finally, the evolution of the temperature-induced martensite zTIM is driven by �owe=oz, which can be in-

terpreted as the difference of the free energy between the austenite and the martensite phase (cf. Eqs. (24)

and (39)). The introduced proportionality relations do not describe microscopic effects in detail. However,

the macroscopic consequences of the microscopic phenomena are approached by these phenomenological

relations.With these steps the structure of the material model is explained. In the following subsections, specifi-

cations of the free energy and evolution equations for the internal variables Ed, Ydi , and zTIM are presented.


2.2. Free energy

With Eq. (3) the general structure of the free energy function was introduced. In the following the free

energy function will be specified in detail.

2.2.1. Elastic part of the free energy

The elastic part we of the free energy describes the energy storage due to the following characteristic

effects: small deformations of the crystal lattice because of mechanical and thermal loads lead to energy

storage in the material structure. Furthermore, two phases coexist during the phase transitions i.e. this

part of the free energy should account for a mixture of the austenite and martensite phase. Thus, we

introduce the following elastic part of the free energy function (cf. Raniecki and Bruhns, 1991; Helm,

2001):

we ¼ wweðh;Ee; zÞ

¼ lðhÞq

EDe � ED

e þ jðhÞ2q

ðSpEeÞ2 � 3aðhÞjðhÞ

qðSpEeÞðh � h0Þ þ

Z h

h0

cd0ð�hhÞd�hh þ uA0 þ zDu0

� hZ h

h0

cd0ð�hhÞ�hh

d�hh

"þ gA

0 þ zDg0

#: ð19Þ

Here, j denotes the bulk modulus, l the shear modulus, a the linear expansion coefficient, h0 the reference

temperature, and cd0 a part of the specific heat capacity (cf. Section 2.5). The material parameters

Du0 ¼ uM0 � uA0 and Dg0 ¼ gM0 � gA

0 are the initial values of the internal energy u0 and entropy g0 for the

austenite and martensite phase (superscript A stands for austenite and M for martensite). The extension of

the present theory to different thermoelasticity parameters for the austenite and the martensite phase is no

significant problem. However, for simplicity we do not regard the different thermoelasticity constants of

the austenite and martensite phase (cf. Bertram, 1982; Delobelle and Lexcellent, 1996; Huo and M€uuller,1993). In general, the material parameters in the proposed theory j, l, a, and cd0 may depend on tem-

perature.

2.2.2. Inelastic part of the free energy

The inelastic part ws of the free energy describes the energy storage due to inelastic deformations. In the

present model, only the energy storage due to the development of internal stresses will be taken into ac-

count. Other storage mechanisms are neglected, such as interfacial energies and crystal defects. The energy

storage is described by means of a quadratic function:

ws ¼ wwsðh;YeiÞ ¼Xn

i¼1

ciðhÞ2q

Yei � Yei : ð20Þ

Therein, the ciðhÞ are temperature-dependent material parameters. The material parameters ci must be non-

negative because they play the role of elasticity constants i.e. ws has to be positive definite. Consequently,

the free energy function (20) is always non-negative. If a more accurate description of the material be-

haviour is required, a more general isotropic tensor function may be chosen for ws.

2.2.3. Implications of the free energy function

According to the potential relations (8) and the definition (9) for the internal stresses, we have the stress

relation

T ¼ 2lðhÞEDe þ jðhÞðSpEeÞ1� 3aðhÞjðhÞðh � h0Þ1; ð21Þ


the entropy relation

g ¼ 3aðhÞjðhÞq

ðSpEeÞ þZ h

h0

cd0ð�hhÞ�hh

d�hh þ gA0 þ zDg0 �

1

qdlðhÞdh

EDe � ED

e � 1

2qdjðhÞdh

ðSpEeÞ2

þ 3

qdaðhÞdh

jðhÞ�

þ aðhÞ djðhÞdh

�ðSpEeÞðh � h0Þ þ

1

2q

Xn

i¼1

dciðhÞdh

Yei � Yei ; ð22Þ

and the internal stress tensor

Xe ¼Xn

i¼1

X�i ¼Xn

i¼1

ciðhÞYei ¼Xn

i¼1

ciðhÞ½Ed � Ydi : ð23Þ

In Eq. (23) the decomposition (2) is applied in order to show the dependence on the internal variable Ydi .

We see that the temperature-dependent material parameters lead to additional terms in the entropy

function.

By assumption the inelastic part ws of the free energy does not depend on the martensite fraction z.Therefore, the term ow=oz is just the difference between the free energies of the two phases:

owe

oz¼ Du0 � hDg0 ¼ Dw: ð24Þ

If different thermoelasticity properties are regarded for the austenite and martensite phase, the derivative

owe=oz is much more complicated. Basic thermodynamical considerations show (see Section 2.4) that the

material parameters Du0 and Dg0 must be non-positive: Du0 6 0 and Dg0 6 0.

2.3. Evolution equations

In Section 2.1, basic proportionality relations were formulated in order to fulfil the dissipation inequality

(17) in the sense of sufficient conditions. These relations are now completed to evolution equations. The

influence of the process history on the present material response is modelled by use of scalar multipliersincluding case distinctions.

2.3.1. Evolution equation for the internal variable Ed

The stress, necessary to continue the martensitic phase transitions is strongly temperature-dependent.This follows from experiments reported e.g. by Funakubo (1987) and Shaw and Kyriakides, 1995. These

experiments are nearly isothermal monotonous loading and unloading processes, carried out at different

temperature levels. The temperature-dependence of the phase transition stress is also visible in multiaxial

experiments (cf. Lim and McDowell, 1999). Furthermore, elastic and inelastic domains can be distin-

guished. In order to model these material properties, a temperature-dependent yield function is introduced:

f ðTD; h;XÞ ¼ kTD � Xk �ffiffiffi2

3

rkðhÞ: ð25Þ

In the context of the normality rule the yield function (25) implies a proportionality relation of the form

(18)1. Furthermore, the yield function describes a symmetric tension–compression behaviour, observed e.g.

in Lim and Mcdowell (1995) on polycrystalline NiTi. Of course, there are other experiments (e.g. Lim and

McDowell, 1999; Raniecki et al., 2001) on NiTi and theoretical investigations (e.g. Patoor et al., 1996)

indicating that shape memory alloys behave different in tension and compression. In order to model such a

tension–compression asymmetry, another appropriate yield function should be chosen. In particular, theinternal variable Xh, a part of X ¼ Xe þ Xh, describes the experimentally observed temperature-dependence


of the phase transition stress (cf. Funakubo, 1987; Shaw and Kyriakides, 1995; Lim and McDowell, 1999;

Helm and Haupt, 2001). In contrast to this, kðhÞ models the height of the hysteresis.

In the present model the evolution of the internal variable Ed is assumed as

_EEd ¼ kd

ofoT

¼ kdN; ð26Þ

i.e. the inelastic strain rate is proportional to the normal

N ¼ TD � X

kTD � Xk¼ TD � Xe � Xh

kTD � Xe � Xhkð27Þ

on the yield surface. In the flow rule (26) the material function kd is called an inelastic multiplier. The choice

of the tensor-valued variable N in the flow rule (26) is important for the results of the model in the case of

multiaxial loading paths. Already in Graesser and Cozzarelli (1994), a proportionality relation similar to

(26) was chosen, however, formulated in

T� aED

kEDkand not like to Eq. (27) with Xh according to Eq. (13). Here, a is a scalar factor depending on the de-

formation history. In Souza et al. (1998), and later also in Juh�aasz et al. (2000), the internal variable Ed is

used in the flow rule. However, these models apply an indicator function in order to define elastic and

inelastic regions.

For the inelastic multiplier, the following constitutive relation is assumed, distinguishing between the

different deformation mechanisms:

kd ¼1

gdðhÞf

rdðhÞ

� �mðhÞ

A ! ~MM for Dw > 0; f > 0; zSIM < 1;seff > kXhk; and Ed �NP 0;

~MM ! A for Dw > 0; f > 0; zSIM > 0;and Ed �N < 0;

M ! ~MM for f > 0; zTIM > 0; zSIM < 1;and Ed �NP 0;

~MM ! M for f > 0; Dw < 0; zTIM < 1;zSIM > 0 and Ed �N < 0;

0 all other cases:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð28Þ

Therein, seff is a scalar measure for the effective stress state:

seffðTD;XeÞ ¼ kTD � Xek: ð29ÞIn this constitutive equation, case distinctions allow for the different deformation mechanisms:

A ! ~MM: The martensitic phase transition from austenite into martensite requires Dw > 0 (this means that

the temperature is high enough to cause martensitic phase transitions), a positive value of theyield function f > 0 (the stress state is outside the elastic region), zSIM < 1 (austenite is available),

as well as seff > kXhk and Ed �NP 0 (loading conditions).~MM ! A: The retransformation from detwinned martensite into austenite needs similar case distinctions.

However, the loading condition is replaced by an unloading condition: Ed �N < 0.

M ! ~MM: The orientation of the martensite twins is possible if the yield function is positive f > 0, tempe-

rature-induced martensite (identical with twinned martensite) is available, zTIM > 0. The orienta-

tion process is going on if the martensite is not completely orientated zSIM < 1, and finally the

loading condition Ed �NP 0 is fulfilled.


~MM ! M: The reorientation occurs outside of the elastic region f > 0, the temperature is lower than a char-

acteristic value Dw < 0, and stress-induced martensite is available (zTIM < 1 and zSIM > 0).

Experiments (cf. Helm and Haupt, 2001) under stress- and strain-control show that the material be-haviour of NiTi shape memory alloys is rate-dependent in the sense of viscous effects. Viscous material

properties are incorporated by means of the inelastic multiplier of Perzyna-type (cf. Perzyna, 1963). The

idea of this constitutive relation to describe viscoplasticity traces back to a work of Hohenemser and Prager

(1932). As a consequence, the material model describes temperature-dependent relaxation and creep phe-

nomena, because of gdðhÞ, rdðhÞ, and mðhÞ. Additionally, the experimentally observed rate-dependent effects

due to thermomechanical coupling phenomena accompanied by heat conduction are represented by the

model because of the introduced internal variable Xh.

The evolution equation (26) for the inelastic strain tensor and the yield function (25) requires a carefulconsideration in view of the beginning of an inelastic process: Due to the internal variable Xh according to

Eq. (13) both the flow rule and the yield function are singular at kEdk ¼ 0. This is the starting point of the

martensitic phase transition and simultaneously the termination point of the retransformation. In the case

of the retransformation (Eq. (28), ~MM ! A) the singularity is not of interest, because the inelastic strain rate

vanishes at kEdk ¼ 0. However, the singularity must be discussed in the case of the martensitic phase

transition starting from kEdk ¼ 0. In the case of differential equations, approximation techniques (e.g.

implicit Euler method) are often used in order to discuss the properties of differential equations. To this

end, the evolution equation (26) may solved by means of an implicit Euler method:

iþ1Ed � iEd

Dt¼ iþ1kd

iþ1TD � iþ1X

kiþ1TD � iþ1Xk¼ iþ1kd

iþ1TD � iþ1X

iþ1f þffiffi23

qkðiþ1hÞ

: ð30Þ

At the beginning of the phase transition, iEd ¼ 0 is valid. Consequently, the approximated evolution

equation (30) leads via the norm operation to an important relation:

kiþ1Edkiþ1kdDt

¼ 1: ð31Þ

Furthermore, application of Eq. (30) under consideration of Eq. (14) as well as Eq. (31) leads to

iþ1f

"þ

ffiffiffi2

3

rkðiþ1hÞ

#iþ1Ed þ kiþ1Edkiþ1Xh ¼ kiþ1Edk½iþ1TD � iþ1Xe; ð32Þ

and inserting Eq. (13) allows us to calculate the final relation,

iþ1f

"þ

ffiffiffi2

3

rkðiþ1hÞ

�þ qhDwðiþ1hÞi

cd

�#iþ1Ed ¼ kiþ1Edk iþ1TD

�� iþ1Xe

�: ð33Þ

Applying again the norm to both sides of (33) and passing to the limit Dt ! 0, the martensitic phase

transition begins if the yield function

f ¼ kTD � Xek �ffiffiffi2

3

rkðhÞ

�þ qhDwðhÞi

cd

�ð34Þ

is positive. Furthermore, Eqs. (33) and (31) lead to an equation to determine iþ1Ed,

iþ1Ed ¼ iþ1kdDtiþ1TD � iþ1Xe

kiþ1TD � iþ1Xek: ð35Þ


This is identical with the evolution equation

_EEd ¼ kd

TD � Xe

kTD � Xek; ð36Þ

which, however, is only valid for the first time step (i.e. iEd ¼ 0 at the beginning of an inelastic process)

irrespective which kind of integration scheme is applied. Finally, the time discretization of Eq. (11) leads

under consideration of the initial value izSIM ¼ 0 to

iþ1zSIM ¼iþ1kdffiffi

32

qcd

; ð37Þ

and consequently for Dt ! 0 to the evolution equation

_zzSIM ¼ kdffiffi32

qcd

; ð38Þ

valid for the first time step only. Thus, the beginning of the phase transition at kEdk ¼ 0 is governed by a

system of constitutive equations, consisting of the differential equations (36) and (38) in combination with

the yield function (34).

2.3.2. Evolution equation for the internal variable zExperimental observations suggest that the fraction of martensite depends on the thermal and me-

chanical loading history. In Eq. (10) the basic decomposition of the fraction of martensite was introduced.

Accordingly, two constitutive equations are necessary to represent the path-dependencies.

In the case of the stress-induced phase fraction zSIM, only the algebraic function (11)1, depending on the

internal variable Ed, is required. The path-dependencies result from the evolution equation (26) for the

inelastic strain tensor Ed.

On the other hand the temperature-induced fraction of martensite zTIM depends on the temperaturehistory. In order to describe the temperature-dependent evolution of zTIM, the following evolution equation

is assumed, where the case distinctions correspond to the hysteresis behaviour sketched in Fig. 1:

_zzTIM ¼

�_zzSIM

M ! ~MM for f > 0; zTIM > 0; zSIM < 1

and Ed �NP 0;~MM ! M for f > 0; DwðhÞ6 0; zTIM < 1;

zSIM > 0 and Ed �N < 0;

� j _hhjMs �Mf

DwjDwj

A ! M for _zzSIM ¼ 0; _hh < 0; h6MsðseffÞ;f < 0 and z < 1;

� j _hhjAf � As

DwjDwj

M ! A for _zzSIM ¼ 0; _hh > 0; h PAsðseffÞand z > 0;

0 for all other cases:

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð39Þ

Fig. 1. Left: fraction of martensite versus temperature; right: effective stress state versus temperature.


It is physically reasonable that the characteristic start and finish temperatures of the phase transition are

functions of the effective stress, defined in Eq. (29). Of course, such a dependency is difficult to be exper-

imentally verified. However, the experiments shown in Nishimura et al. (1996) support this concept. In the

first case of the evolution equation (39), the relation describes the orientation and reorientation process. Asa result of this, the increase of the stress-induced martensite leads to an decrease in the temperature-induced

martensite and vice versa. The next two cases represent the variation of the temperature-induced martensite

during the temperature variation. The stress-induced martensite does not vary in these cases (_zzSIM ¼ 0).

2.3.3. Evolution equation for the internal variables Ydi

The internal variables Ydi were introduced to model the dissipation effects which occur during the

evolution of internal stress fields Xei . As a consequence of the residual inequality (17), the proportionality_YYdi � Xei is the basis for the structure of the evolution equation:

_YYdi ¼ nniðq1; . . . ; qnÞXei : ð40ÞTherein, ni ¼ nniðq1; . . . ; qkÞ are always non-negative functions or functionals of internal variables q1; . . . ; qk.Here, we choose functions

ni ¼ nniðh; _ssd; zSIMÞ ¼ h_zzSIMi0 exphh� biðhÞz

ciðhÞSIM

iþ h � _zzSIMi0

i biðhÞciðhÞ

_ssd ð41Þ

with

_ssd ¼ffiffiffi2

3

rk _EEdk: ð42Þ

The ni depend on h, _ssd, and zSIM in order to describe the strong increase of stress at the end of the phasetransition plateau during thermomechanical loading (cf. Shaw and Kyriakides, 1995; Helm and Haupt,

2001). In Eq. (41), biðhÞ, biðhÞ, and ciðhÞ are temperature-dependent material parameters. The combination

of the evolution equation (40) with the constitutive relation (23) using the decomposition (2) leads to an

evolution equation for the internal stress Xei :

_XXei ¼ ciðhÞ _EEd þ1

ciðhÞdciðhÞdh

_hh

�� h_zzSIMi0 exp

hh� biðhÞz

ciðhÞSIM

iþ h � _zzSIMi0

ibiðhÞ_ssd

�X�i : ð43Þ

For the special case of temperature-independent material parameters and bi ¼ ci ¼ 0, the well-known

Armstrong–Frederick evolution equation is obtained which is often used in metal plasticity (e.g. Chaboche,

1977; Haupt, 2002).

2.4. Thermodynamic consistency

With the introduction of the free energy in Section 2.2 and the evolution equations in Section 2.3 the

material model was completely defined. However, the positiveness of the internal dissipation (17) has not

yet been proved.

In order to prove thermodynamic consistency and to give a better understanding of the material model,

the three deformation mechanisms are regarded separately. These are the stress-induced phase transi-tion, the orientation and reorientation of the martensite twins and the temperature-induced phase transi-

tion.

• Stress-induced phase transition: In the case of stress-induced phase transitions only a part of the material

model is active: the evolution equation (26) for the inelastic strain tensor, the constitutive relation (11),

describing the coupling between the martensite fraction and the inelastic strain tensors and the evolution


equations for the internal variables Ydi according to Eq. (40). Consequently, the residual internal dissi-

pation inequality,

qd ¼Xn

i¼1

niXei � Xei þ kdkT� XkP 0; ð44Þ

is satisfied, as long as ni is non-negative. This is the case if the material parameter functions biðhÞ are non-negative. The material parameter functions ciðhÞ are non-negative (cf. Section 2.2.2). This guarantees a

non-negative free energy function (20). Furthermore, kd is non-negative by definition according to Eq.

(28).

• Orientation and reorientation of martensite twins: During the orientation and reorientation of martensite

twins phase transitions cannot occur. Consequently, _zz vanishes. Furthermore, Dw is negative and as aresult of this Xh is reduced to 0. The dissipation inequality (7) together with Eqs. (8), (9), and (26) leads

to the restriction,

qd ¼Xn

i¼1

niXei � Xei þ kdkT� XekP 0; ð45Þ

which is likewise fulfilled for all these thermomechanical processes.

• Temperature-induced phase transition: In the last case, inelastic deformations do not occur. Conse-

quently, only the evolution equation for the temperature-induced martensite (39) has to be regarded:

qd ¼qj _hhjjDwjMs�Mf

for the case A ! M;qj _hhjjDwjAf�As

for the case M ! A:

(ð46Þ

Due to Ms > Mf and Af > As, the dissipation d is always non-negative.

Furthermore, the residual dissipation inequality for the case of temperature-induced phase transitions,

qd ¼ �qowe

oz_zzTIM ¼ �q Du0ð � hDg0Þ_zzTIM P 0; ð47Þ

determines the sign of the material parameters Du0 and Dg0. This inequality is equivalent to

Du0 � hDg0 P 0 for hPAsðseffÞ and Du0 � hDg0 6 0 for h6MsðseffÞ; ð48Þ

because of the different signs of _zzTIM during the martensitic transformation (A ! M; _zzTIM > 0) and theretransformation (M ! A; _zzTIM < 0). These two inequalities imply Du0 6 0 and Dg0 6 0. As a result of

this the inequalities Du0 6 0 and Dg0 6 0 lead to the correct signs for Dw ¼ qðDu0 � hDg0Þ, which de-

termine the direction of the evolution equation of zTIM in Eq. (39).

As a result of these three cases, the material model is thermomechanically consistent for arbitrary

thermomechanical processes, as long as the material parameters fulfil the discussed restrictions. Further-

more, the material model is thermodynamically consistent at the starting point of the stress-induced phase

transition from austenite to martensite (kEdk ¼ 0). In this case the presented model is similar to our formermodel, presented in Helm and Haupt (1999).

2.5. Heat conduction equation

Here, we will discuss the implications of the material model in view of the heat conduction equation. The

equation of heat conduction is a consequence of the energy balance, i.e. the first Law of thermodynamics,

_ee ¼ ðw þ hgÞ� ¼ � 1

qdivqþ r þ 1

qT � _EE: ð49Þ


Therein, e is the internal energy. The heat conduction equation is derived from the energy balance (49) in two

steps: In the first step the free energy function (3), the additive decompositions of E (Eq. (1)) and Ed (Eq. (2))

as well as the potential relations (8) and the defining equation for the internal stress field (9) is inserted:

_gg þ 1

qdivq� r ¼ 1

qT � _EEd �

owoz

_zz�Xn

k¼1

owoYei

� _YYei : ð50Þ

In the second step the time derivative of the entropy relation (8) is introduced into Eq. (50), which leads to

the equation of heat conduction:

�ho2w

oh2|fflfflffl{zfflfflffl}cdðh;Ee ;z;Ye1

;...;Yen Þ

_hh þ 1

qdivq� r ¼ h

o2wohoEe

� _EEe|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}�we

þ 1

qT � _EEd|fflfflfflffl{zfflfflfflffl}wd

� owoz

� ho2wohoz

� �_zzþ

Xn

k¼1

owoYei

� ho2w

ohoYei

� �� _YYei

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ws

:

ð51ÞOn the right-hand side of the heat conduction equation (51), various terms with a definite physical meaning

are identified: The first term �we represents the thermoelastic coupling phenomena. The second term wd isthe inelastic i.e. dissipative stress power. In contrast to this, the last term ws describes the energy storage and

release during inelastic deformations.

Due to the temperature-dependence of the material parameters the heat conduction equation (51) looks

rather complicated. For example, the specific heat capacity at constant deformation, cdðh;Ee; z;Ye1 ; . . . ;YenÞ, is a function of the temperature and all internal variables. In order to restrict our attention to main

phenomena, only temperature-independent material parameters are regarded in this subsection. Then, the

heat conduction equation (51) assumes under application of (19) and (20) a more compact form:

cd0 _hh þ 1

qdivq� r ¼ � 3aj

qhSp _EEe þ

1

qT � _EEd � Du0 _zz

"þ 1

q

Xn

i¼1

Xei � _YYei

#: ð52Þ

This formulation of the heat conduction equation allows us to discuss the main deformation mecha-

nisms:

• Orientation and reorientation of the martensite twins: During the orientation and reorientation of the

martensite twins, the fraction of martensite is constant: _zz ¼ 0. Applying the evolution equations (26)

and (40) as well as the yield function (25), the heat conduction equation (52) implies

cd0 _hh þ 1


qhSp _EEe þ

1

qf

"þ

ffiffiffi2

3

rk

#kd þ

Xn

i¼1

ni

qXei � Xei : ð53Þ

The first term on the right-hand side is the known thermoelastic coupling effect. The second and last

terms describe the energy dissipation through inelastic deformations and the production of internal

stresses. In the second term, f can be interpreted as a scalar measure for the overstress describing the

viscous behaviour. For slow processes we have f ! 0 asymptotically; in this case onlyffiffiffiffiffiffiffiffiffiffiffið2=3Þ

pkkd

contributes to the dissipation.

• Stress-induced phase transitions: In the case of the stress-induced phase transitions, the combination of

the heat conduction equation (52) with the evolution equations (26) and (40) as well as the yield function

(25) leads to

cd0 _hh þ 1


qhSp _EEe þ

1

qf

"þ

ffiffiffi2

3

rk

#kd þ

Xn

i¼1

ni

qXei � Xei � hDg0 _zzSIM: ð54Þ


In contrast to the orientation and reorientation of the martensite twins, a further term is present in

the heat conduction equation (54). During the phase transition from austenite to martensite the

term �hDg0 _zzSIM leads to a positive heat production, because Dg0 is always non-positive. Conversely, the

retransformation leads to an negative heat production. The influence of the term �hDg0 _zzSIM is dominantin comparison to the remainder of the right-hand side.

• Temperature-induced phase transitions: During the temperature-induced phase transitions, inelastic de-

formations cannot occur. Consequently, the heat conduction equation (52) reduces to

cd0 _hh þ 1


qhSp _EEe � Du0 _zzTIM: ð55Þ

In this case, �Du0 _zzTIM is positive during the phase transition from austenite to martensite and negativeduring the reverse transformation, because of the always non-positive value of Du0.

This discussion of the heat conduction equation shows, that the material model is able to represent the

main thermomechanical coupling phenomena of shape memory alloys (cf. Fig. 5 or Helm and Haupt, 2000).

3. Numerical studies

In the remainder part of the article numerical simulations of uniaxial as well as biaxial stress–strain

processes under strain-, stress-, and temperature-control are outlined. The material model, a system of

ordinary differential equations, is solved numerically by means of an explicit Runge–Kutta method. Inorder to compare the predictions of the model with the experimental data presented in Helm (2001) and

Helm and Haupt (2001), the initial-value problem is solved for a rod under simple tension and compression.

Moreover, combined tension and torsion of a thin-walled tube is considered. In this case the stress and

strain tensors have the following simple structure if the small strain theory is assumed:

T ¼0 0 0

0 0 s0 s r

24

35 E ¼

eq 0 0

0 eq 12c

0 12c e

24

35: ð56Þ

Here, r is the normal stress, s the shear stress, e the normal strain, c the shear strain and eq the hoop strain.

The choice of the material parameters for the numerical simulations is depicted in Table 1. These material

parameters are not the result of a systematic procedure of identification: they have been chosen for an

approximate description of NiTi shape memory alloys in order to demonstrate the ability of the constitutivetheory to predict the main phenomena of process-dependence and thermomechanical coupling.

3.1. Mechanical loading paths

If experiments on shape memory alloys are carried out with a slow strain rate, temperature variations ofless than 1 K are observed (e.g. Helm and Haupt, 2001). Hence, most of the presented simulations in this

subsection are based on isothermal conditions. Only the simulation depicted in Fig. 5 is based on adiabatic

boundary conditions in order to study the thermomechanical coupling phenomena.

In the first simulation, we investigate the so-called step test under strain-control at a temperature of 300 K.

In this test, the maximal strain is stepwise increased from emax ¼ 0:01 in the first cycle to emax ¼ 0:06 in the 6th

cycle. The strain rate is _ee ¼ �0:0001 s�1. After the last cycle a further cycle is carried out with 12 hold times of

600 s at e ¼ 0:005; 0:015; . . . ; 0:055. Fig. 2 shows the stress response and the fraction of martensite in de-

pendence of the given strain path. The comparison of Fig. 2 with the experimental results depict in Fig. 3(a)shows a good agreement. More details above the cited experimental results are published in Helm and Haupt


(2001). The following main observations are represented by the constitutive model in accordance to

the experiment: the pseudoelastic hysteresis, the stress increase at the end of the phase transition plateau,

relaxation phenomena and retransformation processes during the unloading path. The fraction of martensite

is reasonably predicted.

In the next simulation a strain rate of _ee ¼ �0:0001 s�1 with a maximum strain of emax ¼ �0:06 at threedifferent temperatures (h ¼ 240, 270, and 300 K) is regarded. The results are depicted in Fig. 4. Due to the

internal variable Xh the material model represents the pseudoelasticity, the pseudoplasticity as well as the

transition range between pseudoelasticity and pseudoplasticity. As a consequence of the proposed evolution

equation for the internal stresses the reorientation occurs during the first unloading process in the range of

pseudoplasticity at a positive stress level. This kind of behaviour was also experimentally observed (cf.

Helm and Haupt, 2001). It is similar to the known Bauschinger effect, observed in metals under plastic

deformations.

This material behaviour is observed at nearly isothermal properties. In addition, strong thermo-mechanical coupling phenomena occur in shape memory alloys and influence the material behaviour

Table 1

Choice of material parameters

Parameter Symbol Value Unit

Compression modulus j 43,000 MPa

Shear modulus l 19,800 MPa

Mass density q 6,400 kg/m3

Linear expansion coefficient a 1:06� 10�5 1/K

Viscosity gd 2� 107 MPa/s

Exponent m 3 –

Parameter rd 1 MPa

Shear modulus for Xe1 c1 14,000 MPa

Limitation term for Xe1 b1 700 –

Parameter for Xe1 b1 10 –

Parameter for Xe1 c1 10 –

Energy difference Du0 )12,375 J/kg

Entropy difference Dg0 )46.875 J/kgK

Yield value k 50 MPa

Width of the hysteresis cd 0.05 –

Martensite start MsðseffÞ ðqDu0 � cA$~MMðffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pseff � kÞÞ=ðqDg0Þ K

Martensite finish Mf ðseffÞ Ms þ 20 K K

Austenite start AsðseffÞ ðqDu0 � cA$~MMðffiffiffiffiffiffiffiffiffiffiffið3=2Þ

pseff þ kÞÞ=ðqDg0Þ K

Austenite finish AfðseffÞ As þ 20 K

Fig. 2. Step test.


(e.g. Leo et al., 1993; Shaw and Kyriakides, 1995; Lim andMcDowell, 1999; Helm and Haupt, 2001). In Fig.

5 the results of a simulation at three different initial temperatures h0 ¼ 240, 270, and 300 K under adiabaticconditions are presented. The simulation is based on a strain rate of _ee ¼ �0:0001 s�1. The temperature

Fig. 3. Experimental observations on NiTi shape memory alloys (according Helm and Haupt (2001)): (a) step test, (b) proportional

tension–torsion experiment, (c) square-shaped non-proportional experiment (d) butterfly-shaped non-proportional experiment.

Fig. 4. Material behaviour at different temperatures.


evolution is calculated solving the heat conduction equation of Section 2.5 with adiabatic conditions, q ¼ 0

and r ¼ 0. In the case of pseudoplasticity, observed at h0 ¼ 240 K, only dissipation effects contribute to thetemperature increase of approximately 10 K at the end of the deformation path. As a consequence of the

chosen material parameter kðhÞ ¼ const. according to Table 1, the stress–strain behaviour is nearly identical

to Fig. 4. If the same strain path is carried out at h0 ¼ 270 K with an initial austenite phase, the transition

Fig. 5. Adiabatic mechanical loading and unloading at h0 ¼ 240, 270, and 300 K.


region between pseudoplasticity and pseudoelasticity is present. Moreover, at h0 ¼ 300 K pure pseudo-

elasticity occurs. As a result of the occurring heat production, the temperature increase is approximately 40

K. In each cycle, the temperature increases during the phase transition from austenite to martensite and

decreases as a consequence of the retransformation. Moreover, the occurring dissipation effects cause afurther increase of temperature for each cycle; this is clearly seen at the states e ¼ 0. As a consequence of the

introduced internal variable Xh according Eq. (13), the model represents the increasing stress necessary to

stimulate the phase transition if the temperature increases and vice versa.

The response of the material model to multiaxial deformation paths are of particular importance. In the

first simulation, a radial deformation path at 300 K is considered. In this deformation path, the normal and

shear strains are proportional to each other, e ¼ xc, with a constant proportionality factor x. In Fig. 6, the

proportionality factor is varied between 1 (pure tension) and 0 (pure torsion). Each loading cycle with one

proportionality factor is carried out during a time interval of 600 s. The results of the material model arephysically plausible and similar to the experimental observations shown in Fig. 3(b) (cf. Helm and Haupt,

2001).

In the next simulation, a non-proportional deformation path at 300 K is regarded. Here, the ratio be-

tween the shear strain and the normal strain is varied at four points in such a way that the result is a square-

shaped deformation path in the strain space (cf. Fig. 7). Each loading section lasts 60 s, i.e. the duration of

the full experiment is 240 s. The calculated stress path due to the given strain path corresponds to the

experimental observations depicted in Fig. 3(c) (cf. Helm and Haupt, 2001) in the sense that all charac-

Fig. 6. Proportional loading path.


teristic observations are depicted. In particular, the interaction between the normal and shear stress due to

the normal and shear component of the loading path is adequately described.

A further non-proportional deformation path is shown in Fig. 8. Here, a butterfly-shaped deformation

path at 300 K is prescribed. The full deformation path is also carried out in 240 s. Related experimental

data can be found in Fig. 3(d) (cf. Helm, 2001). The stress response of Fig. 8 is in fair agreement with these

experimental data.

In the experiments of Lim and McDowell (1999), the direction of the transformation strain rate is ex-perimentally identified in a circle-shaped strain controlled deformation path (Lim and McDowell, 1999).

The authors investigate the capability of a typical yield function in combination with a normality rule and

in the context of some micromechanical studies. They conclude that only the micromechanical theory

represents these experimental observations very well. In context of Lim and McDowell (1999), it is very

interesting to study the properties of the constitutive equations introduced in Section 2. In a further sim-

ulation, a strain controlled deformation path is calculated: first, the specimen is pulled with a strain rate of_ee ¼ 0:001 s�1. Thereafter, two circle-shaped deformation paths are carried out. Only the second circle-

shaped deformation path as well as the corresponding stress response are depicted in Fig. 9. As a conse-quence of the applied yield function according to Eq. (25), the resulting plot normal stress versus shear

stress is a circle too. The numbers – allow to assign the different strain states with the resulting stress

states. We see that the experimentally observed phase shift between the strain and stress path is also

predicted by the material model. Of course, the experimentally observed deviation from an ideal circle

Fig. 7. Square-shaped deformation path.


cannot be described with the present theory. Such an description requires the application of an other yieldfunction. In the diagram of the stress response the directions of the transformation strain rate _EEd, predicted

by Eq. (26), are depicted at the eight points. It seems noteworthy that the developed theory represents the

Fig. 8. Butterfly-shaped deformation path.

Fig. 9. Circle-shaped deformation path.


experimental observations of Lim and McDowell (1999) very well although a very simple von Mises-type

yield function is applied. The main reason for this property of the model lies in the internal variable Xh,

introduced according to Eq. (13). This influences significantly the direction of the transformation strain

rate.Summarizing, we can state that proportional and non-proportional loading paths are quite reasonably

represented by the present constitutive theory. Moreover, if the material parameters are identified from the

experimental data applying optimization algorithms, a more quantitative approach will be possible.

3.2. Thermomechanical loading paths

In this section we demonstrate the ability of the constitutive model to represent the one-way shape

memory effect and the two-way shape memory effect due to external loads. In the case of the one-wayeffect the stress is first increased at a temperature of 200 K with a stress rate of _rr ¼ 6 MPa/s during 60 s.

Thereafter, the stress is decreased within 60 s until zero. The result of the loading and unloading process is

the pseudoplastic material behaviour. Through the deformation process the martensite twins are orien-

tated. Therefore, the fraction of martensite z is constant, _zzTIM is negative and zSIM increases. During the

Fig. 10. One-way shape memory effect.

Fig. 11. Two-way shape memory effect.


temperature increase ( _hh ¼ 2 K/s for 60 s) the phase transition from detwinned martensite into austenite

takes place. This phase transformation is accompanied by an annihilation of the available inelastic strains,

which establishes the so-called shape memory effect. During the following cooling process ( _hh ¼ �2 K/s), we

have the retransformation from austenite into twinned martensite. The inelastic strains do not developagain. The result of this simulation illustrates Fig. 10.

In contrast to the one-way shape memory effect the two-way shape memory effect occurs if a certain

stress level is applied during the temperature variation. In the simulated case, the stress increases at a

constant temperature of 200 K with a stress rate of _rr ¼ 6 MPa/s. The stress response is pseudoplastic.

Thereafter the temperature is increased with a rate of _hh ¼ 2 K/s up to 370 K. Consequently, the phase

transition from detwinned martensite into austenite occurs and the inelastic deformations vanish simul-

taneously. In the further part of the loading process the temperature decreases with a temperature rate of_hh ¼ �2 K/s. This leads to a retransformation from austenite into detwinned martensite due to the appliedstress field, accompanied with inelastic deformations. Fig. 11 depicts the result of the numerical simulation.

4. Conclusions

In this article, a phenomenological material model to represent the multiaxial shape memory behaviour is

proposed. The material model is based on a free energy function and evolution equations for internal

variables. The internal variables have a definite physical meaning in relation to processes occurring in the

microstructure. Furthermore, all material parameters are regarded as temperature-dependent. In particular,the evolution equations for the internal stresses allow us to represent the material behaviour close to ex-

perimental observations. Furthermore, the choice of the flow rule in connection with the case distinctions

leads to the representation of the multiaxial material behaviour. The complete system of constitutive

equations is compatible with the 2nd Law of thermodynamics in the sense of the Clausius–Duhem in-

equality. The constitutive model represents the main phenomena of the thermomechanical behaviour as

they are experimentally observed. Of course, mainly the qualitative tendencies are represented, because the

numerical simulations are not yet based on material parameters which are identified from experimental data

using optimization procedures, i.e. more detailed validations of the theory are possible after quantitativeparameter identification. This identification together with a thermomechanically coupled finite element

analysis of specific structures and technical processes will be the objective of future work.

Acknowledgement

We gratefully acknowledge the support of this work by the Deutsche Forschungsgemeinschaft (DFG).

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Shape memory behaviour: modelling within continuum thermomechanics

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