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Research Paper

A 3-D constitutive model for pressure-dependent

phase transformation of porous shape memory alloys

M.J. Ashrafia, J. Arghavania, R. Naghdabadia,b,n, S. Sohrabpoura

aDepartment of Mechanical Engineering, Sharif University of Technology, Tehran, IranbInstitute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 30 June 2014

Received in revised form

15 November 2014

Accepted 22 November 2014

Available online 29 November 2014

Keywords:

Shape memory alloys

Porous materials

Pressure dependencyPhase transformation

Multiaxial loading

a b s t r a c t

Porous shape memory alloys (SMAs) exhibit the interesting characteristics of porous

metals together with shape memory effect and pseudo-elasticity of SMAs that make them

appropriate for biomedical applications. In this paper, a 3-D phenomenological constitutive

model for the pseudo-elastic behavior and shape memory effect of porous SMAs is

developed within the framework of irreversible thermodynamics. Comparing to micro-

mechanical and computational models, the proposed model is computationally cost

effective and predicts the behavior of porous SMAs under proportional and non-

proportional multiaxial loadings. Considering the pressure dependency of phase transfor-

mation in porous SMAs, proper internal variables, free energy and limit functions are

introduced. With the aim of numerical implementation, time discretization and solution

algorithm for the proposed model are also presented. Due to lack of enough experimental

data on multiaxial loadings of porous SMAs, we employ a computational simulation

method (CSM) together with available experimental data to validate the proposed

constitutive model. The method is based on a 3-D nite element model of a representative

volume element (RVE) with random pores pattern. Good agreement between the numerical

predictions of the model and CSM results is observed for elastic and phase transformation

behaviors in various thermomechanical loadings.

&2014 Elsevier Ltd. All rights reserved.

1. Introduction

The research on the behavior of smart materials has been

rapidly increasing thanks to their innovative applications.

Among different types of smart materials, shape memory

alloys (SMAs) have two unique features known as pseudo-

elasticity and shape memory effect observed both in dense

and porous SMAs. NiTi which is the most widely used SMA,

exhibits good corrosion resistance and biocompatibility.

Therefore, it can be utilized in several applications, e.g., as

actuators in different mechanisms and as stents, implants and

devices for orthodontic and endodontic applications in med-

ical industry (Yamauchi et al., 2011;Yoneyama and Miyazaki,

2009;Lagoudas, 2008).

Moreover, porous shape memory alloys benet from

porous metal characteristics such as light weight and energy

http://dx.doi.org/10.1016/j.jmbbm.2014.11.023

1751-6161/&

2014 Elsevier Ltd. All rights reserved.

nCorresponding author at: Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran.

Tel.:98 21 6616 5546; fax:98 21 6600 0021.

E-mail address: [email protected] (R. Naghdabadi).

j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 4 2 ( 2 0 1 5 ) 2 9 2 3 1 0
http://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023mailto:[email protected]://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.jmbbm.2014.11.023&domain=pdfhttp://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023http://dx.doi.org/10.1016/j.jmbbm.2014.11.023



absorption (Lefebvre et al., 2008;Gibson and Ashby, 1997). There-

fore, porous SMAs, categorized as new complex architectured

materials, can be used in various applications such as lightweight

structures, biomedical implants, lters, heat exchangers and

energy (shock and vibration) absorbers (Zhao et al., 2006;

Bansiddhi et al., 2008;Xiong et al., 2008). In biomedical applica-

tions (such as articial bone, dental implant and intervertebral

disc), a relatively high porosity level (up to 70%) is required while

in structural applications a low porosity level (below 40%) is

normally required (Wen et al., 2010;DeGiorgi and Qidwai, 2002).

Also, recent researches (Shishkovsky, 2012a,b) have utilized the

shape memory effect of porous SMAs for drug delivery.

In the above applications, the porous SMA structure is part

of a load-bearing structural element and experiences thermal

and mechanical loadings. Therefore, understanding the ther-

momechanical behavior of porous SMAs and development of

a constitutive model are essential to design a porous SMA

structure efciently.

From the modeling point of view, different approaches have

been adopted in the literature. The micromechanical averaging

technique has been utilized by several authors (Qidwai et al.,

2001; Entchev and Lagoudas, 2002, 2004;Nemat-Nasser et al.,

2005; Zhao and Taya, 2007) to investigate the mechanical

response of porous SMAs. Recently, Zhu and Dui (2011) and

Liu et al. (2014)have used this approach to study the behavior

of porous SMAs. They utilized a transformation function

considering the effect of hydrostatic stress as well as the

tensioncompression asymmetry.

Another approach for porous SMAs modeling assumes

periodic distribution of pores which leads to unit cell analysis

(Liu et al., 2012;Qidwai et al., 2001). However, this assumption

deviates signicantly from the irregularity of the real micro-

structure and will overestimate material response especially

for highly porous materials. Moreover, the porosity shape

effects on the strength of porous SMAs were studied by

Qidwai et al. (2001)andZhao and Taya (2007)assuming regular

distribution of porosity. They concluded that shape of porous

microstructure mostly inuences the local repartition of stress/

strain and to some extent the macroscale behavior of porous

SMAs. However, in most of the real microstructures, pores are

randomly dispersed and have various shapes.

Dening a representative volume element (RVE) with ran-

domly distributed pores has also been employed to simulate

the mechanical behavior of porous materials. In this regard,

DeGiorgi and Qidwai (2002) used a 2-D RVE to describe the

mesoscopic behavior of porous SMAs under axial loadings.

Also, Panico and Brinson (2008) developed a dense SMA

constitutive model with permanent inelasticity effects along

with a 3-D RVE to simulate the behavior of porous SMAs.

The macro-scale phenomenological modeling is another

approach for modeling porous SMAs behavior. While there

are several researches on phenomenological modeling of dense

SMAs (see e.g.,Souza et al., 1998;Bo and Lagoudas, 1999;Panico

and Brinson, 2007;Arghavani et al., 2010among others), in the

case of porous SMAs, there are few works. Sayed et al. (2012)

presented a two-phase constitutive model for porous SMAs

which consists of a dense SMA phase and a porous plasticity

phase. The model incorporates the pseudo-elastic and pseudo-

plastic behaviors simultaneously. However, volumetric strain

occurs only during plastic deformation (not during phase

transformation). Based on the GursonTvergaardNeedleman

formulation, Olsen and Zhang (2012) proposed a constitutive

model for shape memory alloys with micro-voids. The model

can reproduce phase transformation strain (volumetric and

deviatoric parts), as well as plastic strain, but not simulta-

neously. Also, Matrejean et al. (2013) have calibrated the

material parameters of a dense SMA model in order to be used

as material parameters for porous SMAs.

Based on the reviewed literature, the general approach is

directed towards using constitutive models with lower computa-

tional cost. In fact, in the design stage, a large number of

simulations should be performed; therefore, a model with low

computational cost is appreciated. The focus of these studies has

been on pseudo-elastic behavior of porous SMAs under uniaxial

loadings using micromechanical-based or phenomenological

models. However, due to various applications of pseudo-

elasticity and shape memory effect of porous SMAs, together

with their pressure dependency, effective modeling of porous

SMAs under general multiaxial loading is still a subject of interest.

In this work, we propose a phenomenological constitutive

model for phase transformation behavior of porous SMAs. We

employ a limit function for phase transformation of porous

SMAs which is pressure dependent. This model is also capable

to describe the dense SMA behavior when appropriate material

parameters are utilized. For the analysis of real structures, nite

element method (FEM) is utilized. In this regard, we develop a

time discretization method and solution algorithm for the

proposed model to be adopted into a nite element code for

the analysis of realistic applications. Due to lack of enough

experimental data on porous SMAs under multiaxial loadings, a

method combining the experimental results and computational

simulations is employed to validate the proposed constitutive

model for porous SMAs. The model can predict shape memory

effect and pseudo-elastic behavior of porous SMAs under

proportional as well as non-proportional multiaxial loadings.

In the following, a thorough description of the constitutive

model is presented in Section 2. Time discretization and

solution algorithm for the proposed model is also discussed.

In Section 3, a computational method for simulation of

porous SMAs is presented and validated by comparing the

results with the available experimental data. The material

parameter identication and numerical results for different

thermomechanical loadings are the subjects of Section 4.

Finally, inSection 5, we discuss and conclude the results.

2. A constitutive model for porous shapememory alloys

In this section, a phenomenological constitutive model is pro-

posed for macroscopic behavior of porous SMAs within the

framework of continuum thermodynamics. We should high-

light that experimental evidences reveal that porous SMAs

exhibit the following behaviors: pseudo-elasticity, shape mem-

ory effect, pressure dependency and permanent strain.

Similar to dense SMAs, porous SMAs exhibit the pseudo-

elasticity and shape memory effect (see e.g., Bernard et al.,

2012; Aydogmus and Bor, 2012; Shishkovsky, 2012a; Scalzo

et al., 2009 among others). It is well known in the litera-

ture that porous materials behavior are pressure dependent

j o u r n a l o f t h e m e c h a n i c a l b e h a v i o r o f b i o m e d i c a l m a t e r i a l s 4 2 ( 2 0 1 5 ) 2 9 2 3 1 0 293



(Altenbach and Ochsner, 2010; Gibson and Ashby, 1997).

Therefore, mechanical behavior of porous SMAs depends on

hydrostatic pressure as well. In other words, from the

macroscale viewpoint, the inelastic strain in porous SMAs

contains both deviatoric and volumetric parts (Olsen and

Zhang, 2012;Zhu and Dui, 2011). Another behavior, observed

in both dense and porous SMAs, is evolution of permanent

strain during phase transformation (Hosseini et al., 2014;

Bram et al., 2011; Malecot et al., 2006). The evolution of

permanent strain mainly depends on the microstructure of

the produced material and level of stress loading. For exam-

ple,Kohl et al. (2011)produced 51% porous NiTi samples with

pseudo-elastic behavior (no permanent strain) under 140 MPa

compressive loading. Meanwhile, in many applications the

loading is controlled within this limit to avoid permanent

strain evolution in porous SMAs.

Therefore, in this study we develop a constitutive model

for porous SMAs considering the effects of pseudo-elasticity,

shape memory effect and pressure dependency. Developing

such a model, it can then be extended to consider permanent

strain effects. In the following, the constitutive equations are

rst developed and time-discrete form and solution algo-

rithm are then presented.

2.1. Constitutive model development

In order to develop a successful constitutive model, suitable

and physically motivated terms for internal variables, limit and

free energy functions should be presented. Such a model should

capture physical features and lead to a simple numerical

procedure. We now propose a general phenomenological con-

stitutive model for porous SMAs which incorporates pseudo-elasticity and shape memory effect under general thermome-

chanical loadings. In the following, we neglect permanent

strain and develop a constitutive model focusing on pseudo-

elasticity, shape memory effect and pressure dependency.

Assuming small strains, we consider the additive strain

decomposition:

el tr 1

where , el and tr are total, elastic and transformation strains,

respectively. Recalling that from a macroscopic viewpoint, por-

ous SMAs show both deviatoric and volumetric transformation

strains, we now decompose the transformation strain as

tr etr

tr

31 2

where etr and tr represent the deviatoric and volumetric

transformation strains, respectively and 1 is the second-order

identity tensor. We also decompose the total strain as:

e

31 3

where e and are the deviatoric and volumetric strains,

respectively.

To develop a constitutive model for porous SMAs, we

consider the deviatoric and volumetric strains e, and

the absolute temperature T as control variables, while the

deviatoric and volumetric transformation strains etr and tr

are considered as internal variables. We now introduce the

Helmholtz free energy function as follows:

e; ; etr; tr; T el chtrth 4

where el, ch, tr and th are thermoelastic strain energy,

chemical energy, transformation strain energy and free energy

due to temperature change, respectively. It should be noted

that we do not consider a fully thermomechanical coupled

model and, as stated, temperature is a control variable.In order to derive the constitutive equations and thermo-

dynamic forces, the second law of thermodynamics should

be satised. Starting from the free energy function presented

in (4), the mechanical dissipation energy Dm is expressed

using the ClausiusDuhem inequality as follows:

Dm : _ _ _T

Z0 5

where is the stress tensor, is the entropy and dot super-

script indicates the derivative of a quantity with respect to

time. Substituting(3) and (4) into (5), we obtain

Dm s

e : _e p

_

T

_T

etr

: _etr

tr_

trZ0 6

where we have also decomposed the stress into its devia-

toric and volumetric parts s and p as

s p1 7

Following standard arguments, we can derive the consti-

tutive equations and thermodynamic forces as follows:

s

e; p

;

T; Xs

etr; Xp

tr 8

whereXs and Xp are the thermodynamic forces associated to

the deviatoric and volumetric transformation strains, respec-

tively. Therefore, the mechanical dissipation inequalityreduces to

Dm Xs : _etr Xp_

trZ0 9

From the macroscale viewpoint, the phase transformation

occurs in porous SMAs both under shear and hydrostatic

loadings. We therefore introduce the following limit function:

F Xeq R 10

whereRis the radius of the elastic domain and the equivalent

thermodynamic forceXeq is dened as

Xeq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJXs J2 X

2p

1 =6

s 11

where the parameter shows the dependency of phase

transformation on hydrostatic pressure and the norm opera-

tor is dened as JAJ ffiffiffiffiffiffiffiffiffiffiffiffiAijAji

p . The limit function (11) is

simple, convex and similar to the yield function already

proposed byDeshpande and Fleck (2000) for plastic behavior

of metal foams.

To satisfy the second law of thermodynamics or the

dissipation inequality (9), we choose the followingow rules

for the internal variables:

_etr _ F

Xs _

Xs1 =6

Xeq12

_

tr

_

F

Xp _

Xp1 =6

Xeq 13




We note that the limit function F is convex and assures

the dissipation inequality (9). Moreover, the Lagrange multi-

plier _ satises the KuhnTucker conditions:

_Z0; Fr0; _F 0 14

We now introduce an equivalent transformation strain

rate _

tr

eq which is work conjugate to the equivalent thermo-dynamic force. To this end, we use the following denition:

Xeq _treq Xs : _e

tr Xp_tr

15

Substituting(11)(13) into (15), the equivalent transforma-

tion strain rate _treq can be derived as

_treq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 =6

J_etr J2 1

_

tr 2 s

16

Motivated from (16), an equivalent transformation strain

tensor tr is dened as a function of internal variablesetr and

tr such that J_tr

J _treq

tr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 =6 q

etr

ffiffiffiffiffi1

3

r tr1

! 17

Thanks to the denition of the equivalent transformation

strain tensor, we now introduce an appropriate form of the

Helmholtz strain energy function for porous SMAs. Here, we

assume linear hardening during phase transformation and

linear dependency of chemical energy on temperature:

el12K

tr 2

GJeetr J2 3KTT0 ;

chTTmh iJtr

J;

tr12hJ

trJ

2 L Jtr

J

;

th u0 T0

c TT0 Tln T=T0

18

whereK and G are the bulk and shear moduli of porous SMA,

respectively, h is a material parameter dening the phase

transformation hardening. Also, Tm is a reference tempera-

ture associated with the starting temperature of transforma-

tion, a material parameter related to the dependency of the

critical stress on the temperature and h iis the positive part

function. In addition, and c are the thermal expansion

coefcient and the heat capacity, respectively; while u0and 0are internal energy and entropy at reference temperatureT0.

Moreover, we introduce the material parameter L corre-

sponding to the maximum transformation strain reached at

the end of the transformation during a uniaxial test. To

satisfy such a constraint, the function L is introduced as

L Jtr

J

0 if Jtr JrL

1 otherwise

( 19

Substituting (18) into (4), we obtain the Helmholtz freeenergy function as

e; ; etr; tr; T

12K tr

2GJeetr J2 3KTT0

TTmh iJtr

J

12h Jtr

J2 L J

trJ

u0 T0

c TT0 Tln T=T0

20

Fig. 1 Computational model for a porous SMA sample with 8000 elements (porosity is represented by white elements):

(a) 10% porosity and (b) 40% porosity.

Table 1Proposed constitutive model for porous SMAs inthe time-continuous frame.

External variables: e; ; TInternal variables: e tr; tr

Material parameters: K; G; h;; T0; ; L; R;

Constitutive equations and thermodynamic forces:

s 2G e etr

; p K tr 3TT0

Xp

tr p p ; Xs

etr s s

Xeq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJXs J2 X

2p

1 =6

s ; Jtr J

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 =6

Jetr J2 1 tr 2 r

s 1 =6 etr

Jtr

JTTmh i hetr

p 1 =6

tr

Jtr

JTTmh i h

tr

Limit function:

F Xeq R

Evolution equations:

_etr _ F

Xs _

Xs1 =6

Xeq

_tr

_ F

Xp _

Xp1 =6

Xeq

KuhnTucker conditions: _Z0; Fr0; _F 0




Therefore, constitutive equations and thermodynamic

forces (8) may be expressed as

s

e 2G eetr

;

p

K tr 3TT0

;

T

0 3KTTmh iJ

tr

JTTm cln T=T0

; 21

Xs

etr ss;

Xp

tr pp

where the tensor s and the scalarp are

s 1 =6 etr

Jtr

JTTmh i he

tr

22

p 1 =6

tr

Jtr

JTTmh i h

tr

23

The variable results from the saturation function sub-

differential and is dened as

0 0.5 1 1.5 2 2.5 3 3.5 420

0

20

40

60

80

100

Axial strain [ % ]

Axialstress[MPa]

1000

8000

27000

0 0.5 1 1.5 2 2.5 3 3.5 410

0

10

20

30

40

50

60

70

80

90

Axial strain [ % ]

Axialstress[M

Pa]

Fig. 2 Convergence study for the CSM of a 60% porous SMA sample: (a) mesh renement and (b) different random mesh

patterns.

Table 2 Material parameters of dense SMA model adopted fromZhao et al. (2005).

E (GPa) h(MPa) (MPa K1) Tm (K) L (%) R(MPa)

68 0.33 17,280 4.6 296 4.9 86 0

0 1 2 3 4 50

200

400

600

800

1000

1200

1400

Strain [ % ]

Stress[MPa]

dense SMA, Zhao et al., 2005 (experiment)

dense SMA, present work (CSM)

0 1 2 3 4 50

100

200

300

400

500

600

700

800

900

1000

Strain [ % ]

Stress[MPa]

13% porous SMA, Zhao et al., 2005 (experiment)

13% porous SMA, present work (CSM)

Fig. 3 Comparison of the CSM results with experiments under uniaxial loading: (a) stressstrain curve for dense SMA and

(b) stressstrain curve for 13% porous SMA.




0 if Jtr JoL

Z0 if Jtr J L

( 24

Briey, compared to dense SMA models, we introduced a

new volumetric internal variable and a pressure dependent

convex limit function. Moreover, to introduce the Helmholtz

free energy function, we dened an equivalent transformation

strain tensor based on the limit function. The evolution of the

internal variables (volumetric and deviatoric parts) was deter-

mined using an associative ow rule.

We remark that when the parameter 0, the model degen-

erates to a dense SMA model1 similar to the model originally

proposed bySouza et al. (1998) and improved by Auricchio and

Petrini (2004)andArghavani (2011a,b).The proposed porous SMA

model in the time-continuous frame is summarized inTable 1.

2.2. Time discretization and solution algorithm

Assuming to be given the state sn; pn; etrn;

trn at time tn, the

actual total strain e; and temperature T at time tn1, the

updated values can be computed using an implicit backward

Euler method. It should be noted that for notation simplicity

here, and in the following, we drop the subindex n 1 for all

variables computed at timetn1. The discretized constitutive

equations are as follows:

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

Strain [ % ]

Stress[MPa]

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

Strain [ % ]

Stress[MPa]

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

Strain [ % ]

Stress[MPa]

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

Strain [ % ]

Stress[MPa]

Fig. 4 Comparison of the CSM results with the micromechanical averaging approach and the unit cell FEM under uniaxial

loading for (a) dense SMA, (b) 10% porous SMA, (c) 20% porous SMA, and (d) 40% porous SMA.

Table 3 Material parameters of dense SMA model adopted from Entchev and Lagoudas (2002).

E(GPa) h(MPa) (MPa K1) Tm (K) L (%) R (MPa)

70 0.33 1550 5.72 303 7.0 68.6 0




0 1 2 3 4 50

50

100

150

200

250

300

350

400

Strain [ % ]

Stress[MPa]

dense SMA

10% porous SMA

20% porous SMA

40% porous SMA

60% porous SMA

0 1 2 3 4 50

20

40

60

80

100

120

140

160

180

200

Strain [ % ]

Stress[MPa]

dense SMA

10% porous SMA

20% porous SMA

40% porous SMA

60% porous SMA

Fig. 5 Stressstrain curves of dense and porous SMAs with different porosities under uniaxial loading: (a) pseudo-elastic

behavior (T310 K) and (b) shape memory effect (T220 K).

Table 4 Material parameters of dense SMA model adopted fromSittner et al. (1995).

E (GPa) h (MPa) (MPa K1) Tm (K) L (%) R(MPa)

53 0.36 1000 2.1 245 4.0 72 0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

100

200

300

400

500

600

Volumetric strain [ % ]

Hydrostaticstress[MPa]

dense SMA

10% porous SMA

20% porous SMA

40% porous SMA

60% porous SMA

0 1 2 3 4 5 6 70

50

100

150

200

250

300

350

400

450



dense SMA

10% porous SMA

20% porous SMA40% porous SMA

60% porous SMA

Fig. 6 Hydrostatic stressvolumetric strain curves of dense and porous SMAs with different porosities under hydrostatic

loading: (a) pseudo-elastic behavior (T310 K) and (b) shape memory effect (T220 K).

Table 5 Material parameters determined for the proposed model.

Porosity(%) E(GPa) h (MPa) (MPa K1) Tm (K) L (%) R(MPa)

10 45.2 0.35 922 1.91 245 4.0 61.2 0.095

20 37.8 0.33 851 1.56 245 4.1 52.3 0.195

40 22.2 0.29 715 0.91 245 4.2 33.5 0.581

60 8.64 0.23 346 0.38 245 4.8 13.9 1.140




p

K tr

s

e 2G eetr

Xs s 1 =6

etr

Jtr

JTTmh i hetr

Xpp 1 =6

tr

Jtr

JTTmh i h

tr

etr etrn Xs

1 =6

Xeq

tr trn Xp

1 =6

XeqF FXs; Xp Xeq Rr0

Jtr

JrL

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

25

where n is the consistency parameter that is time

integrated over the interval tn; t. To solve the time-discrete

constitutive equations, we use an elastic predictorinelastic

corrector procedure. The algorithm assumes the step is

elastic and evaluates elastic trial state, in which the inter-

nal variable remains constant. The admissibility of the trial

functions with (25)7 is then veried. If the trial states are

admissible, the step is elastic; otherwise, the step is inelasticand the internal variables have to be updated through

integration of the evolution equations. To this end, we

assume the step is unsaturated (Jtr JoL) and rewrite con-

stitutive equations (25) in the residual form as follows:

RXs Xs sTR 2G Xs

1 =6

Xeq

1 =6

TTmh i etr

Jtr

J hetr

0

RXp Xp pTR K

Xp1 =6

Xeq

1 =6

TTmh i

tr

Jtr

J htr

0

R

Xeq R 0 26

0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

140

160

180

200

Axial strain [ % ]

Axialstress[MPa]

10%CSM

10%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

140

160

180

200

Axial strain [ % ]

Axialstress[MPa]

20%CSM

20%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0

50

100

150

Axial strain [ % ]

Axialstress[MPa]

40%CSM

40%Proposed Model

0 1 2 3 4 5 6

0

10

20

30

40

50

60

70

80

90

Axial strain [ % ]

Axialstress[MPa]

60%CSM

60%Proposed Model

Fig. 7 Comparison of stressstrain curves for shape memory effect in porous SMAs (T220 K) under uniaxial loading for

different porosities: (a) 10%, (b) 20%, (c) 40%, and (d) 60%.

1From the mathematical viewpoint, lim-0Xeq J Xs J and

upon substitution of (23) and (21)5

into (13), it is concluded that_tr O ; therefore lim-0 _treq J _e

trJ .




If the solution of non-linear equations(26)are admissible

with (25)8 the step is unsaturated; otherwise, we rewrite

Eqs.(25)with the unknown variable as follows:

RXs Xs sTR 2GXsXeq

1 =6

TTmh i etr

Jtr

J hetr

0

RXp Xp pTR K

XpXeq

1 =6

TTmh i

tr

Jtr

J htr

0

R Xeq R 0

R J tr JL 0 27

where sTR andpTR are trial states for deviatoric and hydrostatic

stresses, respectively. In order to solve the nonlinear equations

(26) and(27), we employ the NewtonRaphson method described

in Appendix. For more details on the solution algorithm, see

Arghavani et al. (2011a)andAuricchio and Petrini (2004).

3. Computational simulation

The available experimental data on porous SMAs are not

sufcient and are limited to uniaxial loadings. Therefore, they

cannot be used to study the transformation strain in a purehydrostatic loading. To this end, we employ a computational

simulation method (CSM) to calibrate the proposed model

(Table 1) for the behavior of porous shape memory alloys.

In the CSM, we use a dense SMA constitutive model and a

3D nite element mesh (in which porosities are modeled as

elements with very low stiffness) to study the behavior of

porous SMAs. In this regard, we rst present the computa-

tional simulation method (CSM). The CSM results are then

validated using the available experimental data in compres-

sion loading. Finally, the CSM results in different loading

conditions are utilized to calibrate the proposed constitutive

model for thermomechanical behavior of porous SMAs.

0 0.5 1 1.50

50

100

150

200

250

300

350

400

450


Hydrostaticpressur

e[MPa]

10%CSM

10%Proposed Model

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

50

100

150

200

250

300


Hydrostaticpressur

e[MPa]

20%CSM

20%Proposed Model

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120


Hydrostaticpressure[MPa]

40%CSM

40%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

15

20

25

30

35

40



60%CSM

60%Proposed Model

Fig. 8 Comparison of hydrostatic pressurevolumetric strain curves for shape memory effect in porous SMAs (T220 K)

under hydrostatic loading for different porosities: (a) 10%, (b) 20%, (c) 40%, and (d) 60%.



http:///reader/full/a-3-d-constitutive-model-for-pressure-dependent-phase-transformation-of-porous-shape-memo 10/19

3.1. Computational simulation method (CSM)

Due to the fabrication methods of porous SMAs, the pores

are not regularly distributed and the pore size varies

between 100 and 500 m. Therefore, methods which assume

regular distribution predict the critical stress of phase

transformation more than reality. In the present method,the pores are randomly distributed which could be a good

estimation if the pore sizes are rened adequately. In this

regard, we consider an RVE which is constructed from a cube

with sides of 1 mm. The commercial nite element software

ABAQUS was used for modeling. A regular mesh made of

8000 eight-node brick elements (20 20 20) is generated

and the porous microstructure is simulated by randomly

assigning elastic material properties with negligible stiffness

to a number of these elements according to the porosity

volume fraction. This modeling approach was also utilized

by DeGiorgi and Qidwai (2002) and Panico and Brinson

(2008), where they employed a dense SMA model. For the

purpose of comparison, here we implement the proposed

model for porous SMAs a0and also the simplied model

for dense SMAs 0in a user dened subroutine UMAT in

ABAQUS.

Different RVEs of porous SMAs with porosities of 1060%

were prepared.Fig. 1shows the RVEs with porosities of 10%

and 40% where the porous microstructure is represented by

the white elements. We highlight that although the porositypattern may be different from the real one, the macroscale

behavior of porous SMAs is of good accuracy. We checked the

convergence of the CSM results by mesh renement as well

as different random pore patterns for the porosity range of

interest. Fig. 2 shows the results of pseudo-elastic behavior

for the case with maximum porosity (60% porous sample

under axial loading). It is observed from Fig. 2a that the

results with 8000 and 27,000 elements are almost the same.

Therefore, the model with 8000 elements used in this study is

of good accuracy. Also, Fig. 2b explores the results for three

different random mesh patterns which show that different

mesh patterns have no considerable effect on macro-scale

response of the sample. We remark that, since the thermal

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

300

350

Axial strain [ % ]

Axialstress[MPa]

10%CSM

10%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

300

350

Axial strain [ % ]

Axialstress[M

Pa]

20%CSM

20%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.550

0

50

100

150

200

250

Axial strain [ % ]

Axialstress[MPa]

40%CSM

40%Proposed Model

0 1 2 3 4 520

0

20

40

60

80

100

120

Axial strain [ % ]

Axialstress[MPa]

60%CSM

60%Proposed Model

Fig. 9 Comparison of stressstrain curves for porous SMAs (T310 K) under uniaxial loading for different porosities: (a) 10%,

(b) 20%, (c) 40%, and (d) 60%.




expansion is a secondary effect, we set the thermal expan-

sion coefcient to zero in the rest of the paper.

3.2. Validation of the computational approach

In this section, we rst utilize the experimental results by

Zhao et al. (2005) to validate the computational simulation

approach. We also compare the CSM results with those

predicted by other modeling methods. Entchev and

Lagoudas (2002) and Qidwai et al. (2001) using micromecha-

nical averaging approach and unit cell nite element method,

respectively.

The material parameters of dense SMA reported inTable 2

are determined using dense SMA experiment on porous NiTi

specimens inZhao et al. (2005).Fig. 3shows the stressstrain

curves of experiment results in comparison with the CSM

results for dense and 13% porous SMA. Good agreement is

observed particularly in the forward phase transformation.

We now compare the CSM results with other modeling

approaches already discussed. We consider the

micromechanical averaging approach and the unit cell nite

element method. Entchev and Lagoudas (2002) and Qidwai

et al. (2001)used the same dense SMA model and parameters

Table 3 and studied porous SMA behavior using these two

methods. The CSM results for different porosities are now

compared with the two modeling approaches. Fig. 4 com-

pares stressstrain curves for dense and porous SMAs with

porosities of 10%, 20% and 40%. In low porosities (10% and

20%) the CSM and the two approaches exhibit identical

results. However, in higher porosities (40%) the results of

the micromechanical approach, the unit cell FEM, and the

CSM are not similar. We should highlight that for high

porosities, the micromechanical approach as well as the unit

cell FEM overestimate the critical stress for transformation

since the pore morphology of the real material is not periodic.

Since the pores in real porous SMAs are usually random, the

results of random mesh RVEs are closer to the real porous

SMA behavior (DeGiorgi and Qidwai, 2002).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6100

0

100

200

300

400

500

600

700


Hydrostaticpressu

re[MPa]

10%CSM

10%Proposed Model

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.850

0

50

100

150

200

250

300

350

400

450


Hydrostaticpressu

re[MPa]

20%CSM

20%Proposed Model

0 0.5 1 1.5 2 2.5 350

0

50

100

150

200



40%CSM

40%Proposed Model

0 1 2 3 4 510

0

10

20

30

40

50

60

70



60%CSM

60%Proposed Model

Fig. 10 Comparison of hydrostatic pressurevolumetric strain curves for porous SMAs (T310 K) under hydrostatic loading

for different porosities: (a) 10%, (b) 20%, (c) 40%, and (d) 60%.




3.3. Uniaxial and hydrostatic loadings

In the following, we report the material parameters adopted

from the experimental results ofSittner et al. (1995) and also

used by Auricchio and Petrini (2004) for a dense Cu-based

SMA in Table 4 and study the pseudo-elastic behavior and

shape memory effect of porous SMAs under uniaxial and

hydrostatic loadings. Porous RVEs have been generated with

different porosities of 10%, 20%, 40% and 60%.

First, the response of the computational model is studied

under uniaxial loading. The RVE is subjected to axial strain

with maximum of 4%. The stressstrain curves for different

porosities are demonstrated inFig. 5a and b at high and low

temperatures, respectively. The results show that the critical

stress and hardening of the transformation decrease by

increasing the porosity. Moreover, the maximum transforma-

tion strain for porous SMAs increases with porosity under

uniaxial loading.

We then study the material behavior under hydrostatic

loading via the CSM. The hydrostatic stress versus volumetric

strain, , is plotted in Fig. 6 for porous SMAs with different

porosities. The simulations were carried out at high and low

temperatures T310 K and 220 K) to show the pseudo-elastic

behavior and shape memory effect of porous SMAs, respec-

tively. Martensitic phase transformation is due to deviatoric

stress. However, in porous SMAs (with random pores) the

hydrostatic loading locally generates deviatoric stresses.

These local stresses induce local transformation strain which

can be observed in the global behavior of the porous RVE.

Therefore, porous SMAs exhibit transformation strain even

under pure hydrostatic loading. While the transformation

stress decreases with porosity increase, the maximum of

volumetric transformation strain increases. Moreover, the

hardening of porous SMAs in hydrostatic loading during

transformation substantially decreases with porosity

increase.

The above results reveal that although the porous SMA

samples do not exhibit complete linear hardening during

phase transformation, the behavior is linear in a consider-

able portion of phase transformation. Also, according to low

temperature results (Fig. 5b and 6b), the recoverable strain

(either uniaxial or volumetric) in porous SMAs increases

0 1 2 3 4 50

50

100

150

200

250

300

350

400

Equivalent strain [ % ]

Equivalentstress[MPa]

10%CSM

10%Proposed Model

0 1 2 3 4 50

50

100

150

200

250

300

350

400



20%CSM

20%Proposed Model

0 1 2 3 4 50

50

100

150

200

250


Equivalentstress[MP

a]

40%CSM

40%Proposed Model

0 1 2 3 4 5 60

20

40

60

80

100

120


Equivalentstress[MP

a]

60%CSM

60%Proposed Model

c

Fig. 11 Comparison of equivalent stressstrain curves for porous SMAs (T310 K) under biaxial loading for different

porosities: (a) 10%, (b) 20%, (c) 40%, and (d) 60%.




with porosity. Moreover, the hardening coefcient is large for

lower porosities under hydrostatic loading (Fig. 6). We high-

light that in the phase transformation region, a considerable

part of the RVE still deforms elastically.

It is concluded from the CSM and experimental results

that porous SMAs have the major features of SMAs. In uniaxial

loading, due to the stress concentration and triaxiality around the

pores, phase transformation starts at lower stresses compared to

500 0 500200

150

100

50

0

50

100

150

200

p [ MPa ]

s[MPa]

10% 20% 40% 60%

B

A G

E F

CD

2 1.5 1 0.5 0 0.5 1 1.5 28

6

4

2

0

2

4

6

8


Shear

strain[%]

10%Proposed Model

10%CSM

3 2 1 0 1 2 3

6

4

2

0

2

4

6


Shear

strain[%]

20%Proposed Model

20%CSM

5 0 5

6

4

2

0

2

4

6


Shearstrain[%]

40%Proposed Model

40%CSM

6 4 2 0 2 4 68

6

4

2

0

2

4

6

8


Shearstrain[%]

60%Proposed Model

60%CSM

Fig. 12 Non-proportional shearhydrostatic loading (square shaped) for different porosities of porous SMAs atT310 K:

(a) schematic of boundary value problem, (b) shearpressure loading prole, shear strainvolumetric strain outputs for (c) 10%

porous SMA, (d) 20% porous SMA, (e) 40% porous SMA, and (f) 60% porous SMA.




dense SMAs. Moreover, in contrast to dense SMAs, phase

transformation also occurs under pure hydrostatic loading.

4. Numerical results of the constitutive modelfor porous SMAs

In this section, the method employed for material parameter

identication is rst explained. The model and CSM results

are then compared and different multiaxial (proportional and

non-proportional) loadings are studied to show the capability

of the proposed model in predicting the porous SMA behavior

under general thermomechanical loadings.

4.1. Material parameter identication

In order to determine the material parameters for the proposed

model, we use the CSM results presented in Section 3.3. The

parameters of elastic behavior (Eand), limit function (Rand),

hardening parameter (h) and maximum transformation strain (L)

are determined using low temperature (martensite phase) CSM

results under uniaxial and hydrostatic loadings (Figs. 5b and6b).

Employing high temperature (austenite phase) uniaxial results of

the CSM, the temperature parameter () is also determined

(Fig. 5b). Following this method, the material parameters are

determined for different porosities as reported inTable 5.

4.2. Comparison of the model and CSM results

In this section, we use the material parameters identied in

Table 5and compare the predictions of the proposed model with

the CSM results. First, the results of shape memory effect at low

temperature (T220 K) are presented.Fig. 7compares the model

and CSM results for porous SMAs with porosities of 1060% under

uniaxial loading which are in good agreement for different por-

osities. Fig. 8 compares the model and CSM results for porous

SMAs with porosities of 1060% under hydrostatic loading. It

reveals that pressure dependency increases with porosity and

volumetric change of about 4% occurs for highly porous SMAs.

However, the accuracy of the model results is higher for higher

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

300


Equivalentstress

[MPa]

10%CSM

10%Proposed Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250


Equivalentstress

[MPa]

20%CSM

20%Proposed Model

0 1 2 3 4 50

20

40

60

80

100

120

140

160



40%CSM40%Proposed Model

0 1 2 3 4 5 60

10

20

30

40

50

60

70

80

90



60%CSM60%Proposed Model

Fig. 13 Comparison of equivalent stressstrain curves for porous SMAs (T220 K) under biaxial loading for different

porosities: (a) 10%, (b) 20%, (c) 40%, and (d) 60%.




porosities. It is due to the difference between hardening behavior

under uniaxial and hydrostatic loadings for low porosities. The

results show the capability of the model in describing the shape

memory behavior of porous SMAs under uniaxial and hydrostatic

loadings.

In the following, we verify the model predictions for pseudo-

elastic behavior at high temperatures.Fig. 9compares the model

and CSM results for porous SMAs with porosities of 1060%

under uniaxial loading. The model results are in good agreement

with the CSM results in forward and reverse transformations.

Also, the hardening behavior under uniaxial loading is captured

by the model especially for lower porosities. Moreover, Fig. 10

compares the model and CSM results for porosities of 1060%

under hydrostatic loading. The results show the capability of the

model in predicting pseudo-elastic behavior. Similar to the low

temperature results, the hardening behavior under hydrostatic

loading is predicted more accurately for higher porosities. It is

noted that although the material parameters, except , were

determined using shape memory effect curves (Figs. 5b and6b),

the model results are in good agreement with the CSM results.

While the proposed model employs a single continuum

element, the RVE of the CSM is non-homogeneous (pores and

SMA) and consists of thousands of elements. Therefore, the

computational cost is greatly reduced by utilizing the porous

SMA constitutive model.

4.3. The model predictions for general thermomemchanical

loadings

In this section, we compare the model predictions and the CSM

results for the behavior of porous SMAs under general thermo-

mechanical loadings. It is noted that the computational cost is

considerably reduced when the proposed model is employed

instead of the CSM. In the following, we study the model results

for pseudo-elasticity and shape memory effects of porous SMAs.

4.3.1. Pseudo-elastic behavior

To show the capability of the model in different loadings,

pseudo-elastic behavior of porous SMAs are studied under

proportional and non-proportional loadings. First, the results

for a proportional loading atT310 K are presented. InFig. 11

01

23

4

220

240

260

280

300

320

0

50

100

150

200

Strain[%]

Temperature[K]

Temperature[K]

Stres

s[MPa]

10%Proposed Model

10%CSM

01

23

4

220

240

260

280

300

320

0

50

100

150

200

Strain[%]

Stres

s[MPa]

20%Proposed Model

20%CSM

01

23

4

220

240

260

280

300

320

0

50

100

150

Strain[%]

Temperature[K]

Stress[MPa]

40%Proposed Model

40%CSM

01

23

45

220

240

260

280

300

320

0

20

40

60

80

100

Strain[%]

Temperature[K

]

Stress[MPa]

60%Proposed Model

60%CSM

Fig. 14 Comparison of the model and CSM results for shape memory effect of porous SMAs under uniaxial loading for

different porosities.




equivalent stressstrain curves2 of biaxial loading is com-

pared with the CSM results for different porosities from 10%

to 60%. The comparison reveals that the model and CSM

results are in good agreement. The model predicts acceptable

results at different porosities for forward and reverse trans-

formations as well as saturated condition.

To show the pressure dependency of porous SMAs in more

detail, we now present the model and CSM results at T310 K

for a boundary value problem of a porous SMA cube under

shearhydrostatic loading as shown inFig. 12. As demonstrated

inFig. 12a, we x one edge of the cube and a shear load (s) is

applied on the opposite edge. Moreover, we simulate hydrostatic

pressure by applying normal load (p) on otherve edges of the

cube. The square-shaped loading prole ABCDEFGA is presented

inFig. 12b for different porosities. To present reasonable results,

different amplitudes of the loading prole is considered for

different porosities; the amplitudes decrease with porosity

increase due to load bearing capability of porous SMAs. Axial

strainvolumetric strain outputs for the model and CSM are

shown inFig. 12cf for porosities of 1060%. The results show

good agreement between the model and CSM for different

porosities. Therefore, the model is capable of predicting behavior

under general non-proportional loading.

Moreover, there is a coupling between shear and volumetric

responses in all porous SMA samples. This coupling has also

been observed in non-proportional loadings with inelastic

strains. In other words, the loading induces both volumetric

and shear transformation strains thus coupling between shear

and volumetric responses is observed. Specically, shear loading

induces volumetric strain in paths BC, DE and FG; while

hydrostatic loading induces shear strain in paths CD and EF.

We may also notice that in several paths, one of the strain

responses reaches a maximum and then goes down. For

example, in path CD in which only hydrostatic stress changes

shear strain reaches a maximum and then decreases.

4.3.2. Shape memory effect

Here, we show the model predictions for shape memory

effect of porous SMAs. First, equivalent stressstrain curves

00.5

11.5

2

220

240

260

280

300

320

0

100

200

300

400

500

600

Volumetricstrain[%

]Volumetri

cstrain[%]

Temperature[K]

Temperature[K]

Temperature[K]

Temperature[K]


10%Proposed Model

10%CSM

0 0.51

1.5 22.5

3

220

240

260

280

300

320

0

100

200

300

400

500


20%Proposed Model

20%CSM

01

23

4

220

240

260

280

300

320

0

50

100

150

200

Strain[%]

Stress[MPa]

40%Proposed Model

40%CSM

0 12 3

4 56

220

240

260

280

300

320

0

20

40

60

80

Strain[%]

Stress[MPa]

60%Proposed Model

60%CSM

Fig. 15 Comparison of the model and CSM results for shape memory effect of porous SMAs under hydrostatic loading for

different porosities.

2Similar to (11) an d (17), equivalent stress and strain are

dened as

eqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2

JsJ2 p2

1 =6s

; eqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3 1=6 J

eJ2

1

2 s.




of biaxial loading is compared with the CSM results inFig. 13

for porosities of 1060%. The results show good agreement in

different regions (elastic, phase transformation and satura-

tion) for different porosities.

Moreover, we demonstrate the model capability in predicting

the shape memory effect of porous SMAs. The loading consists

of a mechanical loading (uniaxial or hydrostatic) and a thermal

loading. InFigs. 14and15, the porous SMA samples are under

uniaxial and hydrostatic loadings, respectively. The initial tem-

perature is 220 K that is below the reference temperature Tm.

Phase transformation occurs due to the mechanical loading and

by unloading to zero stress, a mechanically unrecoverable strain

remains. However, this strain is recovered after heating the

materials above the austenite nish temperature. Finally, cooling

the strain-free material to the initial temperature does not alter

its strain or stress state.

The results show very good agreement under uniaxial

loading for different porosities as well as under hydrostatic

loading for higher porosities. This agreement is observed in

different regions: elastic, phase transformation, saturation and

recovery due to temperature increase. For the case of hydro-

static loading on low porosity samples (10% and 20%), since a

considerable portion of the sample deforms elastically even

under high levels of stress, the hardening during transforma-

tion is not predicted well. However, qualitative behavior in all

regions and quantitative behavior in the elastic region and

starting of phase transformation are predicted well.

The model and CSM results show that axial strain recov-

ery increases by porosity. A complete axial strain recovery of

about 4.5% is observed for a 60% porous SMA sample (Fig. 14).

Also, under hydrostatic pressure loading, the volumetric

strain recovery increases with porosity. Fig. 15 depicts a

complete volumetric strain recovery of about 5.5% for a 60%

porous SMA sample.

5. Summary and conclusions

Due to the mechanical properties, porous SMAs can be used

effectively in many applications by controlling porosity especially

as biomaterials. In this study, we proposed a phenomenological

constitutive model for porous SMAs which is capable of predict-

ing phase transformation behavior under general proportional

and non-proportional loadings. To develop such a model, proper

internal variables for phase transformation were introduced

according to pressure dependency of these materials. The free

energy and limit functions were then adopted. Due to lack of

enough experimental data, we rst validated a computational

simulation method (CSM) with the available experiments. The

CSM was then employed to calibrate the proposed model. While

utilizing this model signicantly reduces the computational cost

comparing to the CSM, good agreement between the numerical

predictions of the model and CSM results was observed in

various thermomechanical loadings. The results show that the

recoverable strain in porous SMAs increases with porosity under

uniaxial or hydrostatic loadings. Also, due to stress concentration

in porous SMAs, phase transformation starts earlier. Moreover,

the coupling between shear and volumetric responses is obser-

ved in non-proportional loading of porous SMAs.

As a macro-scale model, by adding only one material

parameter () and one internal variable (tr) for porosity effects,

and utilizing a simple pressure dependent limit function, the

proposed model is capable of predicting the behavior of porous

SMAs. In other words, the model is capable of predicting

volumetric and deviatoric parts of transformation strain in

different regions of phase transformation (starting and nishing

of transformation as well as hardening during transformation).

Also, temperature dependent behavior is predicted in this

model. Therefore, due to simplicity together with computa-

tional efciency, the proposed model can be used for simulation

and design of the new complex architectured materials and

biomedical implants made of porous SMAs. By introducing

higher order terms in the energy function and utilizing more

parameters for describing behavior of porous SMAs, it is

possible to capture nonlinear behavior which is observed

especially at higher strains and under hydrostatic as well as

non-proportional loading. The proposed constitutive model can

be extended to include permanent strain, important in the

applications with high stress levels, which is the subject of

future works.

Appendix A

We use the NewtonRaphson method to solve the nonlinear

equations(26) and (27). For simplicity, we focus only on the

case of saturated phase transformation. Eqs. (27) are solved

for nine unknowns (second-order tensorXs and scalarsXp;

and ). The coefcients of the linearized equations are as

follows:

KL

RXsXs RXsXp

RXs RXs

RXpXs R

XpXp R

Xp R

Xp

RXs R

XpR R

RXs R

XpR R

2666666437777775

28

The consistent tangent matrix D can be computed as a

linearization of the stress:

D d

d29

The linearization of volumetric and deviatoric parts of the

stress and strain tensors are

dp K 1dtr

d

d

ds 2G Idetr

de

:de 30

d 1 : dde Idev : d 31

where Idev I13 1 1. To determine the consistent tangent

matrix, the derivatives dtr=d and detr=de should be deter-

mined. Linearization of the two internal variables yields

detr etrXs :dXs etrXp

dXp etrd e

tr d

dtr trXs :dXstrXp

dXptrd

tr d 32

If we now consider equations (27) as functions of

Xs; Xp;; ; eand , the corresponding linearization gives

RXsXs :dXsRXs

Xp dXpRXs

dRXs dRXse :de RXs d 0




RXpXs

:dXsRXpXp

dX2 RXpdR

Xp d R

Xpe :de R

Xp d 0

RXs :dXsR

XpdXp R

dR

dR

e :de R

d 0

RXs :dXs R

XpdXpR

dRd R

e :de R

d 0 33

in which

RXse 2GI; RXs 0

T

RXpe 0; RXp KRe 0; R

0

Re 0; R 0 34

Therefore, using(28) and (31) we may rewrite(33) and (34)

in matrix form as

dXs

dXp

d

d

266664

377775 K 1L

2GIdevK1T

0T

0T

26664

37775 :d 35

Substituting (35) into (32) and the result into (30), we

obtain

ds 2G Idev etrXs

etrXp etr e

tr

h iK

1L

2GIdevK1T

0T

0T

2666437775

0BBB@1CCCA :d

dp K 1T trXs trXp

tr tr

h iK

1L

2GIdevK1T

0T

0T

26664

37775

0BBB@

1CCCA :d 36

We now dene the matrices DK and DE as follows:

DK 1 trXs trXp

tr

tr

h iK

1L

2GIdevK1T

0T

0T

2

6664

3

7775

DE etrXs

etrXp etr e

tr

h iK

1L

2GIdevK1T

0T

0T

26664

37775 37

The consistent tangent matrix may then be expressed as

D K 1 1 DK 2G Idev DE 38

r e f e r e n c e s

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