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Available online at www.sciencedirect.com
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.2014.11.0237/21/2019 A-3-D-constitutive-model-for-pressure-dependent-phase-transformation-of-porous-shape-memory-alloys_2015_Jour
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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
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(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
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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
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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.
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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
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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
Volumetric strain [ % ]
Hydrostaticstress[MPa]
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
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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 .
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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
Volumetric strain [ % ]
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
Volumetric strain [ % ]
Hydrostaticpressur
e[MPa]
20%CSM
20%Proposed Model
0 0.5 1 1.5 2 2.5 30
20
40
60
80
100
120
Volumetric strain [ % ]
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
Volumetric strain [ % ]
Hydrostaticpressure[MPa]
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%.
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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%.
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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
Volumetric strain [ % ]
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
Volumetric strain [ % ]
Hydrostaticpressu
re[MPa]
20%CSM
20%Proposed Model
0 0.5 1 1.5 2 2.5 350
0
50
100
150
200
Volumetric strain [ % ]
Hydrostaticpressure[MPa]
40%CSM
40%Proposed Model
0 1 2 3 4 510
0
10
20
30
40
50
60
70
Volumetric strain [ % ]
Hydrostaticpressure[MPa]
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%.
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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
Equivalent strain [ % ]
Equivalentstress[MPa]
20%CSM
20%Proposed Model
0 1 2 3 4 50
50
100
150
200
250
Equivalent strain [ % ]
Equivalentstress[MP
a]
40%CSM
40%Proposed Model
0 1 2 3 4 5 60
20
40
60
80
100
120
Equivalent strain [ % ]
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%.
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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
Volumetric strain [ % ]
Shear
strain[%]
10%Proposed Model
10%CSM
3 2 1 0 1 2 3
6
4
2
0
2
4
6
Volumetric strain [ % ]
Shear
strain[%]
20%Proposed Model
20%CSM
5 0 5
6
4
2
0
2
4
6
Volumetric strain [ % ]
Shearstrain[%]
40%Proposed Model
40%CSM
6 4 2 0 2 4 68
6
4
2
0
2
4
6
8
Volumetric strain [ % ]
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.
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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
Equivalent strain [ % ]
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
Equivalent strain [ % ]
Equivalentstress
[MPa]
20%CSM
20%Proposed Model
0 1 2 3 4 50
20
40
60
80
100
120
140
160
Equivalent strain [ % ]
Equivalentstress[MPa]
40%CSM40%Proposed Model
0 1 2 3 4 5 60
10
20
30
40
50
60
70
80
90
Equivalent strain [ % ]
Equivalentstress[MPa]
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%.
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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.
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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]
Hydrostaticstress[MPa]
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
Hydrostaticstress[MPa]
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.
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 0 307
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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
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 0308
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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
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