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Multiscale analysis of the effect of interfacial energy on non-monotonic
stress–strain response in shape memory alloys
S. Stupkiewicz, M. Rezaee-Hajidehi, H. Petryk
⇑
Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawin
´skiego 5B, Warsaw 02-106, Poland
article info
Article history:
Received 30 November 2019
Received in revised form 28 March 2020
Accepted 3 April 2020
Available online 14 April 2020
Keywords:
Microstructures
Martensitic transformation
Size effects
Incremental energy minimization
Propagating instabilities
abstract
The effect of formation and evolution of stress-induced martensitic microstructures on macroscopic
mechanical properties of shape memory alloys in the pseudoelastic regime is investigated with account
for size-dependent energy of interfaces. A quantitative relationship is established between the changes in
free energy and dissipation on the interfaces at three microstructural scales and the overall mechanical
characteristic of the material under tensile loading. The multiscale analysis carried out for a polycrys-
talline NiTi shape memory alloy has revealed that the interfacial energy storage and dissipation can
strongly affect the shape and width of the stress–strain hysteresis loop. The predicted non-monotonic
stress–strain response for the material of a selected grain size shows a remarkable similarity to the exper-
imental one extracted from a tensile test of a laminate by Hallai and Kyriakides (2013). By the classical
Maxwell construction, the non-monotonic response for a material element results in a commonly
observed stress plateau for a tensile specimen, which is associated with the propagation of phase trans-
formation fronts. This behaviour is confirmed with striking accuracy by 3D finite-element computations
performed for a macroscopic tensile specimen, in which propagating instability bands are treated
explicitly.
Ó2020 Institute of Fundamental Technological Research PAS. Published by Elsevier Ltd. This is an open
access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
1. Introduction
The unique behaviour of shape memory alloys (SMAs), notably
the shape memory effect and pseudoelasticity, results from the
martensitic phase transformation (Otsuka and Wayman, 1998).
The transformation is reversible in the sense that the overall
inelastic strain can be induced and fully recovered upon adequate
thermomechanical loading. This is accompanied by formation and
evolution of martensitic microstructures at several spatial scales so
that the martensitic transformation in SMA polycrystals is a truly
multi-scale phenomenon. The present paper deals with quasi-
static, stress-induced and nearly isothermal transformation in the
pseudoelastic regime, with the emphasis put on rarely studied
effects of the interfacial energy and dissipation at lower-scale
interfaces.
The overall behaviour of SMAs is described by numerous phe-
nomenological models (e.g., Raniecki et al., 1992; Auricchio and
Petrini, 2002; Lagoudas et al., 2012; Sedlák et al., 2012;
Stupkiewicz and Petryk, 2013; Qiao and Radovitzky, 2016; Jiang
et al., 2016), and many others, where basic properties of SMAs
are assumed at the macroscopic level. However, several important
features of the macroscopic behaviour of SMAs can be predicted by
micromechanical models that typically consider the individual
martensite variants within the differently oriented grains and
apply scale-transition schemes to arrive at the overall response
of a polycrystalline aggregate. A number of related micromechan-
ical models of various complexity and various predictive capabili-
ties can be found in the literature, cf. some representative models
(Siredey et al., 1999; Gall and Sehitoglu, 1999; Šittner and Novák,
2000; Thamburaja and Anand, 2001; Hackl and Heinen, 2008;
Levitas and Ozsoy, 2009; Sengupta and Papadopoulos, 2009;
Stupkiewicz and Petryk, 2010a; Yu et al., 2015; Xiao et al., 2019)
and a review (Cisse et al., 2016).
The feature that is not yet included (to the best knowledge of
the authors) in any existing micromechanical model of polycrys-
talline SMAs is the effect of interfacial energy at lower-scale inter-
faces on the overall non-monotonic response of a material sample.
The entire formation and evolution of microstructure is accompa-
nied by nucleation, propagation and annihilation of interfaces at
multiple scales. Depending on the type, character and relevant spa-
tial scale, each interface is associated with some energy – the inter-
facial energy of the density referred to the area of an interface.
Evolution and ultimately annihilation of an interface may also be
associated with dissipation of the interfacial energy. The related
https://doi.org/10.1016/j.ijsolstr.2020.04.006
0020-7683/Ó2020 Institute of Fundamental Technological Research PAS. Published by Elsevier Ltd.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
⇑
Corresponding author.
E-mail address: hpetryk@ippt.pan.pl (H. Petryk).
International Journal of Solids and Structures 221 (2021) 77–91
Contents lists available at ScienceDirect
International Journal of Solids and Structures
journal homepage: www.elsevier.com/locate/ijsolstr
contributions to the total free energy and dissipation in a macro-
scopic volume element depend on the characteristic dimensions
of microstructural elements, and thus the overall energy balance
includes size-dependent contributions. As a result, the interfacial
energy can govern the overall behaviour of SMAs in a size-
dependent manner, see (Petryk and Stupkiewicz, 2010) for a gen-
eral theory and (Petryk et al., 2010) for its application to a simpli-
fied case of a rank-three laminate. The conclusion that the
interfacial energy can significantly influence the overall response
of SMAs can be traced back to Müller and Xu (1991), but the
respective macroscopic free energy factor has remained unknown.
The aim of this work is to develop a hierarchical multi-scale frame-
work for predictive modelling of the interfacial energy effects in
SMA polycrystals.
The specific focus of this work is on pseudoelastic behaviour of
SMA polycrystals under quasi-static tension, with the restriction to
the stress-induced transformation and isothermal conditions. In
this regime, the material is initially in the austenitic state, and
application of a mechanical load, after an initial stage of elastic
response, induces the martensitic transformation that is accompa-
nied by an overall inelastic strain. Upon unloading, the inelastic
strain vanishes, and the initial undeformed state is recovered.
The pseudoelastic response exhibits then two characteristic fea-
tures, namely the hysteresis and frequently the stress plateau on
the nominal stress–elongation diagram. The origin of the plateau
and hysteresis has been widely discussed in the literature, but
apparently not yet fully clarified. The present paper is intended
to make a progress in this direction, although with the limitation
of the analysis to the tensile loading/unloading.
Nowadays, the aforementioned stress–elongation plateau is, or
should be, no longer understood as a physical property of the
material, but rather as the effect of propagating instability due to
a negative slope on a part of the stress–strain diagram for the mate-
rial. This has been clearly shown in the experiment by Hallai and
Kyriakides (2013), in which the intrinsic material response of NiTi
under tension has been extracted from a laminate of a NiTi strip
and steel face-strips that practically enforce a uniform deformation
of the NiTi strip. Such a uniform deformation cannot be observed
for a free-standing strip because the negative slope of the uniaxial
stress–strain diagram leads to path instability in the form of strain
localization. The transformation proceeds then through propaga-
tion of macroscopic transformation fronts that separate the trans-
formed and untransformed domains. The fronts propagate at an
approximately constant load under end-displacement control,
hence a plateau is observed on the load–elongation diagram which
should not be interpreted as a stress–strain diagram for the mate-
rial. Fig. 1 reproduced from (Hallai and Kyriakides, 2013) shows
both the intrinsic material response and the specimen response.
The former exhibits a negative slope on a significant part of the
stress–strain diagram, the latter shows a load plateau.
The scenario discussed above (from propagating instability to
load plateau) is in fact commonly observed in experiment. Strain
localization and propagation of Lüders-like bands have been
observed in NiTi strips and wires under tension (Shaw and
Kyriakides, 1997; Pieczyska et al., 2006; Churchill et al., 2009;
Zhang et al., 2010; Sedmák et al., 2016), and instabilities leading
to complex transformation patterns have been observed in NiTi
tubes under tension and combined tension–torsion (Sun and Li,
2002; Reedlunn et al., 2014; Bechle and Kyriakides, 2016) and
under bending (Bechle and Kyriakides, 2014; Reedlunn et al.,
2014). The stress plateau has also been observed in numerous
experiments without direct characterization of strain localization.
The stress plateau during the forward transformation under
tension is often accompanied by a lower stress plateau during
the reverse transformation upon unloading, with the related
rate-independent hysteresis (e.g., Shaw and Kyriakides, 1997;
Churchill et al., 2009; Bechle and Kyriakides, 2014; Reedlunn
et al., 2014). This suggests that the respective material stress–
strain response is also non-monotonic, however, it has never been
determined experimentally.
From a mathematical point of view, the effect of a non-
monotonic local response on macroscopic behaviour in 1D setting
has been widely studied in the literature, following
Ericksen (1975). Several macroscopic models of phenomenological
type have also been developed for SMAs and implemented numer-
ically in order to simulate the related phenomena in 1D setting
(Chang et al., 2006; Alessi and Bernardini, 2015; León Baldelli
et al., 2015; Rezaee-Hajidehi and Stupkiewicz, 2018) and in
2D/3D setting (Duval et al., 2011; Jiang et al., 2017a; 2017b;
Badnava et al., 2018; Rezaee-Hajidehi et al., 2020). A simplified
form of the non-monotonic local response is then usually adopted,
often a piecewise-linear one, although the actual response may
exhibit a visible non-linearity, cf. Fig. 1. Note that a non-
monotonic local response results also from phase-field approaches
applicable to SMA single crystals (e.g. Esfahani et al., 2018; Tu
˚ma
et al., 2018). The models mentioned above involve some kind of
regularization (gradient enhancement, non-local formulation, or
viscous effects) that introduces a characteristic length-scale into
the model and improves the robustness of the corresponding com-
putational schemes. As an exception, the only regularization in the
model of Jiang et al. (2017a,b) is that due to the natural 3D effects
at the macroscopic transformation fronts.
In this paper, a multiscale framework for a quantitative analysis
of the effect of interfacial energy on non-monotonic stress–strain
response of pseudoelastic SMA polycrystals is developed, appar-
ently for the first time. The framework involves three essentially
independent models, two of them are micromechanical models
themselves, and the multi-scale coupling between the models is
performed in a hierarchical manner, i.e. the results of the model
at a lower scale are transferred to the model at a higher scale in
the form of constitutive functions or parameters that are identified
using the lower-scale data. The multiscale framework is schemat-
ically shown in Fig. 2, and the structure of the paper is presented
below. The whole approach is built on the previous extensive work
by the authors, which is referred to in the sequel. Each specific
model used in this work relies on the incremental energy mini-
mization, and the related framework that includes the interfacial
energy effects is briefly recalled in Section 2.
At the lowest scale, we apply the model of a spherical sub-grain
with a rank-two laminate of twinned martensite plates within the
austenite matrix to determine the size-dependent interfacial
energy contributions coming from the interfaces at three scales:
at the twin boundaries, at the austenite–martensite interfaces,
Fig. 1. Nominal stress–elongation response measured for a NiTi strip under tension
and the intrinsic material response extracted from a laminate with steel face-strips,
after Hallai and Kyriakides (2013) (reproduced with permission from Elsevier).
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
78
and at the boundary of the sub-grain. The model is adopted from
(Stupkiewicz and Petryk, 2010b) and is briefly presented in Sec-
tion 3.1; the basic equations are provided in Appendix A. In this
model, the results of yet another micromechanical model (which
is not indicated in Fig. 2 nor discussed in detail in the paper) are
utilized, namely the estimate of the elastic micro-strain energy at
the austenite–twinned martensite interface in NiTi is taken from
(Stupkiewicz et al., 2012). The outcome of the sub-grain model,
i.e. closed-form expressions for the size-dependent interfacial
energy contributions, constitutes then the input to the model of
a SMA polycrystal to be considered at a higher scale.
At the intermediate scale, the overall response of a SMA poly-
crystal is obtained by applying a respective multiscale model that
is enhanced with the interfacial energy effects. This is presented in
Section 4. First, in Section 4.1, the interfacial energy effects are
introduced to the model of a single grain, which is done in a quite
general setting without specifying the explicit form of the bulk free
energy and dissipation contributions. The interfacial energy contri-
butions are evaluated for a representative spherical sub-grain, and
the respective results are taken from Section 3.1. This general
framework is then combined with a multiscale model of a SMA
polycrystal. Here, the bi-crystal aggregate model
(Stupkiewicz and Petryk, 2010a) is employed, which is briefly
introduced in Section 4.2 and the equations are provided in
Appendix B. Application of the complete scheme to polycrystalline
NiTi under uniaxial tension is then reported in Section 4.3. The pre-
dicted stress–strain response is non-monotonic during both load-
ing and unloading and exhibits a significant dependence on the
grain size, which is due to the interfacial energy effects.
Considering that the non-monotonic stress–strain response is
expected to cause the path instability and strain localization, we
finally carry out 3D finite-element simulations of the tension test
in order to study the related effects quantitatively. To this end, in
Section 5, the macroscopic finite-strain model of pseudoelasticity
(Stupkiewicz and Petryk, 2013) is extended in order to accurately
represent the intrinsic material response predicted by the
micromechanical model in Section 4. Specifically, by following
the hierarchical multi-scale approach, the dissipation and free
energy functions are fitted to the respective data resulting from
the lower-scale analysis. To efficiently treat the instabilities and
nonuniform transformation patterns, the finite-element imple-
mentation of the model employs the gradient regularization that
has been recently developed by Rezaee-Hajidehi et al. (2020).
The finite-element simulations deliver the expected load–elonga-
tion response with two plateaus, as well as the transformation pat-
tern in the form of a growing martensite band with inclined
macroscopic transformation fronts.
2. Incremental energy minimization framework
It is well known that the stress-induced martensitic transfor-
mation in SMAs in the pseudoelastic regime proceeds on the
microscopic scale by formation and evolution of laminate
microstructures composed of crystallographic variants of marten-
site. In view of a large number of possible combinations of different
variants of martensite, the micromechanical modelling of SMAs
requires a method of selection of their physically favourable com-
bination for a single grain. Under the assumption of isothermal
conditions and an additional symmetry restriction imposed on dis-
sipation, this is offered by the incremental energy minimization
method (Petryk, 2003) extended to the interfacial contributions
to free energy and dissipation by Petryk and Stupkiewicz (2010).
On the time scale adopted in the rate-independent modeling, a
negative increment of the interfacial energy, as in Eq. (3) below,
is locally related to the annihilation of interfaces. This is associated
with a sudden release of the interfacial energy that can hardly be
reverted into the bulk free energy. Therefore, in the model it con-
tributes predominantly to the rate-independent dissipation
(Petryk and Stupkiewicz, 2010).
In outline, the incremental energy minimization framework
with interfacial energy effects on different scales indexed by s
can be summarized as follows (Petryk and Stupkiewicz, 2010):
(i) Split of the total free energy into bulk and interfacial
contributions,
U¼U
v
þU
i
;U
i
¼X
s
U
i
s
:ð1Þ
Fig. 2. Hierarchical multiscale modelling approach for the analysis of interfacial energy effects in SMA polycrystals.
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
79
(ii) Split of the dissipation increment into bulk and interfacial
contributions,
D
D¼
D
D
v
þ
D
D
i
;
D
D
i
¼X
s
D
D
i
s
:ð2Þ
(iii) Negative increments of interfacial energy contribute to
dissipation,
D
D
i
s
¼
j
s
h
D
U
i
s
i;0
j
s
1;hwi¼ wif w>0;
0ifw0:
ð3Þ
(iv) Microstructure evolution is determined by succesive mini-
mization of the incremental energy supply,
D
E¼
D
Uþ
D
D
!min subject to kinematical constraints:ð4Þ
The aforementioned scheme can be applied at any level of a
hierarchical model, either separately or jointly for all levels of a
polycrystalline aggregate. The theory is described by
Petryk (2003) and Petryk and Stupkiewicz (2010). Further details
of the micromechanical model are discussed in the next sections.
3. Compound interfacial energy in a spherical sub-grain
3.1. Interfacial energy contributions at three scales (Stupkiewicz and
Petryk, 2010b)
The model developed by Stupkiewicz and Petryk (2010b) is
employed here in order to estimate the interfacial energy contribu-
tions in a spherical domain. With reference to the micromechanical
model of a polycrystalline aggregate discussed in Section 4, con-
sider a spherical domain of diameter dthat is identified with a
sub-grain, i.e. a part of a grain occupied by a single family of inter-
nally twinned martensite plates (habit-plane variants) within the
austenite matrix, see Fig. 3. The microstructure is assumed suffi-
ciently fine so that, in view of the assumed periodicity, it is fully
characterized by specifying the volume fraction of martensite
g
,
the twin spacing h
tw
and the plate thickness M(or equivalently
the plate spacing H¼M=
g
).
Following Stupkiewicz and Petryk (2010b), by exploiting the
assumptions of separation of scales and periodicity of the
microstructure, the total free energy density (per unit volume of
the sub-grain) is decomposed into bulk and interfacial
contributions,
/ð
e
;
g
;M;h
tw
Þ¼/
v
ð
e
;
g
Þþ/
i
ð
g
;M;h
tw
Þ;
/
i
¼/
i
tw
þ/
i
am
þ/
i
gb
;ð5Þ
where
e
is the average strain in the sub-grain in the small-
deformation setting. We note that the entire interfacial contribution
/
i
does not depend on the strain
e
. Assumption (3), which we adopt
here, means that interfacial energy release is not fully converted
into the bulk elastic energy but is at least partially dissipated. The
interfacial contribution to dissipation is expressed through the neg-
ative increments of the interfacial contributions to the free energy,
and thus it does not depend on the strain either. Accordingly, by
applying the incremental energy minimization (4), the evolution
of the microstructural length parameters (here, Mand h
tw
) along
a transformation path parameterized by
g
can be determined by
minimizing the interfacial energy contributions independently of
the bulk part. The corresponding model developed for the spherical
sub-grain under consideration is briefly described below, and the
set of respective equations is provided in Appendix A. For more
details the reader is referred to Stupkiewicz and Petryk (2010b).
The analyzed microstructure, cf. Fig. 3, involves three types of
interfaces, namely the twin boundaries, the austenite–martensite
interfaces and the sub-grain boundary, each associated with the
corresponding spacial scale indicated by the subscript
s¼tw;am;gb. The interfacial energy contributions from the three
scales are specified by the respective densities /
i
s
per unit volume
of the sub-grain, cf. Eqs. (A.2)–(A.4).
Two sources of interfacial energy are accounted for, namely the
atomic-scale energy of phase boundaries and the energy of elastic
micro-strains. The atomic-scale interfacial energy is considered at
the twin boundaries and at the austenite–martensite interfaces,
and the corresponding energy densities per unit area are denoted
by
c
a
tw
and
c
a
am
;respectively. The transformation-induced changes
in the atomic-scale energy of grain (sub-grain) boundaries are
assumed negligible, and hence the respective atomic-scale interfa-
cial energy is not considered.
The energy of elastic micro-strains results from local incompat-
ibility of transformation strains. At the austenite–twinned marten-
site interfaces, the individual martensite variants are usually not
compatible with the austenite, while compatibility in the average
sense can be achieved due to twinning (Bhattacharya, 2003). The
energy of the elastic strains that accommodate the local incompat-
ibility, when integrated over the volume and referred to the nom-
inal interface area, is interpreted as an interfacial energy denoted
by
c
e
am
. Note that this energy is not a material parameter. It can
be shown that
c
e
am
is proportional to the twin spacing h
tw
, thus
c
e
am
¼C
e
am
h
tw
;where C
e
am
is a size-independent energy factor that
can be determined by energy minimization for each representative
microstructure along with the corresponding corrugated interface
shape. This approach has been developed in a series of papers cited
in (Stupkiewicz et al., 2012), where specific numerical results for
representative microstructures for a NiTi alloy can be found. Elastic
micro-strain energy is also associated with the martensite plates
that terminate at the sub-grain boundary, and the related energy
density /
i
gb
is proportional to the plate spacing H, cf. Eq. (A.4).
At the initial stage of transformation (
g
0), nucleation of the
first martensitic plates is associated with the increase of the inter-
facial energy, and hence the interfacial contribution to dissipation
is then equal to zero, cf. Eq. (3). Energetically optimal microstruc-
tural parameters h
0
tw
and M
0
, cf. Eq. (A.5), corresponding to the ini-
tial stage of transformation are then obtained by minimizing the
free energy alone, see Eq. (A.6) for the respective formula for the
compound interfacial energy at
g
0. Note the characteristic
square-root scaling rule for the parameters h
0
tw
and M
0
in
Eq. (A.5) with the characteristic lengths ‘
am
and ‘
gb
defined in
terms of the interfacial energy parameters, which is typical for
rank-one laminates (Khachaturyan, 1983). The subsequent evolu-
tion of the plate thickness M¼
g
His obtained by minimizing /
i
Fig. 3. Spherical sub-grain with periodic rank-two laminated microstructure and
interfacial energy at three scales.
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
80
for given
g
, while the twin spacing is kept constant, h
tw
¼h
0
tw
;cf.
Eqs. (A.7)–(A.8).
The predictive capabilities of the approach described above
have been confirmed, both qualitatively and quantitatively, by
the phase-field computations carried out by Tu
˚ma et al. (2016)
with no a-priori assumptions concerning the microstructure.
Specifically, the problem of a cylindrical grain with an austenite–
twinned martensite microstructure has been analyzed, and it has
been shown that the phase-field computations agree reasonably
well with the square-root scaling of M, as predicted by the present
approach, although with a somewhat different coefficient.
3.2. Application to NiTi alloy
The procedure developed by Stupkiewicz and Petryk (2010b)
and summarized in Section 3.1 and Appendix A is now applied to
a spherical sub-grain of a NiTi alloy. The computations are here
performed for three sub-grain sizes d¼13;38;113 mm that corre-
spond to the three grain diameters d
gr
¼20;60;180 mm considered
in Section 4.3.
To estimate the elastic micro-strain energy factor C
e
am
;we use
the results of the computations carried out by Stupkiewicz et al.
(2012) for the eight crystallographically distinct microstructures
at the austenite–twinned martensite interface in NiTi, and we
adopt the smallest value of this factor predicted for microstructure
C-I-2, namely C
e
am
¼9:7 MJ/m
3
, which is close to the earlier simple
estimate in Stupkiewicz and Petryk (2010b). The estimate of the
elastic micro-strain energy at the sub-grain boundary is given by
Eq. (A.4) and results from a simplified micromechanical model
which, unlike the model used to estimate C
e
am
;assumes elastic iso-
tropy and small-strain kinematics. The corresponding material
parameters are the elastic shear modulus
l
¼26:3 GPa, Poisson’s
ratio
m
¼0:33 and shape-strain vector modulus b¼0:1343;as in
(Stupkiewicz and Petryk, 2010b). Likewise, following
Stupkiewicz and Petryk (2010b), the atomic-scale interfacial ener-
gies are assumed equal to
c
a
tw
¼0:014 J/m
2
and
c
a
am
¼0:3 J/m
2
.
Parameters
j
s
are assumed to be close to unity, specifically,
j
tw
¼1;
j
am
¼0:95;and
j
gb
¼0:9.
In Fig. 4, the results of computations are presented in terms of
the evolution of plate thickness Mduring a complete forward
and reverse transformation cycle and in terms of the number of
plates per sub-grain, treated here as a continuous variable equal
to d/H. The predicted microstructural parameters at the initial
stage of transformation, cf. Eq. (A.5), are provided in Table 1.
Fig. 5 shows the individual interfacial contributions /
i
s
to the
free energy density, as predicted for d¼13 mm. Qualitatively, the
results corresponding to the other sub-grain diameters are the
same, the difference is only in the actual values of the individual
contributions. It can be seen that as long as /
i
am
and /
i
gb
increase,
and thus there is no related interfacial dissipation, the two contri-
butions are equal to each other, so that /
i
am
¼/
i
gb
for 0
g
0.5
during loading and for 0.5
g
1 during unloading. This feature
results from minimization of the interfacial contribution to the free
energy in the absence of the related dissipation (cf.
Stupkiewicz and Petryk, 2010b). For 0.5
g
1 during loading
and for 0
g
0.5 during unloading, the interfacial dissipation
defined by Eq. (A.8) is present in the incremental energy minimiza-
tion scheme (cf. Section 2). It prevents the number of plates and
hence the interfacial energy /
i
am
to decrease within some period
after the maximum if
j
gb
<
j
am
< 1, until eventually /
i
am
falls to
zero.
The sum of the two contributions, /
i
am
þ/
i
gb
, can be well
approximated by a quadratic function of the volume fraction
g
,
as indicated by the dashed line in Fig. 5. At the same time, the free
energy contribution of twin interfaces, /
i
tw
, is proportional to
g
. The
results of the present model can thus be summarized in the form of
the following useful formulae that will be exploited in Section 4,
/
i
tw
¼
g
/
max
tw
;/
i
am
þ/
i
gb
4
g
ð1
g
Þ/
max
amþgb
;ð6Þ
where
/
max
tw
¼
2
c
a
tw
h
0
tw
;/
max
amþgb
¼
C
e
am
h
0
tw
M
0:5
þ
c
a
am
M
0:5
þ
32
ffiffi
2
p
2
C
e0
gb
M
0:5
d
;
M
0:5
¼
M
0
ffiffiffiffiffiffiffiffiffiffiffi
32
ffiffi
2
p
p:ð7Þ
Fig. 4. Evolution of microstructural length parameters predicted for a spherical sub-grain of diameter dduring a complete forward–reverse transformation cycle: (a) plate
thickness M, (b) number of plates per sub-grain d/H. Solid lines correspond to the forward transformation, dashed lines correspond to the reverse transformation.
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
81
The closed-form formulae for the coefficients /
max
tw
and /
max
amþgb
have
been derived by Stupkiewicz and Petryk (2010b), and the values of
those parameters corresponding to the three sub-grain diameters
considered are provided in Table 1.
The total interfacial contribution to the thermodynamic driving
force for phase transformation is
f
i
¼@/
i
@
g
/
max
tw
4ð12
g
Þ/
max
amþgb
;@f
i
@
g
8/
max
amþgb
>0:ð8Þ
It follows that for the austenite-to-martensite transformation dur-
ing loading, the transformation driving force f
i
increases linearly
with
g
, which corresponds to the transformation stress decreasing
linearly with
g
. The latter property is frequently introduced as a
constitutive assumption into phenomenological models of pseudoe-
lastic transformation, with the proportionality factor taken arbitrar-
ily. In contrast, the proportionality factor here takes the value
expressed by the closed-form formula (7) derived from the interfa-
cial energy considerations alone. This remarkable conclusion will
have its consequences below when studying a more complex beha-
viour of a polycrystalline aggregate.
4. Transition to the scale of a polycrystalline aggregate
4.1. Interfacial energy contributions for a single grain
A typical scheme of micromechanical modelling of a SMA poly-
crystal involves a certain model of overall behaviour of a single
grain that is combined with a suitable micro-macro transition from
the scale of a single grain to that of a polycrystal. Representative
examples include (Thamburaja and Anand, 2001; Hackl and
Heinen, 2008; Sengupta and Papadopoulos, 2009; Stupkiewicz
and Petryk, 2010a; Xiao et al., 2019). The variables involved in
the constitutive description of a single grain, in addition to the
overall stress and strain, include the volume fractions of marten-
site variants, usually represented by the so-called habit-plane vari-
ants (HPVs) that account for the compatibility of average
transformation strains at austenite–martensite interfaces
(Bhattacharya, 2003). Such a model does not involve any character-
istic dimension of the microstructure and thus is incapable of
describing size effects. Our aim here is to develop a model that,
in a simplified manner, accounts for the size-dependent interfacial
energy effects by exploiting the results of the model described in
Section 3.
Assume that the size-independent bulk contribution to the
overall Helmholtz free energy density of a single grain in a poly-
crystalline aggregate depends on the average strain
e
in the grain
and on the volume fractions
g
k
,k¼1;...;N;of martensite variants
(HPVs) in the grain, thus
/
v
¼
/
v
ð
e
;
g
k
Þ. As a rule, only several
g
k
will be nonzero in a given grain. We further assume that the
microstructure that develops within the grain during the stress-
induced martensitic transformation divides the grain into several
sub-grains such that each sub-grain is occupied by one family of
parallel, internally twinned martensite plates corresponding to
one HPV. Since individual volumes of the sub-grains can hardly
be determined, a simplifying assumption is introduced that the
interfacial energy contributions are only estimated for a represen-
tative spherical sub-grain, of diameter dbeing a specified fraction
of a diameter d
gr
of the actual grain. Consequently, the volume
fraction of the parallel martensite plates within the representative
sub-grain is assumed equal to the average volume fraction of
martensite within the whole grain,
g
¼P
k
g
k
. As a result, the total
free energy density per unit volume of the grain can be written in
the following form
/ð
e
;
g
k
Þ¼
/
v
ð
e
;
g
k
Þþ
/
i
ð
g
Þ;
/
i
ð
g
Þ¼/
i
tw
ð
g
Þþ/
i
am
ð
g
Þþ/
i
gb
ð
g
Þ;
g
¼P
k
g
k
1;
ð9Þ
where the interfacial energy contributions /
i
s
are determined using
the model described in Section 3, with the sub-grain diameter das
the input parameter. Recall that the model delivers the evolution of
microstructural parameters and the related interfacial energies
along a transformation path parameterized by the volume fraction
of martensite, and thus each individual interfacial energy contribu-
tion /
i
s
can be expressed as a function of
g.
The complete evolution problem for a grain is now again formu-
lated as the minimization problem for the incremental energy sup-
ply
D
E, cf. Section 2. In a strain-controlled process, the overall
strain
e
is prescribed and minimization is performed with respect
to the volume fractions
g
k
. The incremental minimization problem
then reads
D
Eð
e
;
g
k
Þ!min
g
k
;
D
E¼
D
/
v
þ
D
/
i
þ
D
D
v
þ
D
D
i
;ð10Þ
where
D
D
i
;the interfacial contribution to dissipation increment, is
expressed in terms of the increments of individual interfacial free
energy contributions according to Eq. (A.8). The bulk contributions
/
v
and
D
D
v
are here arbitrary.
The results reported in Section 3.2 show that /
i
am
and /
i
gb
attain
maximum at
g
¼0:5;while /
i
tw
is proportional to
g
and thus
reaches maximum for
g
¼1. Accordingly, during the forward
Fig. 5. Results for NiTi: interfacial contributions to free energy (d¼13 mm). The
dashed line depicts the approximation of /
i
am
þ/
i
gb
by a quadratic function, Eq. (6).
Table 1
Interfacial energy contributions and microstructural parameters predicted for a spherical sub-grain of NiTi.
d
gr
[mm] d[mm] h
0
tw
[mm] M
0
[mm] /
max
tw
[MJ/m
3
]/
max
amþgb
[MJ/m
3
]
20 13 0.014 0.14 1.95 2.56
60 38 0.019 0.26 1.45 1.56
180 113 0.026 0.48 1.07 0.96
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
82
transformation (
D
g
>0), the interfacial contribution to dissipation
vanishes as long as
g
0:5. For
g
>0:5;the increments
D
/
i
am
and
D
/
i
gb
are nonpositive and contribute to dissipation according to
Eq. (A.8). Assuming for simplicity that
j
tw
¼
j
am
¼
j
gb
¼1;cf.
Appendix A.3, for
g
>0:5 we have
D
/
i
am
þ
D
D
i
am
¼0 and
D
/
i
gb
þ
D
D
i
gb
¼0 so that the incremental energy supply during
the forward transformation takes the form
D
E¼
D
/
v
þ
D
D
v
þ
D
/
i
tw
þ
D
/
i
am
þ
D
/
i
gb
for
D
g
>0 and 0 <
g
0:5;
D
/
v
þ
D
D
v
þ
D
/
i
tw
for
D
g
>0 and 0:5<
g
1:
8
>
>
>
>
<
>
>
>
>
:
ð11Þ
By a similar argument, during the reverse transformation (
D
g<0),
we have
D
E¼
D
/
v
þ
D
D
v
þ
D
/
i
am
þ
D
/
i
gb
for
D
g
<0 and 1 >
g
0:5;
D
/
v
þ
D
D
v
for
D
g
<0 and 0:5>
g
0:
8
>
>
>
>
<
>
>
>
>
:
ð12Þ
Since minimization (10) does not involve evolution of dimensional
quantities, the approximate explicit formulae (6) for /
i
tw
and
/
i
am
þ/
i
gb
can be used with
gin place of
g
, thus leading to a partic-
ularly simple formulation.
The model of a grain, as presented above, can now be combined
with a suitable micromechanical model of a SMA polycrystal.
4.2. Bi-crystal aggregate model (Stupkiewicz and Petryk, 2010a)
The transition from the level of a crystallographic lattice up to
the scale of a SMA polycrystal undergoing stress-induced marten-
sitic transformation is in this work performed using the bi-crystal
aggregate model (Stupkiewicz and Petryk, 2010a). This multiscale
model is illustrated graphically in Fig. 6, and the underlying basic
equations are provided in Appendix B. The model is presented here
only in outline; more details can be found in the reference, along
with the discussion of the assumptions and results, and compar-
ison with experiment.
A distinctive feature of the model is that the scale transition
from the scale of individual grains to that of a polycrystal is per-
formed by introducing an intermediate scale of bi-crystals, cf.
Fig. 6. The polycrystal is thus treated as an aggregate of randomly
oriented bi-crystals, each bi-crystal being formed by two neigh-
bouring grains separated by a planar interface (grain boundary).
Accordingly, grain interaction is accounted for by enforcing the
compatibility conditions (B.5) on the average strains and stresses
of each grain of the bi-crystal. Introduction of the intermediate
scale of bi-crystals and consideration of the respective grain inter-
action mechanism significantly improves, as shown in
Stupkiewicz and Petryk (2010a), the performance of the simple
Taylor (Voigt) and Sachs (Reuss) grain-to-polycrystal scale transi-
tion schemes when applied as the transition scheme between the
bi-crystals and the polycrystal. As a result, those simple transition
schemes can be used with a higher confidence, and specifically the
Taylor scheme is employed in this work.
A characteristic feature of the model is the quantitative descrip-
tion of the transformation at all scales using only a small number
of parameters that remain to be assumed arbitrarily, which will
allow us to predict the changes in the free energy and dissipation
during transformation. This is done by step-by-step minimization
of the incremental energy supply with respect to the variables at
all levels of the description, cf. Appendix B.
4.3. The effect of interfacial energy on non-monotonic stress–strain
characteristic
A multiscale model of a SMA polycrystal with interfacial energy
effects is now obtained in a straightforward manner by including
in the bi-crystal model the scheme developed in Section 4.1.
Specifically, an enhanced model corresponding to the level of a sin-
gle grain is obtained by combining the bulk contribution
/
v
to the
free energy, cf. Eq. (B.2), and the bulk contribution
D
D
v
to the dis-
sipation function, cf. Eq. (B.4), with the respective interfacial
energy contributions according to the general scheme outlined in
Eqs. (9)–(12). The resulting single-grain model enhanced with
the interfacial energy effects can then be seamlessly employed in
the overall scheme of the bi-crystal model. Importantly, the
quadratic-programming structure of the actual computational
scheme is preserved when the closed-form formulae (6) are used
for /
i
tw
and /
i
am
þ/
i
gb
.
The results reported below have been obtained for a NiTi poly-
crystal for three values of an average grain diameter,
d
gr
¼20;60;180 mm. Following Stupkiewicz and Petryk (2010b),
it has been assumed that the diameter of a representative spherical
sub-grain is such that its volume is equal to one fourth of the vol-
ume of the grain, thus d¼4
1=3
d
gr
. This choice is justified by the
analysis of the results of the bi-crystal aggregate model
(Stupkiewicz and Petryk, 2010a) showing that typically three to
five martensite variants (HPV’s) appear within one grain during
proportional loading. Accordingly, the estimates of the interfacial
energy contributions have been computed in Section 3.2 for three
sub-grain diameters d¼13;38;113 mm. The respective material
parameters used as the input to those computations are provided
in Section 3.2, and the resulting interfacial energy contributions,
which serve as the input to the present computations, are provided
in Table 1.
The remaining material parameters pertinent to the bi-crystal
model are the following. The elastic properties are as in Section 3.2,
and the chemical energy, which is correlated with the temperature
and influences the level of the transformation stress, is assumed
equal to
D
am
/
0
¼19 MPa. The bulk contribution to dissipation is
specified by the critical driving force for transformation
f
c
¼3 MPa and by the critical driving force for martensite reorien-
tation f
r
¼f
c
=2.
The transformation strains
e
t
k
of martensite variants (HPV’s) of
NiTi are determined using the crystallographic theory of marten-
site (Bhattacharya, 2003), and N¼48 variants are used in the pre-
sent (small-strain) computations of the uniaxial tension test, see
Stupkiewicz and Petryk (2010a) for details. A drawing texture is
Fig. 6. Bi-crystal aggregate model: a polycrystal is treated as an aggregate of bi-
crystals, and the corresponding additional level is introduced into the typical
sequential averaging scheme (Stupkiewicz and Petryk, 2010a).
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
83
assumed with the h111ipoles of the cubic austenite preferably
aligned with the drawing direction (along which tension is
applied) such that the angle between the h111ipoles and the draw-
ing axis is randomly chosen between 0 and 30
. Following the pro-
cedure developed by Stupkiewicz and Petryk (2010a), the total of
256 distinct grain orientations has been generated according to
the assumed drawing texture. The grains have been arranged in
pairs in a random manner, and the orientation of the correspond-
ing grain boundary has also been generated in a random manner.
The relative volume fraction of grains in each bi-crystal has been
assumed equal to 0.5. Overall, the bi-crystal averaging has been
performed for an aggregate of 128 randomly generated bi-
crystals. It has been checked that this number is sufficient, and fur-
ther increase of the number of bi-crystals does not influence the
results visibly.
Fig. 7a shows the predicted pseudoelastic response of a repre-
sentative element of a NiTi polycrystal under uniaxial tension. As
expected, due to including the interfacial energy effects, the
response exhibits a significant dependence on the grain size. The
dashed line in Fig. 7a indicates the underlying response predicted
by accounting only for the bulk contributions to the free energy
and dissipation, i.e. in the limit of vanishing interfacial energy con-
tributions for d
gr
?1.
Several effects can be observed in Fig. 7a. With decreasing grain
size, the initiation of transformation is shifted to a higher stress.
The stress must increase to compensate the extra thermodynamic
driving force associated with the interfacial contribution to the free
energy,
f
i
¼@
/
i
=@
g
. At the initial stage of transformation, the
interfacial contribution to the free energy increases, cf. Fig. 5 and
formula (8), hence
f
i
is then negative. The initially negative driving
force
f
i
increases during transformation, which leads to a decrease
of the stress. Averaging over the polycrystalline aggregate causes
the stress to vary nonlinearly with the overall tensile strain. At
the later stage, the further decrease of the stress is suspended
because the negative increments in interfacial free energy are dis-
sipated, and ultimately the stress-strain slope becomes strongly
positive in the final stage of transformation. The final slope tends
to the elastic modulus since martensite detwinning has been
excluded form considerations.
This provides an explanation, coming from the interfacial-
energy considerations, why the stress–strain relationship during
the forward transformation is non-monotonic. For the grain size
small enough, a significant branch of negative slope is observed,
in agreement with the experimental relationship quoted after
Hallai and Kyriakides (2013) in Fig. 1. The maximum stress and
the subsequent stress drop increase with decreasing grain size.
Interestingly, the predicted shape of the upper curve for grain size
d
gr
¼20 mm almost coincides with the experimental curve for the
material shown in Fig. 1 after Hallai and Kyriakides (2013); unfor-
tunately, the underlying average grain size of the material tested
was not provided in the reference. The response during the reverse
transformation upon unloading is similar to that during the for-
ward transformation, except that the stress is lower so that a hys-
teresis loop is observed. The area of the hysteresis loop, which is
equal to the energy dissipated in the complete loading-unloading
cycle (in isothermal conditions), also increases with decreasing
grain size. This illustrates the important effect of the interfacial
contribution to dissipation.
Note that the ‘softening’ response, as predicted by the present
model, cannot be observed in a uniaxial tension experiment unless
special means are taken to avoid the nonuniform deformation, as
in the experiment of Hallai and Kyriakides (2013). Indeed, it is
commonly observed that the stress-induced transformation, par-
ticularly in NiTi in tension, proceeds through nucleation of local-
ized transformation bands followed by propagation of
macroscopic transformation fronts (e.g., Shaw and Kyriakides,
1997; Sun and Li, 2002; Zhang et al., 2010; Sedmák et al., 2016).
This is typically accompanied by a stress plateau that is associated
with propagation of the fronts and does not represent the material
behaviour at a material point.
Fig. 7b shows hypothetical specimen responses corresponding
to the intrinsic material responses shown in Fig. 7a. The uniaxial
stress–strain diagram with a negative slope leads to instability of
a uniform deformation path and hence cannot be reproduced by
the specimen (Petryk, 2000). Here, the classical Maxwell construc-
tion has been used. The plateau segment connects the monotoni-
cally increasing segments of the original curve such that the
areas above and beneath the plateau and limited by the original
stress–strain curve are equal. The plateau stress can be shown to
Fig. 7. Grain-size effect on the stress–strain response of NiTi polycrystal under uniaxial tension: (a) material response, (b) specimen response predicted using the Maxwell
construction.
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
84
be the lowest thermodynamically admissible one for propagation
of the transformation zone in which the original stress–strain
curve is traversed at every material point. Additionally, it has been
assumed here that the hypothetical specimen response is formed
by the loading branch of the original stress–strain curve up to
the maximum stress, followed by an instanteneous stress drop, a
stress plateau corresponding to the Maxwell stress, and the final
hardening part towards the end of transformation. The unloading
part of the response is constructed analogously. The maximum
stress and the subsequent instanteneous stress drop are supposed
to correspond to nucleation of the first martensite band. Similar
features are commonly observed in experiments, although the
actual stress overshoot is highly sensitive to imperfections, mate-
rial inhomogeneity, etc. The minimum stress followed by a stress
increase at the beginning of the reverse transformation would cor-
respond to nucleation of the austenite band in a completely trans-
formed specimen. If the reverse transformation does not require
nucleation of the austenite band, but proceeds by a reverse propa-
gation of an existing macroscopic transformation front, then the
corresponding feature is not expected since no energy barrier must
then be overcome and the plateau can be followed from the
beginning.
It is not evident in advance that the hypothetical behaviour
described above will agree with 3D simulations of the transforma-
tion process. To investigate this, a detailed analysis of strain local-
ization and nonuniform transformation is carried out in the next
section using a phenomenological finite-strain model of
pseudoelasticity.
5. Macroscopic modelling of nonuniform phase transformation
The micromechanical model developed in the previous sections
predicts the stress–strain response that exhibits a negative slope
during stress-induced transformation. The related scenario of
strain localization in the macroscopic specimen has then been ana-
lyzed qualitatively by employing the Maxwell construction, cf.
Fig. 7b. In this section, nonuniform phase transformation is ana-
lyzed quantitatively using a macroscopic phenomenological model
of pseudoelasticity implemented in a finite-element code. The
model is briefly described in Section 5.1, its finite-element imple-
mentation is commented in Section 5.2, and the results of finite-
element computations are reported in Section 5.3.
5.1. Phenomenological finite-strain model of pseudoelastic SMAs
The model used in this work is an extension of the model devel-
oped recently by Rezaee-Hajidehi et al. (2020) by introducing a
gradient regularization to the finite-strain 3D model of pseudoelas-
ticity of Stupkiewicz and Petryk (2013). The gradient regularization
has been introduced in order to adequately treat strain localization
and propagation of macroscopic transformation fronts. The model
is formulated by specifying the free energy function and the dissi-
pation function, and the complete evolution problem is then for-
mulated within the incremental energy minimization framework.
The reader is referred to Stupkiewicz and Petryk (2013) for a
detailed presentation of the constitutive model and to Rezaee-Ha
jidehi et al. (2020) for the aspects related to the gradient regular-
ization. Below we describe very briefly the extensions introduced
to the model in order to accurately represent the intrinsic material
response predicted by the micromechanical model.
The modifications introduced in the present paper concern both
the free energy function and the dissipation function. In general,
the two functions cannot be uniquely fitted using the stress–strain
curve alone. To avoid ambiguity, the contributions coming from
the free energy and dissipation functions have been separated
and fitted individually by adequate postprocessing of the results
of the micromechanical model.
The modification introduced to the free energy function con-
cerns the interaction energy term /
int
(
g
), which in the original
macroscopic model (Stupkiewicz and Petryk, 2013) was adopted
as a quadratic function of
g
, the volume fraction of martensite,
with a constant coefficient playing the role of a hardening or soft-
ening modulus. Actually, the interaction energy influences the
stress–strain response only through its derivative /
0
int
ð
g
Þ. It is thus
convenient to perform the fitting directly for the derivative /
0
int
ð
g
Þ;
rather than for the function /
int
(
g
) itself, and this approach has
been adopted here.
The rate-independent dissipation function in the macroscopic
model is assumed as D
^
ð_
g
Þ¼f
c
j_
g
j;where f
c
> 0 is the critical ther-
modynamic driving force for phase transformation. The modifica-
tion introduced here amounts to specifying the critical driving
force separately for the forward and reverse transformation, and
basically consists in introducing the dependence of the critical
driving force on
g
, thus
D
^
ð_
g
;
g
Þ¼f
þ
c
ð
g
Þh_
g
iþf
c
ð
g
Þh_
g
i;ð13Þ
with the meaning of angular brackets as in Eq. (3).
Each of the constitutive functions /
0
int
ð
g
Þ;f
þ
c
ð
g
Þand f
c
ð
g
Þhas
been fitted using a Bernstein polynomial of degree 12. The impact
of the individual contributions on the pseudoelastic stress–strain
response is illustrated in Fig. 8a for the case of the grain diameter
d
gr
¼20 mm. The adopted procedure yields a good fitting of the
response predicted by the micromechanical model, as shown in
Fig. 8b for the three grain diameters examined in Section 4.3.As
a part of the identification procedure, the transformation strain
in tension has been determined as equal to
e
T
¼0:055.
5.2. Finite-element model
The finite-element formulation of the gradient-enhanced model
relies on the micromorphic-type regularization, as proposed by
Rezaee-Hajidehi and Stupkiewicz (2018), see also Mazière and
Forest (2015). The computer implementation follows exactly that
developed by Rezaee-Hajidehi et al. (2020), where all the details
can be found. The resulting computational model involves, as the
global unknown fields, the displacement field, the micromorphic
counterpart of the volume fraction of martensite, and temperature.
The thermomechanical coupling is introduced to the model
because it provides an additional, physically-based regularization
of the non-monotonic response. Indeed, the thermomechanical
coupling improves the robustness of the computational model,
which is particularly beneficial when the softening is high. The
thermomechanical coupling renders the response rate-dependent
so that the experimentally observed loading-rate effects can also
be studied (e.g., Rezaee-Hajidehi et al., 2020), which, however, is
not pursued here.
In the present computations, isoparametric hexahedral ele-
ments are used with triquadratic shape functions for the displace-
ment and with trilinear shape functions for the remaining
unknowns. The computer implementation is carried out using Ace-
Gen, a symbolic code generation system, and the finite-element
computations are performed using AceFEM, a finite-element code
interfaced with AceGen (Korelc and Wriggers, 2016).
Finite-element computations have been carried out for a dog-
bone specimen loaded in tension. The specimen of the total length
of 60 mm contains the gauge segment of the length L¼24 mm and
uniform cross-section of 5.2 0.15 mm
2
. The specimen width
gradually increases in the transition segments between the gauge
segment and the gripping segments, the latter of the width of
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
85
20 mm. The axial displacement is prescribed at the specimen ends
in a way that no bending moment is transmitted to the specimen
by allowing the gripping segments to rotate. The load is applied
with a low constant nominal loading rate of 10
5
s
1
in order to
obtain a nearly isothermal response. The elongation d/Lis deter-
mined as the averaged relative axial displacement of the two
cross-sections at the ends of the gauge segment normalized by
the gauge length L. A small geometrical imperfection is introduced
in the upper-right part of the gauge segment to trigger a non-
symmetric mode of strain localization.
The finite-element mesh within the gauge length is adopted
such that the in-plane element size is 0.13 mm, and through-the-
thickness element size is 0.15 mm, so that the gauge segment is
discretized into 185 40 1 elements. The in-plane element size
gradually increases towards the gripping segments. The total num-
ber of degrees of freedom exceeds 550 000. For completeness, the
model parameters used in the finite-element computations are
provided in Appendix C.
5.3. Finite-element simulations of uniaxial tension test
Fig. 9 shows the normalized force–elongation diagram com-
puted for the grain diameter d
gr
¼20 mm. Also shown are the snap-
shots illustrating the evolution of the transformation pattern at
selected instants, as indicated by the markers and labels on the
force–elongation diagram. Each snapshot represents the specimen
in the deformed configuration with the colour indicating the vol-
ume fraction of martensite.
Initially, the transformation is nearly uniform within the gauge
segment, and this stage corresponds to the initial ‘hardening’
branch of the intrinsic stress–strain relationship for the material.
At the elongation of about 1% a sudden load drop is observed which
is associated with the nucleation of a thin inclined band of marten-
site (snapshot 1). The band nucleates at the imperfection. The
transformation proceeds then through propagation of macroscopic
transformation fronts at an approximately constant load. The
obtained nominal stress–elongation curve is visibly smoother than
Fig. 8. Fitting of the macroscopic model to the uniaxial-tension response predicted by the micromechanical model: (a) illustration of the individual contributions for the case
of d
gr
¼20 mm; (b) comparison for the three grain sizes (solid and dashed lines correspond to the macroscopic and micromechanical model, respectively).
Fig. 9. Finite-element simulation of uniaxial tension: nominal stress–elongation diagram predicted for d
gr
¼20 mm (left), and snapshots of the evolution of the
transformation pattern at selected instants (right). The colour denotes the volume fraction of martensite varying between 0 (blue) and 1 (red), with the intermediate colours
corresponding to the diffuse interfaces (hardly visible because the interfaces are relatively thin). (For interpretation of the references to colour in this figure legend, the reader
is referred to the web version of this article.)
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
86
the experimental one shown in Fig. 1. The wiggles in the experi-
mental curve may be related to abrupt events, like nucleation of
new interfaces, changes in orientation of interfaces, and formation
of criss-cross patterns, or to material inhomogeneity, e.g., due to
the grain microstructure. In the present computations, the abrupt
events are not observed during loading, but they are observed dur-
ing unloading, thus leading to small wiggles at the initial stage of
unloading, as commented below. On the other hand, material inho-
mogeneity is not included in the model, except for that resulting
from the finite-element discretization. However, the finite-
element mesh is here sufficiently fine, in particular, with respect
to the thickness of the diffuse macroscopic transformation fronts,
so that the related nominal stress fluctuations are very small and
thus not visible.
The macroscopic transformation fronts are diffuse interfaces
that separate the domains of low and high volume fraction of
martensite equal to 0.037 and 0.94, respectively. Those values
are close to the limit values of 0 and 1, and their deviation from
the limit values is thus hardly visible. The end of the load plateau
corresponds to the instant at which the martensite domain extends
over the entire gauge length (snapshot 3). With further elongation,
the load increases which is accompanied by a uniform transforma-
tion within the gauge segment until the transformation is
completed.
Upon unloading, the reverse transformation proceeds through a
reverse motion of the macroscopic transformation fronts. At the
initial stage, small oscillations of the load are observed which are
associated with the development and evolution of a criss-cross
pattern at the macroscopic transformation fronts (snapshot 5).
Subsequently, two inclined interfaces form and propagate in a
stable manner until the martensite band annihilates. The event of
annihilation of the macroscopic transformation fronts is accompa-
nied by a small increase of the load.
It has been checked that the plateau stress, both during the for-
ward and reverse transformation, is in excellent agreement with
the Maxwell stress. This is illustrated for d
gr
¼20 mminFig. 10
in which the horizontal dashed line denotes the Maxwell stress,
determined in the standard way by equating the shaded areas
above and below the Maxwell line. It can be seen from Fig. 10 that
the actual specimen response can be remarkably well approxi-
mated by the hypothetical specimen response obtained by
employing the Maxwell construction, cf. Fig. 7. A minor and
expected difference between the two is that the stress minimum
followed by a sudden stress increase at the beginning of the
reverse transformation is not observed in the actual specimen
response because the reverse transformation proceeds through a
reverse motion of already existing macroscopic transformation
fronts. Another small difference is that the nucleation of the
martensite band starts slightly before the maximum stress point,
and the nucleation stress is influenced by the imperfection. More-
over, the subsequent stress drop is characterized by a finite slope,
which has been found to depend mostly on the specimen length,
while an infinite slope is assumed in the hypothetical response.
The specimen responses computed for the three grain sizes are
shown in Fig. 11. The qualitative features are here the same regard-
less of the grain size. Also, it has been checked that in all cases the
plateau stresses agree very well with the respective Maxwell stres-
ses. As a result, the width of the pseudoelastic hysteresis loop
shows a significant grain-size dependence which is inherited from
the respective intrinsic material responses.
6. Conclusion
A multiscale analysis of pseudoelastic behavior of SMAs has
been carried out, starting from crystallographic lattice rearrange-
ments up to the scale of a polycrystalline specimen. The standard
analysis of a rank-two laminate domain has been enhanced by
including a hierarchy of interfacial energies on three scales: of pla-
nar twin interfaces, of corrugated interfaces between austenite and
twinned martensite, and of domain boundaries. To determine the
actual microstructure evolution among plenty of possibilities, the
incremental energy minimization technique (Petryk, 2003) has
been applied. Upon subsequent passage to the scale of a polycrys-
talline aggregate, it has been shown, apparently for the first time,
that the interfacial energy accumulation and release can be pre-
dominant in predicting a non-monotonic uniaxial stress–strain
response and the accompanying hysteresis. The effects are depen-
dent on the grain size, which has been examined quantitatively for
a NiTi SMA with the conclusion that the smaller the grain size (in
Fig. 11. Finite-element simulations of the nominal stress–elongation response of a
tensile specimen for the three grain sizes.
Fig. 10. Comparison of the stress plateau in the 3D finite-element simulation of the
tension test with the Maxwell stress for the non-monotonic response of the
material for grain size d
gr
¼20 mm. The specimen and material responses are
expressed in terms of the nominal stress and elongation.
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
87
the relevant grain-size range) the stronger the calculated effect.
Finally, the finite-element study of a tensile specimen in the
finite-deformation setting has confirmed that the simple Maxwell
construction for the non-monotonic material response in tension
accurately predicts the plateau stress in loading and another one
in unloading.
The simulated stress–elongation hysteresis loop with two pla-
teaus for a NiTi specimen subjected to uniaxial tensile loading
and unloading is both qualitatively and quantitatively close to that
commonly observed in the quasi-static tension tests performed at
a constant temperature corresponding to the pseudoelastic regime.
Of course, the mean stress level depends significantly on the tem-
perature, and can be straightforwardly fitted in the calculations by
adjusting the value of chemical free energy. In turn, the calculated
width of the hysteresis loop is found to depend on the grain size
through the interfacial energy effects and also directly on the
assumed value of the critical driving force f
c
that specifies the bulk
contribution to dissipation. Validation of the present model in this
respect is more difficult since experimental studies of grain-size
dependence of the hysteresis loop are scarce. An increase of hys-
teresis with decreasing grain size has been observed experimen-
tally in CuAlBe polycrystal (Montecinos et al., 2008) and in
CuAlNi microwires (Chen and Schuh, 2011), and this qualitative
effect is correctly reproduced by our model. On the other hand,
the experimental data for the grain sizes within a nanometer range
(Ahadi and Sun, 2015; Sun and He, 2008) show an opposite effect.
Note, however, that the present model deals with domains of
micrometer size to allow formation of rank-two austenite-
martensite laminates, while different transformation mechanisms
may operate in materials with nanometer-sized grains
(Waitz et al., 2007). At the same time, the shape of the non-
monotonic stress–strain curve in loading predicted for the NiTi ma-
terial of grain size d
gr
¼20 mm shows a remarkable similarity to the
experimental one quoted in Fig. 1 after Hallai and
Kyriakides (2013). As mentioned above, that shape is obtained here
solely from the interfacial energy considerations.
All efforts have been made to leave the number of parameters of
unverifiable value in the analysis as small as possible. In result,
practically only one such parameter, f
c
, has remained. This has
been done at the cost of a number of simplifying assumptions
which have been explained in the text and in the cited references.
These assumptions impose limitations on the applicability of the
present micromechanical approach, for instance, to predominantly
proportional loading paths. Further work in needed to clarify and
possibly overcome such limitations in the future.
Declaration of Competing Interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgement
This work has been partially supported by the National Science
Center (NCN) in Poland through Grant No. 2018/29/B/ST8/00729.
Appendix A. Size-dependent microstructure evolution within a
spherical sub-grain
In this appendix we provide the complete set of the governing
equations of the model of microstructure evolution in a spherical
sub-grain, cf. Fig. 3. The model is briefly described in Section 3.1,
while the detailed derivation can be found in (Stupkiewicz and
Petryk, 2010b).
A1. Interfacial energy at three scales
The total free energy density (per unit volume of the sub-grain)
comprises the interfacial energy contributions from three scales,
/
i
¼X
s
/
i
s
;s¼tw;am;gb;ðA:1Þ
with the contribution of twin interfaces,
/
i
tw
¼
c
a
tw
A
tw
¼2
gc
a
tw
h
tw
;ðA:2Þ
the contribution of austenite–twinned martensite interfaces,
/
i
am
¼ð
c
e
am
þ
c
a
am
ÞA
am
¼2ð
C
e
am
h
tw
þ
c
a
am
Þ
H;ðA:3Þ
and the contribution of the boundary of the laminated spherical sub-
grain,
/
i
gb
¼f
C
e
gb
gHA
gb
¼
3
4
p
a
l
b
22
m
1
m
H
dg2
1sin
p
g
2
2
þð1
g
Þ
2
1cos
p
g
2
2
hi
;
ðA:4Þ
where c
a
tw
and c
a
am
are the atomic-scale interfacial energy densities,
c
e
am
¼C
e
am
h
tw
is the elastic micro-strain energy density, fC
e
gb
gis the
average elastic micro-strain energy factor for the sub-grain bound-
ary, and A
tw
¼2g=h
tw
;A
am
¼2=Hand A
gb
¼6=dare the size-
dependent densities of the corresponding interfaces per unit vol-
ume of the sub-grain. In Eq. (A.4),
l
,
m
are the standard elasticity
constants, bis the magnitude of the so-called shape-strain vector
that characterizes the transformation strain of the martensite
plates, and a¼0:197;cf. (Petryk et al., 2010, Eq. (8)).
A2. Initial stage of transformation
Energetically optimal microstructural parameters h
0
tw
and M
0
at
the initial stage of transformation, i.e. for
g
0, are obtained by min-
imization of /
i
0
¼/
i
0
ð
g
;h
tw
;MÞwith respect to h
tw
and M, which
yields
h
0
tw
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
‘
am
M
0
q;M
0
¼ffiffiffiffiffiffiffiffiffi
‘
gb
d
p;‘
am
¼
c
a
tw
C
e
am
;
‘
gb
¼
C
e
am
h
0
tw
þ
C
a
am
C
e0
gb
=2
;ðA:5Þ
where /
i
0
is the leading term of the Taylor expansion of /
i
with
respect to
g
at g¼0;
/
i
0
¼2
gc
a
tw
h
tw
þ
C
e
am
h
tw
Mþ
c
a
am
Mþ
C
e0
gb
M
2d
!
;
C
e0
gb
¼3
4
p
a
l
b
2
2
m
1
m
:
ðA:6Þ
A3. Microstructure evolution
Microstructure evolution during phase transformation is deter-
mined by the incremental energy minimization. Twin spacing is
assumed constant, h
tw
¼h
0
tw
;and evolution of martensite plate
thickness Mis obtained by minimizing the interfacial part of the
incremental energy supply along a transformation path parameter-
ized by
g
, thus M¼Mð
g
Þdetermined as follows,
D
E
i
ð
g
;M;h
0
tw
Þ!min
M
;
D
E
i
¼
D
/
i
þ
D
D
i
;ðA:7Þ
where, as a specification of Eq. (3), the interfacial contribution to
dissipation is equal to
D
D
i
¼P
s
D
D
i
s
;
D
D
i
s
¼
j
s
h
D
/
i
s
i;0
j
s
1;
s¼tw;am;gb:ðA:8Þ
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
88
Since the interfacial energy release can hardly be expected to be
converted to the elastic bulk energy and is thus likely to be dissi-
pated, it is reasonable to take values of
j
s
closer to 1 than to 0.
Appendix B. Bi-crystal aggregate model
In this appendix we provide the complete set of the governing
equations of the bi-crystal aggregate model. The model, formulated
in the small-strain setting, is briefly commented in Section 4.2, and
the details can be found in (Stupkiewicz and Petryk, 2010a).
B1. Rank-two laminated sub-grain
Martensite is assumed to appear in the form of internally-
twinned plates (i.e. habit-plane variants, HPVs) that form a rank-
two austenite–martensite laminate within the corresponding
sub-grain, cf. Figs. 3 and 6. The overall transformation strains
e
t
k
of the martensite plates are compatible with unstressed austenite,
thus
e
t
k
¼1
2ðb
k
m
k
þm
k
b
k
Þ;k¼1;...;N;ðB:1Þ
where the habit-plane normal m
k
and shape-strain vector b
k
can be
obtained from the classical crystallographic theory of martensite
(Bhattacharya, 2003).
B2. Single grain
The Helmholtz free energy of a grain composed of laminated
sub-grains of uniform elastic properties is assumed in the follow-
ing form,
/ð
e
;
g
k
Þ¼
/
v
ð
e
;
g
k
Þ
¼/
0
þ
g
D
am
/
0
þ1
2ð
e
e
t
ÞLð
e
e
t
Þþ
/
v
int
;ðB:2Þ
along with
e
t
¼X
k
g
k
e
t
k
;
g
k
0;
g
¼X
k
g
k
1;ðB:3Þ
where
g
k
denotes the volume fraction of martensite variant k,
gis
the total volume fraction of martensite in the grain,
D
am
/
0
is the
chemical energy of transformation, and Lis the elastic stiffness ten-
sor. The interaction energy
/
v
int
is here neglected,
/
v
int
¼0;which
corresponds to a constant-stress averaging scheme within a single
grain. As usual, we have
r¼@
/=@
e¼Lð
e
e
t
Þ.
The rate-independent dissipation function for phase transfor-
mation (f
c
> 0) and martensite reorientation (f
r
> 0) is adopted in
the following form,
D
Dð
D
g
k
Þ¼
D
D
v
ð
D
g
k
Þ¼f
c
j
D
g
jþ
1
2
f
r
P
k
j
D
g
k
jj
D
g
j
;
D
g
¼P
k
D
g
k
:ðB:4Þ
The original model of Stupkiewicz and Petryk (2010a) does not
account for the interfacial energy effects, thus the free energy
/
and the dissipation function
D
Dcomprise only the bulk contribu-
tions
/
v
and
D
D
v
;respectively.
B3. Bi-crystal composed of two adjacent grains
The polycrystalline material is considered as an aggregate of bi-
crystals, where each bi-crystal is composed of a pair of neighbour-
ing grains separated by a planar grain boundary of orientation n.
The following compatibility conditions at the grain boundary are
imposed on the (average) strains and stresses in the grains,
e
ð2Þ
e
ð1Þ
¼1
2ðcnþncÞ;ð
r
ð2Þ
r
ð1Þ
Þn¼0;ðB:5Þ
where cis an unknown vector.
The average strain and stress within the bi-crystal are given by
e
b
¼f
e
ð1Þ
þð1fÞ
e
ð2Þ
;
r
b
¼f
r
ð1Þ
þð1fÞ
r
ð2Þ
;ðB:6Þ
and the average free energy and dissipation densities read
/
b
¼f
/
ð1Þ
þð1fÞ
/
ð2Þ
;
D
D
b
¼f
D
D
ð1Þ
þð1fÞ
D
D
ð2Þ
;ðB:7Þ
where fand 1 fdenote the volume fractions of the grains in the
bi-crystal, and f¼
1
2
is assumed in the simulations. An analytical
expression for
/
b
has been derived in (Stupkiewicz and
Petryk, 2010a).
B4. Polycrystal as an aggregate of differently oriented bi-crystals
The macroscopic response of a SMA polycrystal is obtained by
averaging the responses of bi-crystals of random grain orientations
and random grain-boundary orientations, possibly characterized
by a crystallographic texture. The respective averaging operation
is denoted by {} so that the macroscopic strain and stress, and
the macroscopic free energy and dissipation densities are given by
E¼f
e
b
g;R¼f
r
b
g;U¼f
/
b
g;
D
D¼f
D
D
b
g:ðB:8Þ
To close the model, the Taylor averaging scheme is adopted so that
the average strain in each bi-crystal is constrained to be equal to the
macroscopic strain,
e
b
¼E:ðB:9Þ
B5. Incremental energy minimization
For a strain-controlled process, the macroscopic response is
obtained by minimizing the incremental energy supply,
D
E¼
D
Uþ
D
D!min for prescribed
D
E:ðB:10Þ
In view of the Taylor constraint (B.9), the minimization problem can
be solved separately for each bi-crystal,
D
E
b
¼
D
/
b
þ
D
D
b
!min for prescribed
D
e
b
¼
D
E:ðB:11Þ
The minimization problem (B.11) is a non-smooth minimization
problem with 2Nunknown increments of the volume fractions of
martensite variants in the bi-crystal. The problem can be trans-
formed to a quadratic programming problem with 4ðNþ1Þ
unknowns, which can be efficiently solved using the interior-point
method.
Appendix C. Model parameters used in the finite-element
computations
In Table C.1 we provide the parameters of the model, as used in
the finite-element computations in Section 5.3, see Rezaee-Hajide
hi et al. (2020) for the detailed description of the model and its
parameters. The material is assumed isotropic with the elastic
properties, Eand
m
, consistent with those employed in the bi-
crystal model. Considering that only uniaxial tension is studied,
the tension–compression asymmetry is neglected, and the trans-
formation strain in tension
e
T
, equal to that in compression, has
been determined by fitting the response in uniaxial tension, cf. Sec-
tion 5.1. The transformation temperature T
t
is determined such
that the chemical energy at the initial temperature T
0
¼296 K is
equal to that assumed in the bi-crystal model,
D
am
/
0
¼
D
s
ðT
0
T
t
Þ¼19 MPa. The entropy of transformation
D
s*, the density
.
0
, the specific heat c, and the thermal conductiv-
ity coefficient Kare adopted after Rezaee-Hajidehi et al. (2020).
S. Stupkiewicz, M. Rezaee-Hajidehi and H. Petryk International Journal of Solids and Structures 221 (2021) 77–91
89
Parameter Gis related to the gradient regularization and con-
trols the thickness of diffuse interfaces (macroscopic transforma-
tion fronts). A target interface thickness has been assumed as
twice the element size, and parameter Ghas been adjusted accord-
ingly by referring to the analytical solution for the interface profile,
cf. Rezaee-Hajidehi and Stupkiewicz (2018). The value of Gin
Table C.1 corresponds to the case of d
gr
¼20 mm with the actual
nonlinear non-monotonic response approximated by a piecewise-
linear one. Parameter
v
, which is associated with the micromor-
phic regularization, is determined such that the martensite volume
fraction and its micromorphic counterpart are reasonably close one
to the other, cf. (Rezaee-Hajidehi and Stupkiewicz, 2018).
As described in Section 5.1, the constitutive functions /
0
int
ð
g
Þ;
f
þ
c
ð
g
Þand f
c
ð
g
Þhave been fitted using Bernstein polynomials of
degree n¼12 of the form
B
n
ðxÞ¼X
n
m
¼0
b
m
b
m
;n
ðxÞ;b
m
;n
ðxÞ¼ n
m
x
m
ð1xÞ
n
m
;ðC:1Þ
where b
m
,n
(x) are the Bernstein basis polynomials and b
m
are the cor-
responding coefficients. The coefficients b
m
fitted for the three con-
stitutive functions /
0
int
ðgÞ;f
þ
c
ðgÞand f
c
ðgÞare provided in Table C.2
for the representative case of d
gr
¼20 mm.
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Table C.2
Coefficients b
m
of the Bernstein polynomials for dgr ¼20 mm (in MPa).
b
0
b
1
b
2
b
3
b
4
b
5
b
6
b
7
b
8
b
9
b
10
b
11
b
12
/
0
int
ðgÞ8.05 15.36 3:53 30.58 24:27 31.66 17:84 20.22 22:90 13.27 9:65 9:20 1.50
f
þ
c
ðgÞ3.00 3.48 1.45 5.00 2.90 1.14 1.83 6.86 5.13 6.60 12.54 11.98 14.52
f
c
ðgÞ12.61 14.28 11.05 12.23 6.78 5.30 6.44 8.10 4.87 2.96 5.88 4.84 4.95
Table C.1
Material parameters of polycrystalline NiTi.
E
me
T
T
t
D
s*
.
0
cK G
v
[GPa] ½ ½ [K] [MPa/K] [kg/m
3
] [J/(kg K)] [W/(m K)] [Pa m
2
] [MPa]
70 0.33 0.055 217 0.24 6500 440 18 0.062 471
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