A micromechanics-inspired constitutive model for shape-memory alloys that accounts for initiation and saturation of phase transformation

A micromechanics-inspired constitutive model

for shape-memory alloys that accounts for initiation

and saturation of phase transformation

Alex Kelly

, Aaron P. Stebner

, Kaushik Bhattacharya

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA

article info

Article history:

Received 29 May 2015

Received in revised form

2 February 2016

Accepted 17 February 2016

Available online 18 February 2016

Keywords:

Shape-memory alloys

Constitutive model

Phase transformation

abstract

A constitutive model to describe macroscopic elastic and transformation behaviors of

polycrystalline shape-memory alloys is formulated using an internal variable thermo-

dynamic framework. In a departure from prior phenomenological models, the proposed

model treats initiation, growth kinetics, and saturation of transformation distinctly, con-

sistent with physics revealed by recent multi-scale experiments and theoretical studies.

Specifically, the proposed approach captures the macroscopic manifestations of three

micromechanial facts, even though microstructures are not explicitly modeled: (1) In-

dividual grains with favorable orientations and stresses for transformation are the first to

nucleate martensite, and the local nucleation strain is relatively large. (2) Then, trans-

formation interfaces propagate according to growth kinetics to traverse networks of

grains, while previously formed martensite may reorient. (3) Ultimately, transformation

saturates prior to 100% completion as some unfavorably-oriented grains do not transform;

thus the total transformation strain of a polycrystal is modest relative to the initial, local

nucleation strain. The proposed formulation also accounts for tension

–

compression

asymmetry, processing anisotropy, and the distinction between stress-induced and tem-

perature-induced transformations. Consequently, the model describes thermoelastic re-

sponses of shape-memory alloys subject to complex, multi-axial thermo-mechanical

loadings. These abilities are demonstrated through detailed comparisons of simulations

with experiments.

1. Introduction

Shape-memory alloys (SMAs) exhibit unusual macroscopic phenomena including superelasticity, the shape-memory effect,

and actuation. Through these behaviors, inelastic strains on the order of several percent are recovered because they are

accommodated through diffusionless, reversible microstructure rearrangement instead of slip, glide, climb, and other irre-

coverable, plastic mechanisms. These remarkable phenomena are enabled by solid-to-solid phase transformation between a

high symmetry

austenite

phase that is stable at high temperatures, and a low symmetry

martensite

phase that is stable at low

temperatures. The symmetry disparity between the phases allows multiple

variants

of the martensite phase

–

martensite

Contents lists available at

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journal homepage:

www.elsevier.com/locate/jmps

Journal of the Mechanics and Physics of Solids

http://dx.doi.org/10.1016/j.jmps.2016.02.007

0022-5096/

Corresponding author.

E-mail address:

bhatta@caltech.edu (K. Bhattacharya)

Journal of the Mechanics and Physics of Solids 97 (2016) 197

–

224

structures that are identical up to their orientation

–

to form from each austenite structure during transformation. The ability

to form variants or patterns of variants manifests an infinite number of possibilities for macroscopic strains, or shape changes,

during transformation of a polycrystal. As for a liquid

–

solid or liquid

–

gas transformation, a Clausius

–

Clapeyron relation

governs the solid-state phase transformation of SMAs. Thus, the transformation may be induced by changing temperature or

stress. The state of thermo-mechanical loading during transformation will also dictate the shape change. Furthermore, while in

the martensitic state, one variant may switch to another

–

i.e.

reorient

–

in response to applied forces, also resulting in

macroscopic shape changes.

Regardless of whether a given martensite structure was formed directly during transformation or via post-transfor-

mation deformation, upon reverse transformation, all variants of martensite revert back to the same initial austenite

structure, resulting in full shape recovery, or

“

shape-memory.

”

Superelasticity

describes a constant ambient temperature

event in which elastic and transformation strains induced by mechanical loading are completely recovered upon unloading.

Analogously, via

the shape-memory effect

, a SMA mechanically deformed, then unloaded below the critical transformation

temperature, while in the martensite phase, returns to its original shape upon heating to austenite.

Actuation

is achieved by

maintaining an applied load while heating and cooling through transformation; the SMA recovers transformation strain

against the load thereby producing mechanical work.

The unusual properties of SMAs have led to their use in a variety of engineering applications ranging from implantable

medical devices to actuators. We refer the reader to

Otsuka and Wayman (1998)

for a detailed introduction. The increasingly

sophisticated applications of these materials calls for a model that is capable of faithfully describing complex phenomena,

while being simple enough to be used in the engineering design process. This demand has motivated a number of models.

Some of these, like the widely used models of

Auricchio et al. (1997)

Auricchio and Petrini (2004)

and

Lexcellent et al.

(2002),

are phenomenological and adapt frameworks from other subjects like plasticity. Others, like the models of

Qidwai

and Lagoudas (2000)

Panico and Brinson (2007)

Sadjadpour and Bhattacharya (2007b

, and

Chemisky et al. (2011)

in-

troduce internal variables to incorporate some microscopic information. Yet other models like those of

Zaki and Moumni

(2007)

focus on thermomechanical interactions and cyclic loadings. We refer the reader to

Chemisky et al. (2011)

for a

comprehensive survey of the literature. Despite significant advances, a phenomenological model that accurately describes

transformation initiation, growth, and saturation during multiaxial proportional and non-proportional loading remains a

work in progress, and motivates the proposed model.

We now consider superelasticity in some detail to describe the ideas of the current model. To understand the underlying

mechanisms behind this phenomenon, we first consider an ideal

single crystal

at a temperature just above its transformation

temperature. At this temperature, austenite is the stable phase and martensite the metastable phase. As the crystal is

subjected to stress, it initially responds elastically as in loading prior to point A in

Fig. 1

a. However, at some critical stress,

the martensite is stabilized (

Burkart and Read, 1953

;

James, 1986

) and the material begins to transform from austenite to

martensite. This event results in a macroscopic inelastic yield event that appears similar to perfect plasticity (A). Once the

material is fully martensite, the martensite now responds elastically on further loading to some maximum load below the

stress required for detwinning and/or plastic flow (B to C). Upon unloading, the martensite remains the stabilized phase as

long as the stress remains high and unloads elastically (C

–

E). However, at some critical stress, it becomes unstable and

begins to transform to the austenite, and this reverse transformation gives rise to the lower plateau (E to F). Once the

transformation is complete, the austenite unloads elastically.

Hysteresis results from energy dissipated during transformation. For superelasticity, it is often characterized by the

differences between the critical stresses for forward and reverse transformation. Single crystal transformation responses

differ with differing loading directions due to crystallographic anisotropy (

Miyazaki et al., 1984

;

Shield, 1995

). Specifically, a

Fig. 1.

A schematic representation of superelasticity in (a) single crystal and (a) polycrystalline specimens.

A. Kelly et al. / J. Mech. Phys. Solids 97 (2016) 197

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critically resolved stress criterion adequately describes the single crystal anisotropy of transformation (

Miyazaki et al., 1984

;

Shield, 1995

). Asymmetries with respect to loading mode (

e.g.

, tension vs. compression) and the deformation mechanisms

are broadly explained by considering the Gibbs energy, and more specifically the Clausius

–

Clapeyron relation between the

critical stress for transformation, the strain of transformation, and temperature (

Burkart and Read, 1953

;

James, 1986

The situation is considerably more complicated in a

polycrystal

. The transformation strains, microstructures, and re-

sponses to the applied stress of the grains all depend on their orientation. Thus unique grains attempt to deform uniquely.

However, they are not free to do so due to intergranular constraints. Therefore transformation is a complex composite

process in a polycrystal; the subject of many recent experimental and theoretical investigations (see for example,

Sitter

et al., 2010

;

Stebner et al., 2013

). Furthermore, the quantified characteristics of superelastic responses vary significantly with

material, processing, and loading mode. Non-proportional loading provides further complexities (

Thamburaja and Anand,

2002

;

McNaney et al., 2003

;

Richards et al., 2013

). The models we previously mentioned each focus on certain aspects of

these phenomena with varying levels of complexity and fidelity, but an engineering demand for a simple, unified framework

that describes all of these observations still exists.

To understand how the present model aims to address this demand, we now consider a polycrystal completely in the

austenite state, above its transformation temperature, subjected to increasing tensile stress. A characteristic polycrystal su-

perelastic stress

–

strain curve in this circumstance is shown in

Fig. 1

b. The material initially loads elastically, then yields to

transformation at the point marked A, macroscopically similar to the single crystal case. The stress in the polycrystal is largely

uniform during elastic loading except as a result of the elastic anisotropy of the austenite. Thus, if we were to ignore the latter,

the first grains that will transform are those most favorably oriented for transformation according to a resolved stress criterion,

as observed in recent neutron diffraction experiments by

Stebner et al. (2015)

. Using in-situ optical microscopy,

Brinson et al.

(2004)

have also observed that the first appearance of the martensite occurs in isolated regions in well-oriented grains. This

finding is also supported by the mesoscale observations of

Daly et al. (2008)

that deviations in linearity of the stress

–

strain

curve occurs well before the formation of macroscopic transformed regions.

Bhattacharya and Schlömerkemper (2010)

have

recently proved in an idealized setting with uniform modulus that the transformation begins in isolated grains, while Richards

et al. have made the same observation using a full-field micromechanical model (

Richards et al., 2013

). Finally,

Sittner and

Novak (2000)

have noted that the constant stress Sachs or Ruess bound accurately describes the initiation of the transformation

in polycrystals. In summary, transformation

initiation

is governed by well-oriented grains.

Transformation growth kinetics, however, are not. As the best oriented grains begin to transform, whole grains cannot

due to the constraints of the neighbors and the transformation proceeds in isolated regions (

Richards et al., 2013

). Thus, the

internal stress distribution amongst grains becomes extremely heterogeneous as stress is redistributed from ideally oriented

grains where transformation has nucleated to slightly misoriented grains, which subsequently nucleate (

Richards et al.,

2013

;

Paranjape and Anderson, 2014

). These events cause a rapid progression of the transformation within the polycrystal.

This progression is further accelerated by the fact that once transformation has nucleated in a grain, little additional driving

force is needed for martensite growth. Additionally, stress redistribution may change the stress orientation within trans-

formed grains, and these events will cause martensite variants other than those initially formed to be preferred (

Stebner

et al., 2015

). The activity of these mechanisms in tandem gives rise to the stress

–

strain plateau AB.

Macroscopically, single and polycrystal superelasticity have appeared very similar up to point B. However, in loading

beyond the plateau (BC), polycrystals exhibit pronounced hardening (

Fig. 1

b) where most single crystals do not (

Fig. 1

a). In

polycrystals loaded to point B (

Fig. 1

b), transformation is incomplete or non-existent in some of the most poorly oriented

grains. Further transformation in these grains requires increased stress, while grains that have completed transformation

begin to elastically load again. Twinning events are also incited that reorient martensite to a greater extent than in the

plateau region (

Stebner et al., 2015

). All of these mechanisms in combination give rise to the observed hardening BC. As a

result, the transformation of the polycrystal is quite heterogeneous (

Brinson et al., 2004

;

Daly et al., 2008

;

Richards et al.,

2013

;

Stebner et al., 2015

;

Paranjape and Anderson, 2014

), as is the state of internal stress (

Pelton et al., 2015

). As trans-

formation and reorientation proceed, the poorly oriented grains exhibit smaller transformation strain in the direction of

loading, and thus they begin to saturate and lock together eventually leading to a network of fully transformed grains

surrounded by partially transformed and untransformed grains at C (

Brinson et al., 2004

). Thus, the

saturation

of the

transformation is governed by the poorly oriented grains. This phenomenon is also the reason why the constant strain Taylor

or Voigt bound gives a good description of the macroscopic transformation strains (

Bhattacharya and Kohn, 1997

;

Shu and

Bhattacharya, 1998

;

Bhattacharya and Suquet, 2005

Forward loading is continued until some maximum load is reached at point C; further loading evokes plasticity, which is

beyond the scope of the current model. Upon unloading, the material initially unloads elastically. However, at some point,

marked D, the material begins to recover its inelastic strain, initially with some softening and then (point E) perfectly. Upon

the recovery of all inelastic strain, at point F, the material unloads elastically.

An interesting consequence of the fact that the initiation and saturation of the martensitic transformation in polycrystals

are governed by two separate mechanisms is that the critically resolved shear stress criterion or the stress

–

strain aspect of

the Clausius

–

Clapeyron relation fails in a polycrystal after the very first initiation(s) of transformation (

Daly et al., 2007

). The

key heuristic and the point of departure

for the current work is the explicit recognition that the mechanics of initiation and

saturation of the martensitic transformation in polycrystals are two essentially different processes. Therefore, we differ-

A. Kelly et al. / J. Mech. Phys. Solids 97 (2016) 197

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199

entiate between two separate kinematic quantities, the nominal transformation strain in the martensite and the effective

transformation strain of the polycrystal. The former heuristically represents the macroscopic transformation strain one

would see if the grains were allowed to transform independently. This quantity governs the initiation of transformation. The

latter heuristically represents the transformation strain one actually sees limited by intergranular constraints. This quantity

governs the saturation of the transformation. We will see that the reorientation of both stress-induced and thermally

stabilized martensite arises naturally due to competition between these mechanisms. While, for simplicity, the introduction

above focused on superelasticity and stress-induced martensite, the model we present is thermomechanical and also

capable of describing thermal transformations and the resulting shape-memory effect and actuation phenomena.

The present model builds on the framework developed by

Sadjadpour and Bhattacharya (2007b

. Their work, like other

internal variable models in the literature, did not distinguish between initiation and saturation mechanisms. The current

work does, and also differs by adopting a new formulation for transformation surfaces that follows from a formula in-

troduced by

Cazacu et al. (2006)

to describe anisotropic and asymmetric plastic yield surfaces; this change improves the

ability to capture transformation asymmetry and processing-induced material anisotropy of SMAs. The present model has

similarities to those of

Panico and Brinson (2007)

and

Chemisky et al. (2011)

. Those models make a distinction between self-

accommodated and oriented martensite, hence they capture some of the features describe above. However, they do not

completely describe the difference between initiation and saturation.

The kinetics of transformation and reorientation in the model we present are rate-independent. Relative to the slow

loading rates of typical engineering applications (10

and below), these kinetics are largely rate independent

(

Abeyaratne et al., 1996

). Macroscopic rate effects are often seen, but these are often the consequence of the release of latent

heat, which is not inherent to the kinetics. In the present model, apparent rate effects due to thermal energy enter naturally

via the energy balance. On occasion, rate effects are also introduced due to grips, work hardened alloys, etc., but these are

small and are the result of particular circumstances that we do not address in this work. We note, however, that inherent

rate effects are important in dynamic situations, and the model can easily be modified to account for these if necessary (for

example as in

Sadjadpour and Bhattacharya, 2007b

Finally, some (highly worked) specimens of nickel

–

titanium based SMAs (a.k.a.

“

Nitinol

”

) display R-phase, an inter-

mediate phase that forms in between the austenite and martensite phases during thermal cycling (

Otsuka and Wayman,

1998

Duerig and Bhattacharya, 2015

). This phase has a small transformation strain and only appears in engineering ap-

plications with very particular processing and thermomechanical loading conditions. We ignore the R-phase in this model.

We proceed to introduce the model in

Section 2

, analytically demonstrate some features of the model in simplified

settings in

Section 3

, discuss our numerical implementation in

Section 4

, and conduct a parameter study while also

demonstrating the model relative to experimental data in

Section 5

. We conclude with a discussion of desirable future work

with the framework in

Section 6

2. Continuum model

2.1. Kinematics

Consider a macroscopic polycrystalline specimen of a shape-memory alloy, so that each material point corresponds to a

representative volume of material with numerous austenite grains, with possible fine-scale martensite microstructure

within each grain. We denote the macroscopic strain as

and the (absolute) temperature as

.Asin

Sadjadpour and

Bhattacharya (2007b)

, we introduce two additional internal variables that describe the state of transformation. The first is

the

volume fraction

of the martensite

. By definition, this variable satisfies the constraint

≤≤

()

01.

The second is the

nominal transformation strain

of the martensite

, which we define as the average transformation strain

of every region of martensite in every grain in the representative volume. We will assume that

∈

()

where

describes the set of all possible nominal transformation strains that may occur. This set could be computed by

taking averages of local martensite strains over all variants in the representative volume, however we prescribe this phe-

nomenologically in the next section. In subsequent sections of the model formulation,

will dictate the driving force on

transformation and the set

will define initiation events. The overall or

effective transformation strain

is obtained by

multiplying the nominal transformation strain with the volume fraction,

. This overall transformation strain drives

macroscopic behaviors governed by inter-granular compatibility; thus

will play a critical role in saturation. We define a

set

that describes the set of effective transformation strains that may be achieved without hardening/softening influence

from poorly oriented grains in forward/reverse transformation events. The effective transformation strain may take on

values outside of this set with some energy penalty, as mathematically prescribed in the next section.

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2.2. Energy

The Helmholtz free energy density of the system is postulated to be of the form

⎛

⎝

⎜

⎞

⎠

⎟

( )()()

()

εελθελελελελ

θθ

λθ

λελελ

=−

()− +

(− )

−()

+ ()+

+()

()

WLcGGG

,,,

:C :

mmm

s m

The first term is the elastic energy density, which depends quadratically on the elastic strain, i.e., the difference between the

total strain and the effective transformation strain. The second term represents the excess chemical energy density of

martensite which depends on temperature. Note that we have chosen to restrict the model to a moderate range of tem-

peratures close to the thermodynamic transformation temperature

where this relation may be taken as linear, scaling

with the latent heat of transformation,

, relative to the difference between the absolute temperature and the thermo-

dynamic transformation temperature. The third term is the contribution of heat capacity

. Each phase of the material may

exhibit a unique heat capacity as well as a unique tensor of bulk elastic constants,

, thus we adopt a rule of mixtures

approach in defining the composite responses of the two phases at a material point

λλλ

()=

+( − )

()

CC1C,

λλλ

()=

+( − )

()

where the superscripts

and

denote austenite and martensite phase properties, respectively.

The last three terms in the energy

,and

describe the energy densities for initiation, saturation, and growth of

transformation, respectively. Specifically,

reflects internal stored energy related to i

ncompatibility between variants and across

grains; i.e., the barrier to transformation growth. Thus, in a manner analogous to hardening in plasticity, we postulate that

⎧

⎨

⎪

⎩

⎪

ξξξ

()=

∈[

]

+∞

()

λλ

0, 1

else

where

λλ

are constitutive constants. The first line accounts for the fact that the amount of stored energy in the

system can increase with increasing volume fraction due to

the development of internal

stresses, thus for choices of

≠

λλ

experiences some hardening within the set [0,1]. The second line enforces the hard constraint

(1)

The functions

for

are defined over symmetric matrices. To prescribe the sets of nominal and effective trans-

formation strains, we introduce functions of the form

following

Cazacu et al. (2006)

⎛

⎝

⎜

⎞

⎠

⎟

()

∑

ξμξμξ

()=

(

) −

(

)

−

()

αααα

gab

where

is any symmetric matrix,

(

,1,2,

denote the principal values of the matrix argument

are 4th order

tensors satisfying minor and major symmetries and reflecting the anisotropy of the material (choosing

where

is the

4th order identity tensor results in isotropy) and

ααα

aab

are constitutive constants to be prescribed. We assume that

−<

ααα

aab

11,1,

to ensure the convexity and coercivity of

We postulate

⎧

⎨

⎩

()=

∈

+∞

()

else.

where the set of allowable initiation strains is taken as

ξξξ

( )=

( )≤ }

()

:tr 0,

0 .

Eq.

(8)

enforces the constraint

(2)

. The martensite strain may evolve freely with no energy cost within the set

, but

evolution outside of this set is not allowed. We can also give physical meaning to the coefficients in

(7)

. The coefficient

denotes some radius of the set

while the constants

and

control asymmetry in the same way the coefficients

and

do in the formulation of Cazacu et al. (see Fig. 4 of

Cazacu et al., 2006

We note that the expressions

(7)

are different from the polynomials of the invariants

and

used in

Sadjadpour and

Bhattacharya (2007a)

. We found that these formulae restricted the amount of asymmetry that one could prescribe before

We assume that the bulk moduli of the phases (under hydrostatic stress) are equal consistent with experimental observations of SMAs (

Patel and

Cohen, 1953

We introduce a notation where

is generically used to represent the argument of the functions; i.e., it is a placeholder for the internal variables of the

model defined in

(3)

We clarify that the colon in

(

)

:tr

reads

“

such that the trace of

”

where in

(7)

the colon denotes the contraction of the fourth and second order

tensors. Despite the ambiguity, both uses of the colon are obvious in their context.

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the sets of allowable strains became non-convex, especially in anisotropic media. This limitation proved unsatisfactory for

highly textured SMA polycrystals. In contrast,

(7)

enabled greater asymmetries of convex sets, especially for anisotropic

media (

Cazacu et al., 2006

Recall that for saturation, the effect of incompatibility between variants and grains means that continued transformation

strain requires internal stresses and additional stored energy. Thus, unlike the previous hard constraints on the allowable

values of volume fraction and martensite strains, we postulate a soft constraint for saturation such that kinematic hardening

occurs once the allowable set of effective transformation saturation strains is exceeded:

⎧

⎨

⎪

⎩

⎪

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

ξξ

()=

()≤

()=

()

()>

()=

+∞

( ) ≠

()

00,tr0

0, tr

where

and

are the saturation hardening coefficient and exponent, respectively. The function

0 on the set

ξξξ

( )=

( )≤ }⊂

()

:tr 0,

where the final inequality is enforced by the choice of constants

, and

, which have analogous interpretation as their

initiation counterparts.

Finally, we assume the usual Fourier law for heat conduction, which is a linear relation between heat flux

and tem-

perature gradient according to the thermal conductivity,

=− ∇

()

2.3. Balance laws, driving forces and kinetics

We postulate the usual balance laws for equilibrium and energy as well as the second law of thermodynamics (Clausius

–

Duhem inequality)

σσσ

()

div

σε

−∇· +

()

⎜⎟

⎛

⎝

⎞

⎠

θθ

≥∇· −

()

where

s

is the stress,

is the body force,

is the internal energy density,

is the entropy density,

is the absolute

temperature,

is the heat flux and

is the radiative heating.

With the free energy density specified in

(3)

, we can use the balance laws and arguments following

Coleman and Noll

(1963)

to obtain constitutive relations for the stress (

s

) and entropy (

λελε

∂

=()(− )

()

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

=−

∂

=−

()

1ln .

We also obtain the following driving forces as the thermodynamic conjugates to the internal variables

and

. This is a

little tricky since the function

are not smooth. However, it is first instructive to assume that the functions

are

smooth. In such situations,

⎛

⎝

⎜

⎞

⎠

⎟

ελε

σε

θθ

εε

ξξ

=−

∂

=− ( −

)

∂

(−

−

(− )

−()−

∂

−

∂

()

λε

:ln:

λσλ

=−

∂

=−

∂

−

∂

()

ελε

In the non-smooth situation, we interpret these equations using usual methods of convex analysis (

Ekeland and Témam,

1999

) as follows

⎛

⎝

⎜

⎞

⎠

⎟

ελε

σε

θθ

εελελ

−−(− )

∂

(−

−

(− )

−()∈ ∂( )+∂()

()

:ln:

m s m

λσλελλε

− +

∈∂( )+∂ ( )

()

dGG

dev

s m

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202

where

σσ

=−

dev

and

∂

denotes the sub-differential of the

To close the system, we relate the driving forces to the relevant rates using kinetic relations. In the rate-dependent

smooth setting, we postulate

λε

=()

()

λλε

K d

for given functions

and

. However, in this paper, we are interested in the rate-independent case where

and

evolve

freely if and only if the corresponding driving forces reach a critical value. Therefore, the functions

and

are neither

smooth or invertible. So, we specify them in the form of Kuhn

–

Tucker conditions. The rate of change of the internal variables

are constitutive functions of the driving forces such that transformation evolves according to:

λλλ

=||<

≥||≤

≤≤

()

λλλλλ

dd d

0for

if0

and martensite strain:

λλ

≥≤∈

()

dd d

0for

:0,

;

mmm

ελθ

==≥

()

Reset

0 when 0,

where

is the critical driving force for transformation and

is the critical driving force for martensite variant reor-

ientation. We may rewrite these conditions by introducing a dissipation potential

λελλελε

(

)= |

|= (

(

)

()

λλ

DddDD

mmm

Now kinetic equations may be rewritten as

∈∂

()

λλε

dDd D

subject to the reset condition

(25)

Combining

(27)

with

(20)

and

(21)

, we can write the evolution equations for

⎛

⎝

⎜

⎞

⎠

⎟

ελε

σε

θθ

εελελε

∈(− )

∂

(−

−

(− )

( )+

∂ (

)+∂ ( )+∂ (

)

()

LG GGD

:ln:

m s m

λσλελλελ

∈−

+∂( )+∂ ( )+∂ (

)

()

GGD

dev

s m

subject to the reset condition

(25)

We will further discuss the interpretation of these conditions through specific analytic examples of shape memory

behaviors in

Section 3

, and their numerical implementation in

Section 4

. For now, we note that these equations are con-

sistent with the variational structure

∈∂ +∂

(

Mielke, 2005

;

Yang et al., 2006

Finally, since

ε=

we can use

(3)

(12)

, and

(17)

to rewrite the energy balance

(14)

θθ

λλε

=∇· ∇ + +

()

λε

cKr

In summary, the model specifies that the behavior of SMAs is governed by the equilibrium equation

(13)

, the heat

equation

(30)

and the evolution equations

(28)

and

(29)

3. Some features of the model

We now demonstrate some of the relationships between the parameters and mechanisms of the model through analytic

calculations. To enable these analyses, we make a number of assumptions: we assume that the elastic moduli of the two

phases

are equal and isotropic and the specific heat of the two phases

cc c

are equal. We also assume

that the exponent

in the definition of

is infinite so that it becomes the hard constraint

⎧

⎨

⎩

ξξξξ

()=

∈={ ()=

()≤}

+∞

()

0:tr0,0.

else.

We also ignore the critical driving force for martensite reorientation (

). Finally, except for actuation in Section 3.1.3,

we assume that

λλ

(6)

so that

may evolve freely within the interval [0, 1].

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3.1. One dimension

It is instructive to specialize the model to one dimension. We can derive this one-dimensional model from the three

dimensional formulation by assuming uniaxial loading; we reserve this formal derivation for

Section 3.3

. For now, we simply

assume

⎛

⎝

⎜

⎞

⎠

⎟

()

εελθελελωθθ

λελελ

=−+ −

+()+ +()

()

WcGGG

,,,

Eln

mmpimsm

where

ωθ

θθ

()=

(− )

()

and the sets

reduce to the scalar intervals

εεεεεε

=[]≤≤≤

()

,,,

The constraints that these choices impose on the kinematic variables

and

are shown in

Fig. 2

(a).

3.1.1. Superelasticity

Consider an isothermal strain-controlled experiment where the temperature is held constant at a value significantly

higher than the transformation temperature so that

ωθ

()>

. Consider the specimen at rest at zero stress so that it is in

the austenite state with

and

. Now subject the specimen to a monotonically increasing overall tensile strain

(

)

For very small times, the stress is given by

σε

()= (

)

. Since this stress is positive, the nominal transformation strain

takes its maximal tensile value

. This is marked as the point

Fig. 2

a. Following

(18)

, the driving force for transfor-

mation in one-dimension is given by

εεω

=−

, and is initially negative. However, since

is already zero, it can not

decrease any further. Thus the volume-fraction remains zero and the stress increases proportional to the imposed strain

according to E, and we begin to traverse from 0

/0 toward the points marked 1 in

Fig. 2

a/b.

As the applied strain increases, so do the stress and the driving force

till the latter becomes positive and eventually

reaches the value

(i.e.,

εεω

−=

) at the points marked 1 in the figure. At this point, the strain and stress are given as

()

λλ

Now transformation begins and proceeds in such a manner to keep

constant and consequently the stress

s

constant so

that the material traverses from the points marked 1 towards 2 with increasing applied strain. As

increases, the overall

transformation strain

increases and eventually saturates the constraint

εε

≤

at the value

Fig. 2.

The kinematic variables and stress-induced martensite in one dimension. (a) The volume fraction and nominal transformation strain are constrained

to lie in the shaded region. (b) The stress

–

strain curve.

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()

This is indicated by the points 2 in the figure, where

εσ

()

λλ

The transformation is now saturated, and further loading does not lead to any further transformation. Thus, the stress

increases in an affine manner with applied strain and the material traverses towards the points 3 in the figure.

The specimen is then unloaded by monotonically decreasing the applied tensile strain. The transformation is initially

constant, with no change, and the material unloads elastically, traversing towards the points 4 in the figure. The stress and

the driving force decrease monotonically, till the latter becomes negative and eventually reaches the value

−

at points 4.

Here,

εσ

−+

()

λλ

Reverse transformation now begins at points 4 and

decreases as unloading proceeds. The driving force and stress remain

constant from points 4 to 5. The reverse transformation is complete at points 5 when

, and

−+

()

λλ

The material subsequently unloads elastically and the material returns to the origins.

We have an analogous situation in a compressive loading cycle with the analogous quantities obtained by replacing the

superscripts

with

in the formulae

(35)

–

(39)

above.

A series of comments are in order. First note that the material parameters

associated with the set of nominal

transformation strains determine the initiation of transformation (c.f.

(35)

) while the material parameters

associated

with the set of effective transformation strains determine the extent of transformation (c.f.

(37)

). This justifies the subscript

for initiation for the former and the subscript

for saturation for the latter. More importantly, the parameters determining

initiation and saturation are independently prescribed in this model.

Second and related, we can infer the values of transformation strain, the stress hysteresis and mean-value of stress to be

εεεεεε

σσσσσ

≔( −

− )=

≔( +

−

− )=

≔( +

+ )=

()

trans

hyst

,,,,,,

,,,,

, ,,,,

Note that each of them can be independently constitutively prescribed

–

the transformation strains are determined by the

parameters

, the hysteresis by the parameter

and the mean-value of the stress by

, and there is no universal relation

amongst them.

Third, tension and compression are different due to the difference in parameters with superscript

and superscript

This tension-compression asymmetry is consistent with various observations going back to

Burkart and Read (1953)

Fourth, note that the transformation from austenite to martensite does not go to completion, but saturates at a maximum

volume fraction of

εε

. This is consistent with surface observations of

Brinson et al. (2004)

, which have been

recently affirmed to be a bulk phenomena by

Stebner et al. (2015)

Fifth, the value of stress at which the transformation begins and completes or the reverse transformation begins and

completes depend on temperature through

.If

depends linearly in temperature as in

(33)

, then these stresses depend

linearly on temperature consistent with the Clausius

–

Clapeyron relation (

Otsuka and Wayman, 1998

). Further, we can invert

these relations to obtain the values of temperature where the transformation begins at zero stress:

θθ

==−

==+

()

λλ

sfc

Sixth, we see above that

σσ

and

. These are all manifestations of the aforementioned assumptions.

There is no reorientation of martensite because we have assumed one dimension. We shall see in

Section 3.2

that reor-

ientation of martensite can give rise to a situation where

σσσσ

,,,,

. Here, we have also assumed hard constraints

on the sets

(34)

and that

λλ

. The more general assumption or soft constraint

(10)

allows the transformation to

continue to evolve to higher volume fractions with increasing stress so that

σσσσ

,,,,

. Setting

λλ

violates all

three equalities

(40)

. We will study these other choices extensively in Section

. We also assume rate-independent kinetics

(23)

and

(24)

. Rate-dependent kinetics would also violate all three inequalities

(40)

, as was examined by

Sadjadpour and

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205

Bhattacharya (2007b)

. Finally, we have assumed isothermal conditions. Most experiments are not isothermal since the

transformation from austenite to martensite is exothermic through the release of the latent heat. We can estimate the

hardening in the other extremity of adiabatic conditions (

) following

Sadjadpour and Bhattacharya (2007b)

. Under

our constitutive assumption and the additional assumption that the dissipation due to kinetics is small compared to latent

heat,

(30)

reduces to

θλ

()

We can integrate this from time

to obtain

⎛

⎝

⎜

⎞

⎠

⎟

λλ

()

(( )− ( ))

()

exp

or the temperature increases with advancing transformation. Returning now to the loading-unloading cycle under tension,

the volume fraction is initially zero and the temperature is

. As we begin loading from the austenite, the temperature

remains constant and things remain unchanged from before till we begin transformation. Thus, as before,

θθ

θε

(−)

()

As transformation proceeds, the temperature rises. However, the driving force has to remain at the critical value

, and

thus,

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

σε

θθσ

εε

() − ( ()− )=

⇒ ()=

()

−

tdt

LLt

exp

1 .

The transformation saturates at

given in

(36)

; here,

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

εε

−

()

exp

1 .

Thus, hardening due to latent heat of transformation that would occur under adiabatic conditions is given by

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

σσ

θε

−=

−

()

exp

1 .

3.1.2. Shape-memory effect

We now subject our one dimensional specimen to a controlled (infinitely slow) temperature-load cycle to explore the

shape-memory effect. We begin as before with an unloaded specimen at a temperature above

with

and

. This

Fig. 3.

Shape-memory effect in one dimension. (a) Stress

–

strain

–

temperature curve and (b) the volume fraction-nominal transformation strain space.

A. Kelly et al. / J. Mech. Phys. Solids 97 (2016) 197

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206

is marked as the points 1 in

Fig. 3

. We lower the temperature below

and

so that the material transforms to martensite

and

at points 2. We now keep the temperature constant and subject the material to a strain controlled loading un-

loading cycle. The moment that strain, and consequently the stress becomes positive, the nominal transformation strain

takes the value

to optimize the energy and the material traverses from points 2 to points 3 in the figure. Now, the effective

transformation strain is the saturation transformation strain

. Note that in

Fig. 3

, this event happens at a value of stress

equal to

; non-zero values would raise the stress of the martensite reorientation plateau. Further loading takes the

material to points 4, and unloading brings it to 5 (which coincides with 3 in the stress

–

strain

–

temperature space). We now

keep the stress zero and heat the specimen. There is no change in strain or transformation as the material traverses from

points 5 to 6. At points 6, the temperature has reached

so that

λλ

−

and the reverse transformation begins. As the

material traverses from points 6 to points 7, the volume fraction goes from

]

. Once it reaches zero, the

resets

to 0 and the material is in the state indicated by points 8 in the figure. Further heating brings the material back to the

starting points 1.

Note that in the thermal cycle, there is no restriction on the amount of transformation, and the specimen is able to

transform fully to martensite. Further, rate and heat transfer effects would make the curves rounded.

3.1.3. Actuation

We now examine the behavior of the model for actuation by studying a constant stress, temperature cycle. In this

subsection, we assume

and

λλ

. This changes the transformation temperatures to

=−

()

=−

(+ )

()

λλ

(− )

()

λλ

()

instead of

(41)

Consider a material at a temperature

well below

with self-accommodated martensite at the points marked 0 in

Fig. 4

. We begin to apply a uniaxial stress to the material. As soon as it reaches the value

σσ

()

the martensite reorients and the martensite strain

jumps from zero to the value

, i.e., to the points 1 in the figure. Since

, this is also the value of the effective transformation strain. We continue to increase the stress and

remains at

until the material reaches a final stress

s

at points 2 in the figure. The value of the strain at this point is

Fig. 4.

Actuation effect in one dimension. (a) The volume fraction-nominal transformation strain space. (b) Strain (black solid) and volume fraction (red d

ashed)

as a function of temperature. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this pa

per.)

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εεε

==+

()

We now begin to heat the material. The driving force for transformation

(19)

is given by

σε

θθ

=−

(− )

−

()

λλ

dL f

This force is initially positive but eventually becomes negative and reaches the critical value

−

when the temperature

reaches

()

σθ

σε

()= +

−

()

λλ

and the material reaches points 3 in

Fig. 4

. The martensite now begins to convert to austenite,

begins to decrease as

temperature increases and the material traverses from points 3 to 4 in the figure. Note that since the stress is still above

s

the nominal transformation strain

increases to maintain the constraint

εε

. However, notice that the strain remains

at the value

(52)

The driving force is

σε

θθ

=−

(− )

−

()

λλ

dL f

In particular, the temperature continues to increase during reverse transformation to maintain a driving force value of

−

The material reaches points 4 when the nominal transformation strain is

and the volume fraction is

given in

(36)

at a

temperature

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

σθ

θε

σε

()= +

−

()

λλ

Further heating causes further reverse transformation as the material traverses from points 4 to 5 in

Fig. 4

. The nominal

transformation strain remains at

, but the effective transformation strain is

so that the total strain

λεσ

begins

to decrease with

. The driving force is now

σε

θθ

=−

(− )

−

()

λλ

dL f

and the material needs further heating to sustain reverse transformation. It reaches points 5 when

, the temperature

()

σθ

σε

()= +

()

and strain

()

Further heating does not change the state or the strain.

The material reaches the maximum temperature at points 6, and then begins to cool. The driving force

σε

=−

θθ

(− )

is negative; however since

, further reverse transformation does not occur. As the temperature decreases, the driving

force becomes positive and eventually reaches the value

at temperature

()

σθ

σε

()= +

−

()

indicated as points 7. The transformation from austenite to martensite now begins and

begins to increase along with the

strain as the material traverses from points 7 to 8 with decreasing temperature. The driving force now is given by the

expression

(57)

, thus the material needs continued cooling to sustain transformation. It finally reaches the points 8 when

the volume fraction reaches the value

given in

(36)

and the temperature reaches the value

⎛

⎝

⎜

⎛

⎝

⎜

⎞

⎠

⎟

⎞

⎠

⎟

σθ

θε

σε

()= +

−

()

λλ

The strain has now reached the value

(52)

Further heating continues transformation;

increases and the driving force is given by

(55)

. The nominal transformation

This decoupling between phase fraction and strain is a result of the simplifying assumptions in this analytic section including

∞

λλ

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