Control of intrinsic instability of superelastic deformation

Control of intrinsic instability ofsuperelastic deformation

Julia Slutskera,*, Alexander L. Roytburdb

aMaterials Science and Engineering Laboratory, National Institute of

Standards and Technology, Gaithersburg, MD 20899, USAbDepartment of Materials and Nuclear Engineering,

University of Maryland, College Park, MD 20742, USA

Received in final revised form 16 March 2002

Abstract

The superelastic deformation of shape memory alloys has intrinsic instability as a result of

incompatibility between martensite and austenite phases at stress-induced transformation. Thisinstability leads to the thermodynamic stress–strain hysteresis of the superelastic deformation. Ithas been shown that the combination of a shape memory active material with a non-transformingpassive material can decrease and suppress the instability of superelastic deformation. The

stability analysis of superelastic deformation allows one to formulate design principles ofadaptive composites with controlled stress–strain hysteresis and potentially large reversibledeformation.

# 2002 Published by Elsevier Science Ltd.

Keywords: A. Phase transformation; Superelastic deformation; Adaptive composite; Stress–strain hysteresis

1. Introduction

Phase transformations in solids as a rule is accompanied by self-strains and,therefore, result in deformation of transforming crystals. The deformation can bereversible (so-called superelastic) if it proceeds as the evolution of a mixture ofcoherent phases, i.e. there are no dislocations, plastic deformation, or fracture duringthe transformation. However, the preservation of the lattice coherency is a necessarybut not a sufficient condition of the reversible superelastic deformation. It is necessaryalso that the equilibrium heterophase microstructure has been maintained during

International Journal of Plasticity 18 (2002) 1561–1581

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0749-6419/02/$ - see front matter # 2002 Published by Elsevier Science Ltd.

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* Corresponding author.

E-mail address: [email protected] (A.L. Roytburd).

phase transformation. For example, a simple polydomain structure consisting ofperiodically alternating layers of the transforming phases is a typical equilibriummicrostructure that arises during uniformly constrained phase transformation(Kohn, 1991; Roytburd and Slutsker, 1997a,b). Then, the superelastic deformation isa result of changing domain fraction due to nucleation or movement of domaininterfaces. It is reversible if the co-existing phases in a polydomain structure arecompatible.The analysis of deformation due to a constrained phase transformation (Roytburd

and Slutsker, 1997b, 1999a,b) has demonstrated a principle difference between thetransformational deformation of a mixture of compatible phases and a mixture ofincompatible ones. This difference is schematically presented for an uniaxial defor-mation of a single crystal in Fig. 1. For compatible phases the strain- and stress-controlled deformations are thermodynamically reversible and exhibit superelasticbehavior at an equilibrium stress, �0=�f/"0, where �f is the difference between thefree energies of an initial (A) and a product (M) phases (Fig. 1a), "0 is the transfor-mational self-strain along the direction of the uniaxial deformation (Fig. 1b).The free energy of a mixture of incompatible phases is a non-convex function of

an average strain and, consequently, an effective elastic modulus has some negativecomponents (Fig. 1c). It corresponds to the instability of the deformation and to thethermodynamic stress–strain hysteresis under loading and unloading at the stress-controlled deformation (Fig. 1d). This hysteresis does not depend on the rate ofdeformation on the contrary to the kinetic hysteresis which is not shown in thesefigures. Thus, the phase incompatibility results in the intrinsic instability of trans-formational deformation. The uniform displacement-controlled deformation pre-sented in the Fig. 1c is possible if there is no coarsening which destroys themicrostructure uniformity. Non-uniform deformation appears as the formation of aneck. The internal stresses are localized in the neck and the phases become almoststress-free.The uniformity of the heterophase polydomain microstructure can be maintained

if a transforming material is combined with a passive material, i.e. it serves as anactive component of an adaptive composite. Besides that, the formation of a com-posite is an effective way to suppress the thermodynamic instability of superelasticdeformation. As shown before (Roytburd, 1992; Roytburd, 1996a; Wen et al., 1996;Roytburd et al., 1999; Roytburd and Slutsker, 1999c; Slutsker and Roytburd, 1999) anadaptive multilayer composite containing an active polydomain component (Fig. 2)demonstrates a stable superelastic deformation unless the fraction of an active compo-nent does not exceed some critical value. Then, the effective superelastic modulus of thecomposite becomes positive and, consequently, the superelastic deformation is stableunder either displacement control or stress control (Fig. 3).The simplest polydomain microstructure consisting of two domains was considered

in the papers on adaptive composite mentioned above. In this paper the concept ofthe control of superelastic deformation in a composite is expanded to the poly-domain structure arising due to martensitic transformations. Typical martensitephases consist themselves of domains (twins) and since, the equilibrium two-phasemixture has polydomain hierarchical structure. The common point of view is that

1562 J. Slutsker, A.L. Roytburd / International Journal of Plasticity 18 (2002) 1561–1581

the co-existing phases during martensitic transformations, an austenite and amartensite, are compatible due to a special internal twin structure of a martensite.However, it has been shown (Roitburd, 1977, 1978, 1983; Roytburd and Pankova,1985) that the compatibility between the phases is violated during stress- or strain-induced transformations because of the change of the twin structure. Therefore,special deformation effects accompanying the formation of incompatible phasesshould appear in the case of martensitic transformations. These effects are negativemoduli at displacement-controlled deformation and thermodynamic hysteresis at

Fig. 1. Superelastic deformation with compatible and incompatible phases. (a) Temperature dependence

of the free energies of an initial (A) and a product (M) phases. (b) Stress–strain curve for compatible

phases. (c) Displacement-controlled deformation for a uniform mixture of incompatible phases. (d) Stress-

controlled deformation of incompatible phases. E is the Young’s modulus, � is the parameter of incom-

patibility. (e) Displacement-controlled deformation for non-uniform distribution of incompatible phases.

J. Slutsker, A.L. Roytburd / International Journal of Plasticity 18 (2002) 1561–1581 1563

stress-controlled deformation that characterize the intrinsic thermodynamicinstability of superelastic deformation (Roytburd and Slutsker, 1995a,b,c, 2001a;Roytburd, 1996b, 2000).1

We will show in the paper that as in the case of simplest polydomain structure, theinstability of superelastic deformation during martensitic transformations can becontrolled if a transforming material is a component of an adaptive composite. Toemphasize physical effects of the phenomenon and to present the results in a visibleform we will consider transformations with small self-strains and identical elasticproperties of the phases.

2. Multilayer adaptive composite under external stress

Consider a composite consisting of active and passive layers. The elastic energy ofinternal stress due to interaction between layers is

Fig. 2. Multilayer composite containing transformable, active, and non-transformable, passive, layers.

�=h/H is the fraction of the active layer, �=l/L is the fraction of the martensite phase in the active layer,

�=d/D is the fraction of the twin domain in the martensite phase.

1 The transformational deformation of single crystals is considered. However, the effects of instability

should be important for polycrystals also, resulting in the contribution to stress-strain hysteresis at dis-

placement-controlled deformation due to uncorrelated deformations in grains.


eI ¼ � 1� �ð Þ1

2"MG

^"M

G^¼ C

^� C

^~nn ~nnC

^~nn

� ��1

nC^

ð1Þ

� is a fraction of an active layer, C^is a tensor of elastic moduli, ~nn is orientation of

the normal of the interface, "M is a misfit strain characterizing incompatibilitybetween layers.For isotropic materials of layers:

eI ¼ � 1� �ð ÞE

2 1� �2ð Þ"M1� �2

þ "M2� �2

þ2�"M1 "M2

h ið2Þ

where "1M and "2

M are the principal misfit strains in an interface plane, E and � areYoung’s modulus and Poisson’s ratio, respectively.After stress-induced transformation the austenite–martensite microstructure is

formed in the active layers. The Gibbs free energy of the composite becomes:

� ¼ �1

2S^�2 þ eI �; "

M �; �ð Þ� �

þ � f �; �ð Þ � �"0 �; �ð Þ½ � ð3Þ

The first term is a free enthalpy of an initial (austenite) phase under the stress, thelast two terms are the free enthalpy of the unconstrained active layer, � is a fraction

Fig. 3. The stable and unstable superelastic deformation of the composite with simplest microstructure in

the active layer (� is a fraction of an active layer).


of the martensite phase, � is a fraction of a twin domain in the martensite, "0(�, �) isan average transformational self-strain of an active layer, which also determine themisfit between layers "M(�,�).We consider a transformation with lowering of symmetry, i.e. the symmetry of a

product phase (martensite) is lower than the symmetry of an initial phase (auste-nite). Then, the crystallographically different but physically identical variants, ordomains, of a product phase can be formed. For example, at the transformationof a cubic phase into a tetragonal one three different variants are formed withself- strains "0;i, i=1,2,3 (Fig. 4a).

"0;1 ¼ "0

1 0 0

0 �� 0

0 0 ��

0B@

1CA; "0;2 ¼ "0

�� 0 0

0 1 0

0 0 ��

0B@

1CA;

"0;3 ¼ "0

�� 0 0

0 �� 0

0 0 1

0B@

1CA

ð4Þ

Fig. 4. Evolution of an equilibrium microstructure in the active layer at uniaxial deformation. (a) Cubic-

tetragonal transformation in the active layer. (b) Equilibrium microstructure containing a mixture of

austenite and polydomain martensite. The possible paths of evolution of the microstructure are shown:

mixture of austenite and single-domain martensite (A+Msd) and twinned martensite (Mtw).


Besides the transformation from a cubic lattice to the tetragonal one (e.g. in In–Tlalloys), the tetragonal self-strain corresponds to the BCC!FCC transformation if�1/2 and "00.2. With opposite sign it describes FCC!BCC transformationand approximately describes BCC!HCP transformation if "00.1. Martensitictransformations with tetragonal self-strain Eq. (4) are considered hereafter.The product phase (martensite) usually consists of two or more different domains

(twins) forming plane-parallel alternation. The domains are compatible if they con-tact along a plane of symmetry of an initial phase (a twinning plane). If the period ofdomain structure is small in comparison with dimensions of the martensite phase,the martensite can be treated as a special polytwin phase with an average self-strain:

"0 �ð Þ ¼ 1� �ð Þ"0;j þ �"0;i i; j ¼ 1; 2; 3 ð5Þ

where � is a volume fraction of i-domain in (i,j) polytwin. It is assumed hereafterthat � is a minor fraction, i.e. �<1��. The self- strains in Eq. (5) are additivebecause they are small. For example, a polytwin product phase consisting ofdomains 1 and 2 has an average self-strain:

"0 �ð Þ ¼ 1� �ð Þ"0;1 þ �"0;2 ¼

"10 �ð Þ 0 0

0 "20 �ð Þ 0

0 0 "30 �ð Þ

0B@

1CA;

"10 �ð Þ ¼ "0 1� � �þ 1ð Þð Þ

"20 �ð Þ ¼ "0 ��þ � 1þ �ð Þð Þ

"30 �ð Þ ¼ �"0�

ð6Þ

The equilibrium two-phase microstructure, in general, consists of plane-parallellayers of a product phase separated by layers of an initial phase (Roitburd, 1978;Roytburd, 1993; Kohn, 1991; Roytburd and Slutsker, 1999a) (Fig. 4b). If the dif-ference between the elastic moduli of the phases is negligible the free energy of thetwo-phase state is as follow:

f ¼ Df�þ 1=2� 1� �ð Þe "0 �ð Þð Þ ð7Þ

where �f=f 02�f 0

1 is a difference of the free energies of an initial phase ( f 01) and the

product phase ( f 02), � is a volume fraction of the martensite. The energy e("0 (�)) is a

minimum of the energy of internal stresses due to incompatibility between the pha-ses. It is equal to the elastic energy of a thin plane-parallel plate with self-strain "0(�)embedded in an austenite matrix. Provided the eigenvalues of "0 (�0) have differentsigns (e.g. "0

2(�)<0; "01(�)>"0

3(�)>0) the energy is minimum, if the plate is elasticallystressed along x2 so that "22

el =-"02(�) and is oriented along a plane with normal n.

n21 ¼"10 �ð Þ � �"20 �ð Þ

"10 �ð Þ � "30 �ð Þ; n2 ¼ 0; n23 ¼ 1� n21 ð8Þ


where � is the Poison’s ratio. Then only one component of the internal stress �22exists because the uniaxially stressed austenite and martensite are compatible alongthe interface with the normal n. The plate’s energy is

e "0 �ð Þð Þ ¼1

2E "20 �ð Þ� �2

ð9Þ

where E is Young’s modulus along the eigenvector x2. This is the minimum energyof an embedded plate if "0

2(a) is the smallest eigenvalue of two ones with the samesign (Roitburd, 1978). Eqs. (8) and (9) are written for an elastically isotropic mar-tensite. But similar results for a plate with minimum elastic energy can be obtainedfor the anisotropic case too if the Young’s modulus along x3 is not much larger thanalong x1 and x2 (Roitburd and Kosenko, 1976).The elastic energies in Eq. (7) are written in a homogeneous approximation. It

means that a spacing of twins, d, is small in comparison with the thickness of mar-tensite layer, D, and this thickness as well as the thickness of austenite layer, D0, issmall in comparison with dimensions along interfaces between the phases. Then, thepolytwin martensite and the two-phase mixture can be considered as macro-scopically homogeneous phases with the energies (9) and (7). For constrained crys-tals, particularly, for martensite plates and for composite layers, the uniformity ofmicrostructure is dictated by the constraint conditions. In spite of a small scale ofthe considered microstructure ((D+D0)/L<<1, d/D<<1), the free energy [Eq. (7)]does not contain the energy of the interfaces between domains. This energy is neg-ligible if d>> , where is a characteristic thickness of an intertwin interface (formartensitic transformations 1�10 nm) (Roytburd and Slutsker, 1999a,b).The variable self-strain of a polytwin martensite due to a variable domain fraction

supplies an additional degree of freedom (in comparison with a single-domain pro-duct phase). It enables to reduce to zero internal stresses in the two-phase mixture.Without external fields there are 24 geometrically different but crystallographically

and physically equivalent equilibrium martensite plates. The martensite obtained bycooling contains all 24 possible orientations. The external stress removes thisdegeneration. The free enthalpy contains the term � �"0 �ð Þ which expresses thework of the external stress on the self-strain of transformation. The trend to max-imize the absolute value of this term leads to the selection of a few preferable var-iants of the plate or often to one variant. This effect of external stress on amartensitic transformation is well known and has been studied extensively, theore-tically and experimentally, starting with a pioneer work by Patel and Cohen, 1953.But there is also the second stress effect following from the flexibility of internalmartensitic structure. It appears as a changing self-strain of the martensite phase.This effect is studied below. To emphasize the effect we will consider the simplestmicrostructure containing only one set of identical parallel martensitic plates. Thesearchitectures correspond to the minimum of the free energy if the external stress isabsent and corresponds to the minimum of the free enthalpy with maximal work� �"0 �ð Þ.


The equilibrium microstructure, �0, �0 can be found from the equations

@�

@��0 ¼

@�

@��0 ¼ 0 ð10Þ

The stress–strain relation and the effective compliance of the superelasticdeformation are

" ¼ �d�

d��0;�0 S

^ � ¼ �d2�

d�2 �0;�0

ð11Þ

In the following section the uniaxial deformation of the composite is analyzed.

3. Superelastic deformation of the composite under uniaxial stress

We consider now the multilayer composite consisting of the active and passivelayers under an uniaxial external stress applying along x1 (Fig. 4). Our task is toobtain the stress-strain relations and the conditions of stability of the superelasticdeformation of the composite.Since the external stress is uniaxial, the martensitic phase contains a major fraction of

domain 1 which contributes the maximum elongation along the extension axis. Thus,the polydomain martensitic phase should be a mixture of domain 1 and domain 2which provide the maximal work due to transformation strain and minimum misfitwith passive layer. Its self-strain is determined by Eq. (6).Then, the free enthalpy (Gibbs free energy) [Eq. (3)] is equal to

� ¼ �1

2

1

E�2 þ � 1� �ð Þ�2 1

2E"20 M�

1þ �ð Þ2

1þ �ð Þ2� 1� �ð Þ

� �

þ � ��fþ � 1� �ð Þ1

2E"20 ��þ � 1þ �ð Þð Þ

2��"0 1� � 1þ �ð Þð Þ

� �

M ¼1� �ð Þ

2

2 1� �ð Þþ

1þ �ð Þ2

2 1þ �ð Þ

ð12Þ

where s s11.After introducing the dimensionless parameters: ’ ¼

�

E "20;� ¼

�f

E "20;� ¼

�

E"0the

Gibbs free energy can be written as follows:

’ ¼ �1

2�2 þ � 1� �ð Þ

1

2M�

1þ �ð Þ2

1þ �ð Þ2� �� 1ð Þ

� ��2

� � 1� � 1þ �ð Þð Þ� þ ��þ �� 1� �ð Þ1

2��þ � 1þ �ð Þ½ �

ð13Þ


The parameter, � ¼ q T�T0

T0E"0(q is a latent heat, T0 is a temperature of phase equi-

librium), can be considered as an ‘‘effective temperature’’.The first term is the enthalpy of a composite before transformation, the second

term is the misfit elastic energy, the third term is the work of external stress on theself-strain of transformation " ¼ �� 1� � 1þ �ð Þ½ �. The last two terms determine theinternal free energy of the active two-phase layer.The equilibrium microstructure is determined by the equations:

@’

@�¼ �� 1� �ð Þ

1þ �ð Þ2

1þ ��2 1� 2�ð Þ þ � 1þ �ð Þ��þ

�� 1� �ð Þ ��þ � 1þ �ð Þð Þ 1þ �ð Þ ¼ 0 ð14aÞ

@’

@�¼ � 1� �ð Þ M�

1þ �ð Þ2

1þ �2� 1� �ð Þ

� ��þ � � 1þ �ð Þ � 1ð Þ�þ

��þ � 1� 2�ð Þ1

2��þ � 1þ �ð Þ½ � ¼ 0 ð14bÞ

The equilibrium solutions, �0, �0, corresponds to the stable or unstable (saddlepoints) states depending on the sign of D:

D ¼@2’

@�2

@2’

@�2�

@2’

@�@�

� �2

ð15Þ

with

@2’

@�2¼ 2� 1� �ð Þ

1þ �ð Þ2

1þ ��2 þ �� 1� �ð Þ 1þ �ð Þ

2ð16aÞ

@2’

@�2¼ � 1� �ð Þ M�

1þ �ð Þ2

1þ �2� 1� �ð Þ

� �� þ � 1þ �ð Þ½ �

2ð16bÞ

@2’

@�@�¼ �2� 1� �ð Þ

1þ �ð Þ2

1þ �� 1� 2�ð Þ þ � 1� 2�ð Þ ��þ � 1þ �ð Þ½ � 1þ �ð Þþ

� 1þ �ð Þ� ð16cÞ

The average strain and the effective compliance of the composite at the transfor-mation in the active layer is determined by Eqs. (10) and (11) and are as follows:

" ¼ � þ ��0 1� �0 1þ �ð Þ½ � ð17Þ


S� ¼ 1þ �@�0

@�1� �0 1þ �ð Þ½ � � ��0 1þ �ð Þ

@�0

@�ð18Þ

where �0, �0 are determined by Eq. (14a,b).Solution of the set of equations [Eq. (14a,b)] shows that there are three different

paths of the microstructure evolution depending on the temperature, �, and thefraction of the active layer, �.

1. The evolution of austenite-martensite microstructure proceeds through theformation and growth of single-domain martensite, no polydomain martensiticphase is formed.

2. Transformation starts with formation and growth of polydomain martensiticphase. However, the fraction of twin domain, �, disappears before austenitetransformation completes. Further superelastic deformation proceedsthrough the increasing of fraction of single-domain martensite.

3. The transformation starts with formation and growth of polydomainmartensitic phase. As deformation proceeds, the fraction of twin domain, �,decreases. The austenite transformation completes before twin structure inthe martensitic phase disappears and further deformation proceeds throughthe twinning of the martensite (superplastic deformation).

In Fig. 5 the three regimes of microstructure evolution at superelastic deformationis shown in the diagram (�, (1��)). The temperatures, �*, �** which separate differentregimes of microstructural evolution will be found from the analysis presented below.

Fig. 5. Three regimes of evolution of an equilibrium microstructure during superelastic deformation of a

composite (� is an ‘‘effective temperature’’, � is a fraction of an active layer).


The start of transformation is determined by Eqs. (14) and (17) (�0=0):

�1 ¼ ��þ �0 1þ �ð Þ

"1 ¼ �1

�0 ¼1

1þ ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �ð Þ

2þ2�

q1þ �

ð19Þ

At �>�*, where

�� ¼ ��2

2ð20Þ

�0=0 in Eq. (19), i.e. the transformation starts with formation of single-domainmartensite.The strain and the fraction of twin domain, �0, at the beginning of transformation

do not depend on the fraction of an active layer, �. They coincide with ones of theunconstrained active material (Roytburd and Slutsker, 2001a), because there is nomisfit between the passive and active layers prior the transformation.At �<�* the transformation starts with formation of polydomain martensite, but

the polytwin structure disappears (�0=0) at

�0 �3ð Þ ¼

��þ ��1

2�2

�� 2 � 1� �ð Þ1� �ð Þ �� ð Þ

1� �2

�3 ¼ ��1

2�2 þ 1� �ð ÞM

"3 ¼ �3 þ ��0

ð21Þ

If �>�*, �0(�3)<0, i.e. only single-domain martensite can be formed. If �<�**,where

�� ¼�2

2þ 1� �ð Þ

1� �ð Þ �� ð Þ

1� �2ð22Þ

�0(�3)>1, that means the transformation completes before the twin domains dis-appear. The temperature, �**, corresponds to the evolution when the phase trans-formation completes and the twin structure disappears simultaneously. Thetemperature, �**, separates regimes (2) and (3) and the temperature, �* separatesregimes (1) and (2) on the (�, (1��)) plane (Fig. 5).


At �>�**, the transformation completes (�0=1, �0=0) at

�2 ¼ ��1

2�2 þ 1� �ð ÞM

"2 ¼ �2 þ �ð23Þ

At �<�**, the transformation completes before twin structure disappears. In thiscase �0=1, �0>0:

�0 �02

� �¼ 1� �ð Þ

1� �

1þ �þ

�

1þ ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�

1þ �2ð Þþ

1� �ð Þ 1� �ð Þ

1þ �

1� �ð Þ 1� �ð Þ

1þ �þ��

1� �

� �s

�02 ¼ 1� �ð Þ

1þ �

1þ �1� 2�0ð Þ

"02 ¼ �02 þ � 1� �0 1þ �ð Þ½ � ð24Þ

The typical stress–strain curves in each regime for different fraction of the activelayer, �, are shown in Figs. 6–8. The stress–strain curve for unconstraint active material(�=1) is shown for comparison. The negative slope on the stress-strain curves (thenegative effective Young’s modulus) corresponds to the unstable solutions of Eq. (14)(D<0) and stress-controlled deformation proceeds with thermodynamic hysteresis.To obtain a stable composite it is necessary to avoid the negative effective elastic

modulus of superelastic deformation at all stages of transformation. In the analysis

Fig. 6. Stress–strain curves for �>�*, at different fractions of the active layer, �.


Fig. 7. Stress–strain curves for �**<�<�*, at different fractions of the active layer, �.

Fig. 8. Stress–strain curves for �<�** at different fractions of the active layer, �.


presented below we will show how to obtain the conditions of stable superelasticdeformation for each regime of microstructural evolution (�<�**, �**<�<�*,�>�*). The conditions of stability will be obtained at the beginning of transforma-tion (yield stress) and at all stages of transformation (general stability).At �>�* (regime 1) the effective compliance of the superelastic deformation of

composite does not change during deformation and equals to

S�

¼ 1þ�

1� �ð ÞM� �2ð25Þ

The effective modulus (E*=1/S*) is positive, i.e. the composite is stable at

� < 1��2

Mð26Þ

At �� < � < �� (regimes 2) the effective compliance at the beginning of trans-formation is

S�

¼ 1þ �1� �0 1þ �ð Þ½ �

2

1� �ð Þ M�1þ �ð Þ

2

1þ �2�0 1� �0ð Þ

� �� þ �0 1þ �ð Þ½ �

2

� � ð27Þ

where �0 is determined by Eq. (19). Thus, the superelastic deformation of the compositeis stable at the beginning of transformation if

� < 1��þ �0 1þ �ð Þ½ �

2

M�1þ �ð Þ

2

1þ �2�0 1� �0ð Þ

� � ð28Þ

The general stability of superelastic deformation in regime 2 is determined by thesign of D at the point where absolute value of negative effective modulus is maximal.As previous analysis shows (Roytburd and Slutsker, 2001a), the absolute value ofeffective modulus of superelastic deformation is maximum at the point where thefraction of the twin domain disappears, i.e. at �=�3, �0=0, �0=�0(�3). Thus, at�**<�<�*, the stability of the composite can be determined by the sign of D at�=�3 [Eq. (15)] The solution of the equation:

1� �ð ÞM� �2� �

2 1� �ð Þ�0

1þ �þ 1� �0ð Þ

� �� 1� �ð Þ

1þ �ð Þ

1þ �ð Þþ �

� �2

�0 ¼ 0

ð29Þ

where �0 is determined by Eq. (21), corresponds to the region of general stability ofthe composite at f**<�<�*.At � < �� (regime 3), the stability of superelastic deformation at the beginning of

transformation is still determined by Eq. (28) with �0 determined by Eq. (19). The


absolute value of negative modulus is maximum at the transition from phase trans-formation to twinning of martensite (�0

2; "02). Thus, the general stability of super-

elastic deformation at � < �� is determined by the sign of D [Eq. (15)] at �=�02, i.e.

from the solution of the following equation:

1� �ð Þ2 1� �ð Þ

2

1� �2� ��þ �0 1þ �ð Þ½ �

2 1þ 2 1� �ð Þ1

1þ �

� ��

2 1� �ð Þ1þ �ð Þ

1þ �ð Þ1� 2�0ð Þ ��þ �0 1þ �ð Þð Þ ¼ 0 ð30Þ

where �0 is determined by Eq. (24).The regions of stability of the composite are shown in Fig. 9 (�, (1��)). The areas

where the superelastic deformation of the composite is stable at all regimes oftransformation, stable at yield stress (beginning of transformation), and unstable areshown by different color. Fig. 10 shows an example of a superelastic deformation ofa composite at different temperature, �, corresponding to this diagram. The com-posite with a fraction of an active component, �=0.75, is deformed at differenttemperatures. At �=�*, the effective elastic modulus is zero, i.e. the composite isstable. With decreasing temperature the area of instability appears, however, at lowtemperature the composite is stable again. As can be seen from this diagram it ispossible to have composite which is stable at the beginning of transformation, butunstable at the changing regimes of transformation.

Fig. 9. Diagram of stability of superelastic deformation of a composite (� is an ‘‘effective temperature’’, �is a fraction of an active layer). Superelastic deformation is stable; superelastic deformation is

stable at yield stress; superelastic deformation is unstable.


4. Discussion

The problem of the composite containing shape memory materials as an activecomponent has recently attracted great interest for development smart materials andstructures (see e.g. Roytburd et al., 2000). The constitutive behavior of the compo-sites with shape memory component has been investigated using both phenomen-ological (Sottos and Kline, 1996; Boyd and Lagoudas, 1996; Stalmans et al., 1997,Lexcellent et al., 2000) and micromechanics- based modeling methods (Cherkaoi etal., 2000; Kawai, 2000; Lu and Weng, 2000; Chaboche et al., 2001). One of the mainobstacles for use shape memory material for wider engineering applications is thestress–strain hysteresis. In this paper the principles of design of adaptive compositeswith a controlled stress-strain hysteresis are proposed. However, for practical designof the stable composite the following should be taken into account.To obtain the stable superelastic deformation of the composite it is necessary not

only to eliminate negative effective modulus, but also to avoid any changes ofmicrostructure regimes of transformation (since the nucleation of the new micro-structure can create additional instabilities). Therefore, the temperature, �**, atwhich austenite transformation completes and twin structure disappears simulta-neously, is the best suitable for designing the composite with minimum hysteresis.Thus, all fractions of an active layer, �, which lie on the line �** (Fig. 9) are suitablefor designing of the composite without thermodynamic hysteresis. However, inorder to obtain the maximal superelastic deformation it is desirable to have thelarger fraction of an active layer as possible. The point in the diagram at which thesuperelastic deformation of the composite is stable at maximum elongation due to

Fig. 10. Temperature dependence of superlelastic deformation of a composite (�=0.75).


transformation corresponds to (�0=0.15, (1-�)0=0.28, � 0=0.72). The stress-straincurve for �=0.72, �=0.15 is shown in Fig. 11. This curve corresponds to the stablesuperelastic deformation, the austenite and twins are disappears simultaneously andthere are no changing microstructure regimes of transformation. The reversiblesuperelastic deformation equals to 0.8"0. Thus, the constraints imposed by thecomposite architecture make the evolution of the microstructure reversible butsimultaneously decrease the actuated superelastic strain. By optimization of mezos-tructure and microstructure of the composite it is possible to find compromisebetween a magnitude of superelastic deformation and its reversibility. The exampleshown above demonstrates that reversible deformation is only 20% less than maximaltransformational deformation.The optimal deformation can only be obtained if the passive components of

composite have a large limit of elasticity. There are two possibilities which should beexplored: (1) very thin metal passive layers with perfect crystalline structure whichenable to demonstrate elastic deformation up to 5%, (2) polymer passive layers withlarge reversible deformation. The first option leads to design of adaptive nano-composites. The theoretical investigation of deformation of a nanocomposites withthe simplest polydomain martensitic structure of active layers has been done in(Roytburd and Slutsker, 2001b). The second option requires studying of compositeswith substantially different elastic properties of active and passive components. Asour preliminary study (Slutsker and Roytburd, 2001) shows it may be done byexpanding the analysis presented in this paper. Thus, the theoretical approachdeveloped in this paper creates the basis for more general theory and modeling of

Fig. 11. The stable superelastic deformation of a composite with maximal elongation due to transforma-

tion in the active layer (�=0.15, �=0.72).


adaptive nano- and micro-composites. For metal/metal microcomposites the resultsof this work can be applied to design of materials with elastic moduli sensitive tochange of temperature and deformation. For this purpose the stability of deformationat the beginning of transformation is sufficient.The analysis presented in this paper describes the superelastic deformation of the

composite at uniaxial external stress along x1. At the deformation along other axes,the different set of martensite plates will be formed, and the performance of thecomposite will be different. However, the principles of calculation of stability of thesuperelastic deformation of the composite will remain the same.The thermodynamic hysteresis considered in this paper is a part of experimentally

observed stress–strain hysteresis which usually contain strain-rate dependent kineticpart. However, the discussion of relative contributions of thermodynamic andkinetic factors to the stress–strain hysteresis of superelastic deformation is out of theframe of this paper.

5. Conclusions

The intrinsic thermodynamic instability of the superelastic deformation and pos-sibility to control it are explored. The thermodynamic analysis of the uniaxialsuperelastic deformation of an adaptive composite due to martensitic transformationin an active layer shows.1. There are three regimes of microstructure evolution during stress-induced mar-

tensitic transformation depending on temperature, �, and fraction of the active layer, �.At �>��2/2 the evolution of martensite-austenite microstructure proceeds

through the formation and growth of single-domain martensite.At �2

2 þ 1� �ð Þ1��ð Þ ��ð Þ

1��2< � < ��

�2

2 the evolution of microstructure proceedsthrough the formation of polydomain martensite with internal polytwin structure.The internal polytwin structure disappears before phase transformation completesand further deformation proceeds through the transformation of austenite into single-domain martensite.At � < �2

2 þ 1� �ð Þ1��ð Þ ��ð Þ

1��2, the evolution of microstructure starts with formation

of polydomain martensite with polytwin structure. The transformation completesbefore polytwin structure disappears and further deformation proceeds throughtwinning of martensite.2. The effective superelastic modulus of a composite depends on the martensite

microstructure. The modulus is independent on strain if the microstructure containssingle-domain martensite. It depends on stress if the microstructure contains apolytwin martensite. The magnitude of the effective modulus changes dramaticallywith regimes of superelastic deformation. The modulus can be negative, dependingon fractions of the active and passive layers and temperature, which corresponds tothe unstable stress-controlled deformation.3. The diagram of stability of the composite depending on layer’s fraction and

temperature is obtained. This diagram allows one to design superelastic compositewith controlled thermodynamic instability and minimal stress–strain hysteresis.


Acknowledgements

A. Roytburd is grateful to the support of AFOSR. The support of NationalResearch Council Postdoctoral Assistantship is greatly appreciated by J. Slutsker.

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