Thermal effects in the superelasticity of crystalline shape-memory materials

Journal of the Mechanics and Physics of Solids 51 (2003) 1015 – 1058

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Thermal e%ects in the superelasticity of crystalline shape-memory materials Lallit Ananda;∗ , Morton E. Gurtinb a Department

of Mechanical Engineering, Massachusetts Institute of Technology, Room 1-310, 77 Mass. Ave, MIT, Cambridge, MA 02139, USA b Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA Received 5 June 2002; accepted 6 January 2003

Abstract This paper develops a three-dimensional theory for the superelastic response of single-crystal shape-memory materials. Since energetic considerations play a major role in the phase transformations associated with the superelastic response, we have developed the theory within a framework that accounts for the laws of thermodynamics. We have implemented a special set of constitutive equations resulting from the general theory in a 4nite-element computer program, and using this program have simulated the superelastic response of a single crystal Ti–Ni shape-memory alloy under both isothermal and thermo-mechanically coupled situations. Both manifestations of superelasticity—stress–strain response at 4xed temperature and strain–temperature response at 4xed stress—are explored. The single-crystal constitutive-model is also used to discuss the superelastic response of a polycrystalline aggregate with a random initial crystallographic texture. The overall features of the results from the numerical simulations are found to be qualitatively similar to existing experimental results on Ti–Ni. ? 2003 Elsevier Science Ltd. All rights reserved. Keywords: A. Shape-memory alloys; B. Superelasticity; C. Crystal mechanics; D. Thermo-mechanical response

1. Introduction The individual grains in polycrystalline shape-memory alloys (SMAs) can abruptly change their lattice structure under thermomechanical loading. This capability of undergoing solid–solid, di%usionless, displacive phase transformations leads to the ∗

Corresponding author. Tel.: +1-617-253-1635; fax: +1-617-258-8742. E-mail addresses: [email protected] (L. Anand), [email protected] (M.E. Gurtin).

0022-5096/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-5096(03)00017-6

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technologically important properties of superelasticity 1 and shape-memory. In this paper we focus on the superelastic response of shape-memory materials. Superelasticity is a consequence of a stress-induced transformation 2 from austenite to martensite and back when a sample is tested between zero and a 4nite but small strain at a constant ambient temperature above a 4xed temperature usually referred to as the austenite 4nish temperature. The sample experiences little or no permanent deformation in such a strain cycle, giving the impression that the material has undergone only elastic deformation; hence the term superelastic. However, there is dissipation induced by the motion of sharp interfaces between the two material phases. For a given material, the size and shape of the “Gag-type” hysteresis loops usually vary with loading rate and temperature. Another manifestation of the superelastic response of a shape-memory material is the strain–temperature response at 4xed stress; by cycling the temperature over a narrow range of temperatures at a 4xed stress, one can obtain a recoverable strain cycle. For the past 20 years or so there has been substantial activity world-wide to construct suitable constitutive models for shape-memory materials. Several of the existing one-dimensional constitutive models capture the major response characteristics of SMAs reasonably well (e.g. Liang and Rogers, 1990; Abeyaratne and Knowles, 1993; Ivshin and Pence, 1994; Bekker and Brinson, 1997; Govindjee and Hall, 2000). Prominent among the three-dimensional models are those proposed by (e.g., Sun and Hwang, 1993a, b), Auricchio and co-workers (e.g., Auricchio and Taylor, 1997, Auricchio et al., 1997), and Lagoudas and co-workers (e.g., Boyd and Lagoudas, 1996); most such three-dimensional models are isotropic in nature and employ various combinations of the standard stress invariants in formulating the rules for the reversible transformation of austenite to martensite. However, due to their prior thermomechanical processing history, SMAs typically possess strong initial crystallographic textures; for that reason reliable three-dimensional constitutive models are needed to predict the thermomechanical response of components made from initially textured SMAs, under complex multi-axial loading. To this end, in recent years, several constitutive models, based on crystal-mechanics, have been proposed in the literature (cf., e.g., Patoor et al., 1988, 1995, 1996; Siredey et al., 1999; Lu and Weng, 1998; Lim and McDowell, 1999; Gall and Sehitoglu, 1999; Tokuda et al., 1999; Gall et al., 2000; Sittner and Novak, 2000; Entemeyer et al., 2000; Gao et al., 2000; Huang et al., 2000; Govindjee and Miehe, 2001; Kitajima et al., 2002). In typical applications, the strains incurred by superelastic materials are small, but the rotations may be quite large. Unfortunately, most of the existing crystal-mechanical models ignore the e%ects of large rotations. In contrast, Thamburaja and Anand (2001, 2002) have developed a three-dimensional model for polycrystalline shape-memory alloys that allows for 5nite deformations. This model predicts, with reasonable quantitive accuracy, the isothermal superelastic response of an initially textured Ti–Ni shape-memory alloy in tension, compression, torsion, and combined tension–torsion experiments.

1

Also called pseudoelasticity. The high temperature phase in shape-memory materials is called austenite, the low temperature phase martensite; the austenitic phase typically has a higher crystallographic symmetry than the martensitic phase. 2

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Since energetic considerations and temperature changes play a major role in the phase transformations underlying superelastic response, there is a strong need for a thermomechanical model for the superelastic response of SMAs. Accordingly, the goal of this paper is a crystal-mechanical theory that allows for 4nite deformations and thermal inGuences within a consistent thermodynamic framework. The basic kinematical assumption of our theory is a decomposition F = Fe Fp of the deformation gradient into: (i) an inelastic part Fp whose temporal behavior is governed by the generation, growth, and annihilation of austenitic–martensitic 4ne structure; 3 and (ii) an elastic part Fe that represents stretching and rotation of the overall structure. This decomposition is identical in form to that introduced by KrLoner and Lee (Kroner, 1960; Lee, 1969) to discuss 4nite plasticity. In fact, we 4nd it
natural to take as ˙  a second assumption a decomposition of F˙ p Fp−1 whose form   S0 is similar to that generally used to discuss crystal plasticity induced by crystallographic slip, but here the dyads S0 represent the rank-one tensors that typically characterize the austenite–martensite 4ne structures of the equilibrium theory, while the scalar multipliers represent transformation rates. 4 The spatially continuous 4elds that de4ne our theory therefore represent averages meant to apply at length scales which are large compared to the length scales associated with the 4ne structure, thereby allowing us to bypass the diNcult problem of explicitly computing individual microstructures. 5 More speci4cally, in the equilibrium theory of austenite–martensite phase transitions (Ball and James, 1987) the strain-energy minimizers—in the sense of minimizing sequences—generally appear in the form of 4ne structure involving abrupt transitions between austenite and twinned martensite in which the transitions are jumps in the deformation gradient of the form S0 = b0 ⊗ m0 , with b0 a vector that represents the transition strain and m0 a unit vector normal to the habit plane (the plane of the transition). A result of the equilibrium theory is a complete catalog of such rank-one tensors S0 for various shape-memory alloys (cf. Bhattacharya, 1993; Hane and Shield, 1999; James and Hane, 2000). We here consider the complete set of 192 transitions for Ti–Ni discussed by these workers. The plan of this paper is as follows. We develop the general theory in Sections 2–8. In Section 9 we specialize our constitutive equations, and in Section 10 we summarize specialized constitutive equations that should be useful in applications. The partial di%erential equations for the deformation and temperature 4elds are summarized in Section 11. In Section 12, using estimated values of the material parameters for the shape-memory alloy Ti–Ni, we apply the specialized equations to model the superelastic response of single crystals and polycrystals. The two manifestations of superelasticity— stress–strain response at 4xed temperature and strain–temperature response at 4xed stress—are explored. Section 13 contains concluding remarks.

3 We use the superscript p for two reasons: (i) to stress similarities between the present theory and single-crystal plasticity; and (ii) as shorthand for “phase transformation”. 4 This model is not intended to describe a material initially in the martensitic state and inelastic deformation caused by reorientation and de-twinning of the martensitic variants. 5 Cf., e.g., Luskin (1996) for a detailed discussion of the computation of microstructures.

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2. Kinematics 2.1. Kinematics 2.1.1. Decomposition of the deformation gradient We consider a homogeneous crystalline body B identi4ed with the region of space it occupies in a 4xed reference con5guration, and denote by X an arbitrary material point of B. A motion of B is then a smooth one-to-one mapping x = y(X; t) with deformation gradient, velocity, and velocity gradient given by 6 ˙ −1 : ˙ F = ∇y; v = y; L = grad v = FF (2.1) We base the theory on the decomposition F = Fe Fp :

(2.2)

Here, with respect to Fig. 1 and suppressing the argument t: (i) Fp (X) represents the local deformation of referential segments dX to segments dl = Fp (X) dX in the relaxed lattice con5guration due to the generation, growth and annihilation of austenitic/martensitic 4ne-structure in a microscopic neighborhood of X. (ii) Fe (X) represents the mapping of segments dl in the relaxed lattice con4guration into segments dx = Fe (X) dl in the deformed con4guration due to the “elastic mechanisms” of stretching and rotation of the lattice. The relaxed lattice con5guration might therefore be viewed as the ambient space into which an in4nitesimal neighborhood of X is carried by the linear transformation Fp (X)—or into which an in4nitesimal neighborhood of x = y(X) is pulled back by the transformation Fe (X)−1 . We refer to Fp as the 5ne-structural part of F, to Fe as the elastic part of F. By (2:1)3 and (2.2), L = Le + Fe Lp Fe−1

(2.3)

Le = F˙ e Fe−1 ;

(2.4)

with Lp = F˙ p Fp−1 :

As is standard, we de4ne the elastic and inelastic stretching and spin tensors through De = sym Le ;

We = skw Le ;

Dp = sym Lp ; e

e

Wp = skw Lp ; e

p

p

(2.5) p

so that L = D + W and L = D + W . 6 Notation: ∇ and Div denote the gradient and divergence with respect to the material point X in the reference con5guration; grad and div denote these operators with respect to the point x = y(X; t) in the deformed con4guration; a superposed dot denotes the material time-derivative. Throughout, we write Fe−1 = (Fe )−1 , Fp− = (Fp )− , etc. We write sym A and skw A, respectively, for the symmetric, and skew parts of a tensor A. The inner product of tensors A and B is denoted by A · B, and the magnitude of A by √ |A| = A · A.

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Fig. 1. The dark gray regions represent an in4nitesimal neighborhood of X as deformed to the relaxed lattice con4guration via Fp (X), and then to the deformed con4guration via Fe (X). Volume fractions of martensite are computed in the relaxed lattice con4guration; the gray background in that con4guration represents austenite.

2.1.2. Phase transitions In the equilibrium theory of austenite–martensite phase transitions the strain-energy minimizers (in the sense of minimizing sequences) generally appear in the form of 4ne structure involving abrupt transitions between austenite and twinned martensite in which the transitions are jumps in the deformation gradient of the form S0 = b0 ⊗ m0 , with b0 a vector that represents the transition strain and m0 a unit vector normal to the habit plane (the plane of the transition). A result of the equilibrium theory is a catalog of such rank-one tensors S0 for various shape-memory alloys (cf. James and Hane (2000) and the references therein). Here we take, as a basic ingredient of the theory, a system of N such transition tensors S0 = b0 ⊗ m0 ;

 = 1; 2; : : : ; N;

where b0 and m0 , the transformation strain and the habit-plane normal, are constant (not necessarily orthogonal) vectors in the lattice with |m0 | = 1: Note that, unlike the Schmid tensors of crystal plasticity, b0 need not be a unit vector. In fact, |b0 | represents the scalar transformation strain of the th system.

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We denote by  (X; t), 0 6  (X; t) 6 1; the volume fraction of martensite at X associated with the th system at time t. The presumption that the phase transformations take place through nucleation and growth of plate-like volume elements is based on the hypothesis that the evolution of Fp be governed by the transformation rates ˙ via the relation F˙ p = Lp Fp ;

Lp =

N 

˙ S0 :

(2.6)

=1

For system , transformations from austenite to martensite, abbreviated a → m; occur when ˙ ¿ 0; a transformations from martensite to austenite, abbreviated m → a; occur when ˙ ¡ 0. Let =

N 



(2.7)

=1

and assume, as is natural, that 0 6 (X; t) 6 1; where (X; t) represents the total volume fraction of martensite at X. Eq. (2.3) may be written as L = Le +

N 

˙ S ;

(2.8)

=1

or equivalently as ˙ (∇y)F

−1

= F˙ e Fe−1 +

N 

˙ S ;

(2.9)

=1

where S = b ⊗ m ;

b = Fe b0 ;

m = Fe − m0 ;

(2.10)

respectively, are the transformation tensor, the transformation strain, and the habit-plane normal for the th transformation system pushed forward to the deformed con4guration. Note that, by (2.6), tr Lp =

N 

˙ b0 · m0

=1

and thus the transformation process involves local changes in volume.

(2.11)

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For later use, we write  = (1 ; 2 ; : : : ; N )

(2.12)

for the list of martensite volume fractions. 2.1.3. Elastic strain tensor The right polar decomposition of Fe is given by Fe = R e U e ;

(2.13)

where Re is a rotation, while Ue is symmetric, positive-de4nite tensors. As is standard, we de4ne Ce = (Ue )2 = Fe Fe

(2.14)

and refer to Ee = 12 (Ce − 1) = 12 (Fe Fe − 1)

(2.15)

as the elastic strain. 2.2. Frame-indi8erence Changes in frame (observer) are smooth time-dependent rigid transformations of the Euclidean space through which the body moves. We require that the theory be invariant under such transformations, and hence under transformations of the form y(X; t) → Q(t)y(X; t) + q(t);

(2.16)

with Q(t) a rotation (proper-orthogonal tensor) and q(t) a vector at each t. Thus F → QF:

(2.17)

The reference and relaxed con4gurations are independent of the choice of such changes in frame; thus the 4elds , Fp , and Lp are invariant under transformations of form (2.16). This observation, (2.2), (2.15), and (2.17) yield the transformation laws ˙ e; F˙ e → QF˙ e + QF

Fe → QFe ;

Ee → Ee ;

(2.18)

˙  and hence so that Le → QL Q + QQ e

De → QDe Q ;

˙ : We → QWe Q + QQ

(2.19)

3. Principle of virtual power. Macroscopic and microscopic force balances The theory presented here is based on the belief that the power expended by each independent “rate-like” kinematical descriptor be expressible in terms of an associated force system consistent with its own balance. But the basic “rate-like” descriptors, ˙ F˙ e , and ˙ are not independent, as they are constrained by (2.9), and it is namely, y,

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not apparent what forms the associated force balances should take. For that reason, we determine these balances using the principal of virtual power. 3.1. External and internal expenditures of power Throughout we denote by P an arbitrary part (subregion) of the reference body B with n the outward unit normal on 9P. The power expended on P by material or bodies exterior to P results from a macroscopic surface traction s(n), measured per unit area in the reference con4guration, and a macroscopic body force f, measured per unit volume in the reference con4guration, each of whose working accompanies the macroscopic motion of the body. (The body force f is assumed to include inertial forces.) We therefore write the external power in the standard form   Wext (P) = s(n) · y˙ dA + f · y˙ dV (3.1) 9P

P

with s(n) (for each unit vector n) and f de4ned over the body for all time. We assume that power is expended internally by a stress Se work conjugate to F˙ e , and, for each , an internal microforce  work-conjugate to the transformation rates ˙  . We therefore write the internal power in the form    N   ˙ e ˙e dV: (3.2) S ·F +   Wint (P) = P

=1

Here S and  are de4ned over the body for all time. The 4eld  represents an internal microforce, measured per unit volume, associated with the generation, growth, and annihilation of austenitic–martensitic 4ne structures. e



3.2. Principle of virtual power Assume that, at some arbitrarily chosen but 5xed time, the 4elds y and Fe (and ˙ F˙ e , and ˙ as virtual velocities hence F and Fp ) are known, and consider the 4elds y, to be speci4ed independently in a manner consistent with (2.9); that is, denoting the ˜ F˜ e , and ˜ to di%erentiate them from 4elds associated with the actual virtual 4elds by y, evolution of the body, we require that ˜ −1 = F˜ e Fe−1 + (∇y)F

A 

˜ S :

(3.3)

=1

More speci4cally, we de4ne a generalized virtual velocity to be a list ˜ ˜ F˜ e ; ) V = (y; consistent with (3.3). We assume that under a change in frame the 4elds comprising a generalized virtual velocity transform as their nonvirtual counterparts; i.e. ˙ e; F˜ e → QF˜ e + QF

˜ ˜ → :

(3.4)

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Writing





Wext (P; V) =

1023

9P

s(n) · y˜ dA +

P

f · y˜ dV;

   N  e ˜e  ˜ Wint (P; V) = S ·F + dV;   P

(3.5)

=1

respectively, for the external and internal expenditures of virtual power, the principle of virtual power consists of two basic requirements: (V1) Power balance: Given any part P, Wext (P; V) = Wint (P; V)

for all generalized virtual velocities V:

(3.6)

(V2) Frame-indi8erence: Given any part P and any generalized virtual velocity V, Wint (P; V)

is invariant for all changes in frame:

(3.7)

3.3. Macroscopic force and moment balances. Microforce balance To deduce the consequences of the principle of virtual power, assume that (V1) and (V2) are satis4ed. In applying the virtual balance (3.6) we are at liberty to choose any V consistent with constraint (3.3). Consider 4rst the internal power Wint (P; V) under an arbitrary change in frame. In the new frame V transforms to a generalized virtual velocity V∗ , Se to Se∗ , and  to ∗ ; (V2) therefore implies that Wint (P; V) = Wint∗ (P; V∗ ), where, by (3.4),    N  ∗ e∗ e e ∗  ∗ ˙ )+ S · (QF˜ + QF dV  ˜ Wint (P; V ) = P

=1

   N   e∗ e  e ∗  ˜ ˙ )+ Q S · (F˜ + Q QF dV: =   P

(3.8)

=1

Since P, the change in frame, and V are arbitrary, we choose the change in frame such that Q˙ = 0 we 4nd, using (3.7), that Se → QSe ;

 →  :

(3.9)

On the other hand, if we assume that, Q = 1 at the time in question, so that Q˙ is an arbitrary skew tensor, we 4nd that Se Fe = Fe Se : Consider next a generalized virtual velocity with ˜ ≡ 0, so that ˜ p−1 : F˜ e = (∇y)F

(3.10)

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For this choice of V, (V1) yields     s(n) · y˜ dA + f · y˜ dV = Se · F˜ e dV = (Se Fp− ) · ∇y˜ dV: P

9P

P

P

Thus, de4ning S = Se Fp− ;

(3.11)

we may conclude, using the divergence theorem, that   (s(n) − Sn) · y˜ dA + (Div S + f) · y˜ dV = 0: 9P

P

˜ standard variational arguments yield Since this relation must hold for all P and all y, the traction condition s(n) = Sn

(3.12)

and the local force balance Div S + f = 0:

(3.13)

Moreover, (3.10) and (3.11) imply that SF = FS :

(3.14)

Thus S plays the role of the Piola–Kirchho8 stress, and (3.13) and (3.10) represent the local macroscopic force and moment balances. To discuss the microscopic counterparts of these results, choose y˜ ≡ 0. Then the external power vanishes identically,
so that, by (V1), the internal power must also vanish. Further, by (3.3), F˜ e = −  ˜ S Fe . Thus, if we de4ne the resolved stress  through  = (Se Fe ) · S = (SF ) · S ; then (3.5) yields N   ( −  )˜ dV = 0: =1

P

(3.15)

(3.16)

Relation (3.16) must be satis4ed for all ˜ and all P, and a standard argument yields the microforce balance   = 

(3.17)

for each system . This balance arises as a consequence of the arbitrary nature of the transformation rate ˜ and for that reason might be viewed as a force balance for the th system of transformations. Note that the stress Se plays two roles: it generates the standard Piola–kirchho% stress in the macroscopic balance as well as the coupling term between the microscopic and macroscopic balances through its resolved values on the relevant transformation systems.

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3.4. Elastic stress. Internal power revisited Let Te = Fe−1 Se :

(3.18)

Then, since Se Fe = Fe Te Fe , (3.10) yields Te = Te :

(3.19)

Thus Se · F˙ e = Te · (Fe F˙ e ) and, in view of (2.15), Se · F˙ e = Te · E˙ e :

(3.20)

We refer to Te as the elastic stress as it represents a stress work-conjugate to the elastic strain-rate E˙ e . By (3.11), the elastic stress is related to the Piola–Kirchho% stress through the relation S = Fe Te Fp− :

(3.21)

By (2.17), (3.9), and (3.18), Te and  are invariant under a change in frame:

(3.22)

Further, as a consequence of (2.10), (2.14), and (3.15),  = (Ce Te ) · S0 :

(3.23)

Finally, using (3.20) we can express the internal power (3.2) in the form    N  e e   Wint (P) = T · E˙ + dV:  ˙ P

(3.24)

=1

3.5. (Virtual) external microforces We consider the macroscopic body force f to be arbitrarily assignable. In addition, we allow for an arbitrary (generally virtual) external microforce  conjugate to the transformation rate ˙ . Speci4cally, we add the term N    ˙ dV =1

P

to the external power Wext (P) (with an analogous change when virtual velocities are considered). The only change in the results established thus far is that the microforce balance (3.17) is replaced by   −  =   :

(3.25)

We refer to (3.25) as the virtual microforce balance, reserving the term microforce balance for the more standard situation in which  = 0.

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L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

There are two reasons for the introduction of the external microforce  : (i) The procedure used to restrict constitutive equations is based on the requirement that the second law hold in all conceivable processes, irrespective of the diNculties involved in producing such processes in the laboratory. The application of this procedure presumes the availability of external forces that ensure satisfaction of the underlying balance laws in all processes (Coleman and Noll, 1963). (ii) The external microforce allows for a straightforward de4nition of microstability, a notion with strong consequences. The use of external forces to justify kinematical processes is not new. Moreover, the literature is replete with arguments in which various kinematical 4elds are independently varied without consideration of forces needed to incur such variations. 7 The theory presented here makes explicit the external forces needed to support the “virtual processes” used, and, in so doing, ensures that these forces, whether virtual or not, enter the theory in a consistent manner.

4. Balance of energy. Entropy imbalance Let P be an arbitrary part of the body. Let # denote the absolute temperature, let  and  represent the internal energy and entropy densities, measured per unit volume in the reference con4guration, let q denote the heat
9P

P

while the second law takes the form of an entropy imbalance  ˙   q·n r  dV ¿ − dA + dV: P 9P # P #

(4.2)

Thus, since Wext (P) = Wint (P) and since P is arbitrary, we may use (3.24) to obtain local forms of (4.1) and (4.2): ˙ = −Div q + Te · E˙ e +

N 

 ˙ + r;

=1

˙ ¿ −

r 1 1 Div q + 2 q · ∇# + # # #

(4.3)

7 Cf. Drucker (1950), who asserts that: “The concept of work-hardening : : : can be expressed in terms of the work done by an external agency : : :” . See also Rice (1971), who argues that: “: : : one may consider the structural rearrangements and the applied stress or strains as independently prescribable quantities : : :” .

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1027

(valid whether or not the virtual external microforces  are present). It is convenient to introduce the (Helmholtz) free energy =  − #:

(4.4)

Eq. (4.3) then yields the dissipation inequality N

 ˙ + #˙ + 1 q · ∇# − Te · E˙ e −  ˙ 6 0: #

(4.5)

=1

Finally, we note that ; ; ; #; ∇#; and q are invariant under a change in frame;

(4.6)

, , , and # because they are scalar 4elds, ∇# because ∇ is the referential gradient, and q because it is referential.

5. Constitutive theory Writing
for the list
= (1 ; 2 ; : : : ; N ); we consider constitutive equations of the form ˙ = ˆ (Ee ; #; ∇#; ; ); ˙ Te = Tˆ e (Ee ; #; ∇#; ; ); ˙  = (E ˆ e ; #; ∇#; ; ); ˙
=
(E ˆ e ; #; ∇#; ; ); ˙ ˆ e ; #; ∇#; ; ): q = q(E

(5.1)

Note that, by (2:18)3 , (3.22), and (4.6), the constitutive equations (5.1) are frameindi8erent. A basic assumption of the theory is that all “processes” related through the constitutive equations (5.1) be consistent with the dissipation inequality (4.5). Substituting the constitutive equations into the dissipation inequality we 4nd, writing g = ∇#, that     N  ˆ 9ˆ 9 ˆ L 9ˆ 9 e e · g˙ + · + − Tˆ · E˙ + + ˆ · #˙ 9Ee 9# 9g 9˙ =1

+

N  =1



9ˆ −  9



1 ˙ + qˆ · ∇# 6 0: #

(5.2)

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This inequality is to hold for all 5elds Ee (X; t), #(X; t), and (X; t). 8 Bearing in mind ˙ that each of the constitutive functions in (5.2) depends on the variables (Ee ; #; ∇#; ; ), ˙ L , E˙ e , and #˙ (but not ˙ ) appear linearly in (5.2), we see that, since the terms g, their “coeNcients” must vanish. Thus 9 ˆ =9g = 0 and 9 ˆ =9˙ = 0 (for all ), so that ˙ in addition, Tˆ e = 9 ˆ =9Ee and ˆ = −9 ˆ =9#. the free energy is independent of g and ; Thus de4ning 9ˆ ; 9

f = ˆ −

(5.3)

we have the following thermodynamic restrictions: (i) the free energy , the elastic stress Te , and the entropy  are independent of ∇# and ˙ and are related through the constitutive equations = ˆ (Ee ; #; ); Te =

9 ˆ (Ee ; #; ) ; 9Ee

=−

9 ˆ (Ee ; #; ) : 9#

(5.4)

(ii) the microstress  is given by  =

9 ˆ (Ee ; #; ) ˙ + f (Ee ; #; ∇#; ; ) 9

(5.5)

ˆ be for each , where the dissipative microforces f must, with the heat Gux q, consistent with the reduced dissipation inequality N  =1

˙ ˙ − 1 q(E ˙ · ∇# ¿ 0: ˆ e ; #; ∇#; ; ) f (Ee ; #; ∇#; ; ) #

(5.6)

In what follows we assume that the material is strongly dissipative in the sense that, for ∇# = 0 and ˙ = 0, inequality (5.6) is satis4ed with ¿ 0 replaced by ¿ 0. Further, ˙ we assume that, for each , f is independent of g and that qˆ is independent of ; then, taking ∇# = 0 and then ˙ = 0, we obtain the following two inequalities: N 

˙ ˙ ¿ 0 f (Ee ; #; ; )

for ˙ =  0

(5.7)

for ∇# = 0:

(5.8)

=1

and ˆ e ; #; ; ∇#) · ∇# ¡ 0 q(E

8 One can show that our allowance for (possibly virtual) external forces f and  and an external heat supply r allows us to vary these 4elds arbitrarily with the assurance that the basic balances are satis4ed (cf. Gurtin, 2000).

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As is easily veri4ed, the quantity =

N  1  ˙ 1 f  − 2 q · ∇# ¿ 0 # #

(5.9)

=1

represents the entropy production per unit volume, as its integral over a part P is equal to the left side of the entropy imbalance (4.2) minus the right. The 4eld (1=#)f ˙ represents the entropy produced, per unit volume, during temporal changes in the 4ne structure for system . 6. Further consequences of thermodynamics 6.1. Gibbs relations. Entropy balance The constitutive response function b = b (Ee ; #; ) de4ned by b (Ee ; #; ) =

9 ˆ (Ce ; #; ) 9

(6.1)

represents a thermodynamic force conjugate to the martensitic volume fraction  ; as we shall see, this force plays a fundamental role in the theory. In view of (5.4), we have the 5rst Gibbs relation, ˙ = −#˙ + Te · E˙ e +

N 

b ˙ ;

(6.2)

=1

which, with (4.4), yields the second Gibbs relation ˙ = #˙ + Te · E˙ e +

N 

b ˙ :

(6.3)

=1

Using balance of energy (4.3), the second Gibbs relation, and (5.5), we arrive at the entropy balance #˙ = −Div q +

N 

f ˙ + r:

(6.4)

=1

Granted the thermodynamically restricted constitutive relations (5.4), the entropy balance is equivalent to balance of energy. 6.2. Global relations A trivial consequence of the sentence containing (5.9) is that  ˙    q·n r  dV = − dA + dV +  dV: B 9B # B # B

(6.5)

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Further, if we neglect virtual external microforces and assume that the body force f is purely inertial, so that ˙ f · y˙ = −k;

˙ 2; k = 12 |y|

with  the referential density and k the kinetic energy, then, in view of (3.1), balance of energy for B takes the form  B

   ˙ ( + k) dV = − q · n dA + r dV + s(n) · y˙ dA: 9B

B

9B

(6.6)

Since the underlying constitutive relations are consistent with thermodynamics, (6.5) and (6.6) are satis4ed automatically in all processes the body may undergo. Global relations of this form are often useful in discussions of existence, uniqueness, and stability.

7. Mechanical rate-independence. Transformation conditions 7.1. Rate-independence The only rate-dependent constitutive variable is the list ˙ of transformation rates ˙ Suppressing the rate-independent and the relevant response function is f (Ee ; #; ; ). variables (Ee ; #; ), we refer to the theory as mechanically rate-independent if, for each , this function is invariant under all changes of time scale; that is, if ˙ = f () ˙ f ()

for all scalars  ¿ 0;

(7.1)

˙ Note that rate-independence and continuity at ˙ = 0 and all values of the argument . ˙ is independent of . ˙ (Just pass to the limit together imply that, for each , f ()  → 0 in (7.1).) Thus, since a constitutive dependence on ˙ is central to the theory, ˙ continuous at ˙ = 0. In fact, it is central to the theory that we we cannot have f () ˙ to be unde5ned when ˙ = 0, but to have limiting values when ˙ → 0 allow f () with sgn ˙ held 4xed. With this in mind, we assume that, for each , the limits ˙ f () Y+ = lim ˙ →0 ˙ ¿0

and

˙ Y− = − lim f () ˙ →0 ˙ ¡0

exist. A consequence of this assumption and rate-independence is that  +Y+ for ˙ ¿ 0;  ˙ f () = −Y− for ˙ ¡ 0:

(7.2)

(7.3)

As we shall see, the limits Y± represent critical transformation resistances. Of course, ˙ the Y± may also depend on the suppressed arguments (Ee ; #; ). Note that f ()

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1031

depends on ˙ through a dependence on ˙ . When we wish to make this explicit, we ˙ = f (˙ ). 9 write f () To verify (7.3) we simply pass to the limit  → 0 on the right side of (7.1) and use (7.2). Note that the limits Y± are independent of ˙! for ! =  and dependent on ˙ = 0 at most through its sign. Indeed, choose, e.g., an arbitrary ˙ with ˙ ¿ 0, then ˙ = f (): ˙ Y+ = lim f () →0 ¿0

˙ and (hence)  (cf. (5.5)) are unde4ned when ˙ = 0, the reduced Although f () ˙ ˙ are well de4ned dissipation inequality (5.6) remains valid, since the terms f ()  ˙ when  = 0. In fact, a consequence of (5.7) is that Y± ¿ 0: Consistent with the fact that  is unde4ned when ˙ = 0, 10  is considered as indeterminate when ˙ = 0:

(7.4)

This indeterminacy is an essential feature of the theory, as it allows us to consider the microforce balance  =  (or its virtual counterpart) as satis5ed automatically when ˙ = 0. 7.2. Transformation conditions Using (6.1) in conjunction with the microforce balance (3.17), the constitutive equation (5.5), and the rate-independence condition (7.3), we arrive at transformation conditions for the system  asserting that: a → m transformations (˙ ¿ 0) are possible only if  − b = Y+ ;

(7.5)

m → a transformations (˙ ¡ 0) are possible only if  − b = −Y− :

(7.6)

Thus b (=9 ˆ =9 ) represents a backforce for system , 11 while the scalars Y± represent critical transformation resistances. On the other hand, when  − b lies inside the interval (−Y− ; Y+ ) or outside the interval [ − Y− ; Y+ ] we must have ˙ = 0. Next, suppose that the th system is at the threshold for phase transitions as de4ned by (7.5) or (7.6), whichever is relevant, so 9 Y  may also depend on internal state variables such as those used to characterize hardening in ± single-crystal plasticity; while the introduction of such variables is straightforward, their use within the present theory seems unnecessary (cf. the last paragraph of Section 13). 10 Cf. Gurtin (2000, Eq. (78)a). 11 Since b plays a role strictly analogous to the backstress in theories of plasticity.

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L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

that  − b = Y+ and ˙ ¿ 0 or  − b = −Y− and ˙ 6 0. If this system is to remain at this threshold, then we must have ( − b˙ − Y+ )˙ = 0

or

( − b˙ + Y− )˙ = 0;

(7.7)

whichever is relevant. Relations (7.7) represent consistency conditions for the system  to undergo phase transitions. 7.3. Microstability. The
(7.8)

Consider the external microforces  as test forces applied by an external agency to test the stability of the system. Given a state s = (Ee ; #; ) and a list ˙ of transformation rates, we may conclude from (7.3) and (7.8) that  ˙ = [f (s; ˙ ) + b (s) −  (s)]˙ gives the power expended on , per unit volume, by the external agency, and we might regard the material as microstable if such an
agency cannot extract micropower during transitions starting from the state s; i.e., if   ˙ ¿ 0 for any choice of the list ˙ of transformation-rates. Precisely, we de4ne a state s to be microstable if N 

[f (s; ˙ ) + b (s) −  (s)]˙ ¿ 0

=1

˙ or equivalently, by (7.3), if for each , for any choice of ,    +Y+ (s) + b (s) −  (s) ˙ ¿ 0 for all ˙ ¿ 0;    −Y− (s) + b (s) −  (s) ˙ ¿ 0 for all ˙ ¡ 0:

(7.9)

The transition rates ˙ in (7.9) must be consistent with the constraint 0 6  6 1, which rules out rates ˙ ¡ 0 when  = 0 and ˙ ¿ 0 when  = 1. Direct consequences of (7.9) are then the
for 0 ¡  ¡ 1;

for  = 0; −Y− 6  − b

for  = 1:

(7.10)

We henceforth restrict attention to microstable states and hence to processes consistent with the Gag inequalities. These inequalities are illustrated in Fig. 2 for the special case in which the moduli Y± are constant.

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

α α τ -b

1033

typical loading/unloading line

α Y+ martensite 0

1

austenite

ξα

− Y−α austenite/martensite

Fig. 2. Flag diagram illustrating all possible paths. The typical loading/unloading line may lie anywhere in the shaded region.

7.4. Complete transformation conditions We now state three conditions that complete specify the transformational behavior of the material; condition (i) is the content of the 4rst sentence of the paragraph containing (7.7); (ii) and (iii) follow from (7.7) and the Gag inequalities. The three conditions are stated precisely as follows: (i) Elastic-range conditions: if  − b = Y+

then ˙ = 0;

or if  − b = −Y− 

(ii) Conditions for a → m transformations: if  −b then ˙ ¿ 0 and ( − b˙ − Y  ) 0 0;



=Y+ ,

(7.11)



0 6  ¡ 1, and 0 6  ¡ 1, (7.12)

+





(iii) Conditions for m → a transformations: If  − b = 0 ¡  6 1, then ˙ 6 0 and ( − b˙ + Y− ) 1 0; 

(iv) End conditions: If  − b ˙ = 0.



= Y+





and  = 1, or if  − b



−Y− ,

= −Y−



0 ¡  6 1, and (7.13) 

and  = 0, then

Thus if one, beginning in an austenitic state ( = 0), loads the body monotonically, then ˙ = 0 until the onset of the a → m transition for . Once this transition begins,

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L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

(7.12) implies that the transition process continues until  = 1, or until the body is unloaded; i.e., ( − b˙ − Y+ ) ¡ 0. If the body is not unloaded, but instead is loaded monotonically until the time at which  = 1, then, by (iv), at that time ˙ = 0, and with increased monotonic loading the system  will be martensitic ( = 1) and will consequently behave elastically. The possible loading and unloading paths are shown schematically in Fig. 2. An analogous discussion applies to the process of unloading from the austenitic state.

8. Transformation conditions for the rate-dependent theory While shape-memory materials exhibit almost no rate-dependence, it is present, and as such may be useful as a regularization of the underlying equations. 12 For that reason we consider also a rate-dependent constitutive equation for f that reduces to (7.3) in the limit as the rate-dependence, described by the small constant m ¿ 0, tends to zero:  m ˙     +  Y+ for ˙ ¿ 0;   U ˙ = (8.1) f () m  ˙     − Y ˙   U − for  6 0;  where U ¿ 0 is a reference transformation-rate, and Y± ¿ 0 to ensure satisfaction of the dissipation inequality (5.7). In this case  is well de4ned for all transformation-rates and (7.4) is irrelevant. For future use we note that the inverted form of (8.1) is m  f () ˙     for ˙ ¿ 0; +U    Y+  ˙ = (8.2) m  f () ˙      −U  for ˙ 6 0:  Y− In this case the transformation conditions are replaced by  m ˙     for ˙ ¿ 0; +  Y   U +  − b = m  ˙     ˙  −  U Y− for  6 0: 

(8.3)

12 This is true in single-crystal plasticity, where the equations have the same basic structure, cf. Asaro and Needleman (1985).

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1035

Conditions (7.11)–(7.13) as well as the Gag inequalities are inapplicable. The inverted form of (8.3) is    − b 1=m   U  for ( − b ) ¿ 0;   + Y  +  ˙ = (8.4)    1=m  − b   −U  for ( − b ) 6 0:  Y −

9. Constitutive equations appropriate to small elastic strains and small temperature di6erences The theory presented thus far is quite general. We now introduce constitutive equations appropriate for situations in which the elastic strains are small and the temperature close to a 5xed reference temperature #0 . Although the constitutive equations that we shall consider are special, the theory itself remains exact, as no assumptions of an approximative nature are introduced. We consider a free energy of the form ˆ (Ee ; #; ) =

e

(Ee ; #; ) +

th

(#; ) +

p

(#; ):

(9.1)

Here: (i)

e

, the strain energy, is given by e

(Ee ; #; ) = 12 Ee · C()[Ee ] − (# − #0 )A() · C()[Ee ];

(9.2)

where C() is a symmetric, positive-de4nite linear transformations of symmetric tensors into symmetric tensors that represents the elasticity tensor at the reference temperature #0 , while A() is the symmetric thermal expansion tensor at #0 ; (ii) p , the energy of transformation, is given by 13 p

(#; ) =

N T 1  !  ! (# − #T ) + g   ; #T 2

(9.3)

;!=1

here #T , the phase equilibrium temperature, represents the transformation temperature in the absence of stress, T is the latent heat of the a → m transformation at #T , and the scalar constants g! = g! characterize possible energetic interactions between the transformation systems; (iii) th (#), the thermal energy, is given by

 # th ; (9.4) (#; ) = c(# − #0 ) − c# ln #0 with c ¿ 0 a constant speci4c heat. 13 Here we are guided by the one-dimensional development of Abeyaratne and Knowles (1993). An obvious
generalization of (9.3) would be to replace the 4rst term on the right by  (T =#T )(# − #T ) .

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By (5.4), the elastic stress and entropy are given by Te = C()[Ee − (# − #0 )A()];

 # T  = c ln + Ee · C()[A()] −  #0 #T

(9.5)

and by (6.1) the backforce has the form b =

N

 T (# − #T ) + g! ! : #T !=1

As the constitutive equation for the heat Gux we take Fourier’s law q = −K()∇#;

(9.6)

with K(), the conductivity tensor at #0 , positive de4nite and symmetric. Finally, we assume that Y+ = Y− = Y  ≡ constant

(9.7)

˙ remains dependent on ˙ (actually ˙ ) through (7.3) or (8.1). although f ()

10. Summary of the 7nal constitutive equations In this section we summarize the resulting constitutive theory. This theory is meant to characterize small elastic strains in the presence of temperature 4elds close to a 4xed reference temperature #0 , for a material with martensitic–austenitic transformation systems  = 1; 2; : : : ; N de4ned by vector pairs (b0 ; m0 ), |m0 | = 1. The underlying constitutive equations relate the following basic 4elds: F = ∇y; det F ¿ 0 #; # ¿ 0 S; SF = FST q, Fp ,  ; 0 6  6 1 ; 0 6  6 1 Fe = FFp−1 ; det Fe ¿ 0 Ce = Fe Fe Ee = 12 (Ce − 1) Te = Fe−1 SFp

free energy density per unit referential volume deformation gradient absolute temperature Piola–Kirchho% stress heat Gux per unit referential area transformational part of the deformation gradient martensitic volume fraction for the th system total martensitic volume fraction elastic part of the deformation gradient elastic right Cauchy–Green strain elastic strain stress conjugate to Ee .

The constitutive theories for the mechanically rate-independent and the rate-dependent theory are summarized below.

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

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10.1. Mechanically rate-independent theory The free energy is given by 1 (Ee ; #; ) = Ee · C()[Ee ] − (# − #0 )A() · C()[Ee ] 2 +

 N 1  !  ! # T : (# − #T ) + g   + c(# − #0 ) − c# ln #T 2 #0 ;!=1

(10.1) Here C() is the elasticity tensor and A() is the thermal expansion tensor at the reference temperature #0 . The parameter #T is the phase-equilibrium temperature, T is the latent heat of the a → m transformation at temperature #T , and the scalar constants g! = g! characterize energetic interactions between the transformation systems. Also, c is the speci4c heat. The elastic stress–strain relation has the form Te = C()[Ee − A()(# − #0 )];

(10.2)

while the resolved stress and backforce on the th system are given by  = b0 · (Ce Te )m0

(10.3)

and b =

N

 T (# − #T ) + g! ! : #T

(10.4)

!=1

The transformation conditions are  − b = Y   − b = −Y 

for a → m; for m → a;

(10.5) (10.6)

with Y  ¿ 0 the transformation resistance. The
N 

˙ b0 ⊗ m0 ;

(10.7)

=1

with martensitic volume fractions  consistent with 0 6  6 1 and 0 6  6 1. Moreover, (i) if  − b = Y  or if  − b = −Y  , then ˙ = 0; (ii) if  − b = Y  , 0 6  ¡ 1, and 0 6  ¡ 1, then ˙ ¿ 0 and ( − b˙ − Y  ) 0 0;

(10.8)

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(iii) if  − b = −Y  , 0 ¡  6 1, and 0 ¡  ¡ 1, then ˙ 6 0

and

( − b˙ + Y  ) 1 0;

(10.9)

(iv) if  − b = Y  and  = 1, or if  − b = −Y  and  = 0, then ˙ = 0. Finally, we have the entropy relation

 # T  = c ln + Ee · C()[A()] −  #0 #T

(10.10)

together with Fourier’s law q = −K()∇#

(10.11)

for the heat Gux. To complete the constitutive model for a particular material the constitutive parameter/functions that need be speci4ed are {C(); A(); b0 ; m0 ; #T ; T ; g! ; Y  ; c; K()}: 10.2. Rate-dependent theory Here (10.2) – (6.1), (10.10), and (10.11) are as in the mechanically rate-independent theory, but the transformation conditions, Gow rule, and consistency equations of the rate-independent theory are replaced by the rate-dependent
N 

˙ b0 ⊗ m0 ;

=1

  − b 1=m  U ˙ sgn( − b );  =  Y

(10.12)

where U ¿ 0, a reference transformation rate, and m ¿ 0, a small rate-sensitivity parameter, are additional material parameters needed in the rate-dependent theory. 11. Partial di6erential equations for the deformation and temperature 7elds 11.1. Equations referred to the reference con5guration The local force balance for the macroscopic stress is (3.13) supplemented by (3.21), viz. Div S + f = 0;

S = Fe Te Fp− ;

with Te given by (10.2).

(11.1)

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1039

Next, using (10.10) and (10.11), the entropy balance (6.4) has the form c #˙ = Div{K()∇#} +

N

T ˙   ˙ ˙ ˙ + r; # + f () − #Ee · C()[A()] #T

(11.2)

=1

˙ given by with f () ˙ = Y  sgn(˙ ) f ()

(11.3)

for the mechanically rate-independent theory, and m ˙ ˙ = Y   sgn(˙ ) f () U

(11.4)

for the rate-dependent theory. Note that if K, C, and A are independent of , then this balance has the simple form N

T ˙   ˙ ˙ # + f () − #E˙ e · C[A] + r: c #˙ = K · (∇∇#) + #T

(11.5)

=1

11.2. Equations referred to the deformed con5guration The elastic stress is related to the Cauchy stress T = J −1 SF ;

J = det F;

(11.6)

through the relation T = J −1 Fe Te Fe

(11.7)

and in terms of the Cauchy stress the local force balance Div S + f = 0 has the form div T + J −1 f = 0;

(11.8)

where div represents the divergence operator in the deformed con4guration. As before, Te is given by (10.2). Similarly, the heat Gux h, measured per unit area in the deformed con4guration is related to q, which is referential, through h = J −1 Fq:

(11.9)

Thus, since ∇# = F grad #: Fourier’s law (10.11) takes the form h = −J −1 FK()F grad #:

(11.10)

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L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

Thus the counterpart for the deformed con4guration of the term involving the referential divergence in (11.2) is div{J −1 FK()F grad #}: The remaining terms in (11.2) are measured per unit referential volume and hence must be multiplied by J −1 to render them measured per unit volume in the deformed con4guration. Thus the entropy balance (11.2) referred to the deformed con4guration has the form c#˙ = J div{J −1 FK()F grad #} +

N

T ˙   ˙ ˙ # + f () #T =1

˙ − # Ee · C()[A()] + r:

(11.11)

12. Some representative calculations for a Ti–Ni shape-memory alloy The purpose of this section is to present results from numerical simulations of some representative problems involving the superelastic response of the technologically important Ti–Ni shape-memory alloy. The numerical simulations are carried out using the 4nite-element computer program ABAQUS/Explicit (ABAQUS, 2002), for which we have written a user material subroutine to implement our constitutive model. 14 Since we have not attempted to carry out an experimental program to obtain material parameters for a given Ti–Ni shape-memory material, the results presented here are intended to demonstrate only the qualitative features of the theory. All simulations presented below are based on the mechanically rate-independent theory discussed in Section 10.1. For Ti–Ni the high-temperature austenitic phase has a body-centered-cubic superlattice (B2) crystal structure, while the low-temperature martensitic phase is monoclinic. The crystallographic theory of martensite shows that phase transformation in Ti–Ni can occur on 192 possible transformation systems (Hane and Shield, 1999). The components of the 192 transformation systems (m0 ; b0 ) with respect to an orthonormal basis associated with the parent cubic austenite crystal lattice, calculated using the algorithm given in Hane and Shield (1999), are given in Appendix. Much work needs to be done to elucidate the nature and magnitude of the interaction terms g! which contribute to the backforce b in (6.1) for Ti–Ni. In our calculations we ignore the contributions due to interactions between di%erent transformation systems to the backforce, and simply set g! = 0. Further, we also assume that the values of Y  for the 192 systems are all equal and constant, Y  ≡ Y . The dependence of the anisotropic elastic constants and coeNcients of thermal expansion for Ti–Ni on the martensitic volume fraction  is not well-documented in the literature. For simplicity, these are taken to follow the rule of mixtures: C() = 14 Algorithmic details of the numerical implementation are given in Thamburaja and Anand (2001). Central to the algorithm is the use of the method of Anand and Kothari (1996) to determine the amount of transformation on the operative transformation systems in a rate-independent theory.

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1041

(1 − )Ca + Cm , and A() = (1 − )Aa + Am . Also, we have been unable to 4nd experimentally measured values of the anisotropic elastic moduli of single-crystal monoclinic martensite of Ti–Ni. Typically, Young’s Modulus for polycrystalline Ti–Ni in the austenitic state is about two to three times its martensitic value, while Poisson’s ratio is approximately the same for the two states. Guided by this information, we have assumed that single-crystal anisotropic elastic constants of the monoclinic martensite may be treated approximately as those of a cubic material, and that the corresponding values of the sti%nesses, Cij , are one-half as large as those for the austenite. The thermal expansion tensor for the martensite is also taken to be the same as that for a cubic material with Am = m 1. Single crystals of Ti–Ni are hard to grow and not readily available. In our study we shall use the following values for the material parameters for Ti–Ni single crystals estimated by Thamburaja (2002) from experiments on a polycrystalline textured sheet: 15 a a a Elastic moduli for austenite: C11 = 130 GPa, C12 = 98 GPa, C44 = 21 GPa; m m m = 10:5 GPa; Elastic moduli for martensite: C11 = 65 GPa, C12 = 49 GPa, C44 CoeNcients of thermal expansion: a = 11 × 10−6 =K, m = 6:6 × 10−6 =K; Phase equilibrium temperature: )T = 271 K; Latent heat: T = 110 MJ=m3 ; Critical transformation resistance: Y = 4:7 MJ=m3 .

We 4rst consider a single crystal subjected to tension and compression along its [0 0 1]-direction at a 4xed temperature of 300 K under isothermal conditions. The single crystal is modeled using a single eight-noded C3D8R continuum element. Let {e1 ; e2 ; e3 } denote an orthonormal basis of a global coordinate system chosen with the e3 -direction aligned with the [0 0 1] crystallographic direction of the parent cubic lattice. The “hard boundary conditions” imposed on the single element to achieve “simple” tension or compression are (a) the node coincident with the origin of the coordinate system is 4xed in all its degrees of freedom, the second node on the e3 -axis is restricted not to move in the e1 and e2 directions, while the second node on the e1 -axis is restricted not to move in the e2 direction; (b) the bottom four nodes of the element in the (e1 ; e2 )-plane are not allowed to move in the e3 -direction; and (c) the remaining top four nodes in the (e1 ; e2 )-plane are constrained to remain parallel to the original plane, while they are displaced in the ±e3 direction for tension/ compression, at a constant velocity. 16 The resulting Gag-type superelastic stress–strain curves 17 for tension and compression are shown in Fig. 3. In order to emphasize the 15 Such an estimate is not unique, and depends on the precise chemistry and the prior thermomechanical processing history of the polycrystalline material. For example, estimates of single-crystal parameters from a polycrystalline textured rod of Ti–Ni yield the following values for phase transformation parameters: )T = 256 K, T = 100 MJ=m3 , and Y = 8:2 MJ=m3 , Thamburaja and Anand (2001). 16 The response of a single-crystal is very sensitive to small changes in boundary conditions. If the same single-crystal tension experiment is modeled with multiple elements instead of a single element, then one obtains a slightly di%erent response because of the additional degrees of freedom in the multi-element simulation; see Fig. 7. 17

All stress–strain curves in this paper are reported in terms of engineering stress and strain measures.

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L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058 1000

900

TENSION COMPRESSION

800

Stress , MPa

700

600

500

400

300

200

100

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain Fig. 3. Stress–strain curves from simulations of superelastic tension and compression experiments on a Ti–Ni single crystal in the [0 0 1]-direction at 300 K.

pronounced tension-compression asymmetry in the superelastic response for the [0 0 1] orientation, we have plotted the absolute values of stress and strain for the compression simulation. This asymmetry arises because di8erent transformation systems are activated in tension and compression. Recall the transformation conditions:  −

T (# − #T ) = Y #T

 −

T (# − #T ) = −Y #T

for a → m; for m → a:

These conditions imply that as the temperature is increased, the mean level of the superelastic hysteresis loop, which is determined by the value of the backforce b = T =#T (# − #T ), should increase linearly with the temperature. Fig. 4 shows such a temperature dependence for the isothermal superelastic response of a Ti–Ni single crystal subjected to tension along its [0 0 1]-direction at three di%erent temperatures: 315, 300, and 285 K. 18 18 The slight strain-hardening observable in Fig. 4 is due to the “hard” displacement boundary-conditions imposed on the single element calculation to enforce “simple tension”, as well as changes in the resolved forces necessary to activate the operative transformation systems as the specimen is extended; it is not due to self-heating since these results are from isothermal calculations.

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1000

900

800

Stress , MPa

700

600

500

400

300 315 K 300 K 285 K

200

100

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain Fig. 4. Stress–strain curves from simulations of superelastic tension on a Ti–Ni single crystal in the [0 0 1]-direction at three di%erent temperatures: 315, 300, 285 K.

An important manifestation of the superelastic response of a shape-memory material is the strain-versus-temperature response at a 4xed stress level. By cycling the temperature over a narrow range of temperatures at a 4xed stress, one can obtain a recoverable strain cycle. Fig. 5 shows the strain–temperature curve from a simulation for a Ti–Ni single crystal which is 4rst subjected to a 4xed axial stress level of 500 MPa in the [0 0 1]-direction at 320 K, and then subjected to a temperature history where the temperature is decreased linearly from 320 to 280 K in 200 s, and then increased back to 320 K in another 200 s. The material is initially in the austenitic state. The initial elastic strain of ≈ 0:01 is due to the axial stress of 500 MPa. As the temperature is decreased, the strain 4rst decreases slightly due to thermal contraction, and then at ≈ 291 K the austenite to martensite transformation occurs and there is a sudden burst of strain of the order 3.5%. Upon heating, there is at 4rst a slight increase in the strain due to thermal expansion, and then at ≈ 313 K the martensite transforms back to austenite with a sudden strain recovery of the order 3.5%. Transformations that occur can be controlled by the initial stress bias; an increase in the initial stress proportionally increases the austenite-to-martensite as well as the martensite-to-austenite transformation temperatures. We next consider the coupled thermomechanical superelastic response of Ti–Ni single crystals. Recall that the entropy balance (11.2) requires computation of the term J div{J −1 FK()F grad #}. The 4nite element program ABAQUS/Explicit (currently) limits user access to modify the computer code, and allows input of only a single

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Strain

0.04

0.03

m-to-a

a-to-m

0.02

0.01

0 280

285

290

295

300

305

310

315

320

Temperature, K Fig. 5. Strain–temperature curve from simulation of a Ti–Ni single crystal 4rst subjected to a 4xed axial stress level of 500 MPa in the [0 0 1]-direction, and then subjected to a temperature history in which the temperature is decreased linearly from 320 to 280 K in 200 s, and then increased back to 320 K in another 200 s.

Fig. 6. Finite-element mesh for thermo-mechanically coupled simulations of superelastic tension experiments on a Ti–Ni single crystal in the [0 0 1]-direction.

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058 1000

1045

g

900

-4

STRAIN RATE=5× 10 /sec STRAIN RATE=1× 10-4 /sec

f

800 e d

700

Stress , MPa

c b 600

500

400

300 h

200 i j

100 k 0

a l 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain Fig. 7. Stress–strain curves from simulations of superelastic tension experiments on a Ti–Ni single crystal in the [0 0 1]-direction at an initial temperature of 300 K, conducted at nominal strain rates of 1 × 10−4 =s and 5 × 10−4 =s.

scalar value for the thermal conductivity. We have accordingly made several approximations in our calculations. For the cubic austenite the thermal conductivity is isotropic, Ka = !a 1. For the monoclinic martensite the anisotropic thermal conductivities are not known, and as with the elastic constants, these are also approximated as that for cubic materials, Km ≈ !m 1. We shall use a constant value of ! = (!a + !m )=2 in our numerical calculations. Further, since the superelastic strains are small, we neglect the contributions from the J and FF terms, and make the approximation J div{J −1 FK()F grad #} ≈ ! div grad #. Finally, we do not allow for a heat supply ˙ in (11.2). The equation governing the (r ≡ 0) and neglect the term #Ee · C()[A()] change in temperature then becomes N T ˙  ˙ c#˙ ≈ ! div grad # + # + Y | |: (12.1) #T =1

Based on values published in the literature, we use the following values of c = 2:1 MJ=m3 K;

! = 13 W=m K

for Ti–Ni in our calculations. In order to examine the e%ects of heat generation and conduction, we have repeated the calculation for superelastic tension on a Ti–Ni single crystal in the [0 0 1]-direction at an initial temperature of 300 K, when the test is conducted at two nominal strain rates: a “low rate” of 1 × 10−4 =s where the response is expected to be close to that

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Fig. 8. Contours of martensite volume fraction from simulation of superelastic tension conducted at nominal strain rate of 5 × 10−4 =s on a Ti–Ni single crystal in the [0 0 1]-direction at an initial temperature of 300 K. The set of the contours on the left show the transformation from austenite-to-martensite during forward transformation, while those on the right show the transformation from martensite-to-austenite during reverse transformation. The labels (a) – (l) are keyed to the stress–strain curve in Fig. 7. Note that because of the boundary conditions, both the forward and reverse transformations nucleate from the grip-ends and propagate towards the center of the specimen.

for the isothermal case reported previously, Fig. 3; and a slightly “higher rate” of 5×10−4 =s where we expect to see some e%ects of the changes in temperature. The calculations are carried out for a strip of single-crystal Ti–Ni modeled with 357 C3D8RT ABAQUS elements, as shown in Fig. 6. As with the single-element calculations, mechanically “hard boundary conditions” are applied to a%ect “simple” tension on the multi-element mesh. Regarding the thermal boundary conditions, the whole mesh is initially at 300 K. The nodes on the bottom and top faces in the (e1 ; e2 )-planes are kept at 300 K throughout the calculation to simulate massive hard grips which act as constant temperature baths. The heat Gux from the remaining faces of the tension strip due to convection in air is taken to be governed by h=h(#−#0 )n, with a 4lm coeNcient h and a sink temperature #0 . In our calculations we take the 4lm coeNcient to have a value h = 12 W=m2 =K, and a sink temperature of 300 K. Fig. 7 shows the resulting superelastic stress–strain curves. The hysteresis loop corresponding to the lower strain rate is very similar to that obtained from the isothermal single-element calculation, Fig. 3, while the stress–strain curve calculated for the higher strain rate shows a hardening response. Recall that we have set Y ≡ constant and the interaction terms g! to zero

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1047

Fig. 9. Temperature contours from simulation of superelastic tension conducted at nominal strain rate of 5 × 10−4 =s on a Ti–Ni single crystal in the [0 0 1]-direction at an initial temperature of 300 K. The set of the contours on the left show the temperature increase during transformation from austenite-to-martensite, while those on the right show the temperature decrease during transformation from martensite-to-austenite. The labels (a) – (l) are keyed to the stress–strain curve in Fig. 7. During forward transformation the temperature increases by as much as 11 K above the ambient temperature of 300 K due to the exothermic austenite-to-martensite transformation, whereas it decreases by as much as 12 K below the ambient temperature during the reverse endothermic transformation from martensite-to-austenite.

in our calculations. Thus, the hardening response is entirely due to thermal e8ects associated with the phase transformations. Fig. 8 shows evolution of the contours of the martensite volume fraction at representative instances (a) through (l) keyed to Fig. 7, during the forward and reverse transformations for the test at the higher strain rate. Note that because of the boundary conditions, both the forward and reverse transformations start from the grip-ends and propagate as “fronts” towards the center of the specimen. Fig. 9 shows contours of the temperature at the same representative instances during the forward and reverse transformations. During forward transformation the temperature increases by as much as 11 K from the ambient temperature of 300 K due to the exothermic austenite-to-martensite transformation, while it decreases by as much as 12 K from the ambient temperature during the reverse endothermic transformation from martensite-to-austenite. The nucleation and propagation of phase transformation fronts and the associated temperature changes in single crystal Ti–Ni, is qualitatively similar to the results reported by Shaw and Kyriakides (1997) in polycrystalline sheet tensile specimens.

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Fig. 10. (a) {1 1 1} pole 4gure of the near-random initial texture using 729 crystal orientations. (b) Mesh using 729 4nite elements for simulation of superelastic tension and compression experiments on a Ti–Ni polycrystal.

700

600

TENSION COMPRESSION

Stress , MPa

500

400

300

200

100

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain Fig. 11. Stress–strain curves from simulations of superelastic tension and compression experiments on a Ti–Ni polycrystal at 300 K.

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0.07 STRESS, 250 MPa 0.06

Strain

0.05

0.04

a-to-m

m-to-a

0.03

0.02

0.01

0

270

280

290

300

310

320

Temperature, K Fig. 12. Strain–temperature curve from simulation of a Ti–Ni polycrystal 4rst subjected to a 4xed axial stress level of 250 MPa, and then subjected to a temperature history where the temperature is decreased linearly from 325 to 265 K in 200 s, and then increased back to 325 K in another 200 s.

Next, we investigate the result of superelastic simulations for a polycrystalline aggregate with a (near)-random initial texture. In our 4nite-element model of a polycrystal, each element represents one crystal, and a set of crystal orientations which approximate the initial random texture are randomly assigned to the elements. The macroscopic stress–strain responses are calculated as volume averages over the entire aggregate. We have carried out tension and compression simulations for a polycrystalline specimen using a set 729 crystal orientations to represent a “random” initial texture. The {1 1 1}-pole 4gure corresponding to this random texture is shown in Fig. 10a. All other material parameters used in these isothermal simulations were the same as those used in the previous calculations. The initial mesh comprising 729 C3D8R 4nite elements used for these simulations is shown in Fig. 10b. The predicted tension and compression superelastic stress–strain curves are shown in Fig. 11. This 4gure shows that in comparison to the result from the calculation for a single crystal, Fig. 3, the asymmetry between the compression and tension curves for a polycrystalline material with a near-random texture is substantially reduced. 19 Also, unlike the “angular” Gag-type hysteresis loops observed for the single-crystal calculation, the hysteresis loops are now much more rounded because of the successive transformation of the di%erently oriented crystals comprising the polycrystal. 19 Thamburaja and Anand (2001) have shown that for an initially textured polycrystalline rod the asymmetry is not negligible and reversed; that is, the compression curve is higher than the tension curve.

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Finally, Fig. 12 shows the strain–temperature curve from simulation of a Ti–Ni polycrystal 4rst subjected to a 4xed axial stress level of 250 MPa at 325 K, and then subjected to a temperature history where the temperature is decreased linearly from 325 to 265 K in 200 s, and then increased back to 325 K in another 200 s. 20 Again, relative to the corresponding result for a single crystal, Fig. 5, the hysteresis loop is now much more rounded, again because of the successive transformation of the di%erently oriented crystals comprising the polycrystal.

13. Concluding remarks We have developed a thermodynamically consistent theory for the superelasticity of crystalline shape-memory materials. Using a 4nite-element implementation, numerical simulations of some representative problems involving the superelastic response of the shape-memory alloy Ti–Ni are carried out under both isothermal and nonisothermal Table 1 Subset of 24 transformation systems 

m0; 1

m0; 2

m0; 3

b0; 1

b0; 2

b0; 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

−0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 0.4045 −0:4045 0.2153 0.2153 −0:2153 −0:2153 0.4045 −0:4045 0.2153 0.2153 0.4045 −0:4045 0.4045 −0:4045 −0:2153 −0:2153

−0:4045 0.4045 0.2153 0.2153 −0:2153 −0:2153 0.4045 −0:4045 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 0.4045 −0:4045 0.2153 0.2153 −0:2153 −0:2153 0.4045 −0:4045

0.2153 0.2153 −0:4045 0.4045 0.4045 −0:4045 −0:2153 −0:2153 0.2153 0.2153 −0:4045 0.4045 0.4045 −0:4045 −0:2153 −0:2153 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888 −0:8888

0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0638 −0:0638 0.0991 0.0991 −0:0991 −0:0991 0.0638 −0:0638 0.0991 0.0991 0.0638 −0:0638 0.0638 −0:0638 −0:0991 −0:0991

−0:0638 0.0638 0.0991 0.0991 −0:0991 −0:0991 0.0638 −0:0638 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0638 −0:0638 0.0991 0.0991 −0:0991 −0:0991 0.0638 −0:0638

0.0991 0.0991 −0:0638 0.0638 0.0638 −0:0638 −0:0991 −0:0991 0.0991 0.0991 −0:0638 0.0638 0.0638 −0:0638 −0:0991 −0:0991 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568 0.0568

20 All nodes in the mesh are simultaneously subject to this temperature history, and both heat conduction and heat generation are neglected.

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1051

700

600

TENSION, 192 TENSION, 24

Stress , MPa

500

400

300

200

100

0

0

0.01

0.02

0.03

(a)

0.04

0.05

0.06

0.07

Strain 700

600

COMPRESSION, 192 COMPRESSION, 24

Stress , MPa

500

400

300

200

100

0

(b)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Strain

Fig. 13. Comparison of stress–strain curves from simulations using 192 and 24 transformation systems: (a) tension, and (b) compression superelastic experiments on a Ti–Ni polycrystal at 300 K. The absolute values of stress and strain are plotted for the compression curve.

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situations. The two manifestations of superelasticity—stress–strain response at 4xed temperature and strain–temperature response at 4xed stress—have been studied. The results of the numerical simulations are qualitatively similar to existing experimental results on polycrystalline Ti–Ni. In particular, the thermomechanically coupled calculation for a single-crystal strip under tension shows the nucleation and propagation of phase transformation fronts, Figs. 8 and 9. This result is qualitatively similar to that observed in the experiments of Shaw and Kyriakides (1997). We have also examined the e%ects of the time-scales associated with heat generation and conduction on the overall nominal stress–strain curves. As shown in Fig. 7, when the test is conducted at a low nominal strain rate of 1 × 10−4 =s, there is enough time for heat transfer from the specimen to its surroundings and the stress–strain response is close to that for the isothermal case, Fig. 3, with no strain-hardening during the transformations. However, at a slightly higher rate of 5 × 10−4 =s, there is not enough time for the heat (generated/absorbed during the forward/reverse transformations) to completely transfer from the specimen to its surroundings, and the stress–strain curve shows an apparent “strain-hardening” response. Since the calculations were performed with Y constant and the interaction terms g! set to zero, this hardening response is due entirely to thermal e8ects associated with phase transformations. This result is qualitatively similar to that observed in the experiments of Entemeyer et al. (2000) and Thamburaja (2002). Several authors have considered the subset of 24 transformation systems listed in Table 1, rather than the full set of 192 systems listed in Appendix. Fig. 13 shows—for a polycrystalline aggregate with a random initial texture (Fig. 10)—a comparison of the superelastic curves predicted using 24 systems and those predicted using 192 systems. While the two sets of curves are qualitatively similar, the stress–strain curves predicted using 24 systems are higher than those predicted using the full 192 transformation systems of the crystallographic theory of martensite (Hane and Shield, 1999). Acknowledgements LA gratefully acknowledges (a) the help of Prakash Thamburaja with some of the numerical simulations, and (b) the 4nancial support provided by the grant CMS0002930 from NSF. MG gratefully acknowledges the support of the DOE under grant DEFG0101ER25449. Appendix: Transformation systems  1 2 3 4 5

m0; 1

m0; 2

m0; 3

−0:9087 −0:9087 −0:3464 −0:3464 −0:8888

−0:3334 0.3334 0.4334 −0:4334 −0:4045

0.2514 0.2514 −0:8320 −0:8320 0.2153

b0; 1 0.0527 0.0527 0.1211 0.1211 0.0568

b0; 2

b0; 3

−0:0535 0.0535 0.0397 −0:0397 −0:0638

0.1060 0.1060 −0:0257 −0:0257 0.0991

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1053

Appendix (Continued)  6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

m0; 1

m0; 2

m0; 3

b0; 1

b0; 2

b0; 3

−0:8888 −0:3762 −0:3762 −0:8888 −0:8888 −0:3762 −0:3762 −0:9087 −0:9087 −0:3464 −0:3464 −0:3762 −0:3762 −0:8888 −0:8888 −0:3464 −0:3464 −0:9087 −0:9087 −0:3762 −0:3762 −0:8888 −0:8888 −0:3464 −0:3464 −0:9087 −0:9087 −0:3334 0.3334 0.4334 −0:4334 0.5137 0.4045 −0:4045 −0:5137 0.2153 0.2153 −0:7711 −0:7711 0.2514 0.2514 −0:8320 −0:8320 0.7711 0.7711 −0:2153 −0:2153 0.8320 0.8320

0.4045 0.5137 −0:5137 0.2153 0.2153 −0:7711 −0:7711 0.2514 0.2514 −0:8320 −0:8320 0.7711 0.7711 −0:2153 −0:2153 0.8320 0.8320 −0:2514 −0:2514 −0:5137 0.5137 0.4045 −0:4045 −0:4334 0.4334 0.3334 −0:3334 −0:9087 −0:9087 −0:3464 −0:3464 −0:3762 −0:8888 −0:8888 −0:3762 −0:8888 −0:8888 −0:3762 −0:3762 −0:9087 −0:9087 −0:3464 −0:3464 −0:3762 −0:3762 −0:8888 −0:8888 −0:3464 −0:3464

0.2153 −0:7711 −0:7711 −0:4045 0.4045 0.5137 −0:5137 −0:3334 0.3334 0.4334 −0:4334 −0:5137 0.5137 0.4045 −0:4045 −0:4334 0.4334 0.3334 −0:3334 0.7711 0.7711 −0:2153 −0:2153 0.8320 0.8320 −0:2514 −0:2514 0.2514 0.2514 −0:8320 −0:8320 −0:7711 0.2153 0.2153 −0:7711 −0:4045 0.4045 0.5137 −0:5137 −0:3334 0.3334 0.4334 −0:4334 −0:5137 0.5137 0.4045 −0:4045 −0:4334 0.4334

0.0568 0.0638 0.0991 0.1195 0.0485 −0:0216 0.1195 −0:0485 −0:0216 0.0568 0.0991 −0:0638 0.0568 0.0991 0.0638 0.1195 −0:0216 0.0485 0.1195 −0:0216 −0:0485 0.0527 0.1060 −0:0535 0.0527 0.1060 0.0535 0.1211 −0:0257 0.0397 0.1211 −0:0257 −0:0397 0.1195 0.0216 −0:0485 0.1195 0.0216 0.0485 0.0568 −0:0991 0.0638 0.0568 −0:0991 −0:0638 0.1211 0.0257 −0:0397 0.1211 0.0257 0.0397 0.0527 −0:1060 0.0535 0.0527 −0:1060 −0:0535 0.1195 −0:0485 0.0216 0.1195 0.0485 0.0216 0.0568 0.0638 −0:0991 0.0568 −0:0638 −0:0991 0.1211 −0:0397 0.0257 0.1211 0.0397 0.0257 0.0527 0.0535 −0:1060 0.0527 −0:0535 −0:1060 −0:0535 0.0527 0.1060 0.0535 0.0527 0.1060 0.0397 0.1211 −0:0257 −0:0397 0.1211 −0:0257 0.0485 0.1195 −0:0216 0.0638 0.0568 0.0991 −0:0638 0.0568 0.0991 −0:0485 0.1195 −0:0216 0.0991 0.0568 −0:0638 0.0991 0.0568 0.0638 −0:0216 0.1195 0.0485 −0:0216 0.1195 −0:0485 0.1060 0.0527 −0:0535 0.1060 0.0527 0.0535 −0:0257 0.1211 0.0397 −0:0257 0.1211 −0:0397 0.0216 0.1195 −0:0485 0.0216 0.1195 0.0485 −0:0991 0.0568 0.0638 −0:0991 0.0568 −0:0638 0.0257 0.1211 −0:0397 0.0257 0.1211 0.0397 (Appendix continued on next page)

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Appendix (Continued) 

m0; 1

m0; 2

m0; 3

b0; 1

b0; 2

b0; 3

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104

−0:2514 −0:2514 0.4045 0.5137 −0:5137 −0:4045 −0:4334 0.4334 0.3334 −0:3334 −0:7711 0.2153 0.2153 −0:7711 0.2514 0.2514 −0:8320 −0:8320 0.5137 0.4045 −0:4045 −0:5137 −0:3334 0.3334 0.4334 −0:4334 0.4045 0.5137 −0:5137 −0:4045 −0:4334 0.4334 0.3334 −0:3334 −0:2153 0.7711 0.7711 −0:2153 0.8320 0.8320 −0:2514 −0:2514 −0:0008 −0:8138 −0:8589 −0:2779 0.0419 −0:8887 −0:9098 −0:2012

−0:9087 −0:9087 −0:8888 −0:3762 −0:3762 −0:8888 −0:3464 −0:3464 −0:9087 −0:9087 0.5137 0.4045 −0:4045 −0:5137 −0:3334 0.3334 0.4334 −0:4334 −0:7711 0.2153 0.2153 −0:7711 0.2514 0.2514 −0:8320 −0:8320 −0:2153 0.7711 0.7711 −0:2153 0.8320 0.8320 −0:2514 −0:2514 0.4045 0.5137 −0:5137 −0:4045 −0:4334 0.4334 0.3334 −0:3334 −0:8138 −0:0008 −0:2779 −0:8589 −0:8887 0.0419 −0:2012 −0:9098

0.3334 −0:3334 −0:2153 0.7711 0.7711 −0:2153 0.8320 0.8320 −0:2514 −0:2514 −0:3762 −0:8888 −0:8888 −0:3762 −0:9087 −0:9087 −0:3464 −0:3464 −0:3762 −0:8888 −0:8888 −0:3762 −0:9087 −0:9087 −0:3464 −0:3464 −0:8888 −0:3762 −0:3762 −0:8888 −0:3464 −0:3464 −0:9087 −0:9087 −0:8888 −0:3762 −0:3762 −0:8888 −0:3464 −0:3464 −0:9087 −0:9087 0.5811 0.5811 −0:4302 −0:4302 0.4565 0.4565 −0:3629 −0:3629

−0:1060 −0:1060 0.0638 0.0485 −0:0485 −0:0638 −0:0397 0.0397 0.0535 −0:0535 −0:0216 0.0991 0.0991 −0:0216 0.1060 0.1060 −0:0257 −0:0257 0.0485 0.0638 −0:0638 −0:0485 −0:0535 0.0535 0.0397 −0:0397 0.0638 0.0485 −0:0485 −0:0638 −0:0397 0.0397 0.0535 −0:0535 −0:0991 0.0216 0.0216 −0:0991 0.0257 0.0257 −0:1060 −0:1060 0.0934 0.0350 0.0052 0.0901 0.1008 0.0278 0.0010 0.0999

0.0527 0.0527 0.0568 0.1195 0.1195 0.0568 0.1211 0.1211 0.0527 0.0527 0.0485 0.0638 −0:0638 −0:0485 −0:0535 0.0535 0.0397 −0:0397 −0:0216 0.0991 0.0991 −0:0216 0.1060 0.1060 −0:0257 −0:0257 −0:0991 0.0216 0.0216 −0:0991 0.0257 0.0257 −0:1060 −0:1060 0.0638 0.0485 −0:0485 −0:0638 −0:0397 0.0397 0.0535 −0:0535 0.0350 0.0934 0.0901 0.0052 0.0278 0.1008 0.0999 0.0010

0.0535 −0:0535 −0:0991 0.0216 0.0216 −0:0991 0.0257 0.0257 −0:1060 −0:1060 0.1195 0.0568 0.0568 0.1195 0.0527 0.0527 0.1211 0.1211 0.1195 0.0568 0.0568 0.1195 0.0527 0.0527 0.1211 0.1211 0.0568 0.1195 0.1195 0.0568 0.1211 0.1211 0.0527 0.0527 0.0568 0.1195 0.1195 0.0568 0.1211 0.1211 0.0527 0.0527 0.0433 0.0433 −0:0607 −0:0607 0.0375 0.0375 −0:0485 −0:0485

L. Anand, M.E. Gurtin / J. Mech. Phys. Solids 51 (2003) 1015 – 1058

1055

Appendix (Continued) 

m0; 1

m0; 2

m0; 3

b0; 1

b0; 2

b0; 3

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153

−0:0008 −0:8138 −0:8589 −0:2779 0.0419 −0:8887 −0:9098 −0:2012 0.9098 −0:2012 −0:0419 −0:8887 0.8589 −0:2779 0.0008 −0:8138 −0:0419 −0:2012 0.9098 −0:8887 0.0008 −0:2779 0.8589 −0:8138 0.8589 −0:8138 0.0008 −0:2779 0.9098 −0:2012 −0:0419 −0:8887 −0:8589 −0:2779 −0:0008 −0:8138 −0:9098 −0:8887 0.0419 −0:2012 0.0419 −0:8887 −0:9098 −0:2012 −0:0008 −0:2779 −0:8589 −0:8138 0.9098

0.5811 0.5811 −0:4302 −0:4302 0.4565 0.4565 −0:3629 −0:3629 −0:2012 0.9098 −0:8887 −0:0419 −0:2779 0.8589 −0:8138 0.0008 0.4565 0.3629 −0:3629 −0:4565 0.5811 0.4302 −0:4302 −0:5811 −0:2779 0.0008 −0:8138 0.8589 −0:2012 0.9098 −0:8887 −0:0419 0.4302 0.4302 −0:5811 −0:5811 0.3629 −0:4565 −0:4565 0.3629 −0:8887 0.0419 −0:2012 −0:9098 −0:8138 −0:8589 −0:2779 −0:0008 0.3629

−0:8138 −0:0008 −0:2779 −0:8589 −0:8887 0.0419 −0:2012 −0:9098 −0:3629 0.3629 0.4565 −0:4565 −0:4302 0.4302 0.5811 −0:5811 −0:8887 0.9098 −0:2012 −0:0419 −0:8138 0.8589 −0:2779 0.0008 0.4302 0.5811 −0:5811 −0:4302 0.3629 −0:3629 −0:4565 0.4565 −0:2779 −0:8589 −0:8138 −0:0008 −0:2012 0.0419 −0:8887 −0:9098 −0:4565 −0:4565 0.3629 0.3629 −0:5811 0.4302 0.4302 −0:5811 −0:2012

0.0934 0.0433 0.0350 0.0350 0.0433 0.0934 0.0052 −0:0607 0.0901 0.0901 −0:0607 0.0052 0.1008 0.0375 0.0278 0.0278 0.0375 0.1008 0.0010 −0:0485 0.0999 0.0999 −0:0485 0.0010 −0:0010 0.0999 −0:0485 0.0999 −0:0010 0.0485 −0:1008 0.0278 0.0375 0.0278 −0:1008 −0:0375 −0:0052 0.0901 −0:0607 0.0901 −0:0052 0.0607 −0:0934 0.0350 0.0433 0.0350 −0:0934 −0:0433 −0:1008 0.0375 0.0278 0.0999 0.0485 −0:0010 −0:0010 −0:0485 0.0999 0.0278 −0:0375 −0:1008 −0:0934 0.0433 0.0350 0.0901 0.0607 −0:0052 −0:0052 −0:0607 0.0901 0.0350 −0:0433 −0:0934 −0:0052 0.0901 0.0607 0.0350 −0:0934 0.0433 −0:0934 0.0350 −0:0433 0.0901 −0:0052 −0:0607 −0:0010 0.0999 0.0485 0.0999 −0:0010 −0:0485 −0:1008 0.0278 −0:0375 0.0278 −0:1008 0.0375 0.0052 0.0607 0.0901 0.0901 0.0607 0.0052 0.0934 −0:0433 0.0350 0.0350 −0:0433 0.0934 0.0010 0.0485 0.0999 0.0278 −0:0375 0.1008 0.1008 −0:0375 0.0278 0.0999 0.0485 0.0010 0.1008 0.0278 −0:0375 0.0278 0.1008 −0:0375 0.0010 0.0999 0.0485 0.0999 0.0010 0.0485 0.0934 0.0350 −0:0433 0.0901 0.0052 0.0607 0.0052 0.0901 0.0607 0.0350 0.0934 −0:0433 −0:0010 0.0485 0.0999 (Appendix continued on next page)

1056

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Appendix (Continued) 

m0; 1

m0; 2

m0; 3

b0; 1

b0; 2

b0; 3

154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192

−0:2012 −0:0419 −0:8887 0.8589 −0:8138 0.0008 −0:2779 0.5811 0.5811 −0:4302 −0:4302 0.4565 0.4565 −0:3629 −0:3629 0.4565 0.3629 −0:3629 −0:4565 0.5811 0.4302 −0:4302 −0:5811 0.3629 0.3629 −0:4565 −0:4565 0.4302 0.4302 −0:5811 −0:5811 0.4302 0.5811 −0:5811 −0:4302 0.3629 0.4565 −0:4565 −0:3629

−0:3629 −0:4565 0.4565 0.4302 0.5811 −0:5811 −0:4302 −0:0008 −0:8138 −0:8589 −0:2779 0.0419 −0:8887 −0:9098 −0:2012 −0:0419 −0:2012 0.9098 −0:8887 0.0008 −0:2779 0.8589 −0:8138 −0:9098 −0:2012 0.0419 −0:8887 −0:8589 −0:2779 −0:0008 −0:8138 0.8589 −0:8138 0.0008 −0:2779 0.9098 −0:8887 −0:0419 −0:2012

0.9098 −0:8887 −0:0419 −0:2779 0.0008 −0:8138 0.8589 −0:8138 −0:0008 −0:2779 −0:8589 −0:8887 0.0419 −0:2012 −0:9098 −0:8887 0.9098 −0:2012 −0:0419 −0:8138 0.8589 −0:2779 0.0008 −0:2012 −0:9098 −0:8887 0.0419 −0:2779 −0:8589 −0:8138 −0:0008 −0:2779 0.0008 −0:8138 0.8589 −0:2012 −0:0419 −0:8887 0.9098

0.0999 −0:1008 0.0278 −0:0052 0.0350 −0:0934 0.0901 0.0433 0.0433 −0:0607 −0:0607 0.0375 0.0375 −0:0485 −0:0485 0.0375 0.0485 −0:0485 −0:0375 0.0433 0.0607 −0:0607 −0:0433 0.0485 0.0485 −0:0375 −0:0375 0.0607 0.0607 −0:0433 −0:0433 0.0607 0.0433 −0:0433 −0:0607 0.0485 0.0375 −0:0375 −0:0485

−0:0485 −0:0375 0.0375 0.0607 0.0433 −0:0433 −0:0607 0.0934 0.0350 0.0052 0.0901 0.1008 0.0278 0.0010 0.0999 −0:1008 0.0999 −0:0010 0.0278 −0:0934 0.0901 −0:0052 0.0350 0.0010 0.0999 0.1008 0.0278 0.0052 0.0901 0.0934 0.0350 −0:0052 0.0350 −0:0934 0.0901 −0:0010 0.0278 −0:1008 0.0999

−0:0010 0.0278 −0:1008 0.0901 −0:0934 0.0350 −0:0052 0.0350 0.0934 0.0901 0.0052 0.0278 0.1008 0.0999 0.0010 0.0278 −0:0010 0.0999 −0:1008 0.0350 −0:0052 0.0901 −0:0934 0.0999 0.0010 0.0278 0.1008 0.0901 0.0052 0.0350 0.0934 0.0901 −0:0934 0.0350 −0:0052 0.0999 −0:1008 0.0278 −0:0010

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