A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

Transcription

1 Journal of the Mechanics and Physics of Solids 55 (2007) A three-dimensional phenomenological model for martensite reorientation in shape memory alloys M. Panico, L.C. Brinson Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd., Evanston, IL , USA Received 21 April 2006; received in revised form 19 March 2007; accepted 25 March 2007 Abstract In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed. r 2007 Elsevier Ltd. All rights reserved. Keywords: Phase transformation; Shape memory alloys; Simulations; Constitutive behavior; Multiaxial nonproportional loading 1. Introduction Shape memory alloys (SMAs) are materials that are able to recover a large inelastic deformation (up to 10% in some cases) due to a thermomechanical diffusionless phase Corresponding author. Tel.: ; fax: address: cbrinson@northwestern.edu (L.C. Brinson) /$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi: /j.jmps

2 2492 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) transformation between two solid phases: the highly symmetric austenite (A) and the less ordered martensite (M) (Funakubo, 1987; Otsuka and Wayman, 1998). This unique property makes this class of materials very suitable for innovative applications in various fields, ranging from aeronautical to medical device industries. Austenite is a solid phase, usually characterized by a body centered cubic crystallographic structure, which transforms to martensite by means of a lattice shearing mechanism. From a macroscopic point of view the martensite phase can exist in two states: self-accommodated, or twinned, martensite (M t ) and oriented, or detwinned, martensite (M d )(Brinson, 1993). The twinned martensite is formed by simple cooling under no external loading constraints; then, typically 24 variants of equal volume fractions form in a self-accommodated fashion and no significant macroscopic strain is incurred. In contrast, oriented martensite is produced by an applied stress and, consequently, the martensitic variants are preferentially oriented by the direction of the external force. This oriented martensite causes a macroscopic strain and can be formed either from phase transformation of austenite under a mechanical loading or from reorientation of selfaccommodated martensite. At zero applied stress, forward (A-M) and reverse (M-A) transformations are delimited by four characteristic temperatures M f, M s, A s and A f that, for most alloys, are related such that M f om s oa s oa f. The thermomechanical phase transformation produces two unique effects: pseudoelasticity and the shape memory effect. These phenomena are schematically displayed in terms of uniaxial stress strain response in Fig. 1. At a temperature T4A f the material exhibits pseudoelasticity which results in a large non-linear inelastic strain recoverable upon unloading (see Fig. 1a). At a temperature ToA s the material presents the shape memory effect that produces a large permanent strain which may be recovered by heating the sample (see Fig. 1b). In order to determine which crystallographic states (A, M t, M d ) are stable for different conditions of uniaxial stress and temperature, kinetic phase diagrams can be constructed and a linear dependence of the critical transformation stress on temperature is generally observed (Bekker and Brinson, 1997; Tanaka et al., 1995; Tobushi et al., 1991). In the last decades, the interest in the mechanical behavior of SMA has rapidly grown with the increasing number of potential industrial applications. Several computational and experimental efforts have been made at different scales to gain further insight into the Fig. 1. Schematic of pseudoelasticity (a) and shape memory effect (b) for a SMA material.

3 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) processes that characterize the SMA complex material response. The proposed models can be broadly grouped into three categories: micro models, micro macro models and macro models according to the scale at which they attempt to describe the SMA mechanical behavior. Micro-models focus on the description of micro-scale features, such as nucleation, interface motion, twin growth, etc. (Abeyaratne and Knowles, 1990; Ball and James, 1987). They are very useful to understand the fundamental phenomena, but not easily applicable at the structural scale. Micro macro-models use micromechanics and macroscopic thermodynamics to formulate the constitutive equations (Fischer and Tanaka, 1992; Gao et al., 2000; Huang and Brinson, 1998; Huang et al., 2000; Lexcellent et al., 1996; Patoor et al., 1994; Sun and Hwang, 1993). These models appear to have good predictive capabilities and moreover, in some cases they were also able to successfully reproduce reorientation and detwinning of martensite variants (Fang et al., 1998; Gao and Brinson, 2002; Huang and Brinson, 1998; Marketz and Fischer, 1996; Thamburaja, 2005). However, they employ a large number of internal variables which makes computations time intensive, resulting in a difficult application for engineering purposes. Macro-models only use macroscopic quantities to describe the state of the system and they are usually based on phenomenological thermodynamics or direct experimental data fitting (Bekker and Brinson, 1997; Boyd and Lagoudas, 1994; Brinson, 1993; Leclercq and Lexcellent, 1996; Raniecki and Lexcellent, 1994; Raniecki et al., 1992; Tanaka, 1986). Therefore, this last class is the most efficient model for engineering applications since they allow fast computations, but they can only describe the global mechanical response while all the microscopic details are ignored. To date macroscopic phenomenological models have predominantly dealt with onedimensional SMA mechanical behavior. These models have been usually tested and validated in the case of one-dimensional or three-dimensional proportional loading histories, partially due to the lack of experimental data under general loading conditions. Recently, some experimental investigations under general loading conditions have been presented. These consist of combined tension torsion experiments on thin-walled tube specimens (Lim and McDowell, 1999; Sittner et al., 1995), biaxial tension tests (Vivet and Lexcellent, 1999) and biaxial compression tests (Bouvet et al., 2002). These experimental results clearly show the need of incorporating non-proportional multiaxial loading effects into models in order to obtain better predictions of the material response. In such cases, complex phenomena such as variant reorientation, variant coalescence and simultaneous forward and reverse transformation and their associated transformation strain need to be taken into account (Bouvet et al., 2004a). Boyd and Lagoudas (1996) were the first to develop a model which explicitly accounts for reorientation of martensite variants through an opportune strain term. However, their model is not thoroughly tested in the case of non-proportional loading histories and has not been directly compared to experimental results. Recently, some authors (Andra et al., 2001; Auricchio and Petrini, 2002; Juhasz et al., 2002; Souza et al., 1998) have proposed an interesting approach based on a tensor valued internal variable (the transformation strain tensor), which fully describes the state of the martensite structure. The evolution law for this tensorial internal variable is obtained by means of the thermodynamic framework of irreversible processes. These models appear to capture some of the features of the SMA response in a simple one-dimensional setting and in more complex multiaxial conditions. However, further investigations and comparisons with experimental results are still needed. Particularly, the reorientation process of martensite variants does not seem to be

4 2494 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) appropriately modeled due to the assumption of a single tensorial internal variable. Recently, Bouvet et al. (2004a) proposed a phenomenological model for pseudoelasticity of SMAs under multiaxial non-proportional loadings. Their model postulates the existence of two different yield surfaces for forward and reverse transformation and gives good qualitative predictions of the biaxial experiments performed in Bouvet et al. (2002). However, the finite element implementation of this model does not appear straightforward and the evolution of twinned martensite is not considered. The purpose of this work is to propose a phenomenological model which accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is cast within the framework of classical thermodynamics (Rice, 1971) and is able to account for the evolution of both twinned and detwinned martensite. This work is based on the unified model of Leclerq and Lexcellent (1996) and extends their formulation to more general loading conditions. Therefore, in the following a similar notation will be adopted. A distinctive feature of our model is the treatment of parent phase transformation and martensite variant reorientation as two different physical processes which are governed by distinct evolution laws. This paper is organized as follows. In Section 2 we present the model with its assumptions and we derive the governing constitutive equations. In Section 3 several numerical tests are performed for both one-dimensional and multiaxial loading conditions. Here, the model is also tested to reproduce multiaxial nonproportional experiments available in literature. Conclusions and possible future developments are discussed in Section Formulation 2.1. Constitutive assumptions As mentioned, the new SMA constitutive model is based on thermodynamics of irreversible processes; therefore, at each instant the thermodynamical state of a representative volume element is fully defined by a number of external (or control) and internal variables. Accordingly, by defining an appropriate potential it is possible to determine the thermodynamical forces conjugate to both the external and internal variables. Small deformations are assumed and stands for the classical linear strain tensor. Transformation and detwinning of martensite variants happen mainly through a shear process; therefore, a symmetric deviatoric tensor in, which is referred to as inelastic strain tensor, is chosen to represent the average effect of oriented martensite. Then, if tr denotes the trace of a tensor the following condition must be satisfied: tr in ¼ 0. (1) Since we are in the small strain regime the following additive decomposition between elastic and inelastic strains may be applied: ¼ e þ in. (2) We denote with z s and z T, respectively, the volume fraction of detwinned and twinned martensite. Then, the following constraints apply: 0pz s p1; 0pz T p1; 0pz ¼ z s þ z T p1. (3)

5 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Since the macroscopic strain is associated with volume fraction of oriented martensite, we adopt a form for z s similar to that in Andra et al. (2001): z s ¼ in pffiffiffiffiffiffiffi, (4) 3=2 g where g is the maximum uniaxial transformation strain. From Eq. (4) the increment of martensite volume fraction can be written as: _z s ¼ in : _ in pffiffiffiffiffiffiffi 3=2 g in. (5) The increment of in is subdivided into two contributions which, respectively, originate from transformation of the parent phase and reorientation of previously developed oriented martensite: _ in ¼ _ tr þ _ re. (6) We assume that the reorientation strain term does not contribute to increase the martensite volume fraction, but rather only produces a reorientation of the inelastic strain according to the local stress state. Therefore, the condition in : _ re ¼ 0 has to be met and an appropriate evolution law for re will be chosen to serve this purpose (see Section 2.4). Then, Eq. (5) becomes: _z s ¼ in : _ tr pffiffiffiffiffiffiffi 3=2 g in. (7) 2.2. Free energy We choose the elastic strain tensor e and the absolute temperature T as control variables, while z s and z T are the internal variables. The following classical expression for the Helmoltz free energy of the three phase system is adopted (Leclercq and Lexcellent, 1996): cð e ; T; z s ; z T Þ¼ 1 2r e : L : e þ u A 0 TZA 0 z TðDu 0 TDZ 0 Þ z s hdu 0 TDZ 0 iþc v ðt T 0 Þ T ln T þ Dc, ð8þ T 0 where r is the material density, L is the isotropic elasticity tensor which is assumed to be the same for all the phases, u A 0 and u M 0 are the specific free energies of austenite and martensite, Z A 0 and Z M 0 are the specific entropies of austenite and martensite, c v is the specific heat at constant volume, T 0 is the equilibrium temperature between parent and product phase, Dc is the so-called configurational energy originated by the phase mixture and the two energy and entropy differences Du 0 ¼ u A 0 um 0, DZ 0 ¼ Z A 0 ZM 0 have been adopted. u M 0 and Z M 0 are assumed to be the same for both detwinned and twinned martensite. Moreover, in the energy contribution deriving from z s we have introduced the MacCauley brackets which calculate the positive part of their argument, i.e. hxi ¼1=2ðx þjxjþ. This assumption allows the critical stress for nucleation of detwinned martensite to plateau at lower temperatures, as experimentally observed.

6 2496 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) The configurational energy is assumed to depend only on z s according to the following quadratic form: Dc ¼ 1 2 H sz 2 s, (9) where H s is a material parameter which governs the initial hardening during phase transformation Second law of thermodynamics In order to derive the thermodynamic dissipative forces we use the second law of thermodynamics which is written in the form of the Clausius Duhem inequality as: D p ¼ 1 r s : _ c _ Z _T 1 q grad TX0, (10) rt where s is the stress tensor, Z is the specific entropy, q is the heat flux vector and the dot indicates the derivative of a quantity with respect to time. From Fourier s law for heat conduction ðq ¼ k th grad TÞ, with kth a positive thermal isotropic conductivity parameter, we obtain: 1 q grad TX0. (11) rt Therefore, we are left with the following inequality for the mechanical dissipation potential to satisfy: rd p ¼ s : _ r c _ rz _TX0. (12) Using the definition of the free energy in Eq. (8) we can rewrite the Clausius Duhem inequality as: rd p ¼ r qc qt Z _T þ s r qc q e : _ e þ s : _ in r qc _z s r qc _z T X0. ð13þ qz s qz T This inequality must be satisfied for any _ e and _T, therefore we obtain the following state equations: s ¼ r qc q e ¼ L : e, (14) Z ¼ qc qt ¼ c v ln T þ Z A 0 T z hdu 0 TDZ TDZ 0 z s DZ 0 i 0 0 jdu 0 TDZ 0 j. (15) By introducing Eqs. (6) (7) into the inequality (13), we obtain: rd p ¼! s 0 r qc in pffiffiffiffiffiffiffi qz s 3=2 g in : _ tr þ s 0 : _ re r qc qz T _z T X0, (16) where the prime indicates the deviatoric part of a tensor. We have replaced the stress with its deviatoric part on account of Eq. (1). Then, the dissipation density can be

7 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) written as: rd p ¼ X tr : _ tr þ X re : _ re þ X T _z T X0. (17) In Eq. (17) we have introduced the three thermodynamical dissipative forces: in X tr ¼ s 0 r½htdz 0 Du 0 iþh s z s Š pffiffiffiffiffiffiffi 3=2 g in, (18) X re ¼ s 0, (19) X T ¼ rðtdz 0 Du 0 Þ. (20) 2.4. Evolution equations Now, we can complete the model by introducing evolution laws and limit functions for the internal variables. In order to satisfy the second law of thermodynamics, we assume the following evolution equations: _ tr ¼ _ l tr X tr, (21) _ re ¼ _ l re ^I : X re, (22) _z T ¼ l _ T X T, (23) where the l _ a (a ¼ tr, re, T) are positive Lagrange multipliers and the fourth order tensor ^I is defined as: ^I ¼ðI 0 n nþ, (24) with n ¼ in =jj in jj normalized inelastic strain and I 0 fourth order identity deviatoric tensor. ^I is the so-called projection tensor which projects tensors into their transverse parts and satisfies the following conditions (Belytschko et al., 2000): ^I : n ¼ 0; ^I p ¼ ^I 8p. (25) This tensor is positive semi-definite which ensures the positiveness of the dissipation potential for any increment of the internal variables. In (22), ^I then selects the part of the driving force (the deviatoric stress tensor) that is normal to the existing inelastic strain. This formulation is consistent with the idea that reorientation is produced by an orientation difference between the applied stress tensor and the inelastic strain. Moreover, the first of Eq. (25) satisfies the assumption we made in Eq. (7) that the reorientation strain term does not contribute to an increase of martensite volume fraction Update algorithm for inelastic strain For the internal variables tr and re, we assume the following limit functions: F tr ¼ Y tr ðz s Þ, (26) X tr F re ¼ 1 2 X re : ^I : X re Y re, (27)

8 2498 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) where Y re is a material parameter which determine the onset of the reorientation process while Y tr (z s ) is a function that governs the kinetics of the phase transformation. We assume: ( A f z s B f z s lnð1 z s ÞþC f _z s 40; Y tr ðz s Þ¼ A r ð1 z s Þ B r ð1 z s Þ lnðz s ÞþC r for (28) _z s o0; and the A f/r, B f/r, C f/r are coefficients that describe the kinetics of, respectively, forward and reverse transformation. The Lagrange multipliers and the limit functions satisfy the Kuhn Tucker conditions: F a p0; la _ F a ¼ 0; la _ X0; a ¼ tr; re: (29) Then, the time-discrete system of equations to update the inelastic strain has the following expression: s ¼ L : ð in Þ, X tr ¼ s 0 r½htdz 0 Du 0 iþh s z s Š pffiffiffiffiffiffiffi 3=2 g in, X re ¼ s 0, z s ¼ in pffiffiffiffiffiffiffi, 3=2 g in in ¼ in n þ Dl trx tr þ Dl re ^I : X re, (30) F tr ¼ Y tr ðz s Þp0, X tr Dl tr X0; Dl tr F tr ðx tr Þ¼0, F re ¼ 1 2 X re : ^I : X re Y re p0, Dl re X0; Dl re F re ðx re Þ¼0, where Dl tr ¼ l tr ðl tr Þ n, Dl re ¼ l re ðl re Þ n are the time-discrete multipliers. This system of equations can be solved by means of a Newton Raphson scheme. However, X tr is singular for in ¼ 0, that is in the absence of oriented martensite. In this case an asymptotic analysis allows determination of the governing equations for the martensite nucleation process (Helm and Haupt, 2003). For in ¼ 0, reorientation strain contributions are clearly absent since there is no product phase to be oriented. Following the work of Helm and Haupt, the system of equations for the nucleation case can be written as: s ¼ L : ð in Þ, z s ¼ in pffiffiffiffiffiffiffi, 3=2 g

9 in s 0 ¼ Dl tr, (31) s 0 F tr ¼ s 0 r p ffiffiffiffiffiffiffi ½hTDZ 0 Du 0 iþh s z s Š Y tr ðz s Þp0, 3=2 g Dl tr X0; Dl tr F tr ðx tr Þ¼0, and again the Newton Raphson method can be used to find a solution Twinned martensite evolution In order to handle the case of self-accommodated (twinned) martensite, we assume the following limit function: ( X T Y f T F T ¼ ðz TÞ _z T 40; X T Y r T ðz for (32) TÞ _z T o0; where the two functions Y f T ðz TÞ and Y r T ðz TÞ have the following expressions: Y f T ðz TÞ¼c f z T, (33) Y r T ðz TÞ¼Y r T0 þ s þ cr ð1 z T Þ, (34) p with s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 s 0 : s 0, the equivalent Mises stress. We can determine the evolution of the variable z T directly from the consistency conditions; for forward transformation: _F f T ¼ 0 ) _z T ¼ DZ 0 _T c f with _To0, (35) for reverse transformation: _F r T ¼ 0 ) _z T ¼ DZ 0 c r _T þ _ s c r with _z T o0, (36) however, in the current formulation of the model we choose to deal only with the case of pure thermal loading at constant stress; then, Eq. (36) becomes: _z T ¼ DZ 0 _T c r with _T40. (37) In this restricted case the increment of twinned martensite is decoupled from system (30) and can be directly computed from Eqs. (35) and (37). Therefore, after system (30) is solved and the new value of z s is computed, z T can be updated through the following conditions: 8 >< ðdz 0 =c f Þ _T _z T ¼ ðdz 0 =c r Þ _T >: _z s M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) if F f T ðt; ðz TÞ n ÞX0 F r T ðt; ðz TÞ n ÞX0 ðz T Þ n þ z s 41: and _To0; _T40; (38) The last of Eq. (38) constrains the total martensite volume fraction to be always less than one and by relating the increments of z s and z T allows the model to simulate creation of oriented martensite at the expense of self-accommodated martensite.

10 2500 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Parameter identification The model introduces the following 15 material parameters: g, Du 0, DZ 0, H s, r, A f, B f, C f, A r, B r, C r, Y re, c f, Y T0 r, c r plus the two elastic constants E and n which can be easily determined from a uniaxial tension test. Some of these parameters are related to the common quantities that characterize a uniaxial phase kinetics diagram. For example, at the onset of the thermal transformation between austenite and self-accommodated martensite, we have F f T ðz T ¼ 0; T ¼ M 0 s Þ¼0 ) M0 s ¼ Du 0, (39) DZ 0 where M 0 s is the martensite start temperature. At the end of the forward transformation we have F f T ðz T ¼ 1; T ¼ M 0 f Þ¼0 ) M0 f ¼ Du 0 cf (40) DZ 0 DZ 0 with M 0 f martensite finish temperature. In the case of reverse thermal transformation and a free stress state, we have: F r T ðz T ¼ 1; T ¼ A 0 s Þ¼0 ) A0 s ¼ Du 0 þ Y r T0, (41) DZ 0 rdz 0 where A 0 s is the austenite start temperature. At the end of the reverse transformation we have: F r T ðz T ¼ 0; T ¼ A 0 f Þ¼0 ) A0 f ¼ Du 0 þ Y r T0 þ cr (42) DZ 0 rdz 0 DZ 0 with A 0 f austenite finish temperature. In the case of an applied stress: A s ¼ A 0 s þ s and A f ¼ A 0 f rdz þ s. (43) 0 rdz 0 Therefore, we obtain: C A ¼ rdz 0, (44) where C A represents the conventional slope of the transformation lines in the uniaxial phase kinetics diagram. By knowing the parameters Du 0 and DZ 0 through energetic and entropic considerations and the mass density r, Eqs. (40) (42) allow determination of c f, Y r T0, c r as a function of the alloy characteristic temperatures. g can be easily found from a uniaxial pseudoelastic test while H s and the kinetics parameters A f, B f, C f, A r, B r and C r are calibrated in order to reproduce the SMA hardening behavior during forward and reverse transformation. 3. Numerical results In this section we perform several uniaxial and multiaxial numerical tests under stress control to demonstrate the ability of our model to capture the main features of SMA mechanical behavior. In Section 3.4 a qualitative comparison between experimental results

11 and simulations is also performed. The experimental results are the combined tension torsion tests reported by Sittner et al. (1995) on polycrystalline Cu Al Zn Mn Uniaxial tests M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) For all the simulations in this and the next two sections the material is initially in parent phase (z s ¼ z T ¼ 0) and the numerical parameters listed in Table 1 have been adopted. These parameters have not been identified for a specific alloy, but their values are comparable to the ones adopted in similar models (Helm and Haupt, 2003; Leclercq and Lexcellent, 1996) and they correspond to the following characteristic temperatures M 0 f ¼ 306 K, M0 s ¼ 310 K, A0 s ¼ 317 K, A0 f ¼ 319 K. The first numerical test is a classical uniaxial loading simulation performed at different constant temperatures. In particular, the material is symmetrically loaded to 600 MPa in tension and compression for three test temperatures (310, 320 and 330 K), see Fig. 2. The model is able to reproduce the characteristic hysteresis loops of SMAs for both tension and compression. Furthermore, critical transformation stresses increase with temperature, as experimentally observed. However, the material behavior is symmetric for tension and compression. This is in contrast with several experimental results (Bouvet et al., 2004b; Gall et al., 1999; Liu et al., 1998; Vivet et al., 2001) which suggest strong asymmetry and a generalized J 2 J 3 criterion for the initial transformation surface. Possible modifications of the model to include asymmetric responses will be discussed in future work. Fig. 3 illustrates the results for a temperature-driven simulation at constant prescribed stress. The following three stresses have been considered: 300, 450 and 600 MPa, which cause the material to be initially in a fully oriented martensitic state (z s ¼ 1). Fig. 3 displays a temperature vs. strain plot for a thermal cycle where the material is subsequently heated and cooled between 300 and 400 K for the three different initial stresses. As expected, heating produces reverse transformation to occur and the recovery of the initial Table 1 Material parameters adopted in the numerical simulations Parameter Value Unit E 68,400 MPa n 0.36 g Du 0 14,725 J/kg DZ J/kg K H s 100 J/kg r 8000 Kg/m 3 A f 0 MPa B f 10 MPa C f 72.6 MPa A r 0 MPa B r 0 MPa C r 72.6 MPa Y re 50 MPa 2 c f 190 J/kg r Y T MPa c r 95 J/kg

12 2502 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 2. Predictions of the model for tension compression uniaxial loading at constant temperature, where the critical stress for onset of transformation is observed to increase with temperature. Fig. 3. Predictions of the model for temperature-driven transformation at constant applied stress, where the stress levels are such that each case begins with fully oriented martensite. Critical temperatures for transformation increase with load level. strain. On the other hand, on cooling the material transforms back to oriented martensite and the transformation strain is re-obtained. In accordance with experiments, the critical temperatures for forward and reverse transformations increase with the value of the applied uniaxial stress. These simulation results demonstrate the ability of the model to reproduce pseudoelasticity in the case of uniaxial loading conditions. The simulations in Fig. 2 also reproduce the so-called ferroelasticity which allows recovery of the inelastic strain by application of a stress of opposite sign. To conclude these baseline simulations, we consider a situation where the shape memory effect is observed. Fig. 4 describes the result

13 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 4. Numerical simulation of the shape memory effect. of this simulation in a strain stress temperature tri-dimensional space. The material is first loaded at a temperature of 310 K so that it fully transforms to oriented martensite. On unloading the temperature is such that transformation strain is not recovered. However, when the temperature is raised over A f 0 the inelastic strain disappears due to temperaturedriven reverse transformation (shape memory effect). The strain-free state of the system remains unaltered when the initial temperature of 310 K is re-established Multiaxial reorientation tests In order to test the behavior of the model in the case of multiaxial stress states we have performed numerical biaxial tests. The test temperature is 320 K for which a pseudoelastic behavior is expected. In this test the applied axial and shear stresses are moved in the range 7250 MPa with a square-shaped stress history (see Fig. 5a) producing full phase transformation to oriented martensite and subsequent reorientation of the product phase. Figs. 5b and c display the simulation results; in particular, Fig. 5b is a plot of the axial vs. shear strain while Fig. 5c shows the evolution of the inelastic strain components during the simulation. From these results we may notice the coupling of the axial and the shear inelastic responses which has been experimentally observed in the case of non-proportional loading histories. For example, between loading points B and C, the shear stress is increased while holding axial stress constant. During this time the shear strain increases, accompanied by a decrease in axial strain (Figs. 5b and c). Between loading points C and D, the axial stress is decreased through zero to compression; in particular note that during the unloading, the axial strain decreases substantially while the shear orientation strain increases opportunistically due to the maintained constant shear load. Moreover, the model is able to reproduce pseudoelasticity in a biaxial setting since the inelastic strains are completely recovered upon unloading. In order to highlight the importance of using proper modeling tools for SMA materials, in Fig. 5d we have compared the response of our SMA

14 2504 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 5. Biaxial numerical test with a square-shaped stress history at T ¼ 320 K. Stress path shown in (a) with axial vs shear strains in (b) and strain components vs time in (c). Axial and shear strains are illustrated again in (d) along with response of pure elastic and elasto-plastic materials to the same stress history. material model to the cases of a purely elastic material and an elasto-plastic material with kinematic hardening. The applied stress history is the same for the three cases and again is the one illustrated in Fig. 5a. The three material behaviors are completely different; in particular, the elastic response is purely affine to the stress loading path while the elastoplastic response presents a large residual strain upon unloading and substantially less axial shear coupling. A second numerical non-proportional test has been performed at a temperature of 310 K. In this case, the sample has been loaded uniaxially in a first direction (the 1- direction) so that the material fully transforms to detwinned martensite. Fig. 6a shows the stress-strain curve for this first loading step and, as expected, the temperature is such that the produced transformation strain cannot be recovered upon unloading. After this initial pre-straining we have loaded the material uniaxially in an orthogonal direction (the 2- direction) in order to test the complete reorientation process of the detwinned martensite between two perpendicular directions. Fig. 6b shows the stress strain curve for the second loading step; the martensite fully reorients in the 2-direction and the material behavior is

15 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 6. Simulation of the reorientation process of detwinned martensite between two perpendicular directions. Initial stress strain response in the 1-direction in (a), with the reorientation stress-strain response in the 2-direction in (b). The evolution of the inelastic strains over time is shown in (c). non-linear initially due to the initial pre-straining. Fig. 6c illustrates the evolution of the inelastic strains through these two subsequent loading steps. It may be noticed that the model is able to capture the exchange between the components in 11 and in 22 through the two loading phases, while the third component in 33 varies in such a way to keep the tensor in deviatoric Thermomechanical loading tests In this section we perform two numerical tests that illustrate the ability of the model to capture the evolution of self-accommodated (twinned) martensite. First, we want to simulate a detwinning process; therefore, the material is initially in austenitic state at a temperature of 315 K and the temperature is decreased to 300 K, under the martensite finish temperature, so that it fully transforms in twinned martensite (z T ¼ 1). Then, the sample is loaded uniaxially to a stress of 300 MPa in order to produce the reorientation of the martensite variants and the production of a transformation strain. Fig. 7a shows the

16 2506 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 7. Numerical simulation of the detwinning of twinned martensite: (a) loading path on the phase diagram; (b) volume fraction evolutions. Fig. 8. Numerical test for a thermomechanical loading condition: (a) loading path on the phase diagram; (b) volume fraction evolutions. simulation path in a stress temperature plane where the traditional transformation lines are also reported. Fig. 7b displays the evolution of the two volume fractions z s and z T with stress and temperature. When the critical transformation stress is reached detwinned martensite is generated at the expenses of the self-accommodated one. In a second simulation the material is again cooled from 315 to 300 K producing full transformation to twinned martensite. Then, the stress is increased to 50 Mpa and the temperature raised to 325 K in order to cross the reverse transformation strip, as illustrated in Fig. 8a. Fig. 8b shows the simulation results in terms of martensite volume fraction evolutions. It may be noticed that reverse transformation occurs approximately between A s ¼ 320 K and A f ¼ 322 K; therefore, the increase in the austenite start and finish temperature with stress is properly captured by the model.

17 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Comparison between experimental results and simulations In this section we seek to produce a qualitative comparison between the results of our simulations and experimental data available in the literature. For this purpose, we employ the tension torsion experimental results from Sittner et al. (1995). These authors tested polycrystalline Cu Al Zn Mn (Cu 10 wt%pct, Al 5 wt%pct, Zn 5 wt%pct, Mn 80 wt%pct) at room temperature (T ¼ 285 K ¼ A f +25 K) under pure tension, pure shear and a combination of the two types of loading. The tension torsion experiments were performed on thin-wall specimens in both force and strain control with proportional and non-proportional loading paths. For this simulation material parameters have been calibrated to match the uniaxial stress strain response for pure tension and simple shear. Simulations are performed at a temperature of T ¼ 285 K, in agreement with the experiments. Fig. 9a and Fig. 9b show a comparison between simulation results and experimental data, respectively, for simple tension and pure shear. It may be noticed that the model is able to simulate correctly the main characteristics of both uniaxial responses. A good qualitative agreement is obtained with the same parameters for both the tension and shear. After this first calibration step our objective is to simulate the combined non-proportional loading path which was experimentally realized in Sittner et al. (1995) and is graphically illustrated in Fig. 10a. First an axial stress of 240 MPa is applied to the material, then the shear stress is increased to approximately 200 MPa, while the tension is kept constant. After this, first tension and then shear loading are sequentially removed. Note that these magnitudes of stress applied individually do not lead to phase transformation (see Fig. 9), but in combination will result in a significant amount of transformation (z s ffi0.5). Fig. 10b d illustrates the simulation results, respectively, in terms of e g, s e and t g plots and produce a comparison with the corresponding experimental curves. A qualitatively good agreement between experiments and simulations is observed in which the model is able to reproduce the main characteristics of the experimental behavior. In particular, the magnitudes of the inelastic strains achieved during loading and the reversible nature of the Fig. 9. Predictions of the model compared to experimental results from Sittner et al. (1995) for uniaxial tension in (a) and simple shear in (b).

18 2508 M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) Fig. 10. Simulation of the non-proportional biaxial loading path in (a) and comparison with corresponding experimental results (Sittner et al., 1995): axial vs shear strains in (b); axial stress vs axial strain in (c); shear stress vs shear strain in (d). pseudoelastic strains for the applied non-proportional loading path are in agreement. In addition, tension and torsion are strongly coupled not only in the loading phase (when forward transformation occurs), but also during unloading as a result of the reverse transformation. Another feature we want to highlight is the different material response in the loading branch A B C for the simulation in Fig. 10 and the one displayed in Fig. 5. For the two cases the loading through stages A B C is qualitatively similar, but in the case of Fig. 5 the phase transformation is complete along the A B path and the subsequent B C loading segment is pure reorientation. Therefore, along B C the axial transformation strain must decrease in order to accommodate the increasing shear transformation strain. In contrast, in the simulation of Fig. 10 the material response is entirely elastic along A B (as it is evident from Fig. 10c) and transformation begins only at a point along the B C loading segment. Thus, the material response along B C represents an increase of both the axial and shear strain because transformation is generated in the direction of the applied stress which possesses both axial and shear components.

19 We can notice that some of the experimental features are not replicated by the model. For example, in Fig. 10b there is a discrepancy in the behavior among points C and D for the experiment and the simulation. In the model, as axial load begins to decrease, first reorientation occurs leading to an increase of shear strain at the expense of axial strain. In the experiment reverse transformation appears to begin nearly at the point C with no appreciable reorientation. We can also notice in Fig. 10d a crossing of the shear response which is not captured by the model. This crossing may be partially due to the small amount of plastic deformation developed in the experiment, observed by the residual strain upon unloading in the experimental stress strain curves (both Figs. 9 and 10). Irreversible plastic strain is not modeled in our current formulation. It is noted that irreversible plastic strain in the experiment could also lead to the lack of noticeable reorientation between points C and D. 4. Conclusions In this work we have proposed a three-dimensional macroscopic phenomenological model for SMAs with particular emphasis on modeling reorientation of the product phase according to loading direction. The reorientation process of detwinned martensite can be simulated in a simple approximate way which accounts for the effect of non-proportional loading paths on the material response. The model assumes distinct flow rules for the evolution of the transformation strain, which is related to the martensite volume fraction evolution, the reorientation strain, which instead simulates a change in the orientation of the inelastic strain tensor (switching among variants) and for the evolutions of selfaccommodated martensite. This feature is unique to our model with respect to previously proposed formulations (Auricchio and Petrini, 2002; Souza et al., 1998) and allows our simulations to correctly capture the physics of the complete phase diagram, including transformation, reorientation and self-accommodation. Several computational tests in both uniaxial and biaxial loading conditions have been performed and the model was able to reproduce the main aspects of the SMA mechanical behavior which have been experimentally observed. Experimental tests available in the literature (Sittner et al., 1995) have also been simulated and again the model was successful in capturing the main mechanical features. Based on the promising features of this model, further developments of our formulation are warranted. Further refinements suggested by experimental data include capturing the experimentally observed tension compression asymmetry (Bouvet et al., 2004b; Gall et al., 1999; Liu et al., 1998; Vivet et al., 2001); introducing the return point memory effect in order to simulate the internal loops experimentally manifested during cyclic loading (Huo and Muller, 1993; Ortin, 1991); and taking into account irreversible plastic strains generated by cyclic loading. These and other improvements will be considered in future work. In the scope of these current limitations, the proposed model appears as an effective and simple tool to predict SMA response and, implemented in a finite element scheme, it would represent a useful instrument for the design and realization of complex SMA devices. Acknowledgments M. Panico, L.C. Brinson / J. Mech. Phys. Solids 55 (2007) The authors acknowledge the financial support of the National Science Foundation through Grant number DMR and NASA through Grant number NCC / S4 and Los Alamos National Laboratory through grant number P.

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