Journal of the Mechanics and Physics of Solids 49 (2001) 709 – 737
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Polycrystalline shape-memory materials: e'ect of crystallographic texture P. Thamburaja, L. Anand ∗ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 26 May 2000; received in revised form 11 September 2000
Abstract A crystal-mechanics-based constitutive model for polycrystalline shape-memory materials has been developed. The model has been implemented in a 2nite-element program. In our 2niteelement model of a polycrystal, each element represents one crystal, and a set of crystal orientations which approximate the initial crystallographic texture of the shape-memory alloy are assigned to the elements. The macroscopic stress–strain responses are calculated as volume averages over the entire aggregate. Pseudoelasticity experiments in tension, compression, and shear have been performed on an initially textured polycrystalline Ti–Ni alloy. In order to determine the material parameters for Ti–Ni, the stress–strain results from a 2nite-element calculation of a polycrystalline aggregate subjected to simple tension have been 2t to corresponding results obtained from the physical experiment. Using the material parameters so determined, the predicted pseudoelastic stress–strain curves for simple compression and thin-walled tubular torsion of the initially textured Ti–Ni are shown to be in good accord with the corresponding experiments. Our calculations also show that the crystallographic texture is the main cause for the observed tension–compression asymmetry in the pseudoelastic response of Ti–Ni. The predictive capability of the model for the variation of the pseudoelastic behavior with temperature is shown by comparing the calculated stress–strain response from the model against results from experiments of Shaw and Kyriakides (J. Mech. Phys. Solids 43 (1995) 1243) on Ti–Ni wires at a few di'erent temperatures. By performing numerical experiments, we show that our model is able to qualitatively capture the shape-memory e'ect by transformation. We have also evaluated the applicability of a simple Taylor-type model for shape-memory materials. Our calculations show that the Taylor model predicts the macroscopic pseudoelastic stress–strain curves in simple tension, simple compression and tubular torsion fairly well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory c 2001 Elsevier Science Ltd. All rights reserved. materials. Keywords: A. Phase transformation; B. Constitutive behavior; B. Crystal plasticity; C. Finite elements; C. Mechanical testing ∗
Corresponding author. Tel.: +1-617-253-1635; fax: +1-617-258-8742. E-mail address:
[email protected] (L. Anand).
c 2001 Elsevier Science Ltd. All rights reserved. 0022-5096/01/$ - see front matter PII: S 0 0 2 2 - 5 0 9 6 ( 0 0 ) 0 0 0 6 1 - 2
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1. Introduction Shape-memory alloys (SMAs) are 2nding increased use as functional/smart materials for a variety of applications. The individual grains in these polycrystalline materials can abruptly change their lattice structure in the presence of suitable thermo-mechanical loading. This capability of undergoing a solid–solid, di'usionless, displacive phase transformation leads to the technologically important properties of “pseudoelasticity” and “shape-memory.” Currently, the near-equiatomic Ti–Ni alloys are the most popular shape-memory materials for applications. The reversible transformations between the various phases observed upon changing the temperature, under zero stress, in the case of Ti–Ni are as follows. The transformation sequence upon cooling from a high-temperature body-centered-cubic superlattice, B2, austenite phase is 2rst to a rhombohedral R-phase, and then to a monoclinic martensite phase. Upon heating, the reverse transformations take place; however, the R-phase is not observed. Under certain conditions, the R-phase may be suppressed, and the only transformation in a specimen is the cubic-to-monoclinic transition. Under zero stress, shape-memory materials are distinguished by the following four temperatures: ms , martensite start temperature; mf , martensite 2nish temperature; as , austenite start temperature; and af , austenite 2nish temperature. An important characteristic response of shape-memory materials is superelasticity or pseudoelasticity by transformation. This is a consequence of a stress-induced transformation from austenite to martensite and back when a sample is tested in cyclic uniaxial extension, between zero and a 2nite but small (≈ 5%) strain, under quasi-static conditions at a constant ambient temperature above its austenite 2nish temperature, af . There is little or no permanent deformation experienced by the specimen in such a strain cycle; this gives an impression that the material has only undergone elastic deformation, and hence the term superelastic or pseudoelastic. However, there is hysteresis; the mechanism responsible for the hysteresis is the motion of sharp interfaces between the two material phases. For a given material, the size and other qualitative features of “Iag-type” hysteresis loops usually change with the loading rate and the temperature at which the test takes place. The shape-memory e2ect by transformation occurs when a material which is initially austenitic is 2rst tested in isothermal uniaxial extension at a temperature ms ¡ ¡ af . During forward loading, a transformation from austenite to martensite occurs, but upon reversal and unloading to zero stress, the transformation strain does not recover until the temperature is subsequently raised to ¿ af . If the temperature is lower than the martensite 2nish temperature, ¡ mf , the material is initially in the martensitic state. In this condition SMAs can also exhibit pseudoelastic and shape-memory e'ects, but the inelastic strain is caused by reorientation of the martensitic variants, and not by transformation (e.g. Saburi and Nenno, 1981; Miyazaki and Otsuka, 1989; Otsuka and Wayman, 1999). There is substantial activity worldwide to construct suitable constitutive models for shape-memory materials, and several existing one-dimensional constitutive models can capture the major response characteristics of SMAs reasonably well (e.g. Liang and
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Fig. 1. Experimental stress–strain curves for Ti–Ni at = 298 K in simple tension and simple compression. For compression the absolute values of stress and strain are plotted.
Rogers, 1990; Abeyaratne and Knowles, 1993; Ivshin and Pence, 1994; Bekker and Brinson, 1997). Prominent amongst the three-dimensional models are the ones proposed by Sun and Hwang (1993a,b), Boyd and Lagoudas (1994, 1996), Patoor et al. (1996), Auricchio and Taylor (1997), Lu and Weng (1998) and others, but most of these models have been shown to work best for uniaxial loading. It is diLcult to test the applicability of these models in real three-dimensional situations because there is a lack of pedigreed multi-axial experimental data, although some nice experiments have been recently reported by Lim and McDowell (1999). Indeed, the predictions from these models which have been calibrated from data for simple tension, have not even been veri2ed for the case of simple shear. Also, these models do not adequately capture the asymmetry observed between tension and compression pseudoelastic experiments, where it is found that at a given test temperature: (i) the stress level required to nucleate the martensitic phase from the parent austenitic phase is considerably higher in compression than in tension; (ii) the transformation strain measured in compression is smaller than that in tension; and (iii) the hysteresis loop generated in compression is wider (along the stress axis) than the hysteresis loop generated in tension. These major di'erences between the tension and compression response of a Ti–Ni alloy in pseudoelasticity experiments (to be described more fully later) are shown in Fig. 1. It is now well recognized that shape-memory materials derive their unusual and inherently nonlinear and anisotropic properties from the 2ne-scale rearrangements of phases, or “microstructures”, and that the strain produced by the pseudoelasticity e'ect depends on crystal orientations. Specially oriented single crystals of some
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shape-memory materials can produce sizeable strains (≈ 10%) due to phase transformations. In applications, shape-memory materials are typically polycrystalline in nature, and are usually processed by casting, followed by hot-working (drawing for rods and wires, and rolling for sheet) and suitable heat treatments. Polycrystalline SMAs so produced are usually strongly textured, and various researchers have recently emphasized that crystallographic texture is very important in determining the overall properties of SMAs (e.g. Inoue et al., 1996; Zhao et al., 1998; Shu and Bhattacharya, 1998; Gall and Sehitoglu, 1999). Shu and Bhattacharya (1998) have developed an analytical geometric model to estimate the e'ects of initial crystallographic texture on the amount of recoverable strains in SMAs. They show that texture is important in determining the amount of shape-memory e'ect in polycrystals. In particular, they qualitatively show that even though Ti–Ni and Cu–Zn–Al-based SMAs both undergo cubic–monoclinic transformations, and both have similar transformation strains at the single-crystal level, it is the di'erence in the crystallographic texture between the two polycrystalline SMAs in bulk sheet, rod, and wire forms which gives rise to a larger shape-memory e'ect in Ti–Ni. More recently, Gall and Sehitoglu (1999) have studied the stress–strain behavior of polycrystalline Ti–Ni under tension versus compression. They used a micro-mechanical model which incorporates single-crystal constitutive equations and experimentally measured polycrystalline texture into a “self-consistent” model for polycrystals (Patoor et al., 1995) to argue that the tension/compression asymmetry in Ti–Ni shape-memory alloys was related to the initial crystallographic texture of their specimens. 1 The purpose of this paper is to formulate and numerically implement a crystal-mechanics-based constitutive model for shape-memory materials, and to verify whether the model is able to capture the major features of the experimentally measured e'ects of crystallographic texture on pseudoelasticity of a polycrystalline Ti–Ni alloy. The plan of this paper is as follows: In Section 2, we formulate a rate-independent crystal-inelasticity-based constitutive model, where the inelastic deformations occur by phase transformations. We have implemented our constitutive model in the 2nite-element program ABAQUS/Explicit (1999); algorithmic details of the time-integration procedure used to implement the model in the 2nite-element code are given in Appendix A. This computational capability allows us to perform two types of 2nite-element calculations: (i) where a 2nite-element represents a single grain and the constitutive response is given through a single-crystal constitutive mode, and (ii) where a 2nite-element quadrature point represents a material point in a polycrystalline sample and the constitutive response is given through a Taylor-type polycrystal model.
1 We do not completely understand the model and computational procedure used by Gall and Sehitoglu (1999) because suLcient details are not provided by the authors. It appears that these authors consider a model in which the transformation rates are determinable only if the model contains an invertible “interaction energy matrix” (their Eq. (11)). They attribute the magnitude of the tension–compression asymmetry in the pseudoelasticity behavior of their Ti–Ni material to the magnitudes of the terms of the interaction moduli. In contrast, we show in this paper that the major features of the tension–compression asymmetry in textured drawn bars of Ti–Ni are captured even if the interaction matrix is set to zero.
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In Section 3 the results from experimental measurements of crystallographic texture of a Ti–Ni alloy are described. Material parameters appearing in the constitutive model for this alloy have been evaluated. The procedure to determine these parameters from di'erential scanning calorimetric (DSC) techniques, and a room temperature isothermal pseudoelasticity experiment in tension are also outlined in Section 3. We show that our model is able to reproduce the stress–strain curve of the initially textured Ti–Ni alloy in simple tension. Next, with the model so calibrated, we show that the predictions for the stress–strain curves from the model are in good agreement with pseudoelasticity experiments on the same pre-textured material in simple compression, and in thin-walled tubular torsion. To determine the degree to which the initial texture controls the macroscopic asymmetry in the pseudoelastic response in tension and compression, we have also compared the predicted stress–strain responses in simple tension and simple compression if the initial texture is taken to be random. Our calculations show that in this case the response in compression is very similar to that in tension. Accordingly, we conclude that crystallographic texture is the prime cause for the observed tension–compression asymmetry in shape-memory materials. In Section 4, we evaluate the applicability of a Taylor-type model for inelastic deformations by phase transformation. Our calculations also show that such a Taylor-type model is also able to predict reasonably well the macroscopic stress–strain curves. This result is consistent with the mathematical analyses of Bhattacharya and Kohn (1996,1997), who have shown that the Taylor model reasonably accurately predicts the transformation induced strain (the horizontal extent of the hysteresis loop) in polycrystals. In Section 5 we examine the applicability of our model to another Ti–Ni alloy for which Shaw and Kyriakides (1995) have conducted careful experiments at a variety of di'erent temperatures. Unfortunately, these authors do not report on the initial crystallographic texture of their material. However, since they conducted their experiments on drawn Ti–Ni wires, we assume that their material has a texture similar to our drawn rods. We estimate the constitutive parameters for their material from their DSC results, and their stress–strain results from a pseudoelastic tension test at representative temperature. We show that the predictions for the stress–strain response from our constitutive model are in good agreement with the results from their pseudoelastic experiments at a few temperatures, ¿ af . Shaw and Kyriakides (1995) also report on displacement-controlled experiments at temperatures in the range ms ¡ ¡ af . In these experiments the martensite that forms during forward deformation does not completely transform back to austenite upon reversing the deformation and decreasing the stress to zero. Although Shaw and Kyriakides did not subsequently increase the temperature at zero stress to ¿ as to show the shape-memory e'ect, we have numerically simulated such an experiment, and show that our model is able to also capture the shape-memory e'ect by transformation. We close in Section 6 with some 2nal remarks.
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2. Single-crystal constitutive model Although the underlying physics of inelastic deformation due to phase transformations is quite di'erent from that for dislocation-based plasticity of metals, we have developed a constitutive model 2 for deformation of a single crystal of a shape-memory alloy by austenite–martensite phase transformations, by modifying the widely used framework for crystal plasticity by crystallographic slip. The overall inelastic deformation of a crystal is always inhomogeneous at length scales associated with the “microstructures” accompanying the austenite–martensite phase transformations. Thus, for the continuum level of interest here, the inelastic deformation should be de2ned as an average over a volume element that must contain enough lenticular transformed regions to result in an acceptably smooth process. The such smallest volume element above which the inelastic response may be considered smooth, is labeled a representative-volume element. In our model we will not account for the 2ne spatial structure of phase transformation microstructures, and we shall take an entire single crystal as a representative-volume element (RVE). In particular, for later use, we denote the volume fraction of martensite corresponding to the ith austenite–martensite transformation system in a RVE i , with 0 ≤ i ≤ 1. The by i total volume fraction of martensite in a crystal, = i , must also lie in the range 0 ≤ ≤ 1. i A phase transformation from austenite to martensite occurs when ˙ ¿ 0; we call this the forward transformation. A transformation from martensite to austenite occurs i when ˙ ¡ 0; we call this the reverse transformation. Guided by the one-dimensional theory of Abeyaratne and Knowles (1993) for stressinduced phase transformations in elastic solids, we assume that the dissipation accompanying the phase transformation is given by i fi ˙ ≥ 0; (1) i
where fi is the driving force for transformation. Since i is dimensionless, this “force” is actually an energy per unit volume. This dissipation inequality is satis2ed if individually i
fi ˙ ≥ 0;
(2)
we shall assume that Eq. (2) always holds. In the rate-independent model that we develop below, transformation is assumed to be possible if the driving force fi reaches a critical value fci . The critical values fci , are taken to be positive-valued material parameters. The governing variables in the constitutive model are taken as: (i) The Cauchy stress, T. (ii) The deformation gradient, F. (iii) The absolute temperature, . (iv) Crystal transformation systems labeled by integers i. Each potential system is speci2ed by a unit normal m0i to the austenite-twinned martensite interface, the habit plane, and 2
This model is not intended to describe the situation when the material is initially in the martensitic state, and inelastic deformation is caused by reorientation of the martensitic variants.
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a unit vector b0i denoting the average “transformation” direction. In general, b0i is not perpendicular to m0i . The transformation systems {b0i ; m0i } are assumed to be known in the reference con2guration. The amount of transformation strain on each system, iT , is also assumed to be known. These can be calculated using the crystallographic theory of martensite (e.g. Hane and Shield, 1999; James and Hane, 2000). (v) A transformation deformation gradient, Ftr , with det Ftr ¿ 0. This represents the cumulative e'ect of austenite to martensite transformations on the active transformation systems in the crystal. (vi) The critical values for the driving force for transformation fci ¿ 0 on each system; units of energy per unit volume. (vii) The volume fraction of martensite i for each transformation system. −1 The elastic deformation gradient is de2ned by Fe ≡FFtr with det Fe ¿ 0. It describes the elastic distortion of the lattice; it is this distortion that gives rise to the stress T. The stress power per unit volume of the con2guration determined by Ftr is ˙ −1 ): !˙ = (det Fe )T · (FF This stress power may be additively decomposed as !˙ = !˙ e + !˙ p ; where e !˙ e = T∗ · E˙
is the elastic stress power, with Ee ≡ ( 12 ){FeT Fe − 1} and
T∗ ≡ (det Fe )Fe−1 TFe−T ;
(3)
the Green elastic strain measure and the symmetric second Piola–Kircho' stress tensor relative to the con2guration determined by Ftr . The quantity tr !˙ p = (Ce T∗ ) · (F˙ Ftr−1 );
Ce ≡ FeT Fe ;
(4)
is the inelastic stress power. 2.1. Constitutive equation for stress Elastic stretches in metallic single crystals are generally small. Accordingly, the constitutive equation for the stress is taken as the linear relation T∗ = C[Ee − A( − 0 )]
(5)
ˆ ), fourth-order anisotropic elasticity and second-order ˆ ) and A = A(; with C = C(; thermal expansion tensors. Ee and T∗ are the strain and stress measures de2ned in Eq. (3), and 0 a reference temperature. 2.2. Transformation conditions The functional dependence of the driving force for transformation on each transformation system may be generically written as i fi = fˆ ((Ce T∗ ); ; (b0i ; m0i ; iT ); j ):
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The one-dimensional considerations of Abeyaratne and Knowles (1993) and Knowles (1999), when suitably generalized to the present three-dimensional context, lead us to the following expression for the driving force: i i i ij j (6) h ; f = T − C( − T ) − j
where i = b0i · (Ce T∗ )m0i
(7)
is the resolved stress on the transformation system. Note that this is not the resolved shear stress in the classical Schmid sense since b0i is typically not perpendicular to m0i . Note also that unlike the situation for crystallographic slip, where one typically uses the absolute value of the resolved shear stress since slip can occur in either the positive or negative slip direction on a slip plane, here because the underlying atomic arrangements for phase transformation are polar in nature, the signed value of i appears in the driving force for transformation. The scalar parameter C with units of stress per unit change in temperature, controls the temperature dependence of the driving force, and the temperature T ≡ 12 (ms + as );
(8)
is called the phase equilibrium temperature in a stress-free state. ij j Following Patoor et al. (1995), we have formally introduced the term j h to account for interaction e'ects of various systems on the driving force for transformation; the coeLcients hij , with units of stress, are the interaction moduli. In the rate-independent model developed here, transformation is assumed to be possible if the driving force fi reaches a critical value fci . The critical values fci , are i taken to be positive valued material parameters. Since fi and ˙ must have the same sign according to the dissipation inequality (2), the transformation criteria are iam ≡ (fi − fci ) = 0
for austenite to martensite;
(9)
ima ≡ (fi + fci ) = 0
for martensite to austenite:
(10)
2.3. Flow rule The evolution for Ftr is tr F˙ = Ltr Ftr
(11)
with Ltr given by the sum of the transformation rates on all the systems i Ltr = ˙ iT b0i ⊗ m0i : i i The transformation rates ˙ are restricted as follows: i ˙ ≥ 0;
iam ≤ 0
and
i ˙ iam = 0
(12)
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for forward transformation, and i ˙ ≤ 0;
ima ≥ 0
and
i ˙ ima = 0
for reverse transformation. 2.4. Consistency conditions During forward transformations the following consistency conditions: i i ˙ ˙ am = 0 if iam = 0
(13)
must be satis2ed, and the conditions i i ˙ ˙ ma = 0
if ima = 0;
(14)
must be satis2ed during reverse transformations. The consistency conditions serve to i determine the transformation rates ˙ . Following the computational procedure developed by Anand and Kothari (1996) for rate-independent single-crystal plasticity, a time-integration procedure for the constitutive equations for SMA single crystals has been developed. The constitutive equations and the time-integration procedure have been implemented in the 2nite-element program ABAQUS/Explicit (Abaqus, 1999) by writing a “user material” subroutine. Algorithmic details of the time-integration procedure used to implement the model in the 2nite-element code are given in Appendix A.
3. Evaluation of the constitutive model for a polycrystalline Ti–Ni alloy Suitably thermo-mechanically processed and heat-treated Ti–55.96Ni(wt%) drawn rods of 12.70 mm diameter, intended for superelastic applications, were obtained from a commercial source. Experimental measurements of crystallographic texture of the as-received Ti–Ni were carried out by X-ray irradiation using a Rigaku RU 200 di'ractometer. Pole 2gures were obtained by using the Schultz reIection method with copper-K radiation. To process the experimental data, the preferred orientation package (Kallend et al., 1989) was employed. Each measured pole 2gure was corrected for back◦ ground and defocusing, and also extrapolated for the outer 15 . The {1 1 1}; {1 0 0}, and {1 1 0} pole 2gures for the as-received Ti–Ni are shown in Fig. 2. This 2gure also shows our numerical approximation of this measured texture using 729 and 343 unweighted grain orientations. The texture representation using 729 grain orientations is slightly better that that using 343 grain orientations. In most of the calculations reported below we shall use the 729 grain orientation representation of the initial texture. However, we note that the computed stress–strain curves using the set of 343 grain orientations are very close to those using the 729 grain orientations, and the smaller number of orientations may be used for reasons of computational eLciency.
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Fig. 2. (a) Experimentally measured texture in the as-received Ti–Ni rod, (b) its numerical representation using 729 unweighted discrete crystal orientations, and (c) its numerical representation using 343 unweighted discrete crystal orientations.
By using di'erential scanning calorimetric (DSC) techniques, we have determined the transformation temperatures for our Ti–Ni bar-stock, Fig. 3. They are ms = 251:3 K;
mf = 213:0 K;
as = 260:3 K;
af = 268:5 K:
(15)
The volume fraction and temperature dependence of the anisotropic elastic constants of Ti–Ni are not well documented in the literature. Here, for simplicity, the values of the elastic constants are taken to follow the rule of mixtures C(; ) = (1 − )Ca () + Cm (): The elastic constants for the cubic austenite phase in Ti–Ni at room temperature are taken as (Brill et al., 1991) a C11 = 130 GPa;
a C12 = 98 GPa;
a C44 = 34 GPa:
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Fig. 3. Di'erential scanning calorimetry thermogram for Ti–Ni used in experiments.
We have been unable to 2nd experimentally measured values of the anisotropic elastic moduli of single-crystal monoclinic martensite of Ti–Ni. Typically, the Young’s modulus of austenite is about two to three times that of martensite, while the Poisson’s ratios for the material in the two di'erent phases are approximately equal to each other. Guided by this information, we assume that anisotropic elastic constants of the monoclinic martensite may be approximately treated 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: m C11 = 65 GPa;
m C12 = 49 GPa;
m C44 = 17 GPa:
We realize that this description for the composite elastic constants is rather simpli2ed and approximate. However, since the main purpose of this paper is to model the inelastic response characteristics of shape-memory materials, we shall not further pursue the matter of a more re2ned description of the elastic moduli of a two-phase austenite– martensite mixture. For cubic materials, the thermal expansion tensor is isotropic, A=1, with denoting the coeLcient of thermal expansion. Here, the coeLcient of thermal expansion is taken to follow the rule of mixtures = (1 − )a + m with a and m for Ti–Ni in the austenitic and martensitic conditions taken as (Boyd and Lagoudas, 1996): a = 11 × 10−6 =K;
m = 6:6 × 10−6 =K:
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Table 1 Transformation systems i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
[moi ]1
[moi ]2
[moi ]3
[boi ]1
[boi ]2
[boi ]3
−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
0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4878 −0:4878 0.7576 0.7576 −0:7576 −0:7576 0.4878 −0:4878 0.7576 0.7576 0.4878 −0:4878 0.4878 −0:4878 −0:7576 −0:7576
−0:4878 0.4878 0.7576 0.7576 −0:7576 −0:7576 0.4878 −0:4878 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4878 −0:4878 0.7576 0.7576 −0:7576 −0:7576 0.4878 −0:4878
0.7576 0.7576 −0:4878 0.4878 0.4878 −0:4878 −0:7576 −0:7576 0.7576 0.7576 −0:4878 0.4878 0.4878 −0:4878 −0:7576 −0:7576 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343 0.4343
The crystallographic theory of martensitic transformation shows that phase transformation in Ti–Ni can occur on 192 possible transformation systems (Hane and Shield, 1999). However, it is not clear whether all the 192 possible systems are actually operative during thermo-mechanical loading. Here, we shall only consider the 24 transformation systems used by a variety of recent researchers (e.g. Matsumoto et al., 1987; Lu and Weng, 1998; Gall and Sehitoglu, 1999). The components of the 24 transformation systems (m0i ; b0i ) with respect to an orthonormal basis associated with the parent cubic austenite crystal lattice are given in Table 1. The transformation strains iT are the same for all these 24 transformation systems iT ≡ T = 0:1308:
(16)
Recall that the driving force on each transformation system is taken to be fi = iT i − C( − T ) − hij j : j
Using the measured phase transformation temperatures (15), the value for the phase equilibrium temperature, T , is given by T ≡ ( 12 ){ms + as } = 256 K: Much work needs to be done to elucidate the nature and magnitude of the interaction terms hij for Ti–Ni. As a 2rst approximation, in our calculations we ignore the con-
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Fig. 4. Schematic of an isothermal superelastic stress–strain response in simple tension.
tributions due to interactions between di'erent transformation systems to the driving force for transformation, and simply set hij = 0: Further we assume that the critical values of the driving force fci for the 24 systems are all equal and constant: fci = fc ≡ constant: In principle, the values of the stress–temperature coeLcient, C, and the critical value of the driving force, fc , should be determined from experiments performed on single crystals of Ti–Ni. However, such single crystals are diLcult to grow, and we did not have access to single crystals of this material. Instead, the values of C and fc are estimated from experiments on polycrystalline Ti–Ni as follows. Consider an idealized schematic stress–strain curve for a superelastic tensile test on a polycrystalline shape-memory material, Fig. 4. Let am denote the value of the stress at which martensite nucleates from austenite during the forward transformation, and let ma denote the stress level at which austenite nucleates from martensite during the reverse transformation. Following the one-dimensional analysis of Knowles (1999), we de2ne the Maxwell stress by 0 ≡ 12 (am + ma ); and take it to be given by 0 = C poly ( − T );
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where C poly is the one-dimensional polycrystalline counterpart of the parameter C. Thus, if one knows T from independent DSC measurements, and one estimates am and ma from a pseudoelastic tensile test at a temperature , then the macroscopic value of the stress–temperature coeLcient, C poly , for a polycrystalline material is easily estimated. The critical value for the macroscopic driving force, fcpoly , is then given by fcpoly = T (am − 0 ); where T is the value of the transformation strain in tension for a polycrystalline material, see Fig. 4. Estimates of the values of the corresponding quantities (C; fc ) at the single-crystal level are then obtained from C ≈ SF × C poly
and
fc ≈ fcpoly ;
where SF ¡ 1 is an average value of the Schmid factors 3 (≡ i =am ) of the active forward transforming systems. Finite-element calculations (to be discussed shortly) for simple tension on an aggregate of 729 di'erent grain orientations representing a polycrystalline material with the initial crystallographic texture, show that for our Ti–Ni alloy tested in tension, the average value of the Schmid factors is SF ≈ 0:39: Fig. 1 shows the stress–strain curve from a superelastic tension test performed on our initially textured Ti–Ni at 298 K. The experiment was conducted at a low constant true strain rate of 5 × 10−4 s−1 in order to ensure isothermal testing conditions. An extensometer was used to obtain the macroscopic strain in the gage section of the specimen. The experimental stress–strain curve shows some residual deformation after unloading. This is invariably observed in most experiments on commercial polycrystalline SMAs; seldom is there complete recovery. From this 2gure, am ≈ 470 MPa;
ma ≈ 170 MPa;
T ≈ 0:054:
Using, the value T = 256 K, estimated from our DSC experiments we obtain C poly = 7:6 MPa=K;
fcpoly = 8:2 MJ=m3
for the polycrystalline material. Hence, the estimates of the values of the corresponding quantities at the single-crystal level are C ≈ 3:0 MPa=K
and
fc ≈ 8:2 MJ=m3 :
In our 2nite-element model of a polycrystalline aggregate, each 2nite-element represents one crystal, and a set of crystal orientations which approximate the initial crystallographic texture of the shape-memory alloy are assigned to the elements. The macroscopic stress–strain responses are calculated as volume averages over the entire aggregate. Calculations for simple tension on an aggregate of 729 unweighted di'erent grain orientations representing a polycrystalline material with the initial crystallographic texture, Fig. 2, were carried out using these estimates for the material parameters. Fig. 5a shows the initial 2nite-element mesh. The 2nite-element mesh after 6% deforAgain, since b0i is typically not perpendicular to m0i , these are strictly not Schmid factors, but we shall use the terminology here for convenience. 3
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Fig. 5. (a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed mesh at a tensile strain of 6%. Contours of martensite volume fraction are shown. (c) Pseudoelastic stress–strain curve in tension. The experimental data from this test were used to estimate the constitutive parameters. The curve 2t using the full 2nite-element model of the polycrystal is also shown.
mation in tension is shown in Fig. 5b, together with the contours of the martensite volume fraction. The quality of the curve-2t is shown in Fig. 5c. The numerically computed stress–strain response is close to the experimentally observed one. Given the number of approximative assumptions used to arrive at this curve-2t, the result is very encouraging. To summarize, the single-crystal material parameters used to obtain this 2t for our polycrystalline Ti–Ni alloy are a a a Elastic moduli for austenitic: C11 = 130 GPa; C12 = 98 GPa; C44 = 34 GPa. m m m Elastic moduli for martensitic: C11 = 65 GPa; C12 = 49 GPa; C44 = 17 GPa. Coe6cients of thermal expansion: a = 11 × 10−6 =K; m = 6:6 × 10−6 =K.
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Transformation strain: T = 0:1308. Phase equilibrium temperature: T = 256 K. Stress–temperature coe6cient: C = 3:0 MPa=K. Critical driving force: fc = 8:2 MJ=m3 . Interaction moduli: hij = 0:0 MPa. Using these values of the material parameters and the numerical representation of the measured initial texture, Fig. 2, we have also carried out numerical simulations of simple compression and thin-walled tubular torsion, and compared the calculated stress–strain curves against corresponding physical experiments at 298 K. The initial 2nite-element mesh for the simple compression simulation is shown in Fig. 6a. The 2nite-element mesh after 5% compression is shown in Fig. 6b, along with the contours of the martensite volume fraction. The prediction of the stress–strain curve from the constitutive model is shown in Fig. 6c, where it is compared against corresponding experimental measurements. The experimentally measured pseudoelastic stress–strain response is well approximated by the predictions from the constitutive model. To demonstrate the numerically predicted tension–compression asymmetry, we plot the numerical stress–strain curves in tension and compression in Fig. 7a. On comparing the curves in this 2gure with the corresponding experimental curves, Fig. 1, we conclude that the constitutive model captures the following major features of the observed tension–compression asymmetry rather well: • The stress level required to nucleate the martensitic phase from the parent austenitic phase is higher in compression than in tension. • The transformation strain measured in compression is smaller than that in tension. • The hysteresis loop generated in compression is wider (along the stress axis) than the hysteresis loop generated in tension. Fig. 2 shows that the Ti–Ni bar has a strong {1 1 1} texture. Using our constitutive model and the estimated single-crystal material parameters, we have calculated the stress–strain response for a single crystal subjected to tension and compression along its [1 1 1]-direction. The calculated stress–strain response is plotted in Fig. 7b. Comparison of Fig. 7a with b clearly shows that the tension–compression asymmetry in the polycrystalline specimen has its origins in the crystallographic texture of the as-received Ti–Ni bar. To con2rm this conclusion we have repeated the tension and compression simulations for a polycrystalline specimen using a set of 729 crystal orientations representing a “random” texture, instead of the actual crystallographic texture in the Ti–Ni bar shown in Fig. 2. The {1 1 1} pole 2gure corresponding to this random initial texture is shown in Fig. 8a. All other material parameters used in these simulations were the same as those used in the previous calculations. The predicted tension and compression pseudoelastic stress–strain curves using the random texture are shown in Fig. 8b. This result shows that in comparison to the result from the calculation using the actual initial texture, Fig. 7a, there is not much asymmetry between the compression and tension curves. Also, the small asymmetry in the stress levels between tension and compression observed in the calculation using the random initial texture is in the reverse order. The
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Fig. 6. (a) Undeformed mesh of 729 ABAQUS C3D8R elements. (b) Deformed mesh at a compressive strain of 5%. Contours of martensite volume fraction are shown. (c) Pseudoelastic stress–strain curve in compression. The absolute values of stress and strain are plotted. The prediction from the full 2nite-element model for the polycrystal is also shown.
experiments, Fig. 1, and the numerical simulations using the actual initial texture for the rod, Fig. 7a, show that the compression curves are higher than those for tension, whereas the calculation using the random texture shows the reverse trend. Thus, we conclude that crystallographic texture is the prime cause for the observed tension– compression asymmetry in shape-memory alloys. Finally, the initial mesh used in the simulation for tubular torsion is shown in Fig. 9a. The deformed mesh after a shear strain of 9% is shown in Fig. 9b,
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P. Thamburaja, L. Anand / J. Mech. Phys. Solids 49 (2001) 709 – 737
Fig. 7. Comparison of the predicted response from tension and compression simulations to demonstrate the numerically predicted tension–compression asymmetry for (a) the polycrystal material and (b) single crystal oriented in the {111} direction.
together with the contours for the martensite volume fraction. The predicted nominal shear stress–strain curve is shown in Fig. 9c, where it is compared against the corresponding experimentally measured curve from a tubular torsion experiment performed on a thin-walled specimen under a low shearing strain rate of 1 × 10−4 s−1 , at 298 K. The prediction is in good accord with the experiment.
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Fig. 8. (a) {111} pole 2gure corresponding to a random initial texture. (b) Comparison of the predicted stress–strain response in tension and compression using a random initial texture.
4. Taylor model For polycrystalline materials, a widely used averaging scheme is based on the assumption that the local deformation in each crystal is homogeneous and identical to the macroscopic deformation gradient at the continuum material point (Taylor, 1938). The compatibility between crystals is automatically satis2ed in this approximation;
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Fig. 9. (a) Undeformed mesh of 720 ABAQUS C3D8R elements. (b) Deformed mesh at a shear strain of 9%. Contours of martensite volume fraction are shown. (c) Pseudoelastic stress–strain curve in torsion. The prediction from the full 2nite-element model for the polycrystal is also shown.
however, equlibrium holds only inside a crystal, but is violated across crystal boundaries. For such a model, with T(k) denoting the constant Cauchy stress in each crystal, the volume-averaged Cauchy stress is given by T=
N k=1
(k) T(k) ;
(17)
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Fig. 10. Comparison of the Taylor model against experiment and full 2nite-element calculation in (a) simple tension, (b) simple compression, and (c) shear.
where (k) is the volume fraction of each crystal in a representative volume element. When all crystals are assumed to be of equal volume, the stress T is just the number average over all the crystals: T=
N 1 (k) T : N
(18)
k=1
Taylor model simulations in simple tension, simple compression and simple shear were performed using a single ABAQUS-C3D8R element, and using 729 unweighted grain orientations to represent the initial rod-texture, Fig. 2. The material parameters used in the Taylor model simulations were the same as those calibrated for the full 2nite-element simulations of the polycrystal discussed in the previous section. Fig. 10 compares the stress–strain predictions from the Taylor model against the actual experiments, as well as the full 2nite-element calculations. We 2nd that although the Taylor model slightly overpredicts the stress–strain response relative to the full 2nite-element calculations for the three di'erent cases, it provides a reasonably
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accurate and computationally inexpensive method for determining the response of textured Ti–Ni in a multi-axial setting. 5. E&ect of temperature on the deformation of polycrystalline Ti–Ni In this section we examine the applicablity of our model to another Ti–Ni alloy for which Shaw and Kyriakides (1995) have conducted careful isothermal, low strain rate experiments at a variety of di'erent temperatures in the range 255:6 K ≤ ≤ 373:2 K. Unfortunately, these authors do not report on the initial crystallographic texture of their material. However, since they conducted their experiments on drawn Ti–Ni wires with a processing history similar to that of our own material, we assume that their material has a texture which may be approximated 4 by the numerical representations of the texture of our drawn rods, Fig. 2. We estimate the constitutive parameters for their material from their DSC results, and their stress–strain results from a pseudoleastic tension test at representative temperature. The DSC measurements of Shaw and Kyriakides (1995, Fig. 1) yield the following values for the transformation temperatures: ms = 272:2 K;
mf = 203:2 K;
as = 302:7 K;
af = 335:2 K:
(19)
The material parameters for their Ti–Ni wire were calibrated by 2tting the constitutive model to the pseudoelastic tension experiment conducted at 343:2 K using the methodology outlined in Section 3. The quality of the curve-2t is shown in Fig. 11b. The numerical calculations shown in this 2gure correspond to using the full 2nite element model with 729 elements representing 729 grain orientations, as well as a corresponding single-element Taylor model calculation. The full 2nite-element model of the polycrystal was used to estimate the material parameters; the numerically computed stress–strain response from this calculation is close to the experimentally observed one. The single-crystal material parameters used to obtain this 2t for the polycrystalline Ti–Ni wire of Shaw and Kyriakides are 5 a a a Elastic moduli for austenite: C11 = 130 GPa; C12 = 98 GPa; C44 = 34 GPa. m m m Elastic moduli for martensite: C11 = 65 GPa; C12 = 49 GPa; C44 = 17 GPa. Coe6cient of thermal expansion: a = 11 × 10−6 =K, m = 6:6 × 10−6 Transformation strain: T = 0:1308. Phase equilibrium temperature: T = 287 K. Stress–temperature coe6cient: C = 3:0 MPa=K. Critical driving force: fc = 7:7 MJ=m3 . Interaction moduli: hij = 0:0 MPa.
Figs. 11a and c show the response predicted from the constitutive model compared against the tension experiments at two di'erent temperatures: 333.2, and 353:2 K. 4 5
Wire texture is expected to show a sharper {1 1 1} component than the rod texture.
Except for T and fc which have slightly di'erent values, the values of the material parameters for the Ti–Ni of Shaw and Kyriakides are the same as those for our Ti–Ni alloy.
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Fig. 11. Pseudoelastic stress–strain curves in tension (Shaw and Kyriakides, 1995) at three temperatures (a) = 333:2 K, (b) = 343:2 K, and (c) = 353:2 K. Full 2nite-element and Taylor model prediction from the constitutive model are also shown. The material parameters for the Ti–Ni of Shaw and Kyriakides is obtained from the data at 343.2 K.
The experimentally measured temperature variation of the pseudoelastic stress–strain response is well predicted by the constitutive model. Shaw and Kyriakides (1995) also report on displacement-controlled experiments at temperatures in the range ms ¡ ¡ af . In these experiments the martensite that forms during forward deformation does not completely transform back to austenite upon reversing the deformation and decreasing the stress to zero. Although Shaw and Kyriakides did not subsequently increase the temperature at zero stress to show the shape-memory e'ect, we have numerically simulated such an experiment. In our simulation we employ the Taylor model using a single ABAQUS-C3D8R element with 343 grain orientations to represent the initial texture. The calculation was performed by 2rst imposing an isothermal, = 303:2 K, strain-controlled tension to 5% tensile strain, and then reversing the deformation to reach a state of zero stress. As shown in Fig. 12, the numerical prediction from the model for this part of the simulation is close to the experimental measurements of Shaw and Kyriakides. The temperature in the simulation was then linearly ramped up from 303.2 to 305:8 K by imposing the
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Fig. 12. Simulation of shape memory e'ect (Shaw and Kyriakides, 1995): Isothermal stress–strain response for straining at constant strain rate (ABC) at =303:2 K followed by a temperature ramp (CD) to =305:8 K.
temperature ramp on the nodes of the 2nite element. Fig. 12 shows that the model is able to capture the shape-memory e'ect by transformation; it predicts full recovery to the austenite phase after the temperature is increased to 305:8 K.
6. Concluding remarks The crystal-mechanics-based model for SMAs formulated in this paper is in its relatively early stages of development. It needs to be extended in a variety of di'erent ways, some of which are listed below: 1. The model needs to be extended to account for fully coupled thermo-mechanical situations, and also to account for the e'ects of the latent heat released=absorbed during phase transformations. 2. In the calculations reported in this paper, we have ignored the contributions due to interactions between di'erent transformation systems (the hij matrix) to the driving force for transformation. Much work needs to be done to elucidate the nature of these interactions and to determine the magnitudes of these interaction terms. 3. The present model is rate-independent in nature. It needs to be modi2ed to a rate-dependent form which uses a nucleation criterion, together with a kinetic equation which describes the possible time and temperature dependence of austenite/ martensite phase transformations during pseudoelasticity.
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4. The model is able to account for the phenomena of pseudoelasticity and shape-memory e'ects by phase transformations. It needs to be extended to represent these phenomena when the material is initially in the fully martensitic state, and when inelastic strains are caused by reorientation of the martensitic variants, and not by transformation. Acknowledgements The 2nancial support for this work was provided by the National Science Foundation under Grant CMS-9610130. The ABAQUS 2nite-element software was made available under an academic license from HKS, Inc. Pawtucket, R.I. Appendix A. Time-integration procedure In this appendix we summarize the time-integration procedure that we have used for our rate-independent single-crystal constitutive model. With t denoting the current time, Xt is an in2nitesimal time increment, and = t + Xt. The algorithm, which closely follows that developed by Anand and Kothari (1996) for rate-independent crystal plasticity, is as follows: Given: (1) {F(t); F( ); (t); ( )}; (2) {T(t); Ftr (t)}; (3) {b0i ; m0i ; iT ; fci }; (4) the accumulated martensite volume fractions i (t). Calculate: (a) {T( ); Ftr ( )}, and (b) the accumulated martensite volume fractions i ( ), and march forward in time. The steps used in the calculation procedure are Step 1. Calculate the trial elastic strain Ee ( )trial : Fe ( )trial = F( )Ftr (t)−1 ; Ce ( )trial = (Fe ( )trial )T Fe ( )trial ; Ee ( )trial = (1=2){Ce ( )trial − 1}: Step 2. Calculate the e'ective elastic modulus C(t) and thermal expansion A(t): C(t) = {1 − (t)}Ca + (t)Cm ; A(t) = {1 − (t)}Aa + (t)Am : Step 3. Calculate the trial stress T∗ ( )trial : T∗ ( )trial = C(t)[Ee ( )trial − A(t)(( ) − 0 )]: Step 4. Calculate the trial resolved stress i ( )trial on each transformation system. The resolved stress was de2ned as i ( ) = {Ce ( )T∗ ( )} · (b0i ⊗ m0i ). For in2nitesimal elastic : stretches the resolved stress i ( ) may be approximated at by i ( ) = T∗ ( ) · (b0i ⊗ m0i ). Accordingly, the trial resolved stress is calculated as i ( )trial = T∗ ( )trial · (b0i ⊗ m0i ):
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Step 5. Calculate the trial driving force for phase transformation fi ( )trial : i trial ij j i trial i f ( ) = T ( ) − C(( ) − T ) − h (t) (t) : j
Step 6. Determine the set PA of potentially active transformation systems which satisfy fi ( )trial − fci ¿ 0; 0 ≤ i (t) ¡ 1 and 0 ≤ i (t) ¡ 1 i
for the forward austenite to martensite transformation, and fi ( )trial + fci ¡ 0; 0 ¡ i (t) ≤ 1 and 0 ¡ i (t) ≤ 1 i
for the reverse martensite to austenite transformation. Step 7. Calculate Ftr ( ) = 1 + Xj jT (b0j ⊗ m0j ) Ftr (t); j = 1; : : : ; N;
(A.1)
j∈PA
where N is the total number of potentially transforming systems. Of the N potentially active systems in the set PA, only a subset A with elements M ≤ N , may actually be active (non-zero volume fraction increments). This set is determined in an iterative fashion described below. During phase transformation, the active systems must satisfy the consistency conditions fi ( ) ± fci = 0;
(A.2)
where the − sign holds during forward transformation and the + sign holds during reverse transformation, and where i i i ij j f ( ) = T ( ) − C(( ) − T ) − h (t) ( ) : (A.3) j 6
Retaining the terms of 2rst order in Xj , it is straightforward to show that i ( ) = i ( )trial − (b0i ⊗ m0i ) · C(t)[sym(Ce ( )trial (b0j ⊗ m0j ))]Xj jT : (A.4) j∈PA
Using Eqs. (A.3) and (A.4) in the consistency condition (A.2) gives Aij xj = bi ; i ∈ PA
(A.5)
j∈PA
with Aij = iT {hij (t) + jT (b0i ⊗ m0i ) · C(t)[sym(Ce ( )trial (b0j ⊗ m0j ))]}; 6
Terms such (Cm − Ca )
j∈PA
Xj C(t)
and
j∈PA (Am − Aa )
Xj and (Am − Aa )
j∈PA
Xj A(t)
Xj j∈PA j for |X | 1.
(A.6)
are neglected because (Cm − Ca )
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735
bi = fi ( )trial − fci ¿ 0;
and
xi ≡ Xi ¿ 0
for forward transformation; (A.7)
bi = fi ( )trial + fci ¡ 0;
and
xi ≡ Xi ¡ 0
for reverse transformation: (A.8)
Eq. (A.5) is a system of linear equations for the martensite volume fraction increments xj ≡ Xj . The following iterative procedure based on the singular value decomposition (SVD) of the matrix A is used to determine the active transformation systems and the corresponding martensite volume fraction increments (Anand and Kothari, 1996). Calculate x ≡ x+ = A+ b; where A+ is the pseudo-inverse matrix of the matrix A; if the matrix A is not singular, then the pseudo-inverse, A+ , is the true inverse, A−1 . If for any system the solution xj =Xj ≤ 0 when b j ¿ 0 (during forward transformation), then this system is inactive, and it is removed from the set of potentially active systems PA, and a new A matrix is calculated. Similarly, if xj = Xj ≥ 0 when b j ¡ 0 (during reverse transformation), then this system is also inactive, and it is also not included in the set PA used to determine the new A matrix. This iterative procedure is continued until all xj =Xj ¿ 0 for forward transformations, and xj = Xj ¡ 0 for reverse transformations. The 2nal size of the matrix A is M × M , where M is the number of active systems in the set A. Step 8. Update the inelastic deformation gradient Ftr ( ): Xj jT (b0j ⊗ m0j ) Ftr (t): (A.9) Ftr ( ) = 1 + j∈A
Step 9. Update the martensite volume fraction for each system j ( ) and the total martensite volume fraction for the single crystal ( ): j ( ) = j (t) + Xj ; j ( ): ( ) =
(A.10) (A.11)
j
If j ( ) ¿ 1, then set j ( )=1 and if j ( ) ¡ 0, then set j ( )=0. Similarly, if ( ) ¿ 1, then set ( ) = 1 and if ( ) ¡ 0, then set ( ) = 0. Step 10. Update the e'ective elastic modulus C( ) and thermal expansion A( ): C( ) = {1 − ( )}Ca + ( )Cm ; A( ) = {1 − ( )}Aa + ( )Am : Step 11. Compute the elastic deformation gradient Fe ( ) and the stress T∗ ( ): Fe ( ) = F( )Ftr ( )−1 ; T∗ ( ) = C( )[Ee ( ) − A( )(( ) − 0 )]: Step 12. Update the Cauchy stress T( ): T( ) = F e ( ){[det F e ( )]−1 T ∗ ( )}F e ( )T :
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