Coupling between martensitic phase transformations and plasticity: A microstructure-based finite element model
Introduction
The coupling of phase transformations and plasticity is an important phenomenon that enables training and optimization of shape memory alloy performance. However, it can produce undesirable characteristics such as accumulation of remnant deformation, reduced work output, and early fatigue failure during repeated thermo-mechanical cycling. This coupling exists, in part, because both inelastic mechanisms – phase transformations and plasticity – alter the internal stress state and this in turn alters the driving force for each mechanism.
Numerous macroscopic observations qualitatively describe the nature of this coupling. In the pseudoelastic regime, macroscopic (∼cm or larger) single and polycrystalline samples subjected to simple tension or compression often exhibit remnant strain after unloading (Gall and Maier, 2002, Gall et al., 2002; Mckelvey and Ritchie, 2001, Miyazaki et al., 1981, Shaw and Kyriakides, 1995). TEM analysis shows dislocations and martensite (Gall and Maier, 2002). Subsequent heating to greater than austenite finish temperature (θAF) frequently reduces but does not eliminate the remnant strain. The reduction is attributed to the removal of stabilized martensite and the remaining portion is often attributed to plasticity. This portion typically increases with test temperature and also for crystal orientations and textures (e.g., 〈1 1 1〉 in compression) with a comparatively large resolved shear stress on austenite slip systems (Gall et al., 2002; Shaw and Kyriakides, 1995). During thermal cycling, the temperature required to transform martensite to austenite tends to increase with increasing biasing stress. This widens the strain vs. temperature hysteresis loop (Hamilton et al., 2004). The interpretation is that plastic strain accommodates and thus stabilizes the martensite phase.
Macroscopic observations also show a pronounced effect of pre-strain at elevated temperature (θ ≫ θAF), where plasticity in austenite dominates. In particular, compressive pre-straining of Ti–55.94wt%Ni increases the linearity and hardening in subsequent compressive stress–strain response at θ > θAF (Rathod et al., 2006). The interpretation is that pre-straining induces a distribution of internal stress that biases the macroscopic stress for transformation. Further, Miller and Lagoudas (2000) show that during thermal cycling of Ti–50at.%Ni under tensile stress, the cooling portion can induce a tensile vs. compressive strain, depending on the pre-strain history.
Microscopic observations reveal that plasticity and transformation are coupled over a range of length scales. On a polycrystalline scale, Gall et al. (2002) show that upon loading, the grain-to-grain variation in crystal orientation will favor plasticity in some grains and transformation in others. At even smaller length scales, dislocations have been observed at the austenite–martensite interface (Hamilton et al., 2004, Norfleet et al., 2009). In particular, Norfleet et al. (2009) show that during pseudoelastic loading, dislocations are formed by the local stress generated by martensite twin variants and further, these dislocations index to an austenite slip system. Austenite slip is also supported by Sehitoglu et al. (2001), who note that the yield strength in austenite is ∼40% smaller than that for martensite, for a solutionized Ti–51.5at.%Ni at room temperature.
Models that couple plasticity and phase transformations can be categorized by the nature of the coupling and the microstructural scale of the constitutive relations. More elementary models adopt a phenomenological description of the coupling. Some use non-crystallographic descriptions of the processes (e.g., Auricchio et al., 2007, Feng and Sun, 2007, Hartl and Lagoudas, 2008, Jemal et al., 2009, Lee et al., 2009, Taleb and Petit, 2006, Zaki and Moumni, 2007). For example, the amount of retained martensite after unloading is sometimes viewed as a phenomenological function of equivalent plastic strain (Hartl and Lagoudas, 2008). Others use a crystallographic description of transformation (Sun et al., 2008, Sun et al., 2009). Overall, such models tend to be computationally efficient but they are not robust, since the coupling function is limited to specific loading conditions, temperatures, or alloy compositions.
A fundamental advance is to couple these mechanisms via the stress redistribution caused by one mechanism on the other. Such models use a crystallographic description of transformation. Some use stress redistribution as the only coupling mechanism (Pan et al., 2007, Thamburaja, 2005, Wang et al., 2008) and others add phenomenological coupling functions to capture additional effects, such as the effect of dislocation structure on martensite formation (Fischer et al., 2000, Iwamoto, 2004, Kouznetsova and Geers, 2007, Levitas, 1997, Levitas et al., 1998, Levitas and Idesman, 2002).
These more fundamental approaches adopt a range of constitutive descriptions and scales. For example, (Levitas et al., 1998, Levitas and Idesman, 2002 study the propagation of one transformation front in an elasto-plastic medium, appropriate at smaller length scales. Other models (Fischer et al., 2000, Iwamoto, 2004, Kouznetsova and Geers, 2007, Kouznetsova and Geers, 2008, Pan et al., 2007, Sun et al., 2008, Sun et al., 2009, Thamburaja, 2005, Wang et al., 2008) track only the volume fraction of martensite habit plane/Bain strain variants, thus smearing out the detailed configuration. This scale is adopted by Thamburaja, 2005, Pan et al., 2007 to study how martensite detwinning and reorientation couples with plasticity in the martensite phase. The J2-von Mises description of plasticity used in these prior models is advanced by Wang et al. (2008). They adopt a crystal-based description of martensite plasticity, that becomes active only after the austenite–martensite transformation is complete.
Other advances in finite element modeling have pushed toward smaller scales in an effort to capture fundamental transformation processes, but they do not include plasticity. Thamburaja et al. (2009) employ spatial gradient terms in free energy to develop a non-local description of detwinning. A Landau-type description of free energy (Idesman et al., 2008, Levitas et al., 2010, Sun and He, 2008) has been implemented in FEM, and micromechanics models have been developed to track the growth and reorientation of the austenite–martensite interface (Levitas and Ozsoy, 2009).
The present paper is in the class that couples transformation and plasticity via the stress redistribution caused by one mechanism on the other. A rate-dependent crystal plasticity flow law from Peirce et al. (1982) is employed with a rate-independent, crystallographic law from Thamburaja and Anand (2001) for the forward and backward austenite–martensite transformation. The scale is smeared out so that the volume fractions of various martensite habit plates are predicted, but not their detailed spatial arrangement. This work is unique in that it adopts crystallographic descriptions of both processes and allows for their simultaneous operation. Also, it incorporates crystallographic slip systems in austenite rather than martensite, as supported by recent studies of solutionized Ti–50.7at.%Ni crystals following pseudoelastic loading (Norfleet et al., 2009). This combination of features is expected to better capture plastic strain evolution during pseudoelastic loading, including the grain-to-grain competition between transformation and plasticity in polycrystals, and the relatively fast kinetic rate for transformation vs. plasticity.
Section 2 to follow describes the constitutive relation that couples plasticity and transformation at a material point and Section 3 calibrates the model to experimental data for solutionized single crystals of Ti–50.9at.%Ni. Section 4 shows the predicted response of initially stress-free, randomly oriented polycrystals under isothermal tension-testing at θ > θAF and during thermal cycling with a bias stress. Section 5 shows the predicted effects of elevated temperature pre-straining on subsequent pseudoelastic and thermal cycling response. An assessment of the model predictions is provided in Section 6. Overall, the model is suitable at smaller imposed strains, where martensite detwinning is not expected to dominate.
Section snippets
Constitutive relation and finite element model
A goal of the present work is to implement a constitutive relation that relates the total deformation increment at a material point to the current microstructure, stress, and stress increment at that point. The deformation includes contributions from elasticity, thermal expansion, plasticity, and phase transformation from austenite to martensite or vice-versa. The microstructural information at a material point includes the crystallographic orientation of the parent austenite phase, the volume
Material parameters: calibration to single crystal data
Table 1 shows the adopted material parameters for solutionized Ti–50.9at.%Ni single crystals. They are partitioned into three groups.
Elastic–thermal properties Gel-thermal = {CA, CM, Ath-A, Ath-M} are taken from the literatures (Brill et al., 1991, Boyd and Lagoudas, 1996). The approximation CM = CA is made, based on recent first principle calculations that show the elastic moduli of the B2 and B19′ phases to be similar (Wagner and Windl, 2008). This is viewed as more realistic than the assumption CM ∼ ½
Response of an initially stress-free random polycrystal
The finite element model with parameters listed in Table 1 is used to predict the behavior of a solutionized Ti–50.9at.%Ni polycrystal with random grain orientations and an initially stress-free state. Inserting θT, λT, hin, hincom, and fc from Table 1 into Eq. (11) yields a stress-free austenite finish temperature θAF = 274 K and stress-free martensite finish temperature θMF = 229 K. This compares well with the respective experimental values, θAF = 280 K and θMF = 231 K (Gall et al., 2002). Two types of
Effect of pre-straining randomly oriented polycrystals
The predictions thus far show that an initial stress-free polycrystal can develop a residual stress state through stress or thermal cycling, provided plastic deformation is induced. Likewise, experiments show that stress or thermal cycling can alter shape memory response. Further, pre-straining a NiTi shape memory alloy at elevated temperature in the austenitic state is observed to harden and narrow hysteretic stress–strain loops during subsequent room temperature (θ > θAF) testing (Rathod et
Conclusions
A microstructure-based finite element approach is used to study the thermal–mechanical response of NiTi shape memory alloys. The constitutive relation and numerical method of solution are unique in that they couple thermal expansion, anisotropic elasticity, a rate-dependent crystal plasticity approach by Kalidindi et al. (1992), and a rate-independent B2-to-B19′ and B19′-to-B2 phase transformation formalism by Thamburaja and Anand (2001). An important ingredient is that the simultaneous
Acknowledgements
Support from National Aeronautics and Space Administration (Grant No. NNX08AB49A) and the Ohio Super Computer Center (Grant No. PAS676) are gratefully acknowledged. Discussions with R. Vaidyanathan concerning the effects of pre-straining, R. Noebe and S. Padula II concerning the coupling of plasticity and transformation in polycrystals, and S. Bechtel concerning the mechanics are also appreciated.
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