A Simplified Multivariant SMA Model Based on Invariant Plane ...

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will subject the material to multi-dimensional stresses, partial loading and unloading, reorientation of loading, as well as non-uniform temperature conditions. Thus an understanding of <strong>SMA</strong> response to these non-simple loading conditions is needed. In addition, better understanding of the directionality of <strong>SMA</strong> response in general 3-D stress states will open the doors to a new set of potential uses. There are several 3-D <strong>SMA</strong> models currently in the literature (Auricchio and Taylor, 1997; Boyd and Lagoudas, 1994; Boyd and Lagoudas, 1996; Graesser and Cozzarelli, 1994, Patoor, 1994 #26; Lagoudas et al., 1996; Lu and Weng, 1997; Sun and Hwang, 1993a; Sun and Hwang, 1993b). Predominantly, the existing continuum level models approach <strong>SMA</strong> constitutive response by making use of macroscale plasticity theory (often complete with backstresses, flow rules and yield functions). While it is appealing to utilize the vast plasticity model literature and methodology, this approach must be exercised with caution since the <strong>SMA</strong> deformation/stress response is based on different underlying mechanisms than irreversible plastic response. To most accurately represent <strong>SMA</strong> material behavior, the problem is best approached from the fundamental material response up, accounting appropriately for the mechanisms which impart the unique reversible, controllable nature to <strong>SMA</strong>s. Specifically, macroscopic response of <strong>SMA</strong> can be built up from deformation at the variant level for single crystal behavior and polycrystalline response can be based on individual grain (single crystal) response. In recent years, several researchers have proposed <strong>SMA</strong> models based on a micromechanics approach (Goo and Lexcellent, 1997; Huang and Brinson, 1998; Lu and Weng, 1997; Patoor et al., 1994). These models use thermodynamics laws to describe the transformation while using micromechanics to estimate the interaction energy due to the phase transformation. All are based upon transformation strains of individual variants, utilized in varying fashions in the micromechanics formulation. Most of the models handle one variant only and will therefore not allow for loading involving reorientation of martensitic variants. One exception is the <strong>Multivariant</strong> model by Huang and Brinson (Huang and Brinson, 1998) and a further developed version by Gao et al. (Gao et al., 2000), which not only accounts for all variants separately, but also includes the self- 2
accommodated grouping structure for the habit plane variants. Therefore, re-orientation, i.e. transformation from one martensite variant to another martensite variant, is accounted for. Although a wide array of thermomechanical response, including multiaxial loading, can be simulated with the original <strong>Multivariant</strong> <strong>Model</strong> and the results are in good qualitative agreement with the experimental data, a few difficulties in the <strong>Multivariant</strong> model such as high transformation stress and long calculation time prevent it from being practically used. Observation of both stress and temperature induced martensitic transformation reveal that martensite variants tend to form large plates, most of which have invariant plane interfaces with the austenite and reach the grain boundary. Hence the original formulation of the <strong>Multivariant</strong> model in which every variant group (inclusion) is embedded in the austenite phase with numerous other inclusions is not physically representative of material behavior and therefore predicts an interaction barrier to transformation that is too large. In this paper, a simplified <strong>Multivariant</strong> model is developed in which the energy contributed by the incompatibility of inclusion to the matrix is taken to be a small constant. Similar simplified models (with some variation) can be found in literature. However, no systematic study or comparison to experiments with a model such as described here has been performed. Patoor et al. (Patoor et al., 1993a) neglected the interaction between martensite plates and parent phase in the single crystal simulation. However, only forward transformations of orientation dependence etc. were simulated. In the polycrystal case, no intergranular effect was taken into account, and thus stair-like curves were obtained by averaging. Subsequent papers by Patoor et al. (Patoor et al., 1993b; Patoor et al., 1994) use an interaction matrix to account for the interaction between plates and austenite phase in single crystal case, and a self-consistent modulus predicting scheme to obtain better averaging among grains in the polycrystal case. Other micromechanics models (Lexcellent et al., 1996; Lu and Weng, 1997) use different methods to account for the interaction energy, and these will be discussed in more detail in a later section. In many cases, quantitative correlation with experimental results has been difficult to obtain. 3
Page 1: A Simplified <stro
Page 5 and 6: last assumption will be revisited.
Page 7 and 8: many single crystal grains which wh
Page 9 and 10: 1.Experimental data indicate that s
Page 11 and 12: Shaw and Kyriakides, 1995; Shaw and
Page 13 and 14: the future. The same experiments al
Page 15 and 16: There are several notable features
Page 17 and 18: normal, invariant plane shear direc
Page 19 and 20: same change then removed a signific
Page 21 and 22: compromises in calculations of the
Page 23 and 24: Boyd, J.G. and D.C. Lagoudas. 1996.
Page 25 and 26: Otsuka, K. and C.M. Wayman, 1998. S
Page 27 and 28: Table 1. Interaction energy and for
Page 29 and 30: Table 3. Measured and predicted ang
Page 31 and 32: Stress σ 11 (x10 6 Pa) 600 500 400
Page 33 and 34: Stress Σ 11 (x10 6 Pa) 600 500 400
Page 35: Effective Trans. Stress (x10 6 Pa)

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