Recent theoretical advances in the development of functional representations for the free energy of mixing associated with solid-solid phase transformations has made possible the development of a new class of constitutive models based directly on bounds to the relaxed free energy. The present work, which continues that of Govindjee et al. (2002) and Hall and Govindjee (2002), demonstrates the application of the theory to the simulation of the complex behavior exhibited by shape memory alloys. The first three case studies undertaken herein support the validity of the theoretical approach by analyzing subtle aspects of the physics associated with such phase transformations. The final simulation, a torsion tube under combined loading, provides impetus for future experimental and theoretical advancements.
Atkin, R. and Craine, R.1976. “Continuum Theories of Mixtures: Basic Theory and Historical Development,”The Quarterly Journal of Mechanics and Applied Mathematics, 29:209-244.
2.
Aubry, S. , Fago, M. and Ortiz, M.2002. “A Constrained Sequentiallamination Algorithm for the Simulation of Sub-grid Microstructure in Martensitic Materials,”Computer Methods in Applied Mechanics and Engineering (submitted).
3.
Ball, J.1977. “Convexity Conditions and Existence Theorems in Nonlinear Elasticity,”Archive for Rational Mechanics and Analysis, 63:337-403.
4.
Ball, J. and James, R.1987. “Fine Phase Mixtures as Minimizers of Energy,”Archive for Rational Mechanics and Analysis, 100:13-52.
5.
Ball, J. and James, R.1992. “Proposed Experimental Tests of a Theory of Fine Microstructure and the Two-well Problem,”Philosophical Transactions of the Royal Society of London, Series A338(1650):389-450.
6.
Bhattacharya, K. and Dolzmann, G.2000. “Relaxed Constitutive Equations for Phase Transforming Materials,”Journal of the Mechanics and Physics of Solids, 48:1493-1517.
7.
Bhattacharya, K. and Kohn, R.1996. “Symmetry, Texture, and the Recoverable Strain of Shape Memory Alloys,”Acta Metallurgica, 44:529-542.
8.
Chu, C. 1993. Hysteresis and Microstructures: A Study of Biaxial Loading on Compound Twins of Copper-Aluminum-Nickel Single Crystals, PhD Thesis, University of Minnesota.
9.
Also listed under: Applied Mathematical Sciences (Springer-Verlag, New York Inc.); Vol. 78.
10.
Gall, K. and Sehitoglu, H.1999. “The Role of Texture in Tension-Compression Asymmetry in Polycrystalline NiTi,”International Journal of Plasticity, 15:69-92.
11.
Goldsztein, G.2001. “The Effective Energy and Laminated Microstructures in Martensitic Phase Transformations,”Journal of the Mechanics and Physics of Solids, 49:899-925.
12.
Govindjee, S. and Miehe, C.2001. “A Multi-variant Martensitic Phase Transformation Mode: Formulation and Numerical Implementation,”Computer Methods in Applied Mechanics and Engineering, 191:215-238.
13.
Govindjee, S. , Mielke, A. and Hall, G.2002. “The Free-energy of Mixing for n-Variant Martensitic Phase Transformations using Quasi-Convex Analysis,”Journal of the Mechanics and Physics of Solids, 50:1897-1922.
14.
Gurtin, M.1983. “Two Phase Deformations of Elastic Solids,”Archive for Rational Mechanics and Analysis, 84(1):1-29.
15.
Hall, G. and Govindjee, S.2002. “Application of a Partially Relaxed Shape Memory Free Energy Function to Estimate the Phase Diagram and Predict Global Microstructure Evolution,”Journal of the Mechanics and Physics of Solids, 50:501-530.
16.
Hane, K. and Shield, T.1999. “Microstructure in the Cubic to Monoclinic Transition in Titanium-Nickel Shape Memory Alloys,”Acta Metallurgica, 47(9):2603-2617.
17.
Hosford, W.1993. The Mechanics of Crystals and Textured Polycrystals, Oxford University Press, New York.
18.
Ichinose, S. , Funatsu, Y. and Otsuka, K.1985. “Type II Deformation Twinning in γ-1′ Martensite in a Cu-Al-Ni Alloy,”Acta Metallurgica, 33(9):1613-1620.
19.
Inoue, H. , Miwa, N. and Inakazu, N.1996. “Texture and Shape Memory Strain in TiNi Alloy Sheets,”Acta Materialia, 44(12):4825-4834.
20.
James, R.1981. “Finite Deformation by Mechanical Twinning,”Archive for Rational Mechanics and Analysis, 77:143-176.
21.
Kohn, R.1991. “The Relaxation of a Double Well Energy,”Continuum Mechanics and Thermodynamics, 3:193-236.
22.
Novák, V. , Šittner, P., Vokoun, D. and Zárubova, N.1999. “On the Anisotropy of Martensitic Transformations in Cu-Based Alloys,”Materials Science and Engineering, A273-275:280-285.
23.
Otsuka, K. and Shimizu, K.1974. “Twinning in γ′ Cu-Al-Ni Martensite with CU-3Ti Type Ordered Structure,”Transactions of the Japan Institute of Metals, 15:109-113.
24.
Saburi, T. and Nenno, S. 1981. “The Shape Memory Effect and Related Phenomena,” In: Proceedings of an International Conference on Solid Solid Phase Transformations, pp. 1455-1479.
25.
Shield, T.1995. “Orientation Dependence of the Pseudoelastic Behavior of Single Crystals of Cu-Al-Ni in Tension,”Journal of the Mechanics and Physics of Solids, 43(6):869-895.
26.
Shu, Y. and Bhattacharya, K.1998. “The Influence of Texture on the Shape-Memory Effect in Polycrystals,”Acta Materialia, 46(15):5457-5473.
27.
Smyshlyaev, V. and Willis, J.1999. “On the Relaxation of a Three-well Energy,”Proceedings of the Royal Society of London, A455:779-814.
28.
Šittner, P. , Hara, Y. and Tokuda, M.1995. “Experimental Study on the Thermoelastic Martensitic Transformation in Shape Memory Alloy Polycrystal Induced by Combined External Forces,”Metallurgical and Materials Transactions, A26(A):2923-2935.
29.
Šittner, P. and Novák, V.2000. “Anisotropy of Martensitic Transformations in Modeling of Shape Memory Alloy Polycrystals,”International Journal of Plasticity, 16:143-1268.
30.
Šittner, P. , Novák, V., Zárubova, N. and Studnička, V.1999. “Stress State Effect on Martensitic Structures in Shape Memory Alloys,”Materials Science and Engineering, A273-275:370-374.
31.
Wechsler, M. , Lieberman, D. and Read, T.1953. “On the Theory of the Formation of Martensite,”Transactions AIMS Journal of Metals, 197:1503-1515.
32.
Wenk, H.-R.1985. Preferred Orientation in Deformed Metals and Rocks: An Introduction to Modern Texture Analysis, Academic Press, Inc., Orlando, FL.
33.
Zanzotto, G.1996. “The Cauchy-Born Hypothesis, Nonlinear Elasticity and Mechanical Twinning in Crystals,”Acta Crystallographica, A52:839-849.
