Abstract
For certain martensitic phase transformations, one observes a close relation between the width of the thermal hysteresis and the compatibility of two phases. This observation forms the basis of a theory of hysteresis that assigns an important role to the microstructures in the transition layer and their energetics (Zhang et al., Acta Mater 57(15), 4332–4352, 2009). We study microstructures for almost compatible phases in the context of nonlinear elasticity. Using a scalar valued ansatz we show that one expects a transition from uniform to branched patterns for various typical models of the surface energy. We subsequently consider needle-type transition layers and study quantitative differences between hard and soft austenite, and between twins of different martensitic variants.
Similar content being viewed by others
References
Adams R.A., Fournier J.J.F.: Sobolev Spaces. Academic Press, Oxford (2003)
Alberti, G.: Variational models for phase transitions, and approach via Γ-convergence. In: Calculus of Variations and Partial Differential Equations, pp. 95–114. Springer, Berlin (2000)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)
Balandraud X., Delpueyo D., Grédiac M., Zanzotto G.: Almost compatible microstructures in shape memory alloys. Acta Mat. 58(14), 4559–4577 (2010)
Ball J.M.: Mathematical models of martensitic microstructure. Mater. Sci. Eng. A 378, 61–69 (2004)
Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)
Ball J.M., James R.D.: Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Philos. Trans. R. Soc. Lond. A 338, 389–450 (1992)
Bennett C., Sharpley R.: Interpolation of Operators. Academic Press, New York (1988)
Bhattacharya K.: Microstructure of Martensite. Oxford University Press, Oxford (2004)
Bogdan, K., Dyda, B.: The best constant in a fractional Hardy inequality. Math. Nachr. 5–6, 629–638 (2008)
Boullay, Ph., Schryvers, D., Kohn, R.V.: Bending martensite needles in Ni65 Al35 investigated by two-dimensional elasticity and high-resolution transmission electron microscopy. Phys. Rev. B 64, 144105 (2001)
Braess, D.: Finite Elements. Theory, Fast Solvers and Applications in Solid Mechanics. Cambridge University Press, Cambridge (2001)
Braides, A.: Γ-convergence for beginners. Oxford University press, Oxford (2002)
Brenner S., Scott L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Chan, A.: Energieskalierung und Domänenverzweigung bei fest-fest Phasenübergängen mit SO(2)-Invarianz. Diploma thesis, Fachbereich Mathematik, Universität Duisburg-Essen (2007)
Chan, A.: Energieskalierung, Gebietsverzweigung und SO(2)-Invarianz in einem fest-fest Phasenübergangsproblem. PhD thesis, Bonn University (2013)
Chan, A., Conti, S.: Energy scaling and domain branching in solid–solid phase transitions. In: Singular Phenomena and Scaling in Mathematical Models, pp. 243–260. Springer International Publishing, New York (2014)
Chan, A., Conti, S.: Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance. arXiv:1403.6242 [math.AP]
Choksi R., Conti S., Kohn R.V., Otto F.: Ground state energy scaling laws during the onset and destruction of the intermediate state in a type I superconductor. Commun. Pure Appl. Math. 61(5), 595–626 (2008)
Choksi R., Kohn R.V., Otto F.: Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy. Commun. Math. Phys. 201(1), 61–79 (1998)
Chu, C., James, R.: Analysis of microstructures in Cu-14.0%Al-3.9%Ni by energy minimization. Journal de Physique IV 05, 143–149 (1995)
Conti S.: A lower bound for a variational model for pattern formation in shape-memory alloys. Cont. Mech. Thermodyn. 17(6), 469–476 (2006)
Conti, S.: Domain structures in solid–solid phase transitions. In: Alt, H.W., Luckhaus, S., Presutti, E., Salje, E.K.H. (eds.) MFO Report Phase Transitions 2007, pp. 1586–1588. EMS Publishing House (2007)
Cui, J., Chu, Y.S., Famodu, O.O., Furuya, Y., Hattrick-Simpers, J., James, R.D., Ludwig, A., Thienhaus, S., Wuttig, M., Zhang, Z., Takeuchi, I.: Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width. Nat. Mater. 5, 286–290 (2006)
Dal Maso, G.: An Introduction to Γ -Convergence. Birkhäuser, Boston (1993)
DeGiorgi E., Franzoni T.: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 58, 842–850 (1975)
Delville, R.: From functional properties to micro/nano-structures: a TEM study of TiNi(X) shape memory alloys. PhD thesis, Faculteit Wetenschappen, Universiteit Antwerpen (2010)
Delville, R., Kasinathan, S., Zhang, Z., Van Humbeeck, J., James, R.D., Schryvers, D.: Transmission electron microscopy study of phase compatibility in low hysteresis shape memory alloys. Philos. Mag. 90, 177–195 (2010)
Delville R., Schryvers D., Zhang Z., James R.D.: Transmission electron microscopy investigation of microstructures in low-hysteresis alloys with special lattice parameters. Scripta Materialia 60, 293–296 (2009)
Delville, R., Schryvers, D., Zhang, Z., Kasinathan, S., James, R.D.: TEM investigation of microstructures in low-hysteresis Ti50Ni50-x Pd x alloys with special lattice parameters. In: Richter, S., Schwedt, A. (eds.) EMC 2008 14th European Microscopy Congress, Aachen, pp. 413–414. Springer, Berlin (2008)
Delville, R., Shi, H., James, R.D., Schryvers, D.: Special microstructures and twin features in Ti50Ni50-x (Pd,Au) x . Solid State Phenom. 172–174, 105–110 (2011)
Diermeier, J.: Nichtkonvexe Variationsprobleme und Mikrostrukturen, Bachelor’s thesis, Universität Bonn (2010)
Dolzmann G.: Variational Methods for Crystalline Microstructure—Analysis and Computation. Springer, Berlin (2003)
Duerig, T.W., Pelton, A.R.: Ti-Ni shape memory alloys. In: Material Properties Handbook: Titanium Alloys, pp. 1035–1048. ASM (1994)
Duvant G., Lions J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
Dyda B.: The fractional Hardy inequality with a remainder term. Colloq. Math. 122, 59–67 (2011)
Dyda N.: A fractional order Hardy inequality. Ill. J. Math. 48(2), 575–588 (1996)
Ericksen J.L.: Some phase transitions in crystals. Arch. Ration. Mech. Anal. 73, 99–124 (1980)
Frank, R.L., Seiringer, R.: Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008)
Frank, R.L., Seiringer, R.: Sharp fractional Hardy inequalities in half-spaces. In: Rozhkovskaya, T., Laptev, A. (eds.) Around the Research of Vladimir Maz’ya I, International Mathematical Series, vol. 11, pp. 161–167. Springer, New York (2010)
Friesecke G., James R.D., Müller S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)
Grossmann, C., Frenzel, J., Sampath, V., Depka, T., Eggeler, G.: Elementary transformation and deformation processes and the cyclic stability of niti and NiTiCu shape memory spring actuators. Metall. Mater. Trans. A 40, 2530–2544 (2009)
Heinig H.P., Kufner A., Persson L.-E.: On some fractional order Hardy inequalities. J. Inequal. Appl. 1, 25–46 (1997)
James, R.D., Zhang, Z.: A way to search for multiferroic materials with “unlikely” combinations of physical properties. In: Manosa, L., Planes, A., Saxena, A.B. (eds.) The Interplay of Magnetism and Structure in Functional Materials, vol. 79. Springer, Berlin (2005)
Kohn, R.V., Müller, S.: Branching of twins near an austenite-twinned martensite interface. Philos. Mag. A 66, 697–715 (1992)
Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. XLVII, 405–435 (1994)
Kufner A., Persson L.-E.: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003)
Lei C.H., Li L.J., Shu Y.C., Li J.Y.: Austenite-martensite interface in shape memory alloys. Appl. Phys. Lett. 96, 141910 (2010)
Leoni, G.: A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105. American Mathematical Society, Providence (2009)
Li B., Luskin M.: Theory and computation for the microstructure near the interface between twinned layers and a pure variant of martensite. Mater. Sci. Eng. A 273, 237–240 (1999)
Lions, J.-L., Magenes, E.: Problemes aux limites non homogenes et applications, vol. 1. Dunod, Paris (1968)
Loss, M., Sloane, C.: Hardy inequalities for fractional integrals on general domains. J. Funct. Anal. 259(6), 1369–1379 (2010)
Louie, M.W., Kislitsyn, M., Bhattacharya, K., Haile, S.: Phase transformation and hysteresis behavior in Cs1-x Rb x H2PO4. Solid State Ionics 181, 173–179 (2010)
Müller, S.: Variational methods for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), pp. 85–210. Springer, Berlin (1999)
Pitteri, M., Zanzotto, G.: Continuum Models for Phase Transitions and Twinning in Crystals. Chapman and Hall/CRC, London (2002)
Salje, E.K.H.: Phase transitions in Ferroelastic and Co-elastic Crystals: an Introduction for Mineralogists, Material Scientists and Physicists. Cambridge University Press, Cambridge (1990)
Schreiber, C.: Rapport de stage, d.e.a. Freiburg (1994)
Tarta L.: An introduction to Sobolev spaces and interpolation spaces. Springer, Berlin (2007)
Van Humbeeck J.: Shape Memory Alloys: A Material and a Technology. Adv. Eng. Mater. 3(11), 837–850 (2001)
Varadan, V.K., Vinoy, K.J., Gopalakrishnan, S.: Smart Material Systems and MEMS: Design and Development Methodologies. Wiley, Chichester (2006)
Wechsler M.S., Liebermann D.S., Reid T.A.: On the theory of the formation of martensite. J. Metals 197, 1503–1515 (1953)
Zhang, Z.: Special lattice parameters and the design of low hysteresis materials. PhD thesis, University of Minnesota (2007)
Zhang Z., James R.D., Müller S.: Energy barriers and hysteresis in martensitic phase transformations. Acta Mater. 57(15), 4332–4352 (2009)
Zwicknagl, B.: Mathematical analysis of microstructures and low hysteresis shape memory alloys. PhD thesis, University of Bonn (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Mielke
Rights and permissions
About this article
Cite this article
Zwicknagl, B. Microstructures in Low-Hysteresis Shape Memory Alloys: Scaling Regimes and Optimal Needle Shapes. Arch Rational Mech Anal 213, 355–421 (2014). https://doi.org/10.1007/s00205-014-0736-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-014-0736-y