Abstract
A new version of rate-independent generalized plasticity, suitable for the derivation of general thermomechanical constitutive laws for materials undergoing phase transformations, is proposed within a finite deformation framework. More specifically, by assuming an additive decomposition of the finite strain tensor into elastic and inelastic (transformation induced) parts and by considering the fractions of the various material phases as internal variables, a multi-phase formulation of the theory is developed. The concepts presented are applied for the derivation of a three-dimensional thermomechanical model for shape memory alloy materials. The ability of the model in simulating several patterns of the extremely complex behavior of these materials, under both monotonic and cyclic loadings, is assessed by representative numerical examples.
Similar content being viewed by others
References
Abeyaratne R., Bhattacharya K., Knowles J.K.: Strain-energy functions with multiple local minima: modeling phase transformations using finite thermoelasticity. In: Fu, Y., Ogden, R.W. (eds) Nonlinear Elasticity: Theory and Applications, pp. 433–490. Cambridge University Press, Cambridge (2001)
Anand L., Gurtin M.E.: Thermal effects in the superelasticity of crystalline shape-memory materials. J. Mech. Phys. Solids 51, 1015–1058 (2003)
Arndt M., Griebel M., Novác V., Rubíc̆ek T., S̆ittner P.: Martensitic transformation in NiMnGa single crystals: Numerical simulation and experiments. Int. J. Plast. 22, 1943–1961 (2006)
Auricchio F., Taylor R.L., Lubliner J.: Shape-memory alloys: modeling and numerical simulations of the finite strain superelastic behavior. Comput. Methods Appl. Mech. Eng. 171, 387–418 (1997)
Ball J.M., James R.D.: Fine phase mixtures and minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)
Bertram A.: An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plast. 15, 353–374 (1998)
Bhattacharya, K.: Microstructure of Martensite. Why it Forms and How it Gives Rise to the Shape-Memory Effect? Oxford University Press, Oxford (2003)
Boyd J.G., Lagoudas D.C.: Thermomechanical response of shape memory alloy composites. J. Intell. Mater. Syst. Struct. 5, 333–346 (1994)
Brezis H.: On characterization of flow invariant sets. Commun. Pure Appl. Math. 23, 261–263 (1970)
Brinson L.C.: One-dimensional constitutive behavior of shape memory alloys: Thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J. Intell. Mater. Syst. Struct. 4, 229–242 (1993)
Casey J.: Approximate kinematical relations in finite plasticity. Int. J. Solids Struct. 21, 671–682 (1985)
Chan C.W., Chan S.H.J., Man H.C.: 1-D constitutive model for evolution of stress induced R-phase and localized Lüders-like stress-induced martensitic transformation of super-elastic NiTi wires. Int. J. Plast. 32(33), 85–105 (2012)
Ciarlet P.G.: Mathematical Elasticity, Volume 1: Three Dimensional Elasticity, Studies in Mathematics and Its Applications. North–Holland, Amsterdam (1985)
Coleman B.D., Gurtin M.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)
Fosdick R.L., Serrin J.: Global properties of continuum thermodynamic processes. Arch. Rat. Mech. Anal. 59, 97–109 (1975)
Freed Y., Banks–Sills L.: Crack growth resistance of shape memory alloys by means of a cohesive zone model. J. Mech. Phys. Solids 55, 2157–2180 (2007)
Freed, Y., Aboudi, J.: Micromechanical investigation of plasticity—damage coupling of concrete reinforced by shape memory alloy fibers. Smart Mater Struct. 17 art. no. 015046 (2008)
Freed Y., Banks–Sills L., Aboudi J.: On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium. J. Mech. Phys. Solids 56, 3003–3020 (2008)
Fried E., Gurtin M.E.: Dynamic solid–solid transitions with phase characterized by an order parameter. Phys. D. 72, 287–308 (1994)
Green A.E., Naghdi P.M.: A general theory of an elastic–plastic continuum. Arch. Rat. Mech. Anal. 18, 251–281 (1965)
Ivshin Y., Pence T.: A thermodynamical model for a one variant shape memory material. J. Intell. Mater. Syst. Struct. 5, 455–473 (1994)
James R.D., Hane K.F.: Martensitic transformations and shape-memory materials. Acta. Mater. 48, 197–222 (2000)
Jannetti, C., Bassani, J.L., Turteltaub, S.: Three-dimensional finite-element simulation of shape-memory alloys using a thermodynamically—based theory of martensite phase transformations. Seventh United States National Congress on Computational Mechanics (2003)
Kan Q., Kang G.: Constitutive model for triaxial transformation ratcheting Of super-elastic NiTi shape memory alloy at room temperature. Int. J. Plast. 26, 441–465 (2010)
Leclercq S., Lexcellent C.: A general macroscopic description of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 44, 953–980 (1996)
Levitas V.I., Preston D.L.: Thermomechanical lattice instability and phase field theory of martensitic phase transformations, twinning and dislocations at large strains. Phys. Lett. A. 343, 32–39 (2005)
Levitas V.I., Ozsoy I.B.: Micromechanical modeling of stress–induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation. Int. J. Plast. 25, 239–280 (2009)
Lexcellent C., Laydi R.M.: About the choice of a plastic-like model for shape memory alloys. Vietnam J. Mech. VAST 4, 283–291 (2011)
Liang C., Rogers C.A.: One-dimensional thermomechanical constitutive relations for shape memory materials. J. Intell. Mater. Syst. Struct. 1, 207–234 (1990)
Likhachev A.A., Koval Y.N.: On the differential equation describing the hysteresis behavior of shape memory alloys. Scr. Metal. Mater. 27, 223–227 (1992)
Lubliner J.: A simple theory of plasticity. Int. J. Solids Struct. 10, 313–319 (1974)
Lubliner J.: Normality rules in large-deformation plasticity. Mech. Matls 5, 29–34 (1986)
Lubliner, J.: Non-isothermal generalized plasticity. In: Bui, H.D., Nyugen, Q.S. (eds.) Thermomechanical Couplings in Solids, pp. 121–133 (1987)
Lubliner J., Panoskaltsis V.P.: The modified Kuhn model of linear viscoelasticity. Int. J. Solids Struct. 29(24), 3099–3112 (1992)
Lubliner J., Auricchio F.: Generalized plasticity and shape memory alloys. Int. J. Solids Struct. 33, 991–1004 (1996)
Magge C.L.: The Nucleation of Martensite, In Phase Transformations. American Society of Metals, Metals Park (1970)
Malukhin K., Ehmann K.: A model of the kinetics of the temperature—induced phase induced phase transformation in NiTi alloys and its experimental verification. J. Intell. Mater. Syst. Struct. 23, 35–44 (2012)
Martin R.H.: Differential equations on closed subsets of a Banach space. Trans. Am. Math. Soc. A. 251, 356–362 (1973)
Marketz F., Fischer F.D.: Modeling the mechanical behavior of shape memory alloys under variant coalescence. Comput. Mater. Sci. 5, 210–226 (1996)
Meyers A., Xiao H., Bruhns O.: Elastic stress ratcheting and corotational stress rates. Tech. Mech. 23, 92–102 (2003)
Mielke A., Rubíc̆ek T.: A rate-independent model for inelastic behavior of shape-memory alloys. Multiscale Model. Simul. 1, 571–597 (2003)
Müller Ch., Bruhns O.T.: A thermodynamic finite–strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22, 1658–1682 (2006)
Naghdi P.M.: A critical review of the state of finite plasticity. Z. Angew. Math. Phys. (J. Appl. Math. Phys. ZAMP) 41, 315–387 (1990)
Ortiz M.: Plastic yielding as a phase transition. ASME J. Appl. Mech. 66, 289–298 (1999)
Panoskaltsis, V.P.: Mechanics of shape memory alloy materials - Constitutive modeling and numerical implications, pp. 131–166. Intech Publications, (2013)
Panoskaltsis V.P., Bahuguna S., Soldatos D.: On the thermomechanical modeling of shape memory alloys. Int. J. Non Linear Mech. 39, 709–722 (2004)
Panoskaltsis V.P., Polymenakos L.C., Soldatos D.: On large deformation generalized plasticity. J. Mech. Mater. Struct. 3, 441–457 (2008)
Panoskaltsis V.P., Polymenakos L.C., Soldatos D.: Eulerian structure of generalized plasticity: theoretical and computational aspects. J. Eng. Mech. ASCE 134(5), 354–361 (2008)
Panoskaltsis, V.P., Soldatos, D., Triantafyllou, S.P.: Generalized plasticity theory for phase transformations. Procedia Engineering 10 (2011), 3104 – 3108, (Science Direct). 11th International conference on the mechanical behavior of materials, M. Guagliano ed., ICM 11, Milano, Italy, 5–9 June 2011
Panoskaltsis, V.P., Soldatos, D., Triantafyllou, S.P.: A new model for shape memory alloy materials under general states of deformation and temperature conditions. In: Boudouvis, A.G., Stavroulakis, G.E. (eds.) Proceedings of the 7th GRACM international congress on computational mechanics, Athens, Greece, 30 June-2 July 2011
Panoskaltsis, V.P., Polymenakos, L.C., Soldatos, D.: The concept of physical metric in the thermomechanical modeling of shape memory alloys. ASME J. Eng. Mater. Tech. 135(2) (2013)
Ramanathan, G.: Experimental, analytical and computational models for shape memory alloys. M.S. Thesis, Department of Civil Engineering, Case Western Reserve University, Cleveland OH, USA (2002)
Ramanathan, G., Panoskaltsis, V.P., Mullen, R., Welsch, G.: Experimental and computational methods for shape memory alloys. In: Smyth, A. (ed.) Proceedings of the 15th ASCE Engineering Mechanics Conference. Columbia University, NY, 2–5 June 2002
Redheffer R.M.: The theorems of Bony and Brezis on flow invariant sets. Am. Math. Mon. 79, 740–747 (1972)
Saint-Sulpice L., Arbab Chirani C., Calloch S.: A 3D super-elastic model for shape memory alloys taking into account progressive strain under cyclic loadings. Mech. Mater. 41, 12–26 (2009)
Savi M.A., Paiva A., Baĕta-Neves P.A., Pacheco P.M.C.L.: Phenomenological modeling and numerical simulation of shape memory alloys: A thermo-plastic-phase transformation coupled model. J. Intell. Mater. Syst. Struct. 19, 261–273 (2002)
Simo J.C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and multiplicative decomposition Part I: Continuum formulation. Comput Methods Appl. Mech. Eng. 66, 199–219 (1988)
Smallman R.E., Bishop R.J.: Modern Physical Metallurgy and Materials Engineering. Butterworth–Heinemann, Stoneham (2000)
Thamburaja P.: Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys. J. Mech. Phys. Solids 53, 825–856 (2005)
Thamburaja P.: A finite-deformation-based theory for shape-memory alloys. Int. J. Plast. 26, 1195–1219 (2010)
Thamburaja P., Anand L.: Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids 49, 709–737 (2000)
Valanis, K.C.: Irreversible thermodynamics of continuous media, internal variable theory, CISM courses and lectures No. 77, International centre for mechanical sciences. Springer, Wien (1972)
Videnic T., Kosel F., Sajn V., Brojan M.: Biaxial constrained recovery in shape memory alloy rings. J. Intell. Mater. Syst. Struct. 19, 861–874 (2008)
Wayman C.M.: Introduction to the Crystallography of Martensitic Transformation. Macmillan, New York (1964)
Wolf, M., Boettcher, S., Böhm, M.: Phase transformations in steel in the multi-phase case—general modeling and parameter identification. Report 07–02, Zentrum für Technomathematic, Universität Bremen (2007)
Wolf M., Böhm M., Helm D.: Material behavior of steel–Modeling of complex phenomena and thermodynamic consistency. Int. J. Plast. 24, 746–774 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Öchsner.
Rights and permissions
About this article
Cite this article
Panoskaltsis, V.P., Soldatos, D. & Triantafyllou, S.P. On phase transformations in shape memory alloy materials and large deformation generalized plasticity. Continuum Mech. Thermodyn. 26, 811–831 (2014). https://doi.org/10.1007/s00161-013-0312-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-013-0312-y