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A one dimensional variational model of superelasticity for shape memory alloys

Kim Pham

Abstract


In this paper we propose a variational framework for the modeling of superelasticity in shape memory alloys with softening behavior. This model is valid for a class of standard rate-independent materials with a single internal variable. The quasi-static evolution is based on two physical principles: a stability criterion which selects the local minima of the total energy and an energy balance condition which ensures the absolute continuity of the total energy. The stability criterion allows to bypass non-uniqueness issues associated to softening behaviour while the energy balance condition accounts for brutal evolutions at the local levels. We investigate properties of homogeneous and non-homogenous solutions towards this variational evolution problem. Specifically, we show how softening behaviour can lead to instability of the homogeneous states. In this latter case, we show that a stable solution would consist in following the Maxwell line given by the softening behaviour, then resulting in a non-homogeneous evolution.

Keywords


shape memory alloys (SMA); superelasticity; softening; one dimensional model; energetic approach; stability criterion

References


Y. Huo and I. Muller. Nonequilibrium thermodynamics of pseudoelasticity. Continuum Mechanics and Thermodynamics, 5, (3), (1993), pp. 163–204. doi:10.1007/bf01126524.

H. Tobushi, K. Tanaka, T. Hori, T. Sawada, and T. Hattori. Pseudoelasticity of TiNi shape memory alloy: dependence on maximum strain and temperature. JSME International Journal. Ser. A, Mechanics and Material Engineering, 36, (jul, 1993), pp. 314–318. doi:10.1299/jsmea1993.36.3 314.

J. A. Shaw and S. Kyriakides. Thermomechanical aspects of NiTi. Journal of the Mechanics and Physics of Solids, 43, (8), (1995), pp. 1243–1281. doi:10.1016/0022-5096(95)00024-d.

J. A. Shaw and S. Kyriakides. On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Materialia, 45, (2), (1997), pp. 683–700. doi:10.1016/s1359- 6454(96)00189-9.

Q. P. Sun, H. Zhao, R. Zhou, D. Saletti, and H. Yin. Recent advances in spatiotemporal evolution of thermomechanical fields during the solid–solid phase transition. Comptes Rendus Mecanique, 340, (4), (2012), pp. 349–358. doi:10.1016/j.crme.2012.02.017.

Q. P. Sun and K. C. Hwang. Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys–I. Derivation of general relations. Journal of the Mechanics and Physics of Solids, 41, (1), (1993), pp. 1–17. doi:10.1016/0022-5096(93)90060-s.

P. Thamburaja and L. Anand. Polycrystalline shape-memory materials: effect of crystallographic texture. Journal of the Mechanics and Physics of Solids, 49, (4), (2001), pp. 709–737. doi:10.1016/s0022-5096(00)00061-2.

F. Auricchio and E. Sacco. A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite. International Journal of Non-Linear Mechanics, 32, (6), (1997), pp. 1101–1114. doi:10.1016/s0020-7462(96)00130-8.

P. Popov and D. C. Lagoudas. A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. International Journal of Plasticity, 23, (10), (2007), pp. 1679–1720. doi:10.1016/j.ijplas.2007.03.011.

W. Zaki and Z. Moumni. A three-dimensional model of the thermomechanical behavior of shape memory alloys. Journal of the Mechanics and Physics of Solids, 55, (11), (2007), pp. 2455– 2490. doi:10.1016/j.jmps.2007.03.012.

P. Feng and Q.-P. Sun. Experimental investigation on macroscopic domain formation and evolution in polycrystalline niti microtubing under mechanical force. Journal of the Mechanics and Physics of Solids, 54, (8), (2006), pp. 1568–1603. doi:10.1016/j.jmps.2006.02.005.

J. F. Hallai and S. Kyriakides. Underlying material response for Luders like instabilities. International Journal of Plasticity, 47, (2013), pp. 1–12. doi:10.1016/j.ijplas.2012.12.002.

T. Shioya and J. Shioiri. Elastic-plastic analysis of the yield process in mild steel. Journal of the Mechanics and Physics of Solids, 24, (4), (1976), pp. 187–204. doi:10.1016/0022-5096(76)90002-8.

J. Mazars and Y. Berthaud. Une technique expérimentale appliquée au béton pour créer un endommagement diffus et mettre en évidence son caractère unilatéral. Comptes rendus de l’Académie des sciences. Série 2, Mécanique, Physique, Chimie, Sciences de l’univers, Sciences de la Terre, 308, (7), (1989), pp. 579–584.

Z. Song, H. H. Dai, and Q. P. Sun. Propagation stresses in phase transitions of an SMA wire: New analytical formulas based on an internal-variable model. International Journal of Plasticity, (2012). doi:10.1016/j.ijplas.2012.10.002.

J. Ericksen. Equilibrium of bars. Journal of Elasticity, 5, (3), (1975), pp. 191–201.

doi:10.1007/bf00126984.

A. Idesman, V. Levitas, D. Preston, and J.-Y. Cho. Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model. Journal of the Mechanics and Physics of Solids, 53, (3), (2005), pp. 495 – 523. doi:10.1016/j.jmps.2004.10.001.

Q. Sun and Y. He. A multiscale continuum model of the grain-size dependence of the stress hysteresis in shape memory alloy polycrystals. International Journal of Solids and Structures, 45, (13), (2008), pp. 3868–3896. doi:10.1016/j.ijsolstr.2007.12.008.

A. Mielke. Evolution of rate-independent systems. Handbook of differential equations: Evolutionary equations, 2, (2006), pp. 461–559.

B. Bourdin, G. A. Francfort, and J.-J. Marigo. The variational approach to fracture. Journal of Elasticity, 91, (1-3), (2008), pp. 5–148. doi:10.1007/978-1-4020-6395-4.

K. Pham and J.-J. Marigo. The variational approach to damage: I. The foundations. Academie des Sciences. Comptes Rendus. Mecanique, 338, (4), (2010), pp. 191–198.

G. Dal Maso, A. DeSimone, and M. G. Mora. Quasistatic evolution problems for linearly elastic-perfectly plastic materials. Archive for Rational Mechanics and Analysis, 180, (2), (2006), pp. 237–291. doi:10.1007/s00205-005-0407-0.

A. A. Leon Baldelli, C. Maurini, and K. Pham. A gradient approach for the macroscopic modeling of superelasticity in softening shape memory alloys. International Journal of Solids and Structures, 52, (January, 2015), pp. 45–55. doi:10.1016/j.ijsolstr.2014.09.009.

K. Pham. An energetic formulation of a one-dimensional model of superelastic SMA. Continuum Mechanics and Thermodynamics, (2014). doi:10.1007/s00161-014-0346-9.

B. Halphen and Q. S. Nguyen. Sur les matériaux standard généralisés. Journal de Mécanique, 14, (1975), pp. 39–63.

J.-J. Marigo. Constitutive relations in plasticity, damage and fracture mechanics based on a work property. Nuclear Engineering and Design, 114, (1989), pp. 249–272. doi:10.1016/0029- 5493(89)90105-2.

C. Liang and C. Rogers. One-dimensional thermomechanical constitutive relations for shape memory materials. Journal of Intelligent Material Systems and Structures, 1, (2), (1990), pp. 207– 234. doi:10.2514/6.1990-1027.

R. Abeyaratne and J. K. Knowles. A continuum model of a thermoelastic solid capable of undergoing phase transitions. Journal of the Mechanics and Physics of Solids, 41, (3), (1993), pp. 541–571.

Y. Ivshin and T. J. Pence. A thermomechanical model for a one variant shape memory material. Journal of Intelligent Material Systems and Structures, 5, (4), (1994), pp. 455–473. doi:10.1177/1045389x9400500402.

B. D. Coleman and W. Noll. The thermodynamics of elastic materials with heat conduction and viscosity. Archive for Rational Mechanics and Analysis, 13, (1), (1963), pp. 167–178. doi:10.1007/978-3-642-65817-4 9.


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