# American Institute of Mathematical Sciences

April  2013, 6(2): 277-291. doi: 10.3934/dcdss.2013.6.277

## A new "flexible" 3D macroscopic model for shape memory alloys

 1 Dipartimento di Meccanica Strutturale, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy 2 Dipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100 Pavia

Received  November 2011 Revised  January 2012 Published  November 2012

In this paper we introduce a 3D phenomenological model for shape memory behavior, accounting for: martensite reorientation, asymmetric response of the material to tension/compression, different kinetics between forward and reverse phase transformation. We combine two modeling approaches using scalar and tensorial internal variables. Indeed, we use volume proportions of different configurations of the crystal lattice (austenite and two variants of martensite) as scalar internal variables and the preferred direction of stress-induced martensite as tensorial internal variable. Then, we derive evolution equations by a generalization of the principle of virtual powers, including microforces and micromovements responsible for phase transformations. In addition, we prescribe an evolution law for phase proportions ensuring different kinetics during forward and reverse transformation of the oriented martensite.
Citation: Ferdinando Auricchio, Elena Bonetti. A new "flexible" 3D macroscopic model for shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 277-291. doi: 10.3934/dcdss.2013.6.277
##### References:
 [1] J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings, International Journal of Plasticity, 26 (2010), 976-991. doi: 10.1016/j.ijplas.2009.12.003. [2] F. Auricchio and L. Petrini, A three-dimensional models describing stress-temperature induced solid phase transformations. part I: Solution, algorithm and boundary value problems, International Journal of Numerical Methods in Engineering, 6 (2004), 807-836. [3] F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019. [4] E. Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness, Quart. Appl. Math., 61 (2003), 759-81. [5] E. Bonetti, M. Frémond and Ch. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain, Math. Mech. Solids, 11 (2006), 251-75. doi: 10.1177/1081286506040403. [6] H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," Number 5 in North Holland Math. Studies. North-Holland, Amsterdam, 1973. [7] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [8] J. Lubliner, "Plasticity Theory," Macmillan, New York, 1990. [9] J. J. Moreau, "Fonctionelles Convexes," Universià di Roma Tor Vergata Pub., Roma, 2003. [10] A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations, European Journal of Mechanics, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.

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##### References:
 [1] J. Arghavani, F. Auricchio, R. Naghdabadi, A. Reali and S. Sohrabpour, A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings, International Journal of Plasticity, 26 (2010), 976-991. doi: 10.1016/j.ijplas.2009.12.003. [2] F. Auricchio and L. Petrini, A three-dimensional models describing stress-temperature induced solid phase transformations. part I: Solution, algorithm and boundary value problems, International Journal of Numerical Methods in Engineering, 6 (2004), 807-836. [3] F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Computer Methods in Applied Mechanics and Engineering, 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019. [4] E. Bonetti, Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness, Quart. Appl. Math., 61 (2003), 759-81. [5] E. Bonetti, M. Frémond and Ch. Lexcellent, Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain, Math. Mech. Solids, 11 (2006), 251-75. doi: 10.1177/1081286506040403. [6] H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," Number 5 in North Holland Math. Studies. North-Holland, Amsterdam, 1973. [7] M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. [8] J. Lubliner, "Plasticity Theory," Macmillan, New York, 1990. [9] J. J. Moreau, "Fonctionelles Convexes," Universià di Roma Tor Vergata Pub., Roma, 2003. [10] A. C. Souza, E. N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations, European Journal of Mechanics, 17 (1998), 789-806. doi: 10.1016/S0997-7538(98)80005-3.
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