# American Institute of Mathematical Sciences

March  2011, 6(1): 145-165. doi: 10.3934/nhm.2011.6.145

## A rate-independent model for permanent inelastic effects in shape memory materials

 1 Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38100 Povo (Trento), Italy 2 Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany 3 Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia

Received  May 2010 Revised  October 2010 Published  March 2011

This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ones by means of a rigorous $\Gamma$-convergence analysis.
Citation: Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks & Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
##### References:
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Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019.  Google Scholar [10] F. Auricchio and E. Sacco, A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite, Int. J. Non-Linear Mech., 32 (1997), 1101-1114. doi: 10.1016/S0020-7462(96)00130-8.  Google Scholar [11] F. Auricchio, R. L. Taylor and J. Lubliner, Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behaviour, Comput. Mech. Appl. Mech. Engrg., 146 (1997), 281-312. doi: 10.1016/S0045-7825(96)01232-7.  Google Scholar [12] A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys, Math. Models Meth. Appl. Sci. (2010) to appear. Google Scholar [13] Z. Bo and D. C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading. Part III: Evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci., 37 (1999), 1175-1203. doi: 10.1016/S0020-7225(98)00115-3.  Google Scholar [14] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,'' vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.  Google Scholar [15] N. Chemetov, Well-posedness of two-shape memory model, Math. Methods Appl. Sci., 29 (2006), 209-233. doi: 10.1002/mma.672.  Google Scholar [16] P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal., 24 (1995), 1565-1579. doi: 10.1016/0362-546X(94)00097-2.  Google Scholar [17] P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888. doi: 10.1016/0362-546X(92)90228-7.  Google Scholar [18] G. 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Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Phys. D, 72 (1994), 287-308. doi: 10.1016/0167-2789(94)90234-8.  Google Scholar [26] S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. doi: 10.1016/S0045-7825(01)00271-7.  Google Scholar [27] S. Govindjee and E. P. Kasper, A shape memory alloy model for Uranium-Niobium accounting for plasticity, J. Intell. Mater. Syst. Struct., 8 (1997), 815-823. doi: 10.1177/1045389X9700801001.  Google Scholar [28] D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Internat. J. Solids Structures, 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.  Google Scholar [29] K. H. Hoffmann, M. Niezgódka and S. Zheng, Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys, Nonlinear Anal., 15 (1990), 977-990. doi: 10.1016/0362-546X(90)90079-V.  Google Scholar [30] Y. Huo, I. Müller and S. Seelecke, Quasiplasticity and pseudoelasticity in shape memory alloys, in Phase transitions and hysteresis, in Lecture Notes in Math., Vol. 1584, eds. M. Brokate et al (Springer 1994), 87-146.  Google Scholar [31] P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids (2010), to appear. Google Scholar [32] P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Anal. Numer., (2010), to appear. Google Scholar [33] M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructures and its evolution in shape-memory-alloy single cristals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1.  Google Scholar [34] M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., (2010), to appear. Google Scholar [35] D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, L. C. Brinson and X. Gao, Shape memory alloys, Part II: Modeling of polycrystals, Mech. Mater., 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.08.003.  Google Scholar [36] D. C. Lagoudas and P. B. Entchev, Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: Constitutive model for fully dense SMAs, Mech. Mater., 36 (2004), 865-892. Google Scholar [37] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  Google Scholar [38] G. A. Maugin, "The Thermomechanics of Plasticity and Fracture,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.  Google Scholar [39] A. Mielke, Evolution of rate-independent systems, In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Elsevier, (2005), 461-559.  Google Scholar [40] A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41 (2009), 1388-1414. doi: 10.1137/080726215.  Google Scholar [41] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.  Google Scholar [42] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error control for space-time discretizations of a 3D model for shape-memory materials, Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer, 2009. Google Scholar [43] A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 160-182.  Google Scholar [44] A. Mielke and T. Roubíček, A rate independent model for inelastic behaviour of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860.  Google Scholar [45] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.  Google Scholar [46] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. H.-D Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), 117-129. Google Scholar [47] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.  Google Scholar [48] I. Müller, Thermodynamics of ideal pseudoelasticity, J. Phys. IV, C2-5 (1995), 423-431. Google Scholar [49] A. Paiva, M. A. Savi, A. M. B. Braga and P. M. C. L. Pacheco, A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity, Internat. J. of Solid. Struct., 42 (2005), 3439-3457. doi: 10.1016/j.ijsolstr.2004.11.006.  Google Scholar [50] I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials, Control Cybernet., 29 (2000), 341-365.  Google Scholar [51] B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM, Materials Sci. Engrg. A, 438-440 (2006), 454-458. doi: 10.1016/j.msea.2006.01.104.  Google Scholar [52] P. Popov and D. C. Lagoudas, A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite, Int. J. Plasticity, 23 (2007), 1679-1720. doi: 10.1016/j.ijplas.2007.03.011.  Google Scholar [53] B. Raniecki and Ch. Lexcellent, $R_L$ models of pseudoelasticity and their specification for some shape-memory solids, European J. Mech. A Solids, 13 (1994), 21-50. Google Scholar [54] S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation, Int. J. 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##### References:
 [1] T. Aiki, A model of 3D shape memory alloy materials, J. Math. Soc. Japan, 57 (2005), 903-933. doi: 10.2969/jmsj/1158241940.  Google Scholar [2] S. Antman, J. L. Ericksen, D. Kinderlehrer and I. Müller, "Metastability and Incompletely Posed Problems," in the IMA Volumes in Mathematics and its Applications, 3, Springer-Verlag, New York, 1987.  Google Scholar [3] M. Arndt, M. Griebel and T. Roubíček, Modelling and numerical simulation of martensitic transformation in shape memory alloys, Contin. Mech. Thermodyn., 15 (2003), 463-485. doi: 10.1007/s00161-003-0127-3.  Google Scholar [4] M. Arrigoni, F. Auricchio, V. Cacciafesta, L. Petrini and R. Pietrabissa, Cyclic effects in shape-memory alloys: A one-dimensional continuum model, Journal de Physique IV, 11 (2001), 577-582. Google Scholar [5] F. Auricchio, A. Mielke and U. Stefanelli, A rate-independent model for the isothermal quasi-static evolution of shape-memory materials, Math. Models Meth. Appl. Sci., 18 (2008), 125-164. doi: 10.1142/S0218202508002632.  Google Scholar [6] F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations, Internat. J. Numer. Meth. Engrg., 55 (2002), 1255-1284. doi: 10.1002/nme.619.  Google Scholar [7] F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations: Solution algorithm and boundary value problems, Internat. J. Numer. Meth. Engrg., 61 (2004), 807-836. doi: 10.1002/nme.1086.  Google Scholar [8] F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity, Int. J. Plasticity, 23 (2007), 207-226. doi: 10.1016/j.ijplas.2006.02.012.  Google Scholar [9] F. Auricchio, A. Reali and U. Stefanelli, A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties, Comput. Methods Appl. Mech. Engrg., 198 (2009), 1631-1637. doi: 10.1016/j.cma.2009.01.019.  Google Scholar [10] F. Auricchio and E. Sacco, A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite, Int. J. Non-Linear Mech., 32 (1997), 1101-1114. doi: 10.1016/S0020-7462(96)00130-8.  Google Scholar [11] F. Auricchio, R. L. Taylor and J. Lubliner, Shape-memory alloys: Macromodelling and numerical simulations of the superelastic behaviour, Comput. Mech. Appl. Mech. Engrg., 146 (1997), 281-312. doi: 10.1016/S0045-7825(96)01232-7.  Google Scholar [12] A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys, Math. Models Meth. Appl. Sci. (2010) to appear. Google Scholar [13] Z. Bo and D. C. Lagoudas, Thermomechanical modeling of polycrystalline SMAs under cyclic loading. Part III: Evolution of plastic strains and two-way shape memory effect, Int. J. Eng. Sci., 37 (1999), 1175-1203. doi: 10.1016/S0020-7225(98)00115-3.  Google Scholar [14] M. Brokate and J. Sprekels, "Hysteresis and Phase Transitions,'' vol. 121 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996.  Google Scholar [15] N. Chemetov, Well-posedness of two-shape memory model, Math. Methods Appl. Sci., 29 (2006), 209-233. doi: 10.1002/mma.672.  Google Scholar [16] P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys, Nonlinear Anal., 24 (1995), 1565-1579. doi: 10.1016/0362-546X(94)00097-2.  Google Scholar [17] P. Colli and J. Sprekels, Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys, Nonlinear Anal., 18 (1992), 873-888. doi: 10.1016/0362-546X(92)90228-7.  Google Scholar [18] G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'' Birkhäuser-Boston, 1993.  Google Scholar [19] G. Duvaut and J.-L. Lions, "Inequalities in Mechanics and Physics,'' Springer-Berlin, 1976.  Google Scholar [20] F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves, J. Phys. C4 Suppl., 12 (1982), 3-15. Google Scholar [21] F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys, J. Phys. Condens. Matter, 2 (1990), 61-77. doi: 10.1088/0953-8984/2/1/005.  Google Scholar [22] M. Frémond, Matériaux à mémoire de forme, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 304 (1987), 239-244. Google Scholar [23] M. Frémond, "Non-Smooth Thermomechanics,'' Springer-Verlag, 2002.  Google Scholar [24] M. Frémond and S. Miyazaki, "Shape Memory Alloys,'' Springer-Verlag, 1996. Google Scholar [25] E. Fried and M. E. Gurtin, Dynamic solid-solid transitions with phase characterized by an order parameter, Phys. D, 72 (1994), 287-308. doi: 10.1016/0167-2789(94)90234-8.  Google Scholar [26] S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Methods Appl. Mech. Engrg., 191 (2001), 215-238. doi: 10.1016/S0045-7825(01)00271-7.  Google Scholar [27] S. Govindjee and E. P. Kasper, A shape memory alloy model for Uranium-Niobium accounting for plasticity, J. Intell. Mater. Syst. Struct., 8 (1997), 815-823. doi: 10.1177/1045389X9700801001.  Google Scholar [28] D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermomechanics, Internat. J. Solids Structures, 40 (2003), 827-849. doi: 10.1016/S0020-7683(02)00621-2.  Google Scholar [29] K. H. Hoffmann, M. Niezgódka and S. Zheng, Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys, Nonlinear Anal., 15 (1990), 977-990. doi: 10.1016/0362-546X(90)90079-V.  Google Scholar [30] Y. Huo, I. Müller and S. Seelecke, Quasiplasticity and pseudoelasticity in shape memory alloys, in Phase transitions and hysteresis, in Lecture Notes in Math., Vol. 1584, eds. M. Brokate et al (Springer 1994), 87-146.  Google Scholar [31] P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires, Math. Mech. Solids (2010), to appear. Google Scholar [32] P. Krejčí and U. Stefanelli, Well-posedness of a thermo-mechanical model for shape memory alloys under tension, M2AN Math. Model. Anal. Numer., (2010), to appear. Google Scholar [33] M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructures and its evolution in shape-memory-alloy single cristals, in particular in CuAlNi, Meccanica, 40 (2005), 389-418. doi: 10.1007/s11012-005-2106-1.  Google Scholar [34] M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., (2010), to appear. Google Scholar [35] D. C. Lagoudas, P. B. Entchev, P. Popov, E. Patoor, L. C. Brinson and X. Gao, Shape memory alloys, Part II: Modeling of polycrystals, Mech. Mater., 38 (2006), 391-429. doi: 10.1016/j.mechmat.2005.08.003.  Google Scholar [36] D. C. Lagoudas and P. B. Entchev, Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: Constitutive model for fully dense SMAs, Mech. Mater., 36 (2004), 865-892. Google Scholar [37] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99.  Google Scholar [38] G. A. Maugin, "The Thermomechanics of Plasticity and Fracture,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.  Google Scholar [39] A. Mielke, Evolution of rate-independent systems, In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Elsevier, (2005), 461-559.  Google Scholar [40] A. Mielke, L. Paoli and A. Petrov, On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys, SIAM J. Math. Anal., 41 (2009), 1388-1414. doi: 10.1137/080726215.  Google Scholar [41] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality, SIAM J. Numer. Anal., 48 (2010), 1625-1646. doi: 10.1137/090750238.  Google Scholar [42] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error control for space-time discretizations of a 3D model for shape-memory materials, Proc. of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer, 2009. Google Scholar [43] A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl., 17 (2007), 160-182.  Google Scholar [44] A. Mielke and T. Roubíček, A rate independent model for inelastic behaviour of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860.  Google Scholar [45] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.  Google Scholar [46] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, Proc. of the Workshop on Models of Continuum Mechanics in Analysis and Engineering, eds. H.-D Alber, R. Balean and R. Farwig (Shaker-Verlag, 1999), 117-129. Google Scholar [47] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rational Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.  Google Scholar [48] I. Müller, Thermodynamics of ideal pseudoelasticity, J. Phys. IV, C2-5 (1995), 423-431. Google Scholar [49] A. Paiva, M. A. Savi, A. M. B. Braga and P. M. C. L. Pacheco, A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity, Internat. J. of Solid. Struct., 42 (2005), 3439-3457. doi: 10.1016/j.ijsolstr.2004.11.006.  Google Scholar [50] I. 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