Well-posedness of a thermo-mechanical model for shape memory alloys under tension
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, p. 1239-1253

We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.

DOI : https://doi.org/10.1051/m2an/2010024
Classification:  74N30,  74C05,  35K55
Keywords: shape memory alloys, thermo-mechanics, well-posedness, hysteresis operator
@article{M2AN_2010__44_6_1239_0,
     author = {Krej\v c\'\i , Pavel and Stefanelli, Ulisse},
     title = {Well-posedness of a thermo-mechanical model for shape memory alloys under tension},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {6},
     year = {2010},
     pages = {1239-1253},
     doi = {10.1051/m2an/2010024},
     zbl = {pre05835020},
     mrnumber = {2769056},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_6_1239_0}
}
Krejčí, Pavel; Stefanelli, Ulisse. Well-posedness of a thermo-mechanical model for shape memory alloys under tension. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 6, pp. 1239-1253. doi : 10.1051/m2an/2010024. http://www.numdam.org/item/M2AN_2010__44_6_1239_0/

[1] T. Aiki, A model of 3D shape memory alloy materials. J. Math. Soc. Jpn. 57 (2005) 903-933. | Zbl 1076.74041

[2] 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. | Zbl 1068.74588

[3] F. Auricchio and L. Petrini, Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Numer. Methods Eng. 55 (2002) 1255-1284. | Zbl 1062.74580

[4] F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part I: Solution algorithm and boundary value problems. Int. J. Numer. Methods Eng. 61 (2004) 807-836. | Zbl 1075.74599

[5] F. Auricchio and L. Petrini, A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: Thermomechanical coupling and hybrid composite applications. Int. J. Numer. Methods Eng. 61 (2004) 716-737. | Zbl 1075.74598

[6] 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. | Zbl 0906.73006

[7] F. Auricchio, A. Reali and U. Stefanelli, A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, in Topics on Mathematics for Smart Systems (Rome, 2006), World Sci. Publishing (2007) 1-14. | Zbl 1113.74053

[8] F. Auricchio, A. Reali and U. Stefanelli, A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plast. 23 (2007) 207-226. | Zbl 1105.74031

[9] 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. | Zbl 1151.74319

[10] 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. Eng. 198 (2009) 1631-1637.

[11] A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys. Preprint IMATI-CNR 27PV09/20/0 (2009).

[12] M. Brokate and J. Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences 121. Springer-Verlag, New York (1996). | Zbl 0951.74002

[13] P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Anal. 24 (1995) 1565-1579. | Zbl 0832.35066

[14] 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. | Zbl 0766.35062

[15] T.W. Duerig, A.R. Pelton, Eds., SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference. ASM International (2003).

[16] T.W. Duerig, K.N. Melton, D. Stökel and C.M. Wayman, Eds., Engineering aspects of shape memory alloys. Butterworth-Heinemann (1990).

[17] F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves. J. Phys. C4 Suppl. 12 (1982) 3-15.

[18] 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.

[19] 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.

[20] M. Frémond, Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002). | Zbl 0990.80001

[21] S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng. 191 (2001) 215-238. | Zbl 1007.74061

[22] D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct. 40 (2003) 827-849. | Zbl 1025.74022

[23] M. Hilpert, On uniqueness for evolution problems with hysteresis, in Mathematical Models for Phase Change Problems, J.F. Rodrigues Ed., Birkhäuser, Basel (1989) 377-388. | Zbl 0701.35009

[24] 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. | Zbl 0728.35055

[25] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Int. Series Math. Sci. Appl. 8. Gakkotosho, Tokyo (1996). | Zbl 1187.35003

[26] P. Krejčí and U. Stefanelli, Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory wires. Preprint, IMATI-CNR, 12PV09/10/0 (2009).

[27] 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. Materials 38 (2006) 391-429.

[28] V.I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Struct. 35 (1998) 889-940. | Zbl 0931.74059

[29] G.A. Maugin, The thermomechanics of plasticity and fracture, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992). | Zbl 0753.73001

[30] A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys. Adv. Math. Sci. Appl. 17 (2007) 160-182. | Zbl 1138.49014

[31] 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. | Zbl 1201.49011

[32] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality. WIAS Preprint n. 1407 (2009).

[33] A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error control for space-time discretizations of a 3D model for shape-memory materials, in Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer (2009).

[34] I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials. Control Cybernet. 29 (2000) 341-365. | Zbl 1205.74140

[35] B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM. Materials Sci. Eng. A 438-440 (2006) 454-458.

[36] 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. Plast. 23 (2007) 1679-1720. | Zbl 1127.74008

[37] B. Raniecki and Ch. Lexcellent, RL models of pseudoelasticity and their specification for some shape-memory solids. Eur. J. Mech. A Solids 13 (1994) 21-50. | Zbl 0795.73010

[38] S. Reese and D. Christ, Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and finite element implementation. Int. J. Plast. 28 (2008) 455-482. | Zbl 1145.74005

[39] T. Roubíček, Models of microstructure evolution in shape memory alloys, in Nonlinear Homogenization and its Appl. to Composites, Polycrystals and Smart Materials, P. Ponte Castaneda, J.J. Telega, B. Gambin Eds., NATO Sci. Series II/170, Kluwer, Dordrecht (2004) 269-304.

[40] A.C. Souza, E.N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids 17 (1998) 789-806. | Zbl 0921.73024

[41] U. Stefanelli, Analysis of a variable time-step discretization for the Penrose-Fife phase relaxation problem. Nonlinear Anal. 45 (2001) 213-240. | Zbl 0983.65101

[42] P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids 49 (2001) 709-737. | Zbl 1011.74049

[43] F. Thiebaud, Ch. Lexcellent, M. Collet and E. Foltete, Implementation of a model taking into account the asymmetry between tension and compression, the temperature effects in a finite element code for shape memory alloys structures calculations. Comput. Materials Sci. 41 (2007) 208-221.

[44] A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences 111. Springer, Berlin (1994). | Zbl 0820.35004

[45] S. Yoshikawa, I. Pawłow and W.M. Zajączkowski, Quasi-linear thermoelasticity system arising in shape memory materials. SIAM J. Math. Anal. 38 (2007) 1733-1759. | Zbl 1131.35083