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Well-posedness of a thermo-mechanical modelfor shape memory alloys under tension

Published online by Cambridge University Press:  17 March 2010

Pavel Krejčí  and
Ulisse Stefanelli
Pavel Krejčí
Affiliation:
Matematický ústav AV ČR, Žitná 25, 11567 Praha 1, Czech Republic. krejci@math.cas.cz; http://www.math.cas.cz/ krejci/
Ulisse Stefanelli
Affiliation:
IMATI – CNR, via Ferrata 1, 27100 Pavia, Italy. stefanelli@imati.cnr.it; http://www.imati.cnr.it/ulisse
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Abstract

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.

Keywords

Shape memory alloysthermo-mechanicswell-posednesshysteresis operator
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 6 , November 2010 , pp. 1239 - 1253
Copyright
© EDP Sciences, SMAI, 2010

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References

Aiki, T., A model of 3D shape memory alloy materials. J. Math. Soc. Jpn. 57 (2005) 903–933. CrossRef
Arndt, M., Griebel, M. and Roubíček, T., Modelling and numerical simulation of martensitic transformation in shape memory alloys. Contin. Mech. Thermodyn. 15 (2003) 463–485. CrossRef
Auricchio, F. and Petrini, L., 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. CrossRef
Auricchio, F. and Petrini, L., 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. CrossRef
Auricchio, F. and Petrini, L., 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. CrossRef
Auricchio, F. and Sacco, E., 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. CrossRef
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.
Auricchio, F., Reali, A. and Stefanelli, U., A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plast. 23 (2007) 207–226. CrossRef
Auricchio, F., Mielke, A. and Stefanelli, U., A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Meth. Appl. Sci. 18 (2008) 125–164. CrossRef
Auricchio, F., Reali, A. and Stefanelli, U., 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. CrossRef
A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys. Preprint IMATI-CNR 27PV09/20/0 (2009).
M. Brokate and J. Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences 121. Springer-Verlag, New York (1996).
Colli, P., Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Anal. 24 (1995) 1565–1579. CrossRef
Colli, P. and Sprekels, J., Global existence for a three-dimensional model for the thermodynamical evolution of shape memory alloys. Nonlinear Anal. 18 (1992) 873–888. CrossRef
T.W. Duerig, A.R. Pelton, Eds., SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference. ASM International (2003).
T.W. Duerig, K.N. Melton, D. Stökel and C.M. Wayman, Eds., Engineering aspects of shape memory alloys. Butterworth-Heinemann (1990).
Falk, F., Martensitic domain boundaries in shape-memory alloys as solitary waves. J. Phys. C4 Suppl. 12 (1982) 3–15.
Falk, F. and Konopka, P., Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys. J. Phys. Condens. Matter 2 (1990) 61–77. CrossRef
Frémond, M., Matériaux à mémoire de forme. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 304 (1987) 239–244.
M. Frémond, Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002).
Govindjee, S. and Miehe, C., A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng. 191 (2001) 215–238. CrossRef
Helm, D. and Haupt, P., Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct. 40 (2003) 827–849. CrossRef
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.
Hoffmann, K.H., Niezgódka, M. and Zheng, S., Existence and uniqueness to an extended model of the dynamical developments in shape memory alloys. Nonlinear Anal. 15 (1990) 977–990. CrossRef
P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO Int. Series Math. Sci. Appl. 8. Gakkotosho, Tokyo (1996).
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).
Lagoudas, D.C., Entchev, P.B., Popov, P., Patoor, E., Brinson, L.C. and Gao, X., Shape memory alloys, Part II: Modeling of polycrystals. Mech. Materials 38 (2006) 391–429.
Levitas, V.I., Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Struct. 35 (1998) 889–940. CrossRef
G.A. Maugin, The thermomechanics of plasticity and fracture, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992).
Mielke, A. and Petrov, A., Thermally driven phase transformation in shape-memory alloys. Adv. Math. Sci. Appl. 17 (2007) 160–182.
Mielke, A., Paoli, L. and Petrov, A., 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. CrossRef
A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality. WIAS Preprint n. 1407 (2009).
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).
Pawłow, I., Three-dimensional model of thermomechanical evolution of shape memory materials. Control Cybernet. 29 (2000) 341–365.
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.
Popov, P. and Lagoudas, D.C., A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. Int. J. Plast. 23 (2007) 1679–1720. CrossRef
Raniecki, B. and Lexcellent, Ch., R L models of pseudoelasticity and their specification for some shape-memory solids. Eur. J. Mech. A Solids 13 (1994) 21–50.
Reese, S. and Christ, D., Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation. Int. J. Plast. 28 (2008) 455–482. CrossRef
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.
Souza, A.C., Mamiya, E.N. and Zouain, N., Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids 17 (1998) 789–806. CrossRef
Stefanelli, U., Analysis of a variable time-step discretization for the Penrose-Fife phase relaxation problem. Nonlinear Anal. 45 (2001) 213–240. CrossRef
Thamburaja, P. and Anand, L., Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids 49 (2001) 709–737. CrossRef
Thiebaud, F., Lexcellent, Ch., Collet, M. and Foltete, E., 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. CrossRef
A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences 111. Springer, Berlin (1994).
Yoshikawa, S., Pawłow, I. and Zajączkowski, W.M., Quasi-linear thermoelasticity system arising in shape memory materials. SIAM J. Math. Anal. 38 (2007) 1733–1759. CrossRef