# Well-posedness of a thermo-mechanical model for shape memory alloys under tension

Pavel Krejčí; Ulisse Stefanelli

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 6, page 1239-1253
- ISSN: 0764-583X

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topKrejčí, Pavel, and Stefanelli, Ulisse. "Well-posedness of a thermo-mechanical model for shape memory alloys under tension." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1239-1253. <http://eudml.org/doc/250838>.

@article{Krejčí2010,

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.
},

author = {Krejčí, Pavel, Stefanelli, Ulisse},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Shape memory alloys; thermo-mechanics; well-posedness; hysteresis operator; shape memory alloys},

language = {eng},

month = {10},

number = {6},

pages = {1239-1253},

publisher = {EDP Sciences},

title = {Well-posedness of a thermo-mechanical model for shape memory alloys under tension},

url = {http://eudml.org/doc/250838},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Krejčí, Pavel

AU - Stefanelli, Ulisse

TI - Well-posedness of a thermo-mechanical model for shape memory alloys under tension

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/10//

PB - EDP Sciences

VL - 44

IS - 6

SP - 1239

EP - 1253

AB -
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.

LA - eng

KW - Shape memory alloys; thermo-mechanics; well-posedness; hysteresis operator; shape memory alloys

UR - http://eudml.org/doc/250838

ER -

## References

top- T. Aiki, A model of 3D shape memory alloy materials. J. Math. Soc. Jpn.57 (2005) 903–933. Zbl1076.74041
- 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. Zbl1068.74588
- 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. Zbl1062.74580
- 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. Zbl1075.74599
- 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. Zbl1075.74598
- 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. Zbl0906.73006
- 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. Zbl1113.74053
- 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. Zbl1105.74031
- 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. Zbl1151.74319
- 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. Zbl1227.74037
- A.-L. Bessoud and U. Stefanelli, A three-dimensional model for magnetic shape memory alloys. Preprint IMATI-CNR 27PV09/20/0 (2009). Zbl1317.74021
- M. Brokate and J. Sprekels, Hysteresis and phase transitions, Applied Mathematical Sciences121. Springer-Verlag, New York (1996). Zbl0951.74002
- P. Colli, Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Anal.24 (1995) 1565–1579. Zbl0832.35066
- 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. Zbl0766.35062
- 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).
- F. Falk, Martensitic domain boundaries in shape-memory alloys as solitary waves. J. Phys. C4 Suppl.12 (1982) 3–15.
- F. Falk and P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys. J. Phys. Condens. Matter2 (1990) 61–77.
- M. Frémond, Matériaux à mémoire de forme. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre304 (1987) 239–244.
- M. Frémond, Non-smooth Thermomechanics. Springer-Verlag, Berlin (2002). Zbl0990.80001
- 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. Zbl1007.74061
- D. Helm and P. Haupt, Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct.40 (2003) 827–849. Zbl1025.74022
- 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.
- 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. Zbl0728.35055
- 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). Zbl1269.74169
- 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. Materials38 (2006) 391–429.
- V.I. Levitas, Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Struct.35 (1998) 889–940. Zbl0931.74059
- G.A. Maugin, The thermomechanics of plasticity and fracture, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1992).
- A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys. Adv. Math. Sci. Appl.17 (2007) 160–182. Zbl1138.49014
- 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. Zbl1201.49011
- A. Mielke, L. Paoli, A. Petrov and U. Stefanelli, Error estimates for discretizations of a rate-independent variational inequality. WIAS Preprint n. 1407 (2009). Zbl1219.65059
- 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).
- I. Pawłow, Three-dimensional model of thermomechanical evolution of shape memory materials. Control Cybernet.29 (2000) 341–365. Zbl1205.74140
- B. Peultier, T. Ben Zineb and E. Patoor, Macroscopic constitutive law for SMA: Application to structure analysis by FEM. Materials Sci. Eng. A438–440 (2006) 454–458.
- 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. Zbl1127.74008
- B. Raniecki and Ch. Lexcellent, RL models of pseudoelasticity and their specification for some shape-memory solids. Eur. J. Mech. A Solids13 (1994) 21–50. Zbl0795.73010
- 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. Zbl1145.74005
- 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. Zbl1320.74087
- A.C. Souza, E.N. Mamiya and N. Zouain, Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids17 (1998) 789–806. Zbl0921.73024
- U. Stefanelli, Analysis of a variable time-step discretization for the Penrose-Fife phase relaxation problem. Nonlinear Anal.45 (2001) 213–240. Zbl0983.65101
- P. Thamburaja and L. Anand, Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Phys. Solids49 (2001) 709–737. Zbl1011.74049
- 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.
- A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences111. Springer, Berlin (1994). Zbl0820.35004
- 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. Zbl1131.35083

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