Abstract
The specific material properties of shape memory alloys are due to the formation of martensitic microstructures. In this contribution, we develop a strategy to model the material behavior based on energy considerations: we first present narrow bounds to the elastic energy obtained by lamination of the multi-well problem in the monocrystalline case. These considerations are then extended to polycrystals and compared to a convexification bound. Due to the acceptably low difference between convexification lower and lamination upper bound,we use the convexification bound to establish a micromechanical model which, on the basis of physically well motivated parameters such as elastic constants and transformation strains, is able to represent a variety of aspects of the material behavior such as pseudoelasticity, pseudoplasticity and martensite reorientation.
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References
J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100, 1987, 13–52.
T. Bartel and K. Hackl, A novel approach to the modelling of single-crystalline materials undergoing martensitic phase-transformations, Mat. Sci. Eng. A 481, 2008 371–375.
K. Bhattacharya, Microstructure of Martensite – Why It Forms and How It Gives Rise to the Shape-Memory Effect, Oxford University Press, New York, 2003.
C. Bouvet, S. Calloch and C. Lexcellent, A phenomenological model for pseudoelasticity of shape memory alloys under multiaxial proportional and nonproportional loadings, Eur. J. Mech. A-Solids 23, 2004, 37–61.
O.P. Bruno, F. Reitich and P.H. Leo, The overall elastic energy of polycrystalline martensitic solids, J. Mech. Phys. Solids 44, 1996, 1051–1101.
D. Christ and S. Reese, Finite-element modelling of shape memory alloys – A comparison between small-strain and large-strain formulations, Mat. Sci. Eng. A 481, 2008, 343–346.
B. Dacorogna, Quasiconvexification and relaxation of nonconvex problems in the calculus of variations, J. Funct. Anal. 46, 1982, 102–118.
S. Govindjee, K. Hackl and R. Heinen, An upper bound to the free energy of mixing by twincompatible lamination for n-variant martensitic phase transformations, Continuum Mech. Thermodyn. 18, 2007, 443–453.
S. Govindjee, A. Mielke and G.J. Hall, The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Phys. Solids 51, 2003, 763.
S. Govindjee and C. Miehe, A mult-variant martensitic phase transformation model: Formulation and numerical implementation, Comput. Meth. Appl. Mech. Engrg. 191, 2001, 215–238.
S. Grabe and O.T. Bruhns, On the viscous and strain rate dependent behavior of polycrystalline NiTi, Int. J. Solids Struct. 45, 2008, 1876-Ű1895.
K. Hackl and F.D. Fischer, On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. R. Soc. A 464, 2008, 117–132
K. Hackl and R. Heinen, A micromechanical model for pre-textured polycrystalline shape memory alloys including elastic anisotropy, Continuum Mech. Thermodyn. 19, 2008, 499–510.
K. Hackl and R. Heinen, A lamination upper bound to the free energy of n-variant polycrystalline shape memory alloys, J. Mech. Phys. Solids 56, 2008, 2832–2843.
K. Hackl, R. Heinen, W.W. Schmahl and M. Hasan, Experimental verification of a micromechanical model for polycrystalline shape memory alloys in dependence of martensite orientation distributions, Mat. Sci. Eng. A 481, 2008, 347–350.
R. Heinen and K. Hackl, On the calculation of energy-minimizing phase fractions in shape memory alloys, Comput. Meth. Appl. Mech. Engrg. 196, 2007, 2401–2412.
R. Heinen, K. Hackl, W. Windl and M. Wagner, Microstructural evolution during multiaxial deformation of pseudoelastic NiTi studied by first-principles-based micromechanical modeling, Acta Materialia 57, 2009, 3856–3867.
D. Helm and P. Haupt, Shape memory behaviour: Modelling within continuum thermodynamics, Int. J. Solids Struct. 40, 2003, 827–849.
R. Kohn, The relaxation of a double-well energy, Continuum Mech. Thermodyn. 3, 1991, 193–236.
A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Rat. Mech. Anal. 162, 2002, 137–177.
A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Continuum Mech. Thermodyn. 15, 2003, 351–382.
C. Müller and O.T. Bruhns, A thermodynamic finite-strain model for pseudoelastic shape memory alloys, Int. J. Plasticity 22, 2006, 1658–1682.
M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Comput. Meth. Appl. Mech. Engrg. 171, 1999, 419–444.
S. Reese and C. Christ, Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation, Int. J. Plasticity 24, 2008, 455–482.
P. Sedlák, H. Seiner, M. Landa, V. Novák, P. Sittner and L. Mañosa, Elastic constants of bcc austenite and 2H orthorhombic martensite in CuA1Ni shape memory alloy. Acta Materialia 53, 2005, 3643–3661.
V. Smyshlyaev and J. Willis, A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals, Proc. R. Soc. London A 454, 1998, 1573–1613.
V. Smyshlyaev and J. Willis, On the relaxation of a three-well energy, Proc. R. Soc. London A 455, 1998, 779–814.
S. Stupkiewicz and H. Petryk, Modelling of laminated microstructures in stressinduced martensitic transformations, J. Mech. Phys. Sol. 50, 2002, 2303–2331.
M.F.-X. Wagner and W. Windl, Lattice stability, elastic constants and macroscopic moduli of NiTi martensites from first principles, Acta Materialia 56, 2008, 6232–6245.
L. Truskinovsky, About the normal growth apporximation in the dynamical theory of phasetransitions, Continuum Mech. Thermodyn. 6, 1994, 185–208.
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Heinen, R., Hackl, K. (2010). A Micromechanical Model for Polycrystalline Shape Memory Alloys – Formulation and Numerical Validation. In: Hackl, K. (eds) IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials. IUTAM Bookseries, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9195-6_7
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DOI: https://doi.org/10.1007/978-90-481-9195-6_7
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