A robust model of pseudoelasticity in shape memory alloys

SUMMARY A model of pseudoelasticity in shape memory alloys is developed within the incremental energy minimization framework. Three constitutive functions are involved: the Helmholtz free energy and rate-independent dissipation that enter incrementally the minimized energy function, and the constraint function that defines the limit transformation strains. The proposed implementation is based on a unified augmented Lagrangian treatment of both the constitutive constraints and nonsmooth dissipation function. A methodology for easy reformulation of the model from the small-strain to finite-deformation regime is presented. Finite element computations demonstrate robustness of the finite-strain version of the model and illustrate the effects of tension–compression asymmetry and transversal isotropy of the surface of limit transformation strains. Copyright © 2012 John Wiley & Sons, Ltd.

[1]  K. Thermann,et al.  On discretized plasticity problems with bifurcations , 1992 .

[2]  Reza Naghdabadi,et al.  A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys , 2011 .

[3]  L. Brinson,et al.  Shape memory alloys, Part I: General properties and modeling of single crystals , 2006 .

[4]  Tomáš Roubíček,et al.  Modelling of Microstructure and its Evolution in Shape-Memory-Alloy Single-Crystals, in Particular in CuAlNi , 2005 .

[5]  Joze Korelc,et al.  Multi-language and Multi-environment Generation of Nonlinear Finite Element Codes , 2002, Engineering with Computers.

[6]  M. Lambrecht,et al.  Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity , 2004 .

[7]  Ferdinando Auricchio,et al.  Shape-memory alloys: modelling and numerical simulations of the finite-strain superelastic behavior , 1997 .

[8]  Jože Korelc,et al.  Automation of primal and sensitivity analysis of transient coupled problems , 2009 .

[9]  A. Ziółkowski,et al.  Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains , 2007 .

[10]  P. Alart,et al.  A mixed formulation for frictional contact problems prone to Newton like solution methods , 1991 .

[11]  A. Mielke,et al.  A Variational Formulation of¶Rate-Independent Phase Transformations¶Using an Extremum Principle , 2002 .

[12]  Jakub Lengiewicz,et al.  Automation of finite element formulations for large deformation contact problems , 2010 .

[13]  K. Thermann,et al.  Post-critical plastic deformation of biaxially stretched sheets , 1996 .

[14]  T Prakash G. Thamburaja,et al.  Polycrystalline shape-memory materials: effect of crystallographic texture , 2001 .

[15]  S. Stupkiewicz,et al.  Elastic micro-strain energy at the austenite-twinned martensite interface , 2005 .

[16]  A. Bertram,et al.  Relaxation in multi-mode plasticity with a rate-potential , 2005 .

[17]  Etienne Patoor,et al.  Macroscopic constitutive law of shape memory alloy thermomechanical behaviour. Application to structure computation by FEM , 2006 .

[18]  Michael Griebel,et al.  Martensitic transformation in NiMnGa single crystals: Numerical simulation and experiments , 2006 .

[19]  Stefanie Reese,et al.  Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation , 2008 .

[20]  K. Tanaka,et al.  Thermodynamic models of pseudoelastic behaviour of shape memory alloys , 1992 .

[21]  Erwin Stein,et al.  Theory and finite element computation of cyclic martensitic phase transformation at finite strain , 2008 .

[22]  S. Miyazaki,et al.  CRYSTAL STRUCTURE OF THE MARTENSITE IN Ti-49.2 at.%Ni ALLOY ANALYZED BY THE SINGLE CRYSTAL X-RAY DIFFRACTION METHOD , 1985 .

[23]  K. Bhattacharya.  Microstructure of martensite : why it forms and how it gives rise to the shape-memory effect , 2003 .

[24]  Kaushik Bhattacharya,et al.  A micromechanics-inspired constitutive model for shape-memory alloys , 2007 .

[25]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[26]  D. Owen,et al.  Design of simple low order finite elements for large strain analysis of nearly incompressible solids , 1996 .

[27]  S. Stupkiewicz,et al.  Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory , 2010 .

[28]  K. Thermann,et al.  Post-critical plastic deformation in incrementally nonlinear materials , 2002 .

[29]  Stanisław Stupkiewicz,et al.  Finite-strain micromechanical model of stress-induced martensitic transformations in shape memory alloys , 2006 .

[30]  J. Ball,et al.  Fine phase mixtures as minimizers of energy , 1987 .

[31]  E. Sacco,et al.  A 3D SMA constitutive model in the framework of finite strain , 2010 .

[32]  J. Shaw,et al.  Shape Memory Alloys , 2010 .

[33]  Laurent Orgéas,et al.  Stress-induced martensitic transformation of a NiTi alloy in isothermal shear, tension and compression , 1998 .

[34]  C. Lexcellent,et al.  Thermodynamics of isotropic pseudoelasticity in shape memory alloys , 1998 .

[35]  Editors , 1986, Brain Research Bulletin.

[36]  Ken Gall,et al.  The role of texture in tension–compression asymmetry in polycrystalline NiTi , 1999 .

[37]  Alessandro Reali,et al.  On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model , 2011 .

[38]  Michael Ortiz,et al.  Nonconvex energy minimization and dislocation structures in ductile single crystals , 1999 .

[39]  Dimitris C. Lagoudas,et al.  Shape memory alloys, Part II: Modeling of polycrystals , 2006 .

[40]  A. Bertram.  Thermo-mechanical constitutive equations for the description of shape memory effects in alloys , 1983 .

[41]  Dirk Helm,et al.  Shape memory behaviour: modelling within continuum thermomechanics , 2003 .

[42]  Robert L. Taylor,et al.  Multiscale finite element modeling of superelasticity in Nitinol polycrystals , 2009 .

[43]  T. Tadaki,et al.  Shape Memory Alloys , 2002 .

[44]  D. Tortorelli,et al.  Tangent operators and design sensitivity formulations for transient non‐linear coupled problems with applications to elastoplasticity , 1994 .

[45]  O. Bruhns,et al.  A thermodynamic finite-strain model for pseudoelastic shape memory alloys , 2006 .

[46]  Petr Šittner,et al.  Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals , 2000 .

[47]  Stefanie Reese,et al.  A finite element model for shape memory alloys considering thermomechanical couplings at large strains , 2009 .

[48]  D. Helm.  Thermomechanics of martensitic phase transitions in shape memory alloys, I: Constitutive theories for small and large deformations , 2007 .

[49]  Jörn Mosler,et al.  Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening , 2010 .

[50]  Ferdinando Auricchio,et al.  Shape-memory alloys: macromodelling and numerical simulations of the superelastic behavior , 1997 .

[51]  Carsten Carstensen,et al.  Non–convex potentials and microstructures in finite–strain plasticity , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[52]  H. Petryk,et al.  Incremental energy minimization in dissipative solids , 2003 .

[53]  T Prakash G. Thamburaja,et al.  An isotropic-plasticity-based constitutive model for martensitic reorientation and shape-memory effect in shape-memory alloys , 2007 .