Analysis of localization phenomena in Shape Memory Alloys bars by a variational approach

Abstract Localization of the phase transformations in Shape Memory Alloys (SMA) wires are well known. Several experimental and theoretical studies appeared in the last years. In this work the problem is addressed by means of a variational approach within the framework of the modeling of rate-independent materials by the specification of a non-local free energy and a dissipation function, focusing attention on the basic case of isothermal conditions. General expressions are given for a rather broad class of models, whereas a simple model is studied in detail. A full stability analysis of both homogeneous and non-homogeneous solutions is carried out analytically, showing that stable non-homogeneous solutions have necessarily to occur if the bar is longer than an internal length determined by the constitutive parameters. The analysis also shows that snap-back phenomena may occur both in the nucleation and the coalescence phase, depending on another material length which is also function of the number of transformation fronts. This helps to explain why the second stress drop associated to coalescence is much more difficult to observe experimentally. Closed form expressions are given for the phase fraction profiles of both single and multiple localizations as well as nucleation and propagation stresses. A comparison between the prediction of the model with experimental data finally shows a good agreement both in terms of global response and in the spatio-temporal evolution of the transformation domains.

[1]  Jean-Jacques Marigo,et al.  Constitutive relations in plasticity, damage and fracture mechanics based on a work property , 1989 .

[2]  Nihon-Kikai-Gakkai JSME international journal , 1992 .

[3]  A. Mielke Differential, Energetic, and Metric Formulations for Rate-Independent Processes , 2011 .

[4]  Davide Bernardini,et al.  Shape‐Memory Materials, Modeling , 2002 .

[5]  Hisaaki Tobushi,et al.  Pseudoelasticity of TiNi Shape Memory Alloy : Dependence on Maximum Strain and Temperature , 1993 .

[6]  Davide Bernardini,et al.  New micromechanical estimates of the interaction energy for shape memory alloys modeled by a two-phases microstructure* , 2016 .

[7]  Davide Bernardini,et al.  Mathematical Models for Shape-Memory Materials , 2009 .

[8]  T. B. Zineb,et al.  A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures , 2014 .

[9]  J. Shaw,et al.  Thermomechanical aspects of NiTi , 1995 .

[10]  Arun R. Srinivasa,et al.  On the thermomechanics of shape memory wires , 1999 .

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

[12]  John A. Shaw,et al.  Thermodynamics of Shape Memory Alloy Wire: Modeling, Experiments, and Application , 2006 .

[13]  P. Sedlák,et al.  A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations , 2016 .

[14]  J. Marigo,et al.  Gradient Damage Models Coupled with Plasticity and Nucleation of Cohesive Cracks , 2014, Archive for Rational Mechanics and Analysis.

[15]  Stelios Kyriakides,et al.  Initiation and propagation of localized deformation in elasto-plastic strips under uniaxial tension , 1997 .

[16]  S. Kyriakides,et al.  Localization in NiTi tubes under bending , 2014 .

[17]  John A. Shaw,et al.  An Experimental Setup for Measuring Unstable Thermo-Mechanical Behavior of Shape Memory Alloy Wire , 2002 .

[18]  C. Maurini,et al.  A gradient approach for the macroscopic modeling of superelasticity in softening shape memory alloys , 2015 .

[19]  H. Dai,et al.  Propagation stresses in phase transitions of an SMA wire: New analytical formulas based on an internal-variable model , 2013 .

[20]  Han Zhao,et al.  Recent advances in spatiotemporal evolution of thermomechanical fields during the solid-solid phase transition , 2012 .

[21]  Mohamed Haboussi,et al.  Modelling of localization and propagation of phase transformation in superelastic SMA by a gradient nonlocal approach , 2011 .

[22]  Christoph Wehrli,et al.  The Derivation of Constitutive Relations from the Free Energy and the Dissipation Function , 1987 .

[23]  J. Shaw Simulations of localized thermo-mechanical behavior in a NiTi shape memory alloy , 2000 .

[24]  Davide Bernardini,et al.  Models for one-variant shape memory materials based on dissipation functions , 2002 .

[25]  Ingo Müller,et al.  Thermodynamics of pseudoelasticity —an analytical approach , 1993 .

[26]  Christian Lexcellent,et al.  Shape-Memory Alloys Handbook , 2013 .

[27]  Ferdinando Auricchio,et al.  A rate-independent model for the isothermal quasi-static evolution of shape-memory materials , 2006, 0708.4378.

[28]  S. Kyriakides,et al.  Underlying material response for Lüders-like instabilities , 2013 .

[29]  J. Marigo,et al.  Gradient damage models coupled with plasticity: Variational formulation and main properties , 2015 .

[30]  D. Lagoudas Shape memory alloys : modeling and engineering applications , 2008 .

[31]  Jean-Jacques Marigo,et al.  Approche variationnelle de l'endommagement : II. Les modèles à gradient , 2010 .

[32]  Alexander Mielke,et al.  A Mathematical Framework for Generalized Standard Materials in the Rate-Independent Case , 2006 .

[33]  Yongjun He,et al.  Rate-dependent domain spacing in a stretched NiTi strip , 2010 .

[34]  Zhiqiang Li,et al.  Phase transformation in superelastic NiTi polycrystalline micro-tubes under tension and torsion––from localization to homogeneous deformation , 2002 .

[35]  H. Dai,et al.  Phase transitions in a slender cylinder composed of an incompressible elastic material. I. Asymptotic model equation , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[36]  V. Levitas The postulate of realizability: Formulation and applications to the post-bifurcation behaviour and phase transitions in elastoplastic materials-II , 1995 .

[37]  Stelios Kyriakides,et al.  On the nucleation and propagation of phase transformation fronts in a NiTi alloy , 1997 .

[38]  H. Petryk Thermodynamic conditions for stability in materials with rate-independent dissipation , 2005, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[39]  J. Shaw A thermomechanical model for a 1-D shape memory alloy wire with propagating instabilities , 2002 .

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

[41]  P. Feng,et al.  Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force , 2006 .

[42]  Tips and tricks for characterizing shape memory alloy wire: Part 3-localization and propagation phenomena , 2009 .

[43]  Kim Pham Construction et analyse de modèles d'endommagement à gradient , 2010 .

[44]  Jean-Jacques Marigo,et al.  The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models , 2011 .

[45]  A. Mielke,et al.  On rate-independent hysteresis models , 2004 .

[46]  D. Bernardini On the macroscopic free energy functions for shape memory alloys , 2001 .

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

[48]  H. Dai,et al.  Closed-form solutions for inhomogeneous states of a slender 3-D SMA cylinder undergoing stress-induced phase transitions , 2015 .

[49]  A. Maynadier,et al.  Thermomechanical modelling of a NiTi SMA sample submitted to displacement-controlled tensile test , 2014 .

[50]  Nguyen Quoc Son Bifurcation et stabilité des systèmes irréversibles obéissant au principe de dissipation maximale , 1984 .