Modelling of martensitic transformation and plastic slip effects on the thermo-mechanical behaviour of Fe-based shape memory alloys

doi:10.1016/j.mechmat.2008.11.007

Mechanics of Materials

Volume 41, Issue 7, July 2009, Pages 849-856

https://doi.org/10.1016/j.mechmat.2008.11.007 Get rights and content

Abstract

In this paper, we propose a thermo-mechanical three-dimensional constitutive law adapted to Fe-based shape memory alloys. It takes into account the effect of the martensitic transformation and the plastic slip mechanisms and their interaction. The adopted formulation is based on a simplified micromechanical description. The macroscopic behaviour is derived by considering the equivalent homogeneous effect on a representative volume element. The Gibbs free energy expression is defined. Thermodynamic driving forces are then derived and compared to critical forces leading to the constitutive equations solved by Newton–Raphson numerical scheme. Obtained results for thermo-mechanical loadings are compared to experimental ones.

Introduction

Fe-based shape memory alloys present interesting thermo-mechanical properties induced by the coupling of martensitic transformation and plastic slip mechanisms. They have a low cost compared to NiTi or Copper-based alloys, even if their shape memory effect and resistance to corrosion are limited. A wide range of experimental works were carried out in order to characterise their metallurgical properties and thermo-mechanical behaviour Sato et al., 1982, Tamarat et al., 1991, Verbeken et al., 2008. Recently, techniques are proposed in order to improve shape memory effect of these alloys by reducing the grain size with equal channel angular pressing process (Wei et al., 2008) or by the aligning precipitations of second-phase particles realised through ageing after pre-deformation at room temperature (Wen et al., 2007). Corrosion resistance improvement was also proposed by surface treatment (Charfi et al., 2008). By contrast, works dealing on their behaviour modelling, necessary to the optimization of their performances and application design by finite element method, start to emerge with largely one-dimensional formulations Goliboroda et al., 1999, Lazghab and Wu, 2005. In addition, most proposed accurate three-dimensional thermo-mechanical models are largely adapted for Copper-based or NiTi shape memory alloys because they only describe the phase transformation effect. Plastic gliding is not taken into account because it occurs at a high stress level Peultier et al., 2006, Leclercq and Lexcellent, 1996. Other models, based on micromechanical approach, are proposed Bhattacharyya and Weng, 1994, Cherkaoui et al., 2000, Fischer et al., 2000, Petit et al., 2007, Kouznetsova and Geers, 2008. They describe the effect of the inelastic strain induced by phase transformation and plastic gliding and are more adapted to TRIP steels. In fact, they consider a strain-induced martensite and the behaviour is only described for mechanical loading at a constant temperature.

In order to lead to a representative formulation of the Fe-based SMA thermo-mechanical behaviour, a three-dimensional model is proposed. It is based on the formulation developed by Peultier et al. (2006). Major modifications are introduced in order to take into account the plastic gliding effect and its interaction with phase transformation effect. The inelastic strain is induced by phase transformation and plastic gliding. Studies in Fe-based SMA (Fe–Mn–Si) (Chung et al., 1996; Zhang et al., 1998), showed that the thermally induced martensite (self-accommodated) is very limited and negligible compared to the oriented martensite. It needs a high level energy to be activated. Consequently, the transformation strain intensity is assumed stress-oriented. Its direction is governed by the deviatory part of the stress tensor. The behaviour is then described by two scalar internal variables. They correspond to the martensite volume fraction (f) and the amplitude of plastic gliding (γ). They are macroscopic internal variables which allow the implementation of the constitutive equations in a finite element code making it possible to numerically design applications on Fe-based SMA.

The Gibbs free energy is considered with its different contributions (potential energy, elastic energy related to the incompatibilities induced by the martensitic transformation and the plastic slip, and chemical energy related to the martensitic transformation). Its expression is given function of loading variables (stress or strain and temperature), of the material characteristics and the abovementioned internal variables. After simplified scale transition operations, a new expression of this free energy is obtained which depends only on the macroscopic variables. These average operations are carried out on an elementary representative volume. Different interaction energies (inter-granular and intra-granular interactions: between, respectively, martensite variants, slip systems and martensite variants and slip systems) were also expressed according to the macroscopic internal variables by introducing scalar characteristic parameters. The driving forces of transformation and plastic slip are derived from the Gibbs free energy expression. The comparison of these driving forces to critical ones leads to the kind of induced behaviour for a given thermo-mechanical loading level (elasticity, transformation, plasticity, transformation and plasticity). Finally, the expression of the thermo-mechanical constitutive law is derived starting from the Hooke law and the consistency rules corresponding to the driving forces.

Section snippets

Constitutive equations

Let us consider a representative volume element (RVE) of Fe-based SMA. It corresponds to a polycrystalline aggregate. Each grain (single crystal) presents a parent phase, austenite, on which martensite can occur for a thermo-mechanical loading. This can be accompanied by a plastic gliding in the parent phase, Fig. 1. An inelastic strain, E^ine, is then induced by these two diffusionless mechanisms. The compatibility between the total macroscopic strain, E, and the inelastic one is obtained by

Thermodynamic forces

For the considered RVE, the combination of the first and second principle of thermodynamic leads to the following equation (Clausius–Duhem relation): $d L (Σ_{ij}, T, f, γ) - S_{T} dT - E_{ij} d Σ_{ij} \geq 0$ with S_T is the entropy of the RVE induced by the applied thermo-mechanical loading. By replacing the free energy by its expression and taking into account the physic limitation, the combination of Eqs. (13), (14), (15) gives: $(\begin{matrix} (S_{ijkl} Σ_{kl} + E_{ij}^{Ine} - E_{ij}) d Σ_{ij} + (Σ_{ij} - H_{g} E_{ij}^{Ine}) {dE}_{ij}^{Ine} \\ - (Bf + S_{T}) dT + (- {fH}_{V} + \frac{1}{2} γ^{2} H_{S} + \frac{1}{2} γ H_{SV} - B (T - T_{0}) \\ - λ_{0} + λ_{1}) df + (\frac{1}{2} {fH}_{SV} - (1 - f) γ H_{S}) \end{matrix})$

Numerical resolution

The behaviour corresponding to Eq. (23) is nonlinear. The applied loading is considered in an incremental way. For each increment, previous state of stress and internal variable is considered and an increment of temperature and strain is applied. Increments of stress, martensite volume fraction and plastic gliding are determined by solving with the Newton–Raphson scheme the following equations (Simo et al., 2000): $R (Σ, E, T, f, γ, Δ Σ, Δ E, Δ T, Δ f, Δ γ) = [\begin{matrix} Δ G^{f} = Δ F_{m}^{f} - Δ F_{yield}^{f} \\ Δ G^{γ} = Δ F_{m}^{γ} - Δ F_{yield}^{γ} \\ Δ Σ_{ij} - C_{ijkl} (Δ E_{kl} - Δ E_{kl}^{}) \end{matrix}]$

Results and discussion

In order to check and validate the proposed formulation, experimental tests were carried out. They correspond to a serial of thermo-mechanical tests with tensile loading at a given constant temperature followed by an unloading to a zero stress and a heating to a temperature about 200 °C. This last loading step should distinguish the transformation strain (recovery strain) from the plastic one (permanent one). The tensile stress is measured by a force sensor. The tensile strain is measured by an

Conclusions and prospects

In this paper, a thermo-mechanical constitutive model, adapted to Fe-based SMA, is presented. It takes into account the effect of the phase transformation and the plastic gliding. It is based on a micromechanical approach and describes in an averaged and macroscopic way the interactions between successively grains variants and plastic slip systems. Two macroscopic internal variables are introduced: martensite volume fraction and plastic gliding. Transformation and plastic strains are assumed

Acknowledgements

This study is realised in the framework of a cooperative scientific project (CMCU Program No. 04S1117) between two Tunisian research teams and four French ones. It was funded by the Tunisian ministry of national education and the French ministry of foreign affairs. We gratefully acknowledge their financial support.

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Shape Memory Alloys (SMAs) have the main interesting property to recover an inelastic strain induced by martensitic transformation. The initial shape can be recovered directly after unloading or with the application of an additional heating. Iron-based SMAs (Fe-SMAs) are characterized by a high coupling between phase transformation and plastic slip at low temperatures under small stress levels. Thermomechanical constitutive models describing such coupling developed based on small-strains are not suitable for higher loading levels. This motivates the proposed development of a finite-strain constitutive model for Fe-SMAs considering thermomechanical coupling between phase transformation and plastic slip, and by extending the small-strain model within a finite-strain thermodynamical framework in order to describe large strains mainly induced by plastic hardening in Fe-SMA material point. The model here has two internal variables (volume fraction of martensite and the accumulative plastic strain). It is based on the assumption of the local multiplicative split of the deformation gradient into elastic and inelastic parts with a total Lagrangian formulation. The inelastic deformation gradient splits also into a transformation and a plastic parts. The developed model is implemented into the commercial software Matlab. The results obtained for thermomechanical loadings are discussed and, a good agreement with experimental results is also observed.
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By just a small variation in the manufacturing procedure including the heat treatment, micro-structure and the atomic percentage, many changes in the properties of these materials can be observed. Therefore the curiosity to find out the relations between these behaviors, has made these alloys one of the top subjects in the field of the metallurgical researches since their discovery (Mitwally and Farag, 2009; Potapov et al., 1997; Otsuka and Ren, 2005; Miller and Lagoudas, 2001; Jemal et al., 2009; Clingman et al., 2003; Mabe et al., 2006; Melton and Mercier, 1978). SMAs have been extensively implemented and used in industry in recent years due to their unique properties, which are well-known as shape memory effect (SME) and pseudo-elasticity (PE) (Melton, 1999; Liang et al., 1996; Garner et al., 2000; Kumar and Lagoudas, 2008; Hartl and Lagoudas, 2007; Butera, 2008; Kheirikhah et al., 2010; Bil et al., 2013; Jani et al., 2014; David Safranski and Griffis, 2017).
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Over the last three decades, many classical constitutive models were established to describe the unique shape memory effect and superelasticity of SMAs (Brinson, 1993; Leclercq and Lexcellent, 1996; Auricchio et al., 1997; Zaki and Moumni, 2007a,b; Panico and Brinson, 2007; Lagoudas 2008; Lagoudas et al., 2012; Hartl et al., 2010; Chemisky et al., 2011; Arghavani et al., 2011). Meanwhile, some advanced constitutive models were developed to describe the irreversible plastic deformation of SMAs occurred in the austenite phase (Jemal et al., 2009; Hartl and Lagoudas, 2009; Khalil et al., 2012), martensite phase (Yan et al., 2003; Paiva et al., 2005; Hartl and Lagoudas, 2009; Zaki et al., 2010) or austenite-martensite interface (Bo and Lagoudas 1999a,b; Lagoudas and Entchev, 2004; Saint-Sulpice et al., 2009; Rebelo et al., 2011; Chemisky et al., 2014, 2018; Yu et al., 2015). In these models, various deformation mechanisms and their interactions were considered, simultaneously.
Based on the mean-field approach, a micromechanical constitutive model is established to describe the unusual temperature-dependent deformation of Mg–NiTi composites. Firstly, the constitutive equations of two phases are proposed: for NiTi shape memory alloy (SMA), a simplified Hartl–Lagoudas model (Hartl and Lagoudas, 2009) which considering the martensite transformation and plastic deformation is adopted; while, for magnesium (Mg), an elastic-plastic model including a new nonlinear plastic hardening law is employed. The dependence of elastic moduli and yield surfaces of two phases at ambient temperature is addressed. Then, a non-isothermal incremental Mori–Tanaka homogenization method is employed and further extended to describe the interaction between two phases and calculate the macroscopic overall stress-strain response of Mg–NiTi composites. Finally, comparisons between the simulated results and the corresponding experimental ones show that the temperature-dependent deformation of the composites with different volume fractions of NiTi SMA phase can be well captured by the proposed model. Predicted results demonstrate that the unusual temperature-dependent deformation of Mg–NiTi composites originates from the change in the inelastic deformation mechanism of NiTi SMA with the variation of temperature.

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Modelling of martensitic transformation and plastic slip effects on the thermo-mechanical behaviour of Fe-based shape memory alloys

Abstract

Introduction

Section snippets

Constitutive equations

Thermodynamic forces

Numerical resolution

Results and discussion

Conclusions and prospects

Acknowledgements

J. Mech. Phys. Solids

Int. J. Plast.

Mater. Charact.

Int. J. Plast.

Computat. Mater. Sci.

Mech. Mater.

J. Mech. Phys. Solids

Int. J. Solids Struct.

Int. J. Plast.

Int. J. Plast.

Mech. Mater.

Acta Metall.