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Procedia Structural Integrity 2 (2016) 15406-1552

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21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Microstructural modelling of plastic deformation and defects accumulation in FeMn-based shape memory alloys

Margarita E. Evarda*, Aleksandr A. Volkova, Fedor S. Belyaeva, Anna D. Ignatovaa,

Natalia A. Volkovaa

An approach is presented to describe the functional and mechanical behaviour of FeMn-based shape memory alloys undergoing fcc - hcp phase transformation. The multi-variance of the reverse transformation is taken into consideration. The martensitic transformation and the micro plastic deformation due to the pfastic accommodation of martensite are considered on the microscopic level. The micro plastic deformation is described from the point of view of the plastic flow theory. Isotropic hardening and kinematic hardening are taken into account and are related to the densities of scattered and oriented deformation defects. The thermodynamic forces causing growth of martensite and reversible deformation defects are the derivatives of the Gibbs' potential on the; respective internal variables. The macro deformation of the representative volume of the polycryttal is calculated by averaging all micro strains. The results show a good qualitative agreement with available experimental data.

© 22016, PROSTR (Ptocedia Structural ¡Integrity) Hosting by Elsevier Ltd. All rights reserved. Peer-teview ender responsibility of the Scientific Committae ofECF21.

Keywarcfs:FeMn, FeMnSi, shape memory, plaseicile defects.

1. Introduction

Non-trivM projeerties of aTtoys undergoeng martenstoc phase transformations have attracted attention of researcliers aiact des^ners for decades (Funakubo (1987X Otsulka (1998)).. Unusual mechanical properties of shape memory aUoys (SMA) made possible creation of unique products and technologies. SubstantMly it was enhanced by rapid devetoftment of maTeriak stience., espeyMly of new metifods oa aUoymg and thermomechanical treatment allowing production of SMA with preset parameters (Brailovski et a^ 2008). Thus, scrutinized by the end of the

* Corresponding author. Tel.: +7-812-428-4220; fax: +7-812-428-7079.

E-mail aMress: m.evar:@spbu.ru

aSaint Petersburg State University, Saint Petersburg,199034, Russian Federation

Abstract

2452-3216 © 2016, PROSTR (Procedia Structural Integrity) Hosting by Elsevier Ltd. All rights reserved.

Peer-review under responsibility of the Scientific Committee of ECF21.

10.1016/j.prostr.2016.06.196

20th century FeMn-based alloys undergoing reversible fcc-hcp martensitic transformation (Likhachev et al. (1975), Otsuka (1998)) and demonstrating rather small shape memory and corrosion stability, provoked new interest of scientists after the works of Sato (1982) describing the shape memory effect in Fe-Mn-Si alloy. It was shown that the more was the Si content, the more was the recovery ratio. Investigations of three-, four- and multicomponent alloys during the recent decade lead to a great interest in FeMnSi-based alloys in connection with their possible applications (Wang (2007), Li and Dunne (1997), Sawaguchi et al. (2015)). For example, after the proper thermomechanical treatment practically perfect shape memory effect was observed in Fe-Mn-Si-Cr-Ni-Nb-C alloy with recoverable strain up to 3.5 % (Wang (2007)). Substitution of 2 % manganese atoms by Cu and Al atoms results in a growth of both corrosion stability and shape recovery ratio in Fe-30Mn-6Si alloy (Li and Dunne (1997)). It was also shown that the Fe-30Mn-(6-x)Si-xAl (x=0-6 wt.%) alloys demonstrate extremely large fatigue life: 8000 cycles at strain amplitude 2 %, Nikulin et al. (2015). All these properties complemented by good machinability and low price compared to TiNi make FeMnSi-based alloys very attractive for using as working elements of thermomechanical coupling, reinforcing parts and vibration protection devices for large-size structures Sawaguchi et al. (2006), Nikulin et al. (2015)).

For successful and reliable application of FeMn-based SMAs as well as for the prediction of new properties one needs both experimental studies and models allowing calculation of mechanical behavior in various temperature and stress conditions. There exist a number of microscopic and macroscopic models for simulation of the deformation of TiNi-type SMA specimens (Patoor et al. (1996), Huang and Brinson (1998), Evard and Volkov (1999) and others). For Fe-based SMA one cannot find such abundant variety. In the frames of a phenomenological two-level synthetic model Goliboroda et al. (1999) it was supposed that the yield stress of a material in the two-phase state depends on the respective amount of the martensite. A constitutive model for Fe-based SMA was proposed by Khalil et al. Khalil et al. (2012). It describes the effect of the phase transformation, plastic sliding, and their interaction. The internal variables of this model are the volume fraction of martensite and the plastic deformation. In the mentioned works (Goliboroda et al. (1999), Khalil et al. (2012)) the results of description of isothermal deformation behaviour of Fe-based alloy were presented.

One of the specific features of fcc - hcp transformations is the multi-variance of both direct and reverse transformations. In the frames of microstructural model (Evard and Volkov (1997)) we tried to take into account this fact by introducing the assumptions, first, of the existence of a maximum size of martensitic crystal and, second, of the possibility of the reverse transformation of a martensite crystal by a deformation not equal to the inverse of that, by which this martensite crystal appeared. The second assumption means that the principle of the "exactly back" reverse transformation is not valid for fcc - hcp - fcc transformations. In the work (Evard and Volkov (1999)) the multi-variance of the transformation was described in a more physically substantiated way by taking into account the symmetry of the fcc and hcp lattices and the symmetry of the transformation strain tensor. In the present work this approach was used to calculate the phase deformation while the microplastic deformation was calculated alongside with the densities of the scattered and oriented deformation defects similarly how it was done by Volkov et al. (2015) for TiNi alloy.

2. Model

In the frames of the microstructural model the representative volume was considered to consist of grains characterized by orientations ra of the crystallographic axes. In each of these grains there can appear N crystallographically equivalent variants of martensitic crystals. The fcc ^ hcp transformation is realized by one of the three simple shears by 1/6 < 112 > fcc vector on each second {111}fcc plane. These N =12 variants of martensite are characterized by transformation strain tensors Dn and martensite quantities ®n. (n = 1,2,...,N). At the reverse transformation each of three shears 1/3 < 1120 > hcp restores the initial orientation of austenite. Thus, following to Evard and Volkov (1999) one can divide all variants into four triplets (zones) with parameters

^zone = 1 V1 ^ , z = 1, 2, 3, 4,

J n=3 z-2

characterizing the amount of martensite belonging to a zone, (1/4) ®z being the volume fraction of martensite of this zone. Quantities ®n themselves do not have physical meaning of any volume fractions of martensite. Still, ®n serve as the measures of the volumes transformed by the phase deformation Dn.

The approximation of the small-strain theory and the Reuss' hypothesis were used for calculation of the strain tensor e of the representative volume:

e = S/i£gr(Wi), (2a)

£gr = £gr e + £gr T + £gr Ph + £gr MP (2b)

Here f and ssr(ra,) are the volume fraction and the strain of a grain with the orientation of the crystallographic axes ra,-, the sum is taken over all grains and a grain strain egr is considered as the sum of elastic egre, thermal egrT, phase 6grPh and micro plastic egrMP deformations.

The elastic strain egre was calculated by the Hook's law and the use of the "mixture rule" corresponding to the Reuss' approach. : The thermal strain 6grT was calculated in a similar way by the isotropic expansion law.

The phase strain for each martensite variant is the Bain's deformation Dn realizing the transformation of the lattice and (1/N)®n is the weight of the n-th variant in the total phase strain:

£gr Ph = 1/W(£N=i^n Dn). (3)

Micro plastic strains due to the accommodation of martensite are caused by the incompatibility of the phase strains. According to (Volkovetal. (2015)) we assumed that the phase strain of a Bain's variant activates a combination of slips producing a strain proportional to the deviator of the phase strain. Thus, for the total micro plastic strain of a grain one can write:

1 N ...

sMP gr = — Yks pD (4)

N^i n n

1 v n=1

where internal variables snp are measures of the micro-plastic strains, devDn is the deviator part of tensor Dn, k is a material constant setting the scale of the microplastic strain measures snp.

To formulate the evolution equations for the variables ®n and snp we consider the Gibbs' potential of a grain consisting of the two-phases:

G = (1 - ®gr)CA +OnGMn + Gmix , (5)

where GA and GMn are the eigen potentials of austenite and n-th variant of martensite (potentials of the phases if they were not interacting), GmK is the potential of mixing, which is the elastic inter-phase stress energy. The eigen potentials can be written as

Ga=Ga-sa(t-T0)-(T2-T°)2-ej-2Da^^,,a = A-Mn- (6)

where superscript a=A stands for austenite and a=Mn - for the n-th variant of martensite; T0 is the phase equilibrium temperature (i.e. such temperature, at which GA = GMn); G0a and S0a (a=A, Mn) are the Gibbs' potentials and the entropies at stress ct=0 and temperature T=T0; ei^Ta (a=A, Mn) are strains of the phases at ct=0; cnaand Daijki (a=A, Mn) are the specific heat capacities at constant stress and the elastic compliances. For T0 we use an estimate proposed by Salzbrenner and Cohen (1979): T0 = (Ms+Af)/2 (hereinafter Ms, Mf, As, Af are the characteristic temperatures and q0 is the latent heat of the transformation).

To estimate the inter-phase stress energy Gmvc we take into account that it grows with the phase deformations characterized by variables ®n and it is decreases by oriented defects bn. Thus,

cmix = 2 M<£n-6n)2, (7)

where ^=q0((Mf-Ms)/T0).

The thermodynamic force causing transformation by strain Dn is the derivative of the Gibbs' potential on the ®n:

Fn = —N —. (8)

n 3on v '

When transformation is in progress a moving interface experiences a resistance force because of the energy barriers of martensite crystal nucleation and other obstacles. The corresponding dissipative force responsible for the existence of the hysteresis we refer to as the friction force F and assume that it has a constant absolute value and hinders the transformation. So we can write the transformation condition as

Fn = ±Fr, (9)

where the plus sign is for the direct and minus - for the reverse transformation. The value of Ffr is derived from the transformation characteristics: Ffr =q0(Ms-T0)/T0.

In addition, for iron-manganese alloys we have two extra conditions of transformation: (1) direct transformation cannot occur to make the volume fraction of martensite in a grain ®®r>1; (2) martensite fraction in a zone (1/4) ®z cannot become less than zero.

The process of mechanical twinning (reorientation) of martensite "through virtual austenite" was also considered. We proposed that the dissipative force Fr tw for the reorientation differs from that for the transformation. It was suggested that the reorientation of martensite in a grain can occur only if this grain is purely martensitic (®®r = 1) and the condition of the reverse martensitic transformation are not satisfied for every variant of martensite.

Condition (8) is insufficient for the determination of the increments of all internal variables. To find the variation law of variables bn following to Volkov et al. (2015) we formulate the micro-plastic flow conditions similarly to the classic plastic flow condition:

|FnP- Fnpl= Fny, |FnP|> 0, (10)

whereFnp is the generalized force conjugated with the parameters bn (F^ = —N Here Fny and Fnp are the forces describing the isotropic and kinematic hardening.

According to Evard et al. (2006) and Volkov et al. (2015) the deformation defects generated by the microplastic flow we divide in two groups: oriented defects bn and scattered defects fn, suggesting the evolution equations for them in the form:

bn = % — (1 ¡¡f)\bn i^M), fn = \3\ (11)

where p* is a material constant.

It was assumed that the irreversible defects give rise to the isotropic hardening and the reversible ones - to the kinematic hardening. Thus, we relate the defect densities fn and bn to and by closing equations, which were chosen in the simplest linear form

= ayfn, Fnp = apbn, (12)

where ay and ap are material constants.

From condition (9) and (10) and formulae (8), (11), (12) evolution equations relating the increments of the internal variables ®n, bn, fn and e^ to the increments of stress and temperature are derived. Formulae (1) - (4) allow calculating the reversible and irreversible macroscopic strain.

3. Results of simulation

The values of the material constants specifying the elastic, thermal and phase deformation of SMA were chosen to reproduce the functional and mechanical behavior of a FeMn-type SMA. The values of all constants are collected in Table 1.

Table 1. Values of the material constants.

Material constant Value

Characteristic temperatures Mf, Ms, As, Af 320, 370, 470, 520 K

Latent heat q0 -65 MJ/m3

Number of martensite variants N 12

Lattice deformation matrix D by Schumann (1964) (to be symmetrize) È ( -2 1 1 -2 i)

Elastic modulus of austenite EA 200 GPa

Elastic modulus of martensite EM 200 GPa

Poisson's ratio of austenite vA 0.33

Poisson's ratio of martensite vm 0.33

Critical reorientationforce Ffr tw 50 MJ/m

Isotropic hardening factor ay 0.1 MPa

Kinematic hardening factor ap 2 MPa

Oriented defects saturation factor p* 1

All calculations were made for uniaxial tension. The heat expansion strain was neglected. The micro plastic deformation was taken into account with initial critical yield force Fy = 3 MPa which was the same for all martensitic variants. The macroscopic athermal plastic deformation at present is beyond the scope of the model.

The shape memory effect at heating after an active loading in martensitic state is presented in Fig. 1. One should note that the shape memory effect is not perfect even when the microplastic deformation is absent (the dashed line). This is due to the fact that the principle "exactly back" is not valid for the reverse transformation for multi-variant hcp ^ fcc phase transformation. The micro plastic deformation leads to increase of the irreversible strain.

300 400 500 600 temperature, K

Fig. 1. Recovery of strain after an active loading in martensitic state up the stress 200 MPa and unloading. a

10E 50--1---r-■-p-. u->-1-•- I ■ P-

200 400 600 200 400 600

temperature, K temperature, K

Fig. 2. Dependence of strain (a) and density of f-defects (b) on temperature at cooling under the stress100 MPa and

heating without stress.

Fig 3. Dependence of strain on temperature at cooling and heating under the constant stress 100 MPa.

Fig. 4. Stress strain diagrams for the model specimen in martensitic state.

Figure 2a presents the results of simulation of the strain variation at cooling under the stress 100 MPa and heating without an applied stress. The irreversible strain in one cycle decreases with the number of cycles tending to some small value. Accumulations of densities of the scattered defects f for one of the grains is presented in Fig. 2 b.

Figure 3 illustrates the strain variation at thermocycling under a constant stress. Results of modeling of mechanical loading in the martensitic state are presented in Fig. 4.These results qualitatively agree with direct observations of iron-manganese alloys obtained by Likhachev et al. (1975).

4. Conclusion

The developed microstructural model is suitable for description of functional and mechanical properties of FeMn-based shape memory alloys. It allows simulating accumulation of the phase strain at cooling under stress, shape memory effect and martensite reorientation. All the presented results qualitatively agree with the experimental data for FeMn-based alloys. Accounting the athermal plastic deformation and internal stress produced by all kinds

Fig.3. Dependence of strain on temperature at cooling and heating under the constant stress 100 MPa.

of incompatibilities will make possible to receive both qualitative and quantitative agreement with experimental data.

Acknowledgements

This research was supported by the grants of Russian Foundation of Basic Research 15-01-07657and 16-01-00335.. References

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