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

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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 Thermo-mechanical modeling of aaa, high pressure blade of aa an a*, Aleksandr Margarita E. Evarda* A. Volkov Fedor S. Belyaevaa,turbine Anna D. Ignatova , airplane gas Natalia A. turbine Volkovaaa engine

XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal

a aSaint

Russian a Saint b c Saint Petersburg Petersburg State State University, University, Saint Petersburg,199034, Petersburg,199034, Russian Federation Federation

P. Brandão , V. Infante , A.M. Deus *

a

Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa, An is behaviour Portugal An approach approach is presented presented to to describe describe the the functional functional and and mechanical mechanical behaviour of of FeMn-based FeMn-based shape shape memory memory alloys alloys undergoing undergoing c Department of Mechanical Engineering, Instituto Superior Técnico, Universidade Lisboa, Rovisco Pais, 1, 1049-001 Lisboa, fcc hcp transformation. The of reverse transformation is taken into consideration. The martensitic fcc --CeFEMA, hcp phase phase transformation. The multi-variance multi-variance of the the reverse transformation is de taken intoAv. consideration. The martensitic Portugal transformation and the micro plastic deformation due to the plastic accommodation of martensite are considered on the

Abstract Abstract b

transformation and the micro plastic deformation due to the plastic accommodation of martensite are considered on the microscopic level. microscopic level. The The micro micro plastic plastic deformation deformation is is described described from from the the point point of of view view of of the the plastic plastic flow flow theory. theory. Isotropic Isotropic hardening and kinematic hardening are taken into account and are related to the densities of scattered and oriented deformation hardening and kinematic hardening are taken into account and are related to the densities of scattered and oriented deformation Abstract defects. defects. The The thermodynamic thermodynamic forces forces causing causing growth growth of of martensite martensite and and reversible reversible deformation deformation defects defects are are the the derivatives derivatives of of the the Gibbs’ potential on the respective internal variables. The macro deformation of the representative volume of the polycrystal is Gibbs’ potential on the respective The macro deformation theincreasingly representative volume ofoperating the polycrystal is During their operation, moderninternal aircraftvariables. engine components are subjectedof to demanding conditions, calculated by averaging all micro strains. The results show a good qualitative agreement with available experimental data. calculated by averaging all micro strains. The results show a good qualitative agreement with available experimental data. especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent © 2016 2016 The Authors. Authors. Published by Elsevier Elsevier B.V.using the finite element method (FEM) was developed, in order to be able to predict © The by B.V. degradation, one ofPublished which is creep. A model © 2016, PROSTR (Procedia Structural Integrity) Hosting by Elsevier Ltd. All rights reserved. Peer-review under responsibility of the Scientific Scientific Committee of ECF21. ECF21.for a specific aircraft, provided by a commercial aviation Peer-review under responsibility of the Committee the creep behaviour of HPT blades. Flight data Peer-review under responsibility of the Scientific Committee ofrecords ECF21.of(FDR) company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model Keywords:FeMn, FeMnSi, memory, plasticity, needed for the FEM shape analysis, a HPT bladedefects. scrap was scanned, and its chemical composition and material properties were Keywords:FeMn, FeMnSi, shape memory, plasticity, defects. obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a 1. Introduction 1. overall Introduction model can be useful in the goal of predicting turbine blade life, given a set of FDR data.

Non-trivial Non-trivial properties properties of of alloys alloys undergoing undergoing martensitic martensitic phase phase transformations transformations have have attracted attracted attention attention of of © 2016 The Authors. Published by Elsevier B.V. researchers and designers for decades (Funakubo (1987), Otsuka (1998)).. Unusual mechanical properties of shape researchers and designers for decades (Funakubo (1987), Otsuka (1998)).. Unusual mechanical properties of shape Peer-review under responsibility of the Scientific Committee of PCF 2016. memory memory alloys alloys (SMA) (SMA) made made possible possible creation creation of of unique unique products products and and technologies. technologies. Substantially Substantially it it was was enhanced enhanced by by rapid development of materials science, especially of new methods of alloying and thermomechanical rapid development of materials science, especially new 3D methods of alloying and thermomechanical treatment treatment Keywords: High Pressure Turbine Blade; Creep; Finite Elementof Method; Model; Simulation. allowing allowing production production of of SMA SMA with with preset preset parameters parameters (Brailovski (Brailovski et et al., al., 2008). 2008). Thus, Thus, scrutinized scrutinized by by the the end end of of the the

* * Corresponding Corresponding author. author. Tel.: Tel.: +7-812-428-4220; +7-812-428-4220; fax: fax: +7-812-428-7079. +7-812-428-7079. E-mail E-mail address: address: [email protected] [email protected] 2452-3216 2452-3216 © © 2016 2016 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. * Corresponding Tel.: +351of Peer-review under responsibility the Peer-review underauthor. responsibility of218419991. the Scientific Scientific Committee Committee of of ECF21. ECF21. E-mail address: [email protected] 2452-3216 © 2016 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the Scientific Committee of PCF 2016. 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

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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 ω 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 Φn. (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 Φ zone z=

1 3z ∑ Φ n , z = 1, 2, 3, 4, 3 n= 3 z − 2

(1)

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characterizing the amount of martensite belonging to a zone, (1/4) Φz being the volume fraction of martensite of this zone. Quantities Φn themselves do not have physical meaning of any volume fractions of martensite. Still, Φn 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 ε of the representative volume: ε

gr

ε = Σ𝑓𝑓𝑖𝑖 εgr (ωi ), =ε + εgr T + εgr Ph + εgr MP gr e

(2a) (2b)

Here fi and εgr(ωi) are the volume fraction and the strain of a grain with the orientation of the crystallographic axes ωi, the sum is taken over all grains and a grain strain εgr is considered as the sum of elastic εgr e, thermal εgr T, phase εgr Ph and micro plastic εgr MP deformations. The elastic strain εgr e was calculated by the Hook’s law and the use of the “mixture rule” corresponding to the Reuss’ approach. : The thermal strain εgr T 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)Φn is the weight of the n-th variant in the total phase strain: n εgr Ph = 1/𝑁𝑁(∑N n=1 Φn D ).

(3)

Micro plastic strains due to the accommodation of martensite are caused by the incompatibility of the phase strains. According to (Volkov et al. (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 MP gr (4) ε= κε p D N

∑ n =1

n

n

where internal variables εn are measures of the micro-plastic strains, devDn is the deviator part of tensor Dn, κ is a material constant setting the scale of the microplastic strain measures εnp. To formulate the evolution equations for the variables Φn and εnp we consider the Gibbs’ potential of a grain consisting of the two-phases: p

1

M𝑛𝑛 + 𝐺𝐺 mix , 𝐺𝐺 = (1 − Φgr )𝐺𝐺 A + ∑𝑁𝑁 𝑛𝑛=1 Φ𝑛𝑛 𝐺𝐺 𝑁𝑁

(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), Gmix is the potential of mixing, which is the elastic inter-phase stress energy. The eigen potentials can be written as a G= G0a − S0a (T − T0 ) −

cσa (T − T0 ) 2 0 a 1 a − εij (T )σij − Dijkk σij σ kl , a = A, Mn, 2T0 2

(6)

where superscript ɑ=A stands for austenite and ɑ=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 (ɑ=A, Mn) are the Gibbs’ potentials and the entropies at stress σ=0 and temperature T=T0; εij0Ta (ɑ=A, Mn) are strains of the phases at σ=0; cσaand Daijkl (ɑ=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 Gmix we take into account that it grows with the phase deformations characterized by variables Φn and it is decreases by oriented defects bn. Thus, 1

where µ=q0((Mf–Ms)/T0).

𝐺𝐺 mix = µ(𝛷𝛷𝑛𝑛 − 𝑏𝑏𝑛𝑛 )2, 2

(7)



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The thermodynamic force causing transformation by strain Dn is the derivative of the Gibbs’ potential on the Φn: 𝐹𝐹𝑛𝑛 = −𝑁𝑁

𝜕𝜕𝐺𝐺

𝜕𝜕Φ𝑛𝑛

.

(8)

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 Ffr and assume that it has a constant absolute value and hinders the transformation. So we can write the transformation condition as Fn = ±Ffr,

(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 Φgr>1; (2) martensite fraction in a zone (1/4) Φz 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 Ffr 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 (Φgr = 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|> 0, (10) |Fnp – Fnρ| = Fny, 𝜕𝜕𝐺𝐺

p

). Here Fny and Fnρ are the forces whereFnp is the generalized force conjugated with the parameters bn (𝐹𝐹𝑛𝑛 = −𝑁𝑁 𝜕𝜕𝜕𝜕𝑛𝑛 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: 𝑏𝑏̇ 𝑛𝑛 = ε̇ p𝑛𝑛 − (1/β∗ )|𝑏𝑏𝑛𝑛 |ε̇ p𝑛𝑛 𝐻𝐻(𝑏𝑏𝑛𝑛 ε̇ p𝑛𝑛 ),

𝑓𝑓𝑛𝑛̇ = |ε̇ p𝑛𝑛 |

(11)

where β* is a material constant. It was assumed that the irreversible defects give rise to the isotropic hardening and the reversible ones – to the 𝜌𝜌 𝑦𝑦 kinematic hardening. Thus, we relate the defect densities fn and bn to 𝐹𝐹𝑛𝑛 and 𝐹𝐹𝑛𝑛 by closing equations, which were chosen in the simplest linear form y

𝜌𝜌

𝐹𝐹𝑛𝑛 = 𝑎𝑎y 𝑓𝑓𝑛𝑛 , 𝐹𝐹𝑛𝑛 = 𝑎𝑎𝜌𝜌 𝑏𝑏𝑛𝑛 ,

(12)

where ay and aρ are material constants. From condition (9) and (10) and formulae (8), (11), (12) evolution equations relating the increments of the p internal variables Φn, bn, fn and ε𝑛𝑛 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.

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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) Elastic modulus of austenite EA

1 1 1 1 � 1 1 1� 12 −2 −2 −2

Elastic modulus of martensite EM

200 GPa

Poisson’s ratio of austenite νA

0.33

Poisson’s ratio of martensite νM

0.33

Critical reorientationforce Ffr tw

50 MJ/m

Isotropic hardening factor ay

0.1 MPa

Kinematic hardening factor aρ

2 MPa

Oriented defects saturation factor β*

1

200 GPa

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.

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

Fig. 2. Dependence of strain (a) and density of f-defects (b) on temperature at cooling under the stress100 MPa and heating without stress.



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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.

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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 Brailovski, V., Inaekyan, K., Prokoshkin, S., Korotitskiy, A., Glezer, A., 2008. Structure and properties features of nanostructured Ti-Ni SMA after severe plastic deformation and post-deformation annealing, SMST-2007 - Proceedings of the International Conference on Shape Memory and Superelastic Technologies, Tsukuba; Japan, 17-26. Evard, M.E., Volkov, A.E., 1997. Computer simulation of the shape memory effects in Fe-Mn type alloys accounting for the features of the FCC - HCP phase transformation, Proceedings of SPIE - The International Society for Optical Engineering 3345, 178-183. Evard, M.E., Volkov, A.E., 1999. Modeling of martensite accommodation effect on mechanical behavior of shape memory alloys, Proceedings of SPIE - The International Society for Optical Engineering 3687, 330-334. Evard, M. E., Volkov, A. E., 1999. Modeling of martensite accommodation effect on mechanical behaviour of shape memory alloys, J. of Eng. Mater. and Technol. 121, 102-104. Evard, M.E., Volkov, A.E., Bobeleva, O.V., 2006. An approach for modelling fracture of shape memory alloy parts, Smart Structures and Systems 2, 357-363. Funakubo, H. (Ed.), “Shape Memory Alloys”, 1986. Gordon and Breach Science Publishers, New York. Huang M., Brinson L.C., 1998. A multivariant model for single crystal shape memory alloy behaviour, J. Mech. Phys. Solids. 46, 1379 – 1409. Goliboroda, I, Rusinko, K., Tanaka, K., 1999. Description of an iron-based shape memory alloy: thermomechanical behavior in terms of the synthetic model, Computational Material Science, 13, 218–226. Khalil, W., Mikolajczak, A., Bouby, C., Zineb, T.B., 2012. A constitutive model for Fe-based shape memory alloy considering martensitic transformation and plastic sliding coupling: Application to a finite element structural analysis, Journal of Intelligent Material Systems and Structures 23, 1143-1160. Li, H., Dunne, D., 1997. New Corrosion Resistant lron-based Shape Memory Alloys, ISIJ International 37, 605-609. Likhachev, V.A., Rybin, V.V., Sokolov, O.G., Kuzmin, S.L., 1975. “Transformation plasticity and mechanical memory at torsion in ironmanganese steels” .A.F.Ioffe OLFTI RAS (preprint #489), Leningrad. [in Russian]. Nikulin, I., Sawaguchi, T., Ogawa, K., Tsuzaki, K., 2015. Microstructure Evolution Associated with a Superior Low-Cycle Fatigue Resistance of the Fe-30Mn-4Si-2Al Alloy, Metallurgical and Materials Transactions A 46, 5103-5113. Otsuka, K., Wayman, C.M. (Eds), “Shape memory materials”, 1998. Cambridge University Press, New York. Patoor, E., Eberhardt, A., Berveiller M., 1996. Micromechanical modelling of superelasticity in shape memory alloys, J. de Physique IV, C1 6, 277 – 292. Salzbrenner, R.J., Cohen, M., 1979. On the thermodynamics of thermoelastic martensitic transformations, Acta Metallurgica 27, 739-748. Sato, A., Chishima, E., Soma, K., Mori, 1982.T. Shape memory effect in gamma-epsilon transformation in Fe–30Mn–1Si alloy single crystals, Acta Metall. 30, 1177-1185. Sawaguchi, T., Kikuchi, T., Ogawa, K.,et al., 2006. Development of prestressed concrete using Fe-Mn-Si-based shape memory alloys containing NbC, Materials Transactions 47, 580-583. Sawaguchi, T., Nikulin, I. Ogawa, K. et al., 2015. Designing Fe-Mn-Si alloys with improved low-cycle fatigue lives, Scripta Materialia. 99, 4952. Schumann, H., 1964. Einflub von Zugspannungen und die F-martesitbildung in austenitischen Mangamstahlen, Neue Hutte, 4, 223-228. [in German]. Volkov A.E., Belyaev, F.S., Evard, M.E., Volkova, N,A., 2015. Model of the Evolution of Deformation Defects and Irreversible Strain at Thermal Cycling of Stressed TiNi Alloy Specimen, MATEC Web of Conferences 33, #03013. Wang, S.H., Wen, Y.H., Zhang, W., L,i, N., 2007. Improvement of shape memory effect in an Fe-Mn-Si-Cr-Ni-Nb-C alloy by NbC precipitated through ageing after pre-deformation, Journal of Alloys and Compounds 437, 208-210.