Journal of Alloys and Compounds - University of Tennessee...2018/06/10  · recently by nanoindentation measurements [15]. In these experi-ments an indenter with a very sharp tip,

lable at ScienceDirect

Journal of Alloys and Compounds 720 (2017) 466e472

Contents lists avai

Journal of Alloys and Compounds

journal homepage: http: / /www.elsevier .com/locate/ ja lcom

Compositional effects on ideal shear strength in Fe-Cr alloys

Luis Casillas-Trujillo, Liubin Xu, Haixuan Xu*

University of Tennessee, Department of Materials Science and Engineering, Knoxville, TN 37996, USA

a r t i c l e i n f o

Article history:Received 8 August 2016Received in revised form25 March 2017Accepted 15 May 2017Available online 17 May 2017

Keywords:Compositional effectsIdeal shear stressDensity functional theoryMagnetismFeCr

* Corresponding author.E-mail address: [email protected] (H. Xu).

http://dx.doi.org/10.1016/j.jallcom.2017.05.1670925-8388/© 2017 Elsevier B.V. All rights reserved.

a b s t r a c t

The ideal shear strength is the minimum stress needed to plastically deform a defect free crystal; it is ofengineering interest since it sets the upper bound of the strength of a real crystal and connects to thenucleation of dislocations. In this study, we have employed spin-polarized density functional theory tocalculate the ideal shear strength, elastic constants and various moduli of body centered cubic Fe-Cralloys. We have determined the magnetic ground state of the Fe-Cr solid solutions, and noticed thatcalculations without the correct magnetic ground state would lead to incorrect results of the lattice andelastic constants. We have determined the ideal shear strength along the 〈111〉{110} and 〈111〉{112} slipsystems and established the relationship between alloy composition and mechanical properties. Weobserve strengthening in the 〈111〉{110} system as a function of chromium composition, while there is nochange in strength in the 〈111〉{112} system. The observed differences can be explained by the responseof the magnetic moments as a function of applied strain. This study provides insights on how electronicand magnetic interaction of constituent alloying elements may influence the properties of the resultingalloys and their dependence on alloy compositions.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

Fe-Cr alloys, among other iron based alloys, are currently beingused as critical structural materials for various energy applications[1e4]. In addition, they are candidates for structural componentsfor next-generation fission and fusion energy systems [5]. Theseapplications require alloys to work in extremely challenging con-ditions, demanding improved mechanical properties and radiationresistance [6,7]. To achieve optimized performance for these ap-plications, it is essential to fundamentally understand the compo-sitional effects on the mechanical properties of these alloys.

The ideal shear strength (ISS) is defined as the stress that ex-ceeds the limit of elastic stability in a defect-free crystal. It is therequired stress to move layers of atoms on top of each other and isrelevant to the dislocation nucleation process. Therefore the ISS isan important parameter in plasticity and fracture [8,9]. Moreover,during extremely high strain rate events, the ISS represents thelattice resistance to dislocation motion [10,11]. In practice, it is ofengineering interest since the ISS provides an upper limit of thestrength of a material. Studies of the ISS have been performed sincethe 1920's [12e14]. However, the requirement of a defect-free

sample made the experimental determination of the ISS difficult.In fact, experimental values of the ISS have been obtained onlyrecently by nanoindentation measurements [15]. In these experi-ments an indenter with a very sharp tip, usually between 50 nmand 1 mm, is pressed against a material with a low defect density.The volume under the indenter can be considered defect-free. Thevalue of the stress required to initiate deformation is either theideal shear strength or the stress required to nucleate dislocationshomogeneously [16]. Similar deformation processes also happenduring the hardness measurement, except the latter is at a muchlarger length scale.

The requirements of a periodic defect-free crystal make firstprinciples calculations a suitable way to estimate the ISS [17]. Thus,the ideal strength is one of the mechanical properties that arepredictable by computational means [16]. Ab initio computations ofthe ISS of several materials have been performed, mostly on singleelement systems: pure bcc metals [18e20], pure fcc metals, andsemiconductors [21]. For instance, studies on ferromagnetic bcciron have been done by Clatterbuck et al. [22]. Ogata et al. [23]explored ferromagnetic nickel and iron. In addition, calculationshave been done in ceramics, carbides, nitrides [24,25], and recentlyin TiVNbMo high entropy alloys [26]. However, to our knowledgeno studies of magnetic alloy systems have been performed,partially because calculations of the ISS in magnetic systems arechallenging. First, the magnetic ground state of the alloy must be

mailto:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.jallcom.2017.05.167&domain=pdfwww.sciencedirect.com/science/journal/09258388http://www.elsevier.com/locate/jalcomhttp://dx.doi.org/10.1016/j.jallcom.2017.05.167http://dx.doi.org/10.1016/j.jallcom.2017.05.167http://dx.doi.org/10.1016/j.jallcom.2017.05.167

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 467

determined, and secondly, the magnetic moments will experiencefrustration; the magnetic moments of chromium atoms would liketo point in the opposite direction to the iron magnetic moments,and at the same time in the opposite direction to other chromiumatoms, as described by Soderlind et al. [27].

In this work, the compositional effects on ISS in bcc Fe-Cr systemare investigated using density functional theory calculations. Thechallenge associated with theoretical investigation of Fe-Cr hasbeen widely recognized, due to the subtle but critical magneticinteractions in the alloy systems. Specifically, we have determinedthe ISS of magnetic Fe-Cr up 12.9% Cr along the 〈111〉{110} and 〈111〉{112} slip systems. To determine the electronic and magnetic originof the observed trend, the cohesive energy, elastic constants, andseveral elastic moduli of Fe-Cr are also obtained and comparedwithprevious studies.

2. Methodology

2.1. Ideal shear stress

Shear deformation can be applied by two different ways: affineand alias [7]. In the affine shear deformation, all atoms are shiftedparallel to the direction of shearing. Practically, only lattice vectorsare changed, while the fractional coordinates of the atoms withinthe supercell remain the same. In the alias deformation only the toplayer of the cell is initially displaced in the shear direction, while theatoms in all other layers remain in their original positions. Thedisplacement of the top layer initially only influences the layer nextto it, but atomistic relaxation leads to the propagation of thedisplacement from top to bottom. Comparing these two methods,the affine deformation gives a homogeneous deformation of thelattice, while the alias deformation depends on the shear plane. Wehave chosen to use the affine shear, since it offers a homogeneousdeformation, which is well suited to the alloy systems.

Fig. 1. Transformations used to apply the shear deformation. A is the basis vector matrix, D ithe slip system as basis vectors. S¡1 is the inverse transformation, used to convert the systecalculations, where the basis vectors are associated with the slip system. (b) The coordinate scrystal system reference.

In this work, we have calculated the ideal shear strength byapplying incremental affine shears to the crystal along the bcccommon slip systems, 〈111〉{110} and 〈111〉{112}. The ISS t is ob-tained by calculating the energy and volume as a function of thestrain, and taking the derivative of the energy E with respect to thestrain g, shown in Equation (1). The ideal shear strength is themaximum of the stress-strain curve [28].

To apply the shear strain it is convenient to use a coordinatesystem defined by the slip system. This Cartesian coordinate systemhas unit vectors ei(i ¼ 1,2,3), where e1 is pointing in the slip di-rection, and e2 and e3 are perpendicular to e1. This transformationconverts the cubic unit cell to an orthorhombic supercell, with thelattice vectors aa(a ¼ 1,2,3) pointing in the direction of the unitvectors ei. To apply the shear strain in the slip plane perpendicularto a3, the supercell is deformed by applying a deformationmatrix Dwith a non-zero shear element.

We have chosen an alternative way to perform the ISS calcula-tions in order to preserve cubic symmetry. Instead of transformingthe coordinate system to the slip system reference, we transformthe deformation matrix to the cubic crystal system. Fig. 1a showsthe usual transformation used for ISS calculations, a cubic systemwith basis vectors ð100Þ, ð010Þ, and ð001Þ is transformed to a co-ordinate system with basis vectors ð111Þ, ð110Þ, and ð112Þ, withð111Þ pointing in the slip direction, and with vectorsð110Þ, andð112Þ perpendicular to the slip planes. The deformation is appliedto the orthogonal supercell. Fig. 1b is the transformation used inthis work. The process is similar to that described above, but aninverse transformation is applied to the deformation matrix toconvert it to the cubic crystal system.

t ¼ 1V

vEvg

(1)

s the deformation matrix, and S is the transformation matrix to a Cartesian systemwithm back to the cubic crystal system. (a) The coordinate system commonly used for ISSystem used in this work, where the deformation matrix transformed to the to the cubic

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472468

2.2. Elastic properties

The elastic properties of cubic single crystals are described bythe three independent elastic constants C11, C12, and C44. We havecalculated the polycrystalline elastic moduli from their relationshipwith the elastic constants. For cubic solids, the bulk-modulus (B) isequivalent to the polycrystalline bulk-modulus and its relationshipwith the elastic constants is given by Equation (2).

B ¼ ðC11 þ 2C12Þ3

(2)

The polycrystalline shear modulus (G) has been obtained by theHill averaging method [29],

G ¼ ðGV þ GRÞ2

(3)

where the Voigt (GV) and Reuss (GR) bounds are given by Ref. [30],

GR ¼5ðC11 � C12ÞC44

ð4C44 þ 3C11 � 3C12ÞGV ¼

ðC11 � C12 þ 3C44Þ5

(4)

Finally the Young's modulus (E), is connected to B and G byRef. [30],

E ¼ 9BGð3Bþ GÞ (5)

2.3. Computational details

The DFT calculations were carried out using the generalizedgradient approximation (GGA) with the PerdeweBurkeeErnzerhoffunctional [31]. The electroneion interaction is described by theprojector augmented wave method (PAW) [32], and the plane wavebasis energy cutoff was set to 500 eV. The Vienna ab initio packageVASP [33,34] was employed to perform the simulations. For the Fe-Cr simulations, we employed a 3 � 3 � 3 supercell with a 2 � 2 � 2k-point sampling grid. The force on each atom is relaxed to less than0.01 eV/Å. To assess magnetism, spin polarized DFT was employed.The effect of local configuration becomes more pronounced inmagnetic systems due to the frustration of the magnetic moments.We have averaged the results of three different configurationsgenerated using the special quasirandom structures approach (SQS)

Fig. 2. Energy versus lattice constant for non-magnetic, antiferromagnetic, fer

[35]. The search of the SQS was restricted to the 54 atom 3 � 3 � 3cubic supercell using the mcsqs [36] program of the Alloy TheoreticAutomated Toolkit (ATAT) [37].

3. Results and discussion

3.1. Magnetic ground state, lattice constant and cohesive energies ofFe-Cr alloys

Beforewe determine themechanical properties of the alloys, theground state properties are examined first. The magnetic state ofthe binary alloy will be influenced by the magnetic states of theparent elements, and the resulting state is a function of thecomposition of the alloy. In solid solution, it is known that chro-mium atoms will try to align their magnetic moment in theopposite direction to the iron atoms, but at the same time they willalso want to have it opposite to other chromium atoms, leading tomagnetic frustration in the lattice as the chromium concentrationincreases. Olsson et al. showed that for concentrations 〈20% chro-mium, the system is well described with a bcc structure in theferromagnetic state [38e40], with a Curie temperature around900e1500 K [41]. The spinodal decomposition clustering occurs athigh chromium concentrations [42,43]. We focus on the region ofthe phase diagram where Fe-Cr forms single phase solid solutions.At 300� C bcc iron forms a solid solution up to 9% chromium; athigher temperatures chromium is completely soluble in iron for allcompositions [44]. Previous theoretical studies have performedcalculations for the whole compositional range of Fe-Cr alloys. Wehave performed calculations up to 12.9% chromium.

We initially calculated the magnetic ground state for the parentelements, iron and chromium, using collinear spin calculations andconsidering ferromagnetic, antiferromagnetic, and nonmagneticconfigurations. Fig. 2 shows the energy versus lattice constant fordifferent magnetic states of iron (Fig. 2a), and chromium (Fig. 2b).Our results are in agreement with previous experimental andtheoretical results. The magnetic state with the lowest energy forbcc iron is ferromagnetic, which is consistent with [45,46]. Chro-mium possesses a spin density wave antiferromagnetic ordering[47]. We found that the antiferromagnetic configuration possessesthe lowest energy for chromium. In addition, we performed non-collinear spin calculations for both iron and chromium, as can beseen in Fig. 2, the difference in energy between the lowest collinearand non-collinear calculations is very small, ~0.003 eV, which is in

romagnetic and non-collinear calculations for (a) bcc iron, (b) chromium.

Table 1Lattice constant of iron and chromium for different magnetic states.

Antiferromagnetic(Å)

Ferromagnetic(Å)

Non-magnetic(Å)

Non-collinear(Å)

Fe 2.8030 2.8310 2.7501 2.83076Cr 2.8625 2.8376 2.8374 2.8502

Fig. 4. Mixing energy of Fe-Cr. We compare the results obtained in this work withprevious theoretical studies.

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 469

agreement with the values reported by Hafner et al. [48]. However,the non-collinear feature did not change the overall magneticconfigurations. ISS calculations are more sensitive to the energydifference rather than absolute values of energies. Therefore,collinear spin calculations are used in this work, due to the smallenergy difference and computational costs associated with thenon-collinear calculations. Table 1 tabulates the value of the latticeconstant for the different magnetic states from Fig. 2.

Vegard's law establishes a linear trend between the latticeconstants of solid solutions and the concentration of the alloyingelements. This behavior is not observed in the Fe-Cr solid solutions.Experimental measurements of the lattice constant as a function ofcomposition do not follow Vegard's law, and a clear linear trendcannot be established. This behavior can be appreciated in Fig. 3a,but it becomes more evident for the whole compositional range asin Ref. [49]. The non-linear behavior is accentuated in the theo-retical results shown in Fig. 3a with this study's DFT results, and theresults obtained with exact muffin-tin orbitals (EMTO) [50]. Thedashed line in Fig. 3 represents Vegard's law for the DFT data. Thelattice constant obtained from DFT is an underestimation for allcompositions compared with EMTO. However, both methods pre-dict qualitatively the same trend. A maximum in the lattice con-stant is obtained between 7.5 and 10% chromium in bothapproaches. We have identified that the alloy with the largest lat-tice constant corresponds to the configuration with the highestmagnetic moment of iron. We found there is no correlation be-tween the magnetic moment of chromium and the lattice constant.Fig. 3b plots the lattice constant and the magnetic moment of ironas a function of chromium concentration, the maximum value forboth parameters occurs at 9.25% chromium content. A similarbehavior has been reported for Fe-Ni alloys [51] where themaximum in lattice constant corresponds to the compositionwhere the magnetic moments reach their maximum value.

Based on the obtained ground states, the mixing energy as afunction of chromium composition is calculated and shown inFig. 4, and compared with previous results from theoretical calcu-lations for the low chromium concentration regime [38,39,52].Quantitatively the value for each of these studies differs by a small

Fig. 3. (a) Lattice constant as a function chromium concentration, from experimental resultThe dashed line indicates Vegard's law for the DFT calculations. (b) The lattice constant asaverage magnetic moment for iron atoms in the right y axis.

amount, but all of them exhibit the same trend.The good agreement obtained in lattice constant and mixing

energy between the results of this work and previous studies in-dicates our simulations are able to reproduce the ground stateproperties of the Fe-Cr alloys. The obtained relaxed structuresrepresent a good starting point for the ISS and elastic constantcalculations.

3.2. ISS and elastic constants of Fe-Cr

Fig. 5a shows the Fe-Cr energy versus strain curve for differentchromium compositions of the 〈111〉{112} slip system. The ISS re-sults for both 〈111〉{112} and 〈111〉{110} slip systems are summa-rized in Fig. 5b. The ISS shows different behaviors between the twoslip systems. First, it can be noted that the magnitude of the ISS inthe 〈111〉{110} slip system is greater than that of the 〈111〉{112} slipsystem for all concentrations. Secondly, they exhibit a differentbehavior as a function of concentration. An increase in magnitudecan be observed in the 〈111〉{110} system as more chromium atomsare incorporated. The magnitude of the ISS for the 〈111〉{112} slipsystem remains almost constant as a function of chromiumcomposition.

s of Pearson [49], EMTO calculations of Zhang [50], and the DFT results from this work.a function of concentration for the current work is plotted in the left y axis, and the

Fig. 5. (a) Energy versus strain curve of Fe-Cr for different compositions for the 〈111〉{112} slip system. The overall energy of the system decreases as the concentration of chromiumrises. The ideal shear strength will be given by the maximum of the derivative of this curve. (b) Ideal shear stress as a function of chromium concentration for the 〈111〉{110} slipsystem and the 〈111〉{112} slip system.

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472470

In order to determine the origin of the nonlinear behavior of theISS as a function of strain in the 〈111〉{110} system, we carried outcalculations to determine the elastic properties of the alloy systemsto compare the elastic and plastic behavior of the alloys. The elasticconstants were calculated by performing six finite distortions of thelattice and deriving the elastic constants from the strain-stressrelationship [53]. The value of the elastic constants increases asthe concentration of chromium increases, and consequently themagnitude of the elastic moduli also increases, which means theelastic properties of iron are improved as chromium is added to thesystem up to 12.9%. Fig. 6a shows the compositional effect on theelastic constants. We have compared the DFT results from thisstudy with elastic constants obtained by EMTO [50]. In both cases,we observe an increase in an almost linear fashion. In the DFT case,results show an increase in C12 and C44, while a subtle increase isobserved in C11. The increase in value is more prominent for theEMTO results. The shear modulus and Young's modulus are shownas a function of composition in Fig. 6b. We again compare with theEMTO calculations, and with experimental results [54]. The resultsobtained in this study by DFT are in better agreement with theexperimental results than those reported by EMTO. The magni-tudes of the theoretical values are overestimated in comparison

Fig. 6. (a) Elastic constants of Fe-Cr as a function of Cr composition from this work calccomparison between experimental results [54], EMTO and DFT values from this work.

with the experimental values. The magnitude of the moduli in-creases almost linearly as a function of chromium composition. Wetherefore conclude that elastic properties and ISS show differentcompositional dependence and only the plastic deformationinvolving 〈111〉{110} slip system exhibits non-linear behavior.

We subsequently performed magnetic structure analysis andfound that the difference in behavior between the 〈111〉{112} andthe 〈111〉{110} can be rationalized by magnetic response of thesystem. Fig. 7 plots the change in magnetic moment for chromium,iron, and the total moment of the system as a function of chromiumcomposition.We observe a different behavior in the response of thechromium magnetic moments in these two slip systems. In the〈111〉{110} slip system (Fig. 7a) the chromium magnetic momentpresents a maximum, while for the 〈111〉{112} slip system (Fig. 7b)the opposite behavior is observed; the magnetic moment of chro-mium possesses a minimum. More importantly, in the 〈111〉{110}system, the magnetic moment of chromium changes with chro-mium composition; whereas in the 〈111〉{112} system the relativechange of chromium magnetic moment are the same for all com-positions. For the iron atoms, in the 〈111〉{110} slip system (Fig. 7c),similar behaviors are observed for different chromium concentra-tions, while in the 〈111〉{112} slip system (Fig. 7d) it can be observed

ulated using DFT, and previous EMTO calculations [50]. (b) Elastic and shear moduli

Fig. 7. Change in magnetic moment per atom as a function of strain, for chromium in the (a) 〈111〉{110} slip system, (b) 〈111〉{112} slip system; iron (c) 〈111〉{110} slip system, (d)〈111〉{112} slip system; and total magnetic moment of the system (e) 〈111〉{110} slip system, (f) 〈111〉{112} slip system.

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 471

the presence of a maximum for low chromium concentrations andtransitions to two local maximums and a local minimum as thechromium concentration increases. The total magnetic momentshows similar behavior as iron (Fig. 7e and f). In addition, thechromiummagnetic moment is sensitive to the local configuration,as it can be seen from the large error bars in the figure. Based onthese observations, we believe the difference in behavior of ISS canbe attributed to the magnetic response of chromium, since both ISSand change in chromium magnetic moment depend on alloycomposition in the 〈111〉{110} slip system and are independent ofchromium concentration in the 〈111〉{112} system. In comparison,both the magnetic moment of iron and total magnetic moment

change consistently with composition.Based on the above results, the addition of chromium improves

the strength of the alloy, which is consistent with previous studies[50]. It should be noted that solutes in bcc metals could lead tosoftening at very low temperatures [55]. One aspect that has notbeen considered in this study is that chromium atoms could alterthe structure of the dislocation core, giving rise to the ductile-to-brittle transition instead of hardening. Since we are treatingdislocation free crystals, the hardening observed in this study is dueto the changes in electronic and magnetic structure and the strainintroduced by substitutional impurities.

L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472472

4. Conclusions

In this work, we have established the relationship betweenmechanical properties and composition in Fe-Cr alloys, andassessed the impact of magnetism in these properties. We havecalculated the lattice constant, ideal shear stress, elastic constantsand cohesive energies for Fe-Cr (up to 12.9%) solid solutions as afunction of composition. The calculations of the lattice constant as afunction of composition present a maximum at 9.25% chromium,which corresponds to the composition with the maximum value ofthe iron magnetic moment. The addition of chromium to the Fe-Crsystem improves its elastic properties within the compositionalrange considered in this study. The magnitude of the shear andYoung's modulus also increase as more chromium is added to thesystem. Finally, the ISS for the two slip systems in question showdifferent behaviors. The ISS of the 〈111〉{112} slip system is inde-pendent of composition, whereas the ISS of the 〈111〉{110} slipsystem increases with chromium concentration. This difference inbehavior can be attributed to the response of the magnetic momentof chromium to the strain conditions in each of the systems.

Acknowledgements

We are thankful to Par Olsson for providing some of the inputfiles for comparing with previous DFT calculations. The research issponsored by the U.S. Department of Energy Nuclear Energy Uni-versity Program (NEUP) under project number 14-6346. Thisresearch used resources of The National Institute for ComputationalSciences at UT under contract UT-TENN0112.

References

[1] L.K. B�eland, Y.N. Osetsky, R.E. Stoller, H. Xu, Interstitial loop transformations inFeCr, J. Alloys Compd. 640 (2015) 219e225.

[2] C. Heintze, F. Bergner, A. Ulbricht, H. Eckerlebe, The microstructure ofneutron-irradiated FeeCr alloys: a small-angle neutron scattering study,J. Nucl. Mater. 409 (2011) 106e111.

[3] Z. Jiao, G. Was, Novel features of radiation-induced segregation and radiation-induced precipitation in austenitic stainless steels, Acta Mater. 59 (2011)1220e1238.

[4] H. Xu, R.E. Stoller, Y.N. Osetsky, D. Terentyev, Solving the puzzle of ˂100˃interstitial loop formation in bcc iron, Phys. Rev. Lett. 110 (2013) 265503.

[5] A. Hishinuma, A. Kohyama, R. Klueh, D. Gelles, W. Dietz, K. Ehrlich, Currentstatus and future R&D for reduced-activation ferritic/martensitic steels,J. Nucl. Mater. 258 (1998) 193e204.

[6] F. Garner, M. Toloczko, B. Sencer, Comparison of swelling and irradiation creepbehavior of fcc-austenitic and bcc-ferritic/martensitic alloys at high neutronexposure, J. Nucl. Mater. 276 (2000) 123e142.

[7] M. Jahn�atek, J. Hafner, M. Kraj�cí, Shear deformation, ideal strength, andstacking fault formation of fcc metals: a density-functional study of Al and Cu,Phys. Rev. B 79 (2009) 224103.

[8] Y. Umeno, M. �Cerný, Effect of normal stress on the ideal shear strength incovalent crystals, Phys. Rev. B 77 (2008) 100101.

[9] K.J. Van Vliet, J. Li, T. Zhu, S. Yip, S. Suresh, Quantifying the early stages ofplasticity through nanoscale experiments and simulations, Phys. Rev. B 67(2003) 104105.

[10] E.W. Hart, Lattice resistance to dislocation motion at high velocity, Phys. Rev.98 (1955) 1775e1776.

[11] W. Johnston, J.J. Gilman, Dislocation velocities, dislocation densities, andplastic flow in lithium fluoride crystals, J. Appl. Phys. 30 (1959) 129e144.

[12] J. Frenkel, Zur theorie der elastizit€atsgrenze und der festigkeit kristallinischerk€orper, Z. Phys. 37 (1926) 572e609.

[13] A. Kelly, N.H. Macmillan, Strong Solids, Oxford University Press, Walton Street,Oxford OX 2 6 DP, UK, 1986 (1986).

[14] M. Polanyi, The X-ray fiber diagram, Z Phys. 7 (1921) 149e180.[15] A. Gouldstone, H.-J. Koh, K.-Y. Zeng, A. Giannakopoulos, S. Suresh, Discrete and

continuous deformation during nanoindentation of thin films, Acta Mater. 48(2000) 2277e2295.

[16] C. Krenn, D. Roundy, M.L. Cohen, D. Chrzan, J. Morris Jr., Connecting atomisticand experimental estimates of ideal strength, Phys. Rev. B 65 (2002) 134111.

[17] W. Luo, D. Roundy, M.L. Cohen, J. Morris Jr., Ideal strength of bcc molybdenumand niobium, Phys. Rev. B 66 (2002) 094110.

[18] C. Krenn, D. Roundy, J. Morris, M.L. Cohen, Ideal strengths of bcc metals,Mater. Sci. Eng. A 319 (2001) 111e114.

[19] D. Roundy, C. Krenn, M.L. Cohen, J. Morris Jr., The ideal strength of tungsten,

Philos. Mag. A 81 (2001) 1725e1747.[20] W. Xu, J.A. Moriarty, Atomistic simulation of ideal shear strength, point de-

fects, and screw dislocations in bcc transition metals: Mo as a prototype, Phys.Rev. B 54 (1996) 6941.

[21] D. Roundy, M.L. Cohen, Ideal strength of diamond, Si, and Ge, Phys. Rev. B 64(2001) 212103.

[22] D. Clatterbuck, D. Chrzan, J. Morris, The ideal strength of iron in tension andshear, Acta Mater. 51 (2003) 2271e2283.

[23] S. Ogata, J. Li, N. Hirosaki, Y. Shibutani, S. Yip, Ideal shear strain of metals andceramics, Phys. Rev. B 70 (2004) 104104.

[24] S.-H. Jhi, S.G. Louie, M.L. Cohen, J. Morris Jr., Mechanical instability and idealshear strength of transition metal carbides and nitrides, Phys. Rev. Lett. 87(2001) 075503.

[25] Y.-J. Wang, C.-Y. Wang, A comparison of the ideal strength between L1 2 Co 3(Al, W) and Ni 3 Al under tension and shear from first-principles calculations,Appl. Phys. Lett. 94 (2009) 261909.

[26] F. Tian, D. Wang, J. Shen, Y. Wang, An ab initio investigation of ideal tensileand shear strength of TiVNbMo high-entropy alloy, Mater. Lett. 166 (2016)271e275.

[27] P. S€oderlind, J.A. Moriarty, J.M. Wills, First-principles theory of iron up toearth-core pressures: structural, vibrational, and elastic properties, Phys. Rev.B 53 (1996) 14063.

[28] D. Roundy, C. Krenn, M.L. Cohen, J. Morris Jr., Ideal shear strengths of fccaluminum and copper, Phys. Rev. Lett. 82 (1999) 2713.

[29] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. Sect. A65 (1952) 349.

[30] L. Vitos, Computational Quantum Mechanics for Materials Engineers: theEMTO Method and Applications, Springer Science & Business Media, 2007.

[31] J.P. Perdew, J. Chevary, S. Vosko, K.A. Jackson, M.R. Pederson, D. Singh,C. Fiolhais, Erratum: atoms, molecules, solids, and surfaces: applications of thegeneralized gradient approximation for exchange and correlation, Phys. Rev. B48 (1993) 4978.

[32] P.E. Bl€ochl, Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953.[33] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for

metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci.6 (1996) 15e50.

[34] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev.B 47 (1993) 558.

[35] A. Zunger, S.-H. Wei, L. Ferreira, J.E. Bernard, Special quasirandom structures,Phys. Rev. Lett. 65 (1990) 353.

[36] A. Van de Walle, P. Tiwary, M. De Jong, D. Olmsted, M. Asta, A. Dick, D. Shin,Y. Wang, L.-Q. Chen, Z.-K. Liu, Efficient stochastic generation of special qua-sirandom structures, Calphad 42 (2013) 13e18.

[37] A. van de Walle, Alloy Theoretic Automated Toolkit (ATAT), Cal Tech, Pasa-dena, CA, 2008.

[38] P. Olsson, I.A. Abrikosov, L. Vitos, J. Wallenius, Ab initio formation energies ofFeeCr alloys, J. Nucl. Mater. 321 (2003) 84e90.

[39] P. Olsson, C. Domain, J. Wallenius, Ab initio study of Cr interactions with pointdefects in bcc Fe, Phys. Rev. B 75 (2007) 014110.

[40] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, Z.-K. Liu, First-principles study ofbinary bcc alloys using special quasirandom structures, Phys. Rev. B 69 (2004)214202.

[41] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values ofthe Thermodynamic Properties of Binary Alloys, 1973. DTIC Document.

[42] J. Cieslak, S. Dubiel, B. Sepiol, M€ossbauer-effect study of the phase separationin the Fe-Cr system, J. Phys. Condens. Matter 12 (2000) 6709.

[43] S. Tavares, R. De Noronha, M. Da Silva, J. Neto, S. Pairis, 475 C embrittlement ina duplex stainless steel UNS S31803, Mater. Res. 4 (2001) 237e240.

[44] A. Handbook, Alloy Phase Diagrams, vol. 3, ASM International, Materials Park,OH, USA, 1992, 2.48.

[45] J. Kübler, Magnetic moments of ferromagnetic and antiferromagnetic bcc andfcc iron, Phys. Lett. A 81 (1981) 81e83.

[46] V. Moruzzi, P. Marcus, K. Schwarz, P. Mohn, Ferromagnetic phases of bcc andfcc Fe, Co, and Ni, Phys. Rev. B 34 (1986) 1784.

[47] L. Corliss, J. Hastings, R. Weiss, Antiphase antiferromagnetic structure ofchromium, Phys. Rev. Lett. 3 (1959) 211.

[48] R. Hafner, D. Spi�s�ak, R. Lorenz, J. Hafner, Magnetic ground state of Cr indensity-functional theory, Phys. Rev. B 65 (2002) 184432.

[49] W. Pearson, WB Pearson a Handbook of Lattice Spacings and Structures ofMetals and Alloys, vol. 1, Pergamon Press, London, 1958.

[50] H. Zhang, B. Johansson, L. Vitos, Ab initio calculations of elastic properties ofbcc Fe-Mg and Fe-Cr random alloys, Phys. Rev. B 79 (2009) 224201.

[51] G. Cacciamani, A. Dinsdale, M. Palumbo, A. Pasturel, The FeeNi system:thermodynamic modelling assisted by atomistic calculations, Intermetallics18 (2010) 1148e1162.

[52] G. Bonny, D. Terentyev, L. Malerba, The hardening of ironechromium alloysunder thermal ageing: an atomistic study, J. Nucl. Mater. 385 (2009) 278e283.

[53] Y. Le Page, P. Saxe, Symmetry-general least-squares extraction of elastic datafor strained materials from ab initio calculations of stress, Phys. Rev. B 65(2002) 104104.

[54] G. Speich, A. Schwoeble, W.C. Leslie, Elastic constants of binary iron-base al-loys, Metall. Trans. 3 (1972) 2031e2037.

[55] D.R. Trinkle, C. Woodward, The chemistry of deformation: how solutes softenpure metals, Science 310 (2005) 1665e1667.

http://refhub.elsevier.com/S0925-8388(17)31762-0/sref1http://refhub.elsevier.com/S0925-8388(17)31762-0/sref1http://refhub.elsevier.com/S0925-8388(17)31762-0/sref1http://refhub.elsevier.com/S0925-8388(17)31762-0/sref1http://refhub.elsevier.com/S0925-8388(17)31762-0/sref2http://refhub.elsevier.com/S0925-8388(17)31762-0/sref2http://refhub.elsevier.com/S0925-8388(17)31762-0/sref2http://refhub.elsevier.com/S0925-8388(17)31762-0/sref2http://refhub.elsevier.com/S0925-8388(17)31762-0/sref2http://refhub.elsevier.com/S0925-8388(17)31762-0/sref3http://refhub.elsevier.com/S0925-8388(17)31762-0/sref3http://refhub.elsevier.com/S0925-8388(17)31762-0/sref3http://refhub.elsevier.com/S0925-8388(17)31762-0/sref3http://refhub.elsevier.com/S0925-8388(17)31762-0/sref4http://refhub.elsevier.com/S0925-8388(17)31762-0/sref4http://refhub.elsevier.com/S0925-8388(17)31762-0/sref5http://refhub.elsevier.com/S0925-8388(17)31762-0/sref5http://refhub.elsevier.com/S0925-8388(17)31762-0/sref5http://refhub.elsevier.com/S0925-8388(17)31762-0/sref5http://refhub.elsevier.com/S0925-8388(17)31762-0/sref5http://refhub.elsevier.com/S0925-8388(17)31762-0/sref6http://refhub.elsevier.com/S0925-8388(17)31762-0/sref6http://refhub.elsevier.com/S0925-8388(17)31762-0/sref6http://refhub.elsevier.com/S0925-8388(17)31762-0/sref6http://refhub.elsevier.com/S0925-8388(17)31762-0/sref7http://refhub.elsevier.com/S0925-8388(17)31762-0/sref7http://refhub.elsevier.com/S0925-8388(17)31762-0/sref7http://refhub.elsevier.com/S0925-8388(17)31762-0/sref7http://refhub.elsevier.com/S0925-8388(17)31762-0/sref7http://refhub.elsevier.com/S0925-8388(17)31762-0/sref8http://refhub.elsevier.com/S0925-8388(17)31762-0/sref8http://refhub.elsevier.com/S0925-8388(17)31762-0/sref8http://refhub.elsevier.com/S0925-8388(17)31762-0/sref9http://refhub.elsevier.com/S0925-8388(17)31762-0/sref9http://refhub.elsevier.com/S0925-8388(17)31762-0/sref9http://refhub.elsevier.com/S0925-8388(17)31762-0/sref10http://refhub.elsevier.com/S0925-8388(17)31762-0/sref10http://refhub.elsevier.com/S0925-8388(17)31762-0/sref10http://refhub.elsevier.com/S0925-8388(17)31762-0/sref11http://refhub.elsevier.com/S0925-8388(17)31762-0/sref11http://refhub.elsevier.com/S0925-8388(17)31762-0/sref11http://refhub.elsevier.com/S0925-8388(17)31762-0/sref12http://refhub.elsevier.com/S0925-8388(17)31762-0/sref12http://refhub.elsevier.com/S0925-8388(17)31762-0/sref12http://refhub.elsevier.com/S0925-8388(17)31762-0/sref12http://refhub.elsevier.com/S0925-8388(17)31762-0/sref12http://refhub.elsevier.com/S0925-8388(17)31762-0/sref13http://refhub.elsevier.com/S0925-8388(17)31762-0/sref13http://refhub.elsevier.com/S0925-8388(17)31762-0/sref14http://refhub.elsevier.com/S0925-8388(17)31762-0/sref14http://refhub.elsevier.com/S0925-8388(17)31762-0/sref15http://refhub.elsevier.com/S0925-8388(17)31762-0/sref15http://refhub.elsevier.com/S0925-8388(17)31762-0/sref15http://refhub.elsevier.com/S0925-8388(17)31762-0/sref15http://refhub.elsevier.com/S0925-8388(17)31762-0/sref16http://refhub.elsevier.com/S0925-8388(17)31762-0/sref16http://refhub.elsevier.com/S0925-8388(17)31762-0/sref17http://refhub.elsevier.com/S0925-8388(17)31762-0/sref17http://refhub.elsevier.com/S0925-8388(17)31762-0/sref18http://refhub.elsevier.com/S0925-8388(17)31762-0/sref18http://refhub.elsevier.com/S0925-8388(17)31762-0/sref18http://refhub.elsevier.com/S0925-8388(17)31762-0/sref19http://refhub.elsevier.com/S0925-8388(17)31762-0/sref19http://refhub.elsevier.com/S0925-8388(17)31762-0/sref19http://refhub.elsevier.com/S0925-8388(17)31762-0/sref20http://refhub.elsevier.com/S0925-8388(17)31762-0/sref20http://refhub.elsevier.com/S0925-8388(17)31762-0/sref20http://refhub.elsevier.com/S0925-8388(17)31762-0/sref21http://refhub.elsevier.com/S0925-8388(17)31762-0/sref21http://refhub.elsevier.com/S0925-8388(17)31762-0/sref22http://refhub.elsevier.com/S0925-8388(17)31762-0/sref22http://refhub.elsevier.com/S0925-8388(17)31762-0/sref22http://refhub.elsevier.com/S0925-8388(17)31762-0/sref23http://refhub.elsevier.com/S0925-8388(17)31762-0/sref23http://refhub.elsevier.com/S0925-8388(17)31762-0/sref24http://refhub.elsevier.com/S0925-8388(17)31762-0/sref24http://refhub.elsevier.com/S0925-8388(17)31762-0/sref24http://refhub.elsevier.com/S0925-8388(17)31762-0/sref25http://refhub.elsevier.com/S0925-8388(17)31762-0/sref25http://refhub.elsevier.com/S0925-8388(17)31762-0/sref25http://refhub.elsevier.com/S0925-8388(17)31762-0/sref26http://refhub.elsevier.com/S0925-8388(17)31762-0/sref26http://refhub.elsevier.com/S0925-8388(17)31762-0/sref26http://refhub.elsevier.com/S0925-8388(17)31762-0/sref26http://refhub.elsevier.com/S0925-8388(17)31762-0/sref27http://refhub.elsevier.com/S0925-8388(17)31762-0/sref27http://refhub.elsevier.com/S0925-8388(17)31762-0/sref27http://refhub.elsevier.com/S0925-8388(17)31762-0/sref27http://refhub.elsevier.com/S0925-8388(17)31762-0/sref28http://refhub.elsevier.com/S0925-8388(17)31762-0/sref28http://refhub.elsevier.com/S0925-8388(17)31762-0/sref29http://refhub.elsevier.com/S0925-8388(17)31762-0/sref29http://refhub.elsevier.com/S0925-8388(17)31762-0/sref30http://refhub.elsevier.com/S0925-8388(17)31762-0/sref30http://refhub.elsevier.com/S0925-8388(17)31762-0/sref30http://refhub.elsevier.com/S0925-8388(17)31762-0/sref31http://refhub.elsevier.com/S0925-8388(17)31762-0/sref31http://refhub.elsevier.com/S0925-8388(17)31762-0/sref31http://refhub.elsevier.com/S0925-8388(17)31762-0/sref31http://refhub.elsevier.com/S0925-8388(17)31762-0/sref32http://refhub.elsevier.com/S0925-8388(17)31762-0/sref32http://refhub.elsevier.com/S0925-8388(17)31762-0/sref33http://refhub.elsevier.com/S0925-8388(17)31762-0/sref33http://refhub.elsevier.com/S0925-8388(17)31762-0/sref33http://refhub.elsevier.com/S0925-8388(17)31762-0/sref33http://refhub.elsevier.com/S0925-8388(17)31762-0/sref34http://refhub.elsevier.com/S0925-8388(17)31762-0/sref34http://refhub.elsevier.com/S0925-8388(17)31762-0/sref35http://refhub.elsevier.com/S0925-8388(17)31762-0/sref35http://refhub.elsevier.com/S0925-8388(17)31762-0/sref36http://refhub.elsevier.com/S0925-8388(17)31762-0/sref36http://refhub.elsevier.com/S0925-8388(17)31762-0/sref36http://refhub.elsevier.com/S0925-8388(17)31762-0/sref36http://refhub.elsevier.com/S0925-8388(17)31762-0/sref37http://refhub.elsevier.com/S0925-8388(17)31762-0/sref37http://refhub.elsevier.com/S0925-8388(17)31762-0/sref38http://refhub.elsevier.com/S0925-8388(17)31762-0/sref38http://refhub.elsevier.com/S0925-8388(17)31762-0/sref38http://refhub.elsevier.com/S0925-8388(17)31762-0/sref38http://refhub.elsevier.com/S0925-8388(17)31762-0/sref39http://refhub.elsevier.com/S0925-8388(17)31762-0/sref39http://refhub.elsevier.com/S0925-8388(17)31762-0/sref40http://refhub.elsevier.com/S0925-8388(17)31762-0/sref40http://refhub.elsevier.com/S0925-8388(17)31762-0/sref40http://refhub.elsevier.com/S0925-8388(17)31762-0/sref41http://refhub.elsevier.com/S0925-8388(17)31762-0/sref41http://refhub.elsevier.com/S0925-8388(17)31762-0/sref42http://refhub.elsevier.com/S0925-8388(17)31762-0/sref42http://refhub.elsevier.com/S0925-8388(17)31762-0/sref42http://refhub.elsevier.com/S0925-8388(17)31762-0/sref43http://refhub.elsevier.com/S0925-8388(17)31762-0/sref43http://refhub.elsevier.com/S0925-8388(17)31762-0/sref43http://refhub.elsevier.com/S0925-8388(17)31762-0/sref44http://refhub.elsevier.com/S0925-8388(17)31762-0/sref44http://refhub.elsevier.com/S0925-8388(17)31762-0/sref45http://refhub.elsevier.com/S0925-8388(17)31762-0/sref45http://refhub.elsevier.com/S0925-8388(17)31762-0/sref45http://refhub.elsevier.com/S0925-8388(17)31762-0/sref46http://refhub.elsevier.com/S0925-8388(17)31762-0/sref46http://refhub.elsevier.com/S0925-8388(17)31762-0/sref47http://refhub.elsevier.com/S0925-8388(17)31762-0/sref47http://refhub.elsevier.com/S0925-8388(17)31762-0/sref48http://refhub.elsevier.com/S0925-8388(17)31762-0/sref48http://refhub.elsevier.com/S0925-8388(17)31762-0/sref48http://refhub.elsevier.com/S0925-8388(17)31762-0/sref49http://refhub.elsevier.com/S0925-8388(17)31762-0/sref49http://refhub.elsevier.com/S0925-8388(17)31762-0/sref50http://refhub.elsevier.com/S0925-8388(17)31762-0/sref50http://refhub.elsevier.com/S0925-8388(17)31762-0/sref51http://refhub.elsevier.com/S0925-8388(17)31762-0/sref51http://refhub.elsevier.com/S0925-8388(17)31762-0/sref51http://refhub.elsevier.com/S0925-8388(17)31762-0/sref51http://refhub.elsevier.com/S0925-8388(17)31762-0/sref51http://refhub.elsevier.com/S0925-8388(17)31762-0/sref52http://refhub.elsevier.com/S0925-8388(17)31762-0/sref52http://refhub.elsevier.com/S0925-8388(17)31762-0/sref52http://refhub.elsevier.com/S0925-8388(17)31762-0/sref52http://refhub.elsevier.com/S0925-8388(17)31762-0/sref53http://refhub.elsevier.com/S0925-8388(17)31762-0/sref53http://refhub.elsevier.com/S0925-8388(17)31762-0/sref53http://refhub.elsevier.com/S0925-8388(17)31762-0/sref54http://refhub.elsevier.com/S0925-8388(17)31762-0/sref54http://refhub.elsevier.com/S0925-8388(17)31762-0/sref54http://refhub.elsevier.com/S0925-8388(17)31762-0/sref55http://refhub.elsevier.com/S0925-8388(17)31762-0/sref55http://refhub.elsevier.com/S0925-8388(17)31762-0/sref55

Compositional effects on ideal shear strength in Fe-Cr alloys1. Introduction2. Methodology2.1. Ideal shear stress2.2. Elastic properties2.3. Computational details

3. Results and discussion3.1. Magnetic ground state, lattice constant and cohesive energies of Fe-Cr alloys3.2. ISS and elastic constants of Fe-Cr

4. ConclusionsAcknowledgementsReferences