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First-principles study of the ideal strength of Fe3C cementite
Materials Science & Engineering A 572 (2013) 25–29
Contents lists available at SciVerse ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
First-principles study of the ideal strength of Fe3C cementite N. Garvik, Ph. Carrez, P. Cordier n Unite´ Mate´riaux et Transformations, UMR-CNRS 8207, Universite´ Lille 1, Sciences et Technologies, 59655 Villeneuve d’Ascq, France
a r t i c l e i n f o
abstract
Article history: Received 21 January 2013 Received in revised form 11 February 2013 Accepted 13 February 2013 Available online 21 February 2013
The ultimate (ideal) mechanical properties of iron carbide Fe3C (cementite) have been calculated using first-principles calculations and the generalized gradient approximation under tensile and shear loading. Our results confirm that cementite is elastically anisotropic, in particular with a low C44 (18 GPa). We also show that cementite is anisotropic from the point of view of the theoretical strength. In tension, the elastic instability is reached at 16% strain and 22 GPa along [100]. Larger elongation (23%) can be reached when the cementite is pulled along [010] or [001] (the ideal tensile stress is then 20 and 32 GPa, respectively). Results are more contrasting in shear. The low C44 value allows very large shear deformation (up to ca. 40%) to be sustained along [010](001) or [001](010) before the cementite structure becomes unstable. The higher ideal shear stress (ISS), 22.4 GPa, is exhibited for [001](100). Other shear loading conditions lead to ultimate strains in the range 14–22% for ideal shear stresses between 12 and 19 GPa. & 2013 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Cementite Ideal strength Elastic constants
1. Introduction Iron carbide Fe3C cementite is one of the most important strengthening phases in steels. However, only few studies are available on the mechanical properties of cementite [1–4]. Extreme deformation processes, like cold working, are used to improve the strength of materials. In the case of pearlitic steel, the exact mechanisms responsible for this strengthening are still under investigation. Part of the strengthening is certainly due to the refinement of the lamellar structure of pearlitic steels upon straining [5,6]. However, severe plastic deformation also promotes decomposition of cementite and redistribution of carbon in the ferrite [7–9]. In any case, cementite cannot be considered as non-deformable. Under the most severe deformation conditions, it is unlikely, due to the small sizes of the remaining cementite lamellae, that conventional plasticity by nucleation and propagation of dislocations accounts for the deformation of cementite. Dislocation plasticity and fracture being inhibited by the small sizes of the lamellae, the mechanical properties of cementite must be close to the ultimate strength of the material. It thus seems important to better understand the ultimate mechanical properties of cementite in the context of severe plastic deformation. The ultimate mechanical properties of a solid are related to its structural stability as a function of strain. This problem has been originally formalized by Born [10] and Born and Huang [11] by expanding the internal energy of a crystal in a power series of
n
Corresponding author. Tel.: þ33 320 434341. E-mail address: [email protected] (P. Cordier).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.02.028
strain. The structural stability of the solid requires the energy to remain positive. This leads to a set of conditions on the elastic constants. The corresponding criterion, named after Born, describes however the stability of an unstrained solid. Under a finite strain, both the symmetry and the properties of the solid are affected. Wang et al. [12] have proposed a stability criterion based on a Gibbs integral which combines the change in the Helmholtz free energy and the external work done during deformation. Morris and Krenn [13] have shown that the criterion proposed by Wang et al. [12] is compatible with Gibbs’s original formulation of elastic stability (see also [14]). During the last decades, atomistic calculations have offered new possibilities for investigating the theoretical strength of solids with increasing crystal chemistry complexity (see [15] and references inside). Theoretical strengths have been calculated for tensile or shear tests on iron [16–18]. The method consists in applying the desired deformation increment and then to calculate the elastic energy and the relevant Cauchy stress as a function of strain (in the fully relaxed state). The Cauchy stress is determined through the derivative of the free energy as a function of strain [13,19]. The instability corresponds to the maximum of stress (inflexion points in the free energy curve are thus an indication of instability). It must be kept in mind that, strictly speaking, applying only a few low-symmetry distortions yields only an upper bound on the theoretical strength [20–22]. Based on some physical insights (straining along high-order symmetry axis and shearing along possible slip systems), this approach is however considered to represent a good approach to the problem [13] which can be subject to further refinements. In this study, we investigate the theoretical strength of cementite Fe3C. Ab initio calculations of the structure and elastic
26
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
properties of cementite are first presented. Then we calculate the ideal tensile strength (ITS) and ideal shear strength (ISS). Homogeneous strains are applied along high-symmetry directions [100], [010] and [001]. For shear deformations, the shear planes are (100), (010) and (001) which are potential glide planes in the cementite structure [1,3]
2. Computational methods We performed spin polarized DFT calculations [23,24] using the Vienna Ab initio Simulation Package (VASP) [25–28]. The electronic exchange and correlation terms were evaluated using the gradient corrected functional (GGA) [29,30] and electron–ion interactions were treated using the projector augmented-wave method (PAW) [31,32]. For both Fe and C elements, we used the standard PAW pseudo-potentials supplied in VASP. Reference configurations for Fe and C were respectively 3d74s1 and 2s22p2. For all the calculations, plane-wave basis set expansion was limited using a kinetic energy cut-off of 500 eV. The total energy of bulk system was converged with respect to the kinetic energy and the k-point sampling. Using the Monkhorst and Pack scheme [33], we verified that the total energy of bulk Fe3C converged with accuracy better than 0.01 meV/atom by increasing the k-point grid size up to 9 6 10. Ideal tensile or shear tests were performed by applying incremental strain to a single Fe3C unit cell to which periodic boundary conditions were applied. Once strain was applied, we performed a full relaxation of both cell shape (including for instance cross section evolution in response to transversal Poisson effect in case of tensile tests) and atomic positions. Consequently, all the terms of the stress tensor are supposed to be zero except one corresponding to the applied stress, i.e. one sii in case of tensile tests and one sij for shear tests. The structural relaxation was therefore performed until residual stresses reached a value lower than 0.01 GPa.
3. Results 3.1. Ground state properties Before calculating the ultimate mechanical properties of Fe3C, we optimized the equilibrium structure. A unit cell has been built and relaxed for the Pnma configuration, giving rise to the equilibrium lattice parameters a, b and c. The results are displayed in Table 1, where they are compared with available data (both theoretical and experimental). It is shown that the calculations predict the correct Fe3C ground state structure. Then, we have calculated the nine elastic constants of Fe3C by applying small strains (o2%) to the optimized orthorhombic unit cell and by fitting the total energy as a function of strain with a second order polynomial. The results of the elastic constants tensor are given in Table 2. Table 1 Lattice parameters (Pnma) calculated in this study compared to values available in the literature. ˚ a (A)
˚ b (A)
˚ c (A)
Modeling This study [39] [34]
5.038 5.008 5.14
6.720 6.725 6.52
4.485 4.465 4.35
Experimental [40] [41]
5.079 5.090
6.730 6.748
4.510 4.523
3.2. Ideal strength in tension and in shear The ideal strength of cementite is studied by applying increasing strains to a unit cell containing 16 atoms. The evolution of the lattice energy as a function of strain is reported in Fig. 1 for tensile experiments performed along [100], [010] and [001]. All calculations have been performed beyond the onset of mechanical instability. Ultimately, all tensile tests lead to a catastrophic jump of the energy curve. In case of the [010] tensile test, this sudden jump is found to be associated with a substantial structure change (Fig. 2) from Pnma to I4/mmm. For tensile tests along [100] and [001], the discontinuity is not associated with any symmetry changes. Note however that the behavior observed after the onset of instability depends on the experimental conditions. Fig. 3 shows an example corresponding to a tensile test along [100] with a conventional simulation cell corresponding to a single unit cell compared with another calculation where the simulation cell is twice as large. Discontinuities of the energy–strain curves are observed in both cases, but with markedly different behaviors. The structure has also been tested in shear against the three unit cell lattice vectors and in the three (100), (010) and (001) planes. The results of the evolution of the energy as a function of shear strain is shown in Fig. 4. Markedly different curves suggest a very strong anisotropy of cementite. In three cases, sudden jumps in the energy occur after the elastic instability. For [01 0] (001) and [100] (010), these jumps are associated with significant structure modifications. Fig. 5 shows that these changes are due to carbon changing interstitial sites. Two methods can be used to obtain stress–strain curves from our calculations. First, the Cauchy stress can be obtained from the derivative of the energy curve with respect to strain [19]. Alternatively, stresses can be calculated directly from the Hellman– Feynman theorem. Figs. 6 and 7 show the stress–strain curves calculated with both methods. They lead to the same results. The maximum tensile strength is reached for 16% and 23% along [100] and [010] with ideal stresses of 22 GPa and 20 GPa, respectively. Pulling cementite along [001] leads to the highest ideal stress (32 GPa) reached for the largest elongation (24%). Shear tests can be divided into two groups revealing the strong anisotropic behavior of the structure. The first group corresponds to [100](010), [010](100), [001](100) and [100](001) for which the maximum shear stress is reached at relatively low strains (below 22%). For the second group, i.e. [010](001) and [001](010), the structure can be sheared up to 40% before reaching instabilities. Whatever the shear test, ISS values seem to be comparable and are found in a range between 12 GPa and 22 GPa. Table 3 summarizes the results of ideal shear stresses.
4. Discussion Calculating the ground state properties (equilibrium lattice parameters and elastic constants) represents the first test for the calculations. Indeed, Table 2 shows that calculations performed in the literature without spin polarization fail to reproduce the elastic properties of cementite (negative C44 for instance and greater deviations of all values). We have performed test calculations (not presented here) without spin polarization which confirm this observation. Table 1 shows that the equilibrium unit cell and lattice parameters found in this study using spin polarized calculations reproduce satisfactorily the known structure of cementite (the least agreement is found with the study of Ruda et al. [34] which is based on empirical potentials (EAM)). The nine elastic constants calculated here using PAW pseudopotentials are found to be in qualitatively good agreement with prior elastic constants
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
27
Table 2 Elastic constants (in GPa) and moduli (in GPa) calculated in this study.
This study [35] [42] Non-spin polarized
This study [43]
C11
C22
C33
C12
C13
C23
C44
C55
C66
388.4 388 480
343.8 345 443
322.7 322 480
154.6 156 237
145.2 164 236
159.9 162 188
18.1 15 6
132.2 134 149
134.9 134 153
B
G
E
n
217.5 230
93.4 83
222.5 230
0.31 0.29
30
30
˚
20
[100]
[010]
25
[001]
15
ΔE/V (meV/A3)
ΔE/V (meV/A 3)
25
˚
10 5
0.05
0
1x1x1
20 15 2x2x2
10
Engineering Strain 5 Fig. 1. Strain energy as a function of engineering strain for tensile loading along [100], [010] and [001]. The curves are shifted horizontally for clarity.
0.05 0
Engineering strain
Fig. 3. Tensile loading along [100] performed on a simulation cell (1 1 1) corresponding to a single unit cell (red squares) compared to the same calculation where the simulation cell contains 8 unit cells (2 2 2; black circles).
C11 þC33 42C13 Cii 40 (i¼1–6)
Fig. 2. Structural changes associated with the sudden jump in the energy curve occurring at 42% engineering strain during tension along [010]: (a) unstrained cell. and (b) atomic structure exhibited after the sudden jump.
determined using DFT [35]. We confirm that, compared to other elastic constants, C44 is very low, emphasizing the elastic anisotropy of cementite. To allow further comparison with available experimental data, we used the Voigt–Reuss–Hill (VRH) approximation to evaluate macroscopic elastic modulii, namely the bulk (B) and shear (G) modulus and also the Young modulus (E) and the Poisson ratio (n). Once again, using the VRH method, corresponding to an arithmetic mean of Voigt and Reuss average, our calculations reproduce the bulk elastic properties of cementite (Table 1). The knowledge of the elastic constants of cementite represents the first opportunity to assess the mechanical stability of the structure. The conditions for the stability of an orthorhombic structure are that the subdeterminants of the elastic tensor are all positive [10,36]:
C11 þ C22 42C12 C22 þ C33 42C23
The elastic constants displayed in Table 1 satisfy these conditions indicating that the unstrained cementite structure is mechanically stable. The main goal of this study is to investigate the mechanical response of cementite to applied stresses. As such behavior is strain-path dependent, we have chosen to apply tensile and shear loading along crystallographic axes. The most direct information provided by the calculation is the evolution of the strain energy as a function of the applied engineering strain (Figs. 1 and 4). It is shown that in every case, the energy increases with applied strain as we can expect for a stable structure. Every curve also shows an inflexion point. Beyond this point, several kinds of behaviors can occur. The energy may continue to increase continuously, although at a decreasing rate (shear along [001](010)), or even start to decrease markedly as for shear along [010](100) (after ca. 20% engineering strain). However, in most cases, the energy exhibits sudden drops. These drops are sometimes associated with visible structure changes. For instance, tensile loading along [010] triggers a structure change at 42% tensile strain from the Pnma space group to a new structure described within the I4/ mmm space group. Fig. 2 shows that this structure change is associated with an interstitial carbon atom switching from a prismatic site into a tetragonal interstitial site. In case of shear along [010] (001) and [100] (010) the energy drops are associated with jumps of carbon atoms to unoccupied interstitial sites (Fig. 5). However, Fig. 3 shows that the evolution observed after the onset of instability is difficult to capture with the present calculation technique as it depends on the simulation cell
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N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
35
ΔE/V (meV/A3)
30
˚
25
[100](001)
20 15
[100](010)
10 5
0.05 0
Engineering strain 35
˚
[010](001)
25
35 30
20 Stress (GPa)
ΔE/V (meV/A3)
30
Fig. 5. Structural changes associated with the sudden jump in the energy curve occurring at 46% engineering strain during shearing along [010] in (001): (a) unstrained cell and (b) structure exhibited after the sudden jump.
15 10 5
25 20
0.05
0
[001](010)
25
20
20
15 10 0.05
0
Engineering strain Fig. 4. Strain energy as a function of engineering strain for shear loading along (a) [100](010) and [100](001) (b) [010] (100) and [010](001) and (c) [001](100) and [001](010). The curves are shifted horizontally for clarity.
configuration. The search for the stable structure under a given load is not the goal of the present study and would require search algorithms more adapted to structure predictions. In this study, we want to focus on the ultimate strength of cementite. The corresponding stress–strain curves (plotted up to the instability threshold, i.e. the maximum of stress) are shown in tension in Fig. 6 and in shear in Fig. 7. Whereas, experimental studies [37] have highlighted the elastic anisotropy of cementite (calculations of elastic constants confirming this fact), we further demonstrate here that cementite also exhibits anisotropic ultimate properties. Considering the behavior under tensile stresses, we observe that the three strain paths corresponding to pulling
Stress (GPa)
ΔE/V (meV/A3)
Fig. 6. Uniaxial tensile stress–strain curves. Open symbols correspond to the Hellman–Feynman stress and lines correspond to the Cauchy stress obtained from the derivative of the energy curves. The curves are shifted horizontally for clarity and the loading direction is indicated next to each curve.
[001](100)
5
0.05
0
Engineering strain
35
25
[010]
10
Engineering strain
˚
[100]
15
5
[010](100)
30
[001]
15 10
[001](100) [100](001)
[010](001)
[010](100) [100](010)
[001](010)
5 0.05
0
Engineering strain Fig. 7. Shear stress–strain curves. Open symbols correspond to the Hellman– Feynman stress and lines correspond to the Cauchy stress obtained from the derivative of the energy curves. The curves are shifted horizontally for clarity.
tests along the three crystallographic axis lead to different responses. When cementite is pulled along [100], instability occurs at 16% elongation for an ultimate stress of 22.2 GPa. This does not corresponds to the lower tensile stress that can be supported as tension along [010] leads to instability at 19.6 GPa for an elongation of 23%. The greater resistance to tension is shown along [001] as cementite can sustain an elongation of 24% under a stress of 32.1 GPa. The range of responses is even wider under shear loading. The low value of the C44 elastic constant has a strong effect on the ultimate properties as we observe that much larger strains can be reached when cementite is sheared
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
29
Table 3 Ultimate stresses (and associated engineering strains) determined in this study under tensile and shear loading. Tensile tests
[100]
[010]
[001]
Ultimate stress (GPa) Ultimate strain (%)
22.2 16
19.6 23
32.1 24
Shear tests
Ultimate stress (GPa) Ultimate strain (%)
[100](010)
[100](001)
[010](100)
[010](001)
[001](100)
[001](010)
14.4 16
18.2 17
12.6 14
19.6 40
22.4 22
14.2 37
along [001](010) or [010](001). The corresponding shear stresses are however within the range observed for other loading conditions. The lower shear resistance is shown when cementite is solicited along [010](100) (12.6 GPa at 14% strain). The higher strength in shear is found for [001](100) with an instability threshold at 22.4 GPa and 22% engineering strain. The ratio observed of the extreme values is 1.7 for the ideal tensile stresses and 1.6 for the ultimate tensile strains. In shear, this ratio is 1.8 for the ideal tensile stresses and 2.9 for the ultimate tensile strains. Compared to ultimate strengths calculated for iron [16,38], ultimate stresses are found to be higher in cementite in both tension and shear. There is at least a factor 1.5 between the lowest ISS of iron (7.2 GPa for /111S in {112} and 7.8 GPa for /111S in {110} [38]) or ITS (12.7 GPa for /100S [16]) and values reported here for cementite. Note however that all possible loading conditions (even for simple tensile and shear loading) have not been considered here, and that the range of ultimate strengths reported here might be extended. Moreover, in steels, the strength of iron is likely to be controlled by nucleation and propagation of dislocations leading at stress levels significantly lower than the ultimate strength. It must also be remembered that the present approach is not designed to capture dynamic instabilities, such as those caused by ‘soft phonons’ or anharmonic vibrations. The ultimate mechanical properties calculated here refer to the limit of mechanical stability under quasistatic deformation corresponding to some specific strain paths.
5. Concluding remarks In this study, the mechanical stability of cementite has been tested against loading along the crystallographic axis in tension and in shear. Using first-principles calculations, we show that Fe3C cementite exhibits markedly anisotropic ultimate properties. In tension, cementite is slightly weaker when it is strained along [100]. However, the greatest anisotropy is observed in shear. Due to the low value of the C44 elastic constant, cementite can sustain very large strains (up to ca. 40%) when it is sheared along [010](001) or [001](010) before reaching the stability limit. The highest ideal shear strength is also observed for one of these loading conditions ([010](001)). As theoretical strengths represent the ultimate stresses or strains that the material can sustain, their knowledge has important implications for understanding the behavior of cementite under extreme deformation processes where the limits of mechanical stability can be reached. Our results suggest that the morphology and crystal preferred orientations of cementite may play a role in the ultimate hardening properties of this phase in steels.
Acknowledgments Computational resources have been provided by the Centre de Ressources Informatiques, CRI-Lille1, of the University of Lille. The authors warmly thank an anonymous referee and J.D. Whittenberger, journal editor, for crucial improvements of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
G. Sevillano, Mater. Sci. Eng. 21 (1975) 221–225. A. Inoue, T. Ogura, T. Masumoto, Trans. JIM 17 (1976) 149–157. A. Inoue, T. Ogura, T. Masumoto, Scr. Metall. 11 (1977) 1–5. M. Umemoto, Y. Todaka, K. Tsuchiya, Mater. Sci. Forum 426 (2003) 859–864. K. Hono, M. Ohnuma, M. Murayama, S. Nishida, A. Yoshie, Scr. Mater. 44 (2001) 977–983. C. Borchers, T. Al-Kassab, S. Goto, R. Kirchheim, Mater. Sci. Eng. A 502 (2009) 131–138. V.G. Gavriljuk, Mater. Sci. Eng. A 345 (2003) 81–89. Y.J. Li, P. Choi, C. Borchers, S. Westerkamp, S. Goto, D. Raabe, R. Kirchheim, Acta Mater. 59 (2011) 3965–3977. Z. Li, F. Gao, Phys. Chem. Chem. Phys. 14 (2012) 869–876. M. Born, Proc. Cambridge Philos. Soc. 36 (1940) 160. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1956. J. Wang, J. Li, S. Yip, S. Phillpot, D. Wolf, Phys. Rev. B 52 (1995) 12627–12635. J.W. Morris Jr., C.R. Krenn, Philos. Mag. A 80 (2000) 2827–2840. S. Yip, J. Ji, M. Tang, J. Wang, Mater. Sci. Eng. A 317 (2001) 236–240. ˇ ˇ ˇ ´ , P. Sandera, J. Pokluda, M. Cerny M. Sob, J. Comput. Aided Mater. Des. 11 (2004) 1–28. ˇ M. Fria´k, M. Sob, V. Vitek, Philos. Mag. 83 (2003) 3529–3537. ˇ M. Fria´k, M. Sob, V. Vitek, Phys. Rev. B 63 (2001) 052405. D.M. Clatterbuck, D.C. Chrzan, J.W. Morris Jr, Philos. Mag. Lett. 82 (2002) 141–147. D.J. Rasky, F. Milstein, Phys. Rev. B 33 (1986) 2765–2779. V.V. Bulatov, W. Cai, R. Baran, K. Kang, Philos. Mag. 86 (2006) 3847–3859. ˇ ´ , J. Pokluda, Phys. Rev. B 76 (2007) 24115. M. Cerny ˇ ´ , J. Pokluda, Acta Mater. 58 (2010) 3117–3123. M. Cerny P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864–B871. W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133–A1138. G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. G. Kresse, J. Hafner., Phys. Rev. B 49 (1994) 14251. ¨ G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15–50. ¨ G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. P.E. Blochl, Phys. Rev. B 50 (1994) 17953. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. H.J. Monkhorst, J.F. Pack, Phys. Rev. B 13 (1976) 5188. M. Ruda, D. Farkas, G. Garcia, Comput. Mater. Sci. 45 (2009) 550–560. C. Jiang, S.G. Srinivasan, A. Caro, S.A. Maloy, J. Appl. Phys. 103 (2008) 043502. D.C. Wallace, Thermodynamics of Crystals, John Wiley, New-York, 1972. M.A. Weisser, A.D. Evans, S. Van Petegem, S.R. Holdsworth, H. Van Swygenhoven, Acta Mater. 59 (2011) 4448–4457. D.M. Clatterbuck, D.C. Chrzan, J.W. Morris Jr., Acta Mater. 51 (2003) 2271–2283. Z. Lv, F. Zhang, S. Sun, Z. Wang, P. Jiang, W. Zhang, W. Fu, Comput. Mater. Sci. 44 (2008) 690–694. N.M. Popova, Carbide Analysis of Steel, Elliot, Moscow, (1956). F.H. Herbstein, J. Smuts, Acta Crystallogr. 17 (1964) 1331–1332. M. Mookherjee, Am. Mineral. 96 (2011) 1530–1536. H. Ledbetter, Mater. Sci. Eng. A 527 (2010) 2657–2661.
Contents lists available at SciVerse ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
First-principles study of the ideal strength of Fe3C cementite N. Garvik, Ph. Carrez, P. Cordier n Unite´ Mate´riaux et Transformations, UMR-CNRS 8207, Universite´ Lille 1, Sciences et Technologies, 59655 Villeneuve d’Ascq, France
a r t i c l e i n f o
abstract
Article history: Received 21 January 2013 Received in revised form 11 February 2013 Accepted 13 February 2013 Available online 21 February 2013
The ultimate (ideal) mechanical properties of iron carbide Fe3C (cementite) have been calculated using first-principles calculations and the generalized gradient approximation under tensile and shear loading. Our results confirm that cementite is elastically anisotropic, in particular with a low C44 (18 GPa). We also show that cementite is anisotropic from the point of view of the theoretical strength. In tension, the elastic instability is reached at 16% strain and 22 GPa along [100]. Larger elongation (23%) can be reached when the cementite is pulled along [010] or [001] (the ideal tensile stress is then 20 and 32 GPa, respectively). Results are more contrasting in shear. The low C44 value allows very large shear deformation (up to ca. 40%) to be sustained along [010](001) or [001](010) before the cementite structure becomes unstable. The higher ideal shear stress (ISS), 22.4 GPa, is exhibited for [001](100). Other shear loading conditions lead to ultimate strains in the range 14–22% for ideal shear stresses between 12 and 19 GPa. & 2013 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Cementite Ideal strength Elastic constants
1. Introduction Iron carbide Fe3C cementite is one of the most important strengthening phases in steels. However, only few studies are available on the mechanical properties of cementite [1–4]. Extreme deformation processes, like cold working, are used to improve the strength of materials. In the case of pearlitic steel, the exact mechanisms responsible for this strengthening are still under investigation. Part of the strengthening is certainly due to the refinement of the lamellar structure of pearlitic steels upon straining [5,6]. However, severe plastic deformation also promotes decomposition of cementite and redistribution of carbon in the ferrite [7–9]. In any case, cementite cannot be considered as non-deformable. Under the most severe deformation conditions, it is unlikely, due to the small sizes of the remaining cementite lamellae, that conventional plasticity by nucleation and propagation of dislocations accounts for the deformation of cementite. Dislocation plasticity and fracture being inhibited by the small sizes of the lamellae, the mechanical properties of cementite must be close to the ultimate strength of the material. It thus seems important to better understand the ultimate mechanical properties of cementite in the context of severe plastic deformation. The ultimate mechanical properties of a solid are related to its structural stability as a function of strain. This problem has been originally formalized by Born [10] and Born and Huang [11] by expanding the internal energy of a crystal in a power series of
n
Corresponding author. Tel.: þ33 320 434341. E-mail address: [email protected] (P. Cordier).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.02.028
strain. The structural stability of the solid requires the energy to remain positive. This leads to a set of conditions on the elastic constants. The corresponding criterion, named after Born, describes however the stability of an unstrained solid. Under a finite strain, both the symmetry and the properties of the solid are affected. Wang et al. [12] have proposed a stability criterion based on a Gibbs integral which combines the change in the Helmholtz free energy and the external work done during deformation. Morris and Krenn [13] have shown that the criterion proposed by Wang et al. [12] is compatible with Gibbs’s original formulation of elastic stability (see also [14]). During the last decades, atomistic calculations have offered new possibilities for investigating the theoretical strength of solids with increasing crystal chemistry complexity (see [15] and references inside). Theoretical strengths have been calculated for tensile or shear tests on iron [16–18]. The method consists in applying the desired deformation increment and then to calculate the elastic energy and the relevant Cauchy stress as a function of strain (in the fully relaxed state). The Cauchy stress is determined through the derivative of the free energy as a function of strain [13,19]. The instability corresponds to the maximum of stress (inflexion points in the free energy curve are thus an indication of instability). It must be kept in mind that, strictly speaking, applying only a few low-symmetry distortions yields only an upper bound on the theoretical strength [20–22]. Based on some physical insights (straining along high-order symmetry axis and shearing along possible slip systems), this approach is however considered to represent a good approach to the problem [13] which can be subject to further refinements. In this study, we investigate the theoretical strength of cementite Fe3C. Ab initio calculations of the structure and elastic
26
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
properties of cementite are first presented. Then we calculate the ideal tensile strength (ITS) and ideal shear strength (ISS). Homogeneous strains are applied along high-symmetry directions [100], [010] and [001]. For shear deformations, the shear planes are (100), (010) and (001) which are potential glide planes in the cementite structure [1,3]
2. Computational methods We performed spin polarized DFT calculations [23,24] using the Vienna Ab initio Simulation Package (VASP) [25–28]. The electronic exchange and correlation terms were evaluated using the gradient corrected functional (GGA) [29,30] and electron–ion interactions were treated using the projector augmented-wave method (PAW) [31,32]. For both Fe and C elements, we used the standard PAW pseudo-potentials supplied in VASP. Reference configurations for Fe and C were respectively 3d74s1 and 2s22p2. For all the calculations, plane-wave basis set expansion was limited using a kinetic energy cut-off of 500 eV. The total energy of bulk system was converged with respect to the kinetic energy and the k-point sampling. Using the Monkhorst and Pack scheme [33], we verified that the total energy of bulk Fe3C converged with accuracy better than 0.01 meV/atom by increasing the k-point grid size up to 9 6 10. Ideal tensile or shear tests were performed by applying incremental strain to a single Fe3C unit cell to which periodic boundary conditions were applied. Once strain was applied, we performed a full relaxation of both cell shape (including for instance cross section evolution in response to transversal Poisson effect in case of tensile tests) and atomic positions. Consequently, all the terms of the stress tensor are supposed to be zero except one corresponding to the applied stress, i.e. one sii in case of tensile tests and one sij for shear tests. The structural relaxation was therefore performed until residual stresses reached a value lower than 0.01 GPa.
3. Results 3.1. Ground state properties Before calculating the ultimate mechanical properties of Fe3C, we optimized the equilibrium structure. A unit cell has been built and relaxed for the Pnma configuration, giving rise to the equilibrium lattice parameters a, b and c. The results are displayed in Table 1, where they are compared with available data (both theoretical and experimental). It is shown that the calculations predict the correct Fe3C ground state structure. Then, we have calculated the nine elastic constants of Fe3C by applying small strains (o2%) to the optimized orthorhombic unit cell and by fitting the total energy as a function of strain with a second order polynomial. The results of the elastic constants tensor are given in Table 2. Table 1 Lattice parameters (Pnma) calculated in this study compared to values available in the literature. ˚ a (A)
˚ b (A)
˚ c (A)
Modeling This study [39] [34]
5.038 5.008 5.14
6.720 6.725 6.52
4.485 4.465 4.35
Experimental [40] [41]
5.079 5.090
6.730 6.748
4.510 4.523
3.2. Ideal strength in tension and in shear The ideal strength of cementite is studied by applying increasing strains to a unit cell containing 16 atoms. The evolution of the lattice energy as a function of strain is reported in Fig. 1 for tensile experiments performed along [100], [010] and [001]. All calculations have been performed beyond the onset of mechanical instability. Ultimately, all tensile tests lead to a catastrophic jump of the energy curve. In case of the [010] tensile test, this sudden jump is found to be associated with a substantial structure change (Fig. 2) from Pnma to I4/mmm. For tensile tests along [100] and [001], the discontinuity is not associated with any symmetry changes. Note however that the behavior observed after the onset of instability depends on the experimental conditions. Fig. 3 shows an example corresponding to a tensile test along [100] with a conventional simulation cell corresponding to a single unit cell compared with another calculation where the simulation cell is twice as large. Discontinuities of the energy–strain curves are observed in both cases, but with markedly different behaviors. The structure has also been tested in shear against the three unit cell lattice vectors and in the three (100), (010) and (001) planes. The results of the evolution of the energy as a function of shear strain is shown in Fig. 4. Markedly different curves suggest a very strong anisotropy of cementite. In three cases, sudden jumps in the energy occur after the elastic instability. For [01 0] (001) and [100] (010), these jumps are associated with significant structure modifications. Fig. 5 shows that these changes are due to carbon changing interstitial sites. Two methods can be used to obtain stress–strain curves from our calculations. First, the Cauchy stress can be obtained from the derivative of the energy curve with respect to strain [19]. Alternatively, stresses can be calculated directly from the Hellman– Feynman theorem. Figs. 6 and 7 show the stress–strain curves calculated with both methods. They lead to the same results. The maximum tensile strength is reached for 16% and 23% along [100] and [010] with ideal stresses of 22 GPa and 20 GPa, respectively. Pulling cementite along [001] leads to the highest ideal stress (32 GPa) reached for the largest elongation (24%). Shear tests can be divided into two groups revealing the strong anisotropic behavior of the structure. The first group corresponds to [100](010), [010](100), [001](100) and [100](001) for which the maximum shear stress is reached at relatively low strains (below 22%). For the second group, i.e. [010](001) and [001](010), the structure can be sheared up to 40% before reaching instabilities. Whatever the shear test, ISS values seem to be comparable and are found in a range between 12 GPa and 22 GPa. Table 3 summarizes the results of ideal shear stresses.
4. Discussion Calculating the ground state properties (equilibrium lattice parameters and elastic constants) represents the first test for the calculations. Indeed, Table 2 shows that calculations performed in the literature without spin polarization fail to reproduce the elastic properties of cementite (negative C44 for instance and greater deviations of all values). We have performed test calculations (not presented here) without spin polarization which confirm this observation. Table 1 shows that the equilibrium unit cell and lattice parameters found in this study using spin polarized calculations reproduce satisfactorily the known structure of cementite (the least agreement is found with the study of Ruda et al. [34] which is based on empirical potentials (EAM)). The nine elastic constants calculated here using PAW pseudopotentials are found to be in qualitatively good agreement with prior elastic constants
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
27
Table 2 Elastic constants (in GPa) and moduli (in GPa) calculated in this study.
This study [35] [42] Non-spin polarized
This study [43]
C11
C22
C33
C12
C13
C23
C44
C55
C66
388.4 388 480
343.8 345 443
322.7 322 480
154.6 156 237
145.2 164 236
159.9 162 188
18.1 15 6
132.2 134 149
134.9 134 153
B
G
E
n
217.5 230
93.4 83
222.5 230
0.31 0.29
30
30
˚
20
[100]
[010]
25
[001]
15
ΔE/V (meV/A3)
ΔE/V (meV/A 3)
25
˚
10 5
0.05
0
1x1x1
20 15 2x2x2
10
Engineering Strain 5 Fig. 1. Strain energy as a function of engineering strain for tensile loading along [100], [010] and [001]. The curves are shifted horizontally for clarity.
0.05 0
Engineering strain
Fig. 3. Tensile loading along [100] performed on a simulation cell (1 1 1) corresponding to a single unit cell (red squares) compared to the same calculation where the simulation cell contains 8 unit cells (2 2 2; black circles).
C11 þC33 42C13 Cii 40 (i¼1–6)
Fig. 2. Structural changes associated with the sudden jump in the energy curve occurring at 42% engineering strain during tension along [010]: (a) unstrained cell. and (b) atomic structure exhibited after the sudden jump.
determined using DFT [35]. We confirm that, compared to other elastic constants, C44 is very low, emphasizing the elastic anisotropy of cementite. To allow further comparison with available experimental data, we used the Voigt–Reuss–Hill (VRH) approximation to evaluate macroscopic elastic modulii, namely the bulk (B) and shear (G) modulus and also the Young modulus (E) and the Poisson ratio (n). Once again, using the VRH method, corresponding to an arithmetic mean of Voigt and Reuss average, our calculations reproduce the bulk elastic properties of cementite (Table 1). The knowledge of the elastic constants of cementite represents the first opportunity to assess the mechanical stability of the structure. The conditions for the stability of an orthorhombic structure are that the subdeterminants of the elastic tensor are all positive [10,36]:
C11 þ C22 42C12 C22 þ C33 42C23
The elastic constants displayed in Table 1 satisfy these conditions indicating that the unstrained cementite structure is mechanically stable. The main goal of this study is to investigate the mechanical response of cementite to applied stresses. As such behavior is strain-path dependent, we have chosen to apply tensile and shear loading along crystallographic axes. The most direct information provided by the calculation is the evolution of the strain energy as a function of the applied engineering strain (Figs. 1 and 4). It is shown that in every case, the energy increases with applied strain as we can expect for a stable structure. Every curve also shows an inflexion point. Beyond this point, several kinds of behaviors can occur. The energy may continue to increase continuously, although at a decreasing rate (shear along [001](010)), or even start to decrease markedly as for shear along [010](100) (after ca. 20% engineering strain). However, in most cases, the energy exhibits sudden drops. These drops are sometimes associated with visible structure changes. For instance, tensile loading along [010] triggers a structure change at 42% tensile strain from the Pnma space group to a new structure described within the I4/ mmm space group. Fig. 2 shows that this structure change is associated with an interstitial carbon atom switching from a prismatic site into a tetragonal interstitial site. In case of shear along [010] (001) and [100] (010) the energy drops are associated with jumps of carbon atoms to unoccupied interstitial sites (Fig. 5). However, Fig. 3 shows that the evolution observed after the onset of instability is difficult to capture with the present calculation technique as it depends on the simulation cell
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N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
35
ΔE/V (meV/A3)
30
˚
25
[100](001)
20 15
[100](010)
10 5
0.05 0
Engineering strain 35
˚
[010](001)
25
35 30
20 Stress (GPa)
ΔE/V (meV/A3)
30
Fig. 5. Structural changes associated with the sudden jump in the energy curve occurring at 46% engineering strain during shearing along [010] in (001): (a) unstrained cell and (b) structure exhibited after the sudden jump.
15 10 5
25 20
0.05
0
[001](010)
25
20
20
15 10 0.05
0
Engineering strain Fig. 4. Strain energy as a function of engineering strain for shear loading along (a) [100](010) and [100](001) (b) [010] (100) and [010](001) and (c) [001](100) and [001](010). The curves are shifted horizontally for clarity.
configuration. The search for the stable structure under a given load is not the goal of the present study and would require search algorithms more adapted to structure predictions. In this study, we want to focus on the ultimate strength of cementite. The corresponding stress–strain curves (plotted up to the instability threshold, i.e. the maximum of stress) are shown in tension in Fig. 6 and in shear in Fig. 7. Whereas, experimental studies [37] have highlighted the elastic anisotropy of cementite (calculations of elastic constants confirming this fact), we further demonstrate here that cementite also exhibits anisotropic ultimate properties. Considering the behavior under tensile stresses, we observe that the three strain paths corresponding to pulling
Stress (GPa)
ΔE/V (meV/A3)
Fig. 6. Uniaxial tensile stress–strain curves. Open symbols correspond to the Hellman–Feynman stress and lines correspond to the Cauchy stress obtained from the derivative of the energy curves. The curves are shifted horizontally for clarity and the loading direction is indicated next to each curve.
[001](100)
5
0.05
0
Engineering strain
35
25
[010]
10
Engineering strain
˚
[100]
15
5
[010](100)
30
[001]
15 10
[001](100) [100](001)
[010](001)
[010](100) [100](010)
[001](010)
5 0.05
0
Engineering strain Fig. 7. Shear stress–strain curves. Open symbols correspond to the Hellman– Feynman stress and lines correspond to the Cauchy stress obtained from the derivative of the energy curves. The curves are shifted horizontally for clarity.
tests along the three crystallographic axis lead to different responses. When cementite is pulled along [100], instability occurs at 16% elongation for an ultimate stress of 22.2 GPa. This does not corresponds to the lower tensile stress that can be supported as tension along [010] leads to instability at 19.6 GPa for an elongation of 23%. The greater resistance to tension is shown along [001] as cementite can sustain an elongation of 24% under a stress of 32.1 GPa. The range of responses is even wider under shear loading. The low value of the C44 elastic constant has a strong effect on the ultimate properties as we observe that much larger strains can be reached when cementite is sheared
N. Garvik et al. / Materials Science & Engineering A 572 (2013) 25–29
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Table 3 Ultimate stresses (and associated engineering strains) determined in this study under tensile and shear loading. Tensile tests
[100]
[010]
[001]
Ultimate stress (GPa) Ultimate strain (%)
22.2 16
19.6 23
32.1 24
Shear tests
Ultimate stress (GPa) Ultimate strain (%)
[100](010)
[100](001)
[010](100)
[010](001)
[001](100)
[001](010)
14.4 16
18.2 17
12.6 14
19.6 40
22.4 22
14.2 37
along [001](010) or [010](001). The corresponding shear stresses are however within the range observed for other loading conditions. The lower shear resistance is shown when cementite is solicited along [010](100) (12.6 GPa at 14% strain). The higher strength in shear is found for [001](100) with an instability threshold at 22.4 GPa and 22% engineering strain. The ratio observed of the extreme values is 1.7 for the ideal tensile stresses and 1.6 for the ultimate tensile strains. In shear, this ratio is 1.8 for the ideal tensile stresses and 2.9 for the ultimate tensile strains. Compared to ultimate strengths calculated for iron [16,38], ultimate stresses are found to be higher in cementite in both tension and shear. There is at least a factor 1.5 between the lowest ISS of iron (7.2 GPa for /111S in {112} and 7.8 GPa for /111S in {110} [38]) or ITS (12.7 GPa for /100S [16]) and values reported here for cementite. Note however that all possible loading conditions (even for simple tensile and shear loading) have not been considered here, and that the range of ultimate strengths reported here might be extended. Moreover, in steels, the strength of iron is likely to be controlled by nucleation and propagation of dislocations leading at stress levels significantly lower than the ultimate strength. It must also be remembered that the present approach is not designed to capture dynamic instabilities, such as those caused by ‘soft phonons’ or anharmonic vibrations. The ultimate mechanical properties calculated here refer to the limit of mechanical stability under quasistatic deformation corresponding to some specific strain paths.
5. Concluding remarks In this study, the mechanical stability of cementite has been tested against loading along the crystallographic axis in tension and in shear. Using first-principles calculations, we show that Fe3C cementite exhibits markedly anisotropic ultimate properties. In tension, cementite is slightly weaker when it is strained along [100]. However, the greatest anisotropy is observed in shear. Due to the low value of the C44 elastic constant, cementite can sustain very large strains (up to ca. 40%) when it is sheared along [010](001) or [001](010) before reaching the stability limit. The highest ideal shear strength is also observed for one of these loading conditions ([010](001)). As theoretical strengths represent the ultimate stresses or strains that the material can sustain, their knowledge has important implications for understanding the behavior of cementite under extreme deformation processes where the limits of mechanical stability can be reached. Our results suggest that the morphology and crystal preferred orientations of cementite may play a role in the ultimate hardening properties of this phase in steels.
Acknowledgments Computational resources have been provided by the Centre de Ressources Informatiques, CRI-Lille1, of the University of Lille. The authors warmly thank an anonymous referee and J.D. Whittenberger, journal editor, for crucial improvements of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
G. Sevillano, Mater. Sci. Eng. 21 (1975) 221–225. A. Inoue, T. Ogura, T. Masumoto, Trans. JIM 17 (1976) 149–157. A. Inoue, T. Ogura, T. Masumoto, Scr. Metall. 11 (1977) 1–5. M. Umemoto, Y. Todaka, K. Tsuchiya, Mater. Sci. Forum 426 (2003) 859–864. K. Hono, M. Ohnuma, M. Murayama, S. Nishida, A. Yoshie, Scr. Mater. 44 (2001) 977–983. C. Borchers, T. Al-Kassab, S. Goto, R. Kirchheim, Mater. Sci. Eng. A 502 (2009) 131–138. V.G. Gavriljuk, Mater. Sci. Eng. A 345 (2003) 81–89. Y.J. Li, P. Choi, C. Borchers, S. Westerkamp, S. Goto, D. Raabe, R. Kirchheim, Acta Mater. 59 (2011) 3965–3977. Z. Li, F. Gao, Phys. Chem. Chem. Phys. 14 (2012) 869–876. M. Born, Proc. Cambridge Philos. Soc. 36 (1940) 160. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1956. J. Wang, J. Li, S. Yip, S. Phillpot, D. Wolf, Phys. Rev. B 52 (1995) 12627–12635. J.W. Morris Jr., C.R. Krenn, Philos. Mag. A 80 (2000) 2827–2840. S. Yip, J. Ji, M. Tang, J. Wang, Mater. Sci. Eng. A 317 (2001) 236–240. ˇ ˇ ˇ ´ , P. Sandera, J. Pokluda, M. Cerny M. Sob, J. Comput. Aided Mater. Des. 11 (2004) 1–28. ˇ M. Fria´k, M. Sob, V. Vitek, Philos. Mag. 83 (2003) 3529–3537. ˇ M. Fria´k, M. Sob, V. Vitek, Phys. Rev. B 63 (2001) 052405. D.M. Clatterbuck, D.C. Chrzan, J.W. Morris Jr, Philos. Mag. Lett. 82 (2002) 141–147. D.J. Rasky, F. Milstein, Phys. Rev. B 33 (1986) 2765–2779. V.V. Bulatov, W. Cai, R. Baran, K. Kang, Philos. Mag. 86 (2006) 3847–3859. ˇ ´ , J. Pokluda, Phys. Rev. B 76 (2007) 24115. M. Cerny ˇ ´ , J. Pokluda, Acta Mater. 58 (2010) 3117–3123. M. Cerny P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864–B871. W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133–A1138. G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. G. Kresse, J. Hafner., Phys. Rev. B 49 (1994) 14251. ¨ G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15–50. ¨ G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. P.E. Blochl, Phys. Rev. B 50 (1994) 17953. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. H.J. Monkhorst, J.F. Pack, Phys. Rev. B 13 (1976) 5188. M. Ruda, D. Farkas, G. Garcia, Comput. Mater. Sci. 45 (2009) 550–560. C. Jiang, S.G. Srinivasan, A. Caro, S.A. Maloy, J. Appl. Phys. 103 (2008) 043502. D.C. Wallace, Thermodynamics of Crystals, John Wiley, New-York, 1972. M.A. Weisser, A.D. Evans, S. Van Petegem, S.R. Holdsworth, H. Van Swygenhoven, Acta Mater. 59 (2011) 4448–4457. D.M. Clatterbuck, D.C. Chrzan, J.W. Morris Jr., Acta Mater. 51 (2003) 2271–2283. Z. Lv, F. Zhang, S. Sun, Z. Wang, P. Jiang, W. Zhang, W. Fu, Comput. Mater. Sci. 44 (2008) 690–694. N.M. Popova, Carbide Analysis of Steel, Elliot, Moscow, (1956). F.H. Herbstein, J. Smuts, Acta Crystallogr. 17 (1964) 1331–1332. M. Mookherjee, Am. Mineral. 96 (2011) 1530–1536. H. Ledbetter, Mater. Sci. Eng. A 527 (2010) 2657–2661.