- Email: [email protected]
Tension–compression asymmetry in nanocrystalline Cu: High strain rate vs. quasi-static deformation
Computational Materials Science 49 (2010) 260–265
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Tension–compression asymmetry in nanocrystalline Cu: High strain rate vs. quasi-static deformation Avinash M. Dongare a,b,*, Arunachalam M. Rajendran c, Bruce LaMattina d, Mohammed A. Zikry b, Donald W. Brenner a a
Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC, USA Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, USA Department of Mechanical Engineering, University of Mississippi, University, MI, USA d US Army Research Office, Research Triangle Park, NC, USA b c
a r t i c l e
i n f o
Article history: Received 1 March 2010 Received in revised form 25 April 2010 Accepted 4 May 2010 Available online 8 June 2010 Keywords: Molecular dynamics Nanocrystalline metals Tension–compression asymmetry Plastic deformation
a b s t r a c t Large-scale molecular dynamics (MD) simulations are used to understand the yield behavior of nanocrystalline Ni and Cu with grain sizes 610 nm at high strain rates. The calculated flow stress values at a strain rate of 109 s1 suggest an asymmetry in the strength values in tension and compression with the nanocrystalline metal being stronger in compression than in tension. This tension–compression strength asymmetry is observed to decrease with a decrease in grain size of the nanocrystalline metal up to a grain size of 4 nm, after which, a further decrease in grain size results in an increase in the strength asymmetry. The effect of strain rate on the yield behavior of nanocrystalline metals as obtained from MD simulations is discussed and compared with that reported in the literature obtained by molecular statics simulations for quasi-static loading conditions. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Nanocrystalline metals have been extensively studied by computational methods [1–6] to understand the micromechanisms governing their macroscopic mechanical behavior. Deformation mechanisms in loaded nanocrystalline materials can be conveniently classified into intra-grain dynamics, such as dislocation motion and twinning, and inter-grain processes such grain boundary sliding and grain rotation. At sufficiently small grain sizes with diameters under about 20 nm (depending on the material) molecular dynamics simulations and recent experimental studies have shown that inter-grain processes typically dominate deformation. These processes can result in weakening of a metal with decreasing grain size, e.g. inverse Hall–Petch behavior. Nanocrystalline metals with grain sizes in the inverse Hall–Petch regime have gained considerable attention due to their increased strengths during deformation at high strain rates. For example, shock loading of these nanocrystalline metals (strain rates P108 s1) limits the GB sliding mechanism, which can result in ultra-high strengths [7]. As a result, these materials show significant promise for use in extreme conditions (impact/blast) and a fundamental understanding of the
* Corresponding author at: Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC, USA. E-mail address: [email protected] (A.M. Dongare). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.004
deformation and failure of their constituent materials is required to maximize the performance and reliability of these materials. While there has been significant progress in understanding mechanisms of plastic deformation in nanocrystalline metals, understanding the effect of grain size and strain rate on macroscopic deformation behavior at high strain rates is still in its infancy. Computer simulations, with their current capabilities, allow the study of these phenomena and can complement experiments in the design of new materials with superior strength. The performance and reliability of materials is typically analyzed using the commonly used phenomenological yield criteria [8], such as von Mises and Tresca. These conventional yield criteria are based on the assumption that the yielding of the metal is determined by the deviatoric stress and are independent of the pressure and the normal stress. While, these criteria have been found to be appropriate to study deformation behavior in polycrystalline metals, molecular statics (MS) simulations suggest these conventional yield criteria to be inappropriate at grain sizes in the inverse Hall– Petch regime (d 6 15 nm) [9]. For example, deformation studies on the of nanocrystalline Ni (2–4 nm) using molecular static (MS) simulations under quasi-static (Q.S.) loading conditions indicate a strong asymmetry in tensile and compressive strengths that cannot be predicted by the conventional von Mises and Tresca criteria [9,10]. This strength asymmetry is observed to decrease with a decrease in grain size and can be well described using the Mohr– Coulomb criterion typically used to predict deformation behavior
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
in metallic glasses. The MS simulation results suggest that the ultra-fine grains can be considered as amorphous materials at the lower limit of grain size [9]. However, recent results on the yield behavior in nanocrystalline copper with a grain size of 6 nm at a strain rate of 109 s1 suggest the applicability of the von-Mises/ Tresca criteria at these small grain sizes at high strain rates [5]. As a result, the focus of this paper is to understand the difference in the response of the nanocrystalline metal (Ni, Cu) to loading at high strain rates using MD simulations as compared to that observed during quasi-static loading conditions. The strain rates (P109 s1) used here are chosen to reproduce those observed during shock loading to create ultra-high strengths in nanocrystalline metals [7]. The computational details are presented in Section 2. The effect of grain size on tension–compression strength asymmetry at high strain rates is discussed in Section 3 in comparison with that reported in the literature for quasi-static loading, followed by conclusions in Section 4.
2. Computational methods The initial MD system of nanocrystalline Ni and Cu used in this study was created using a Voronoi construction as suggested by Derlet and Van Swygenhoven [11]. Periodic boundary conditions were used in all the three directions, and the system was equilibrated at 300 K for 200 ps to relax the initial structure. The density of the final configuration is 99% of the bulk density of the metal. This procedure constructs a system with random grain orientations containing no textures and a grain-size distribution close to a log-normal distribution [12]. The initial nanocrystalline Cu system with an average grain size of 6 nm is shown in Fig. 1 with the atoms colored according to common neighbor analysis [13]. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination of 12 other than fcc. To study the effect of grain size, nanocrystalline systems with were created
Fig. 1. The initial configuration of nanocrystalline Cu system with an average grain size of 6 nm. The system consists of approximately 1.2 million atoms arranged in 122 grains and each atom is colored according to the common neighbor analysis. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination of 12 other than fcc. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
261
with an average grain size of 2 nm, 3 nm, 4 nm, for Ni and Cu as well as with 6 nm, 8 nm and 10 nm for Cu. The smallest nanocrystalline sample created had at least 122 grains and had a size of at least 16 nm in X, Y and Z directions. This resulted in systems with atoms ranging from a few hundred thousand atoms to 4 million atoms. The MD simulations were carried out using the using the Voter–Chen formulation [14] of the embedded atom method (EAM) potential for copper and with Mishin formulation [15] of the EAM potential for nickel. Both these potentials provide a good description of the unstable and stable stacking fault energies as well as grain boundary energies and therefore is well suited to describe deformation behavior for nanocrystalline Cu and Ni. The elements of the tensor of atomic-level stresses were calculated as
rab ðiÞ ¼
" # 1X a b F ij rij þ M i v ai v bi X0 2 j 1
ð1Þ
where a and b label the Cartesian components, X0 is the atomic volume, Fij is the force on atom i due to atom j, Mi is the mass of atom i, and vi is the velocity of atom i. An effective stress (re) and an effective strain (ee) according to the von Mises criterion for yielding was calculated as 1=2 1 re ¼ ððrx ry Þ2 þ ðry rz Þ2 þ ðrz rx Þ2 þ 6ðr2xy þ r2yz þ r2xz ÞÞ ð2Þ 2 pffiffiffi h i1=2 2 2 2 2 ððex ey Þ þ ðey ez Þ þ ðez ex Þ Þ ee ¼ ð3Þ 3 where rx, ry, and rz are the stresses averaged over the entire system in the X, Y, and Z directions, respectively, and ex, ey, and ez are the engineering strains in the X, Y, and Z directions, respectively. The time step for all of the MD simulation was 2 fs. The temperature was allowed to evolve during the deformation process. 3. Tension–compression asymmetry MD simulations of uniaxial stress loading of the nanocrystalline Ni and Cu sample were carried out at constant strain by deforming the sample in the X direction while maintaining zero stress conditions in the Y and Z directions (ry = rz = 0, and rx = r). At each time step, deformation was achieved by scaling the x coordinate of all the atoms and the periodic box size by a parameter chosen to achieve the desired strain rate (e_ ). The atomic coordinates and periodic box sizes in the Y and Z directions were simultaneously scaled to have a zero stress in the Y and Z directions. The scaling parameter in MD simulations is chosen so as to achieve a strain rate of 109 s1. Based on Eqs. (2) and (3), for the conditions of uniaxial stress loading, the effective von Mises stress (re) reduces to the stress in the loading direction (rx), and the effective strain (ee) reduces to the strain in the loading direction (ex). The effective von Mises stress as a function of effective strain during uniaxial deformation of the nanocrystalline Cu system with a 6 nm grain size in tension and compression at strain rates of 109 s1 is plotted in Fig. 2. The curves are initially linear up to the yield point, after which they start to deviate from elastic behavior. The flow stress (rf) is defined as the peak value of the stress in the stress–strain curve [4,5,9,10]. It can be seen from Fig. 2 that the nanocrystalline Cu is stronger in compression than in tension. The snapshots of the system during MD simulations of deformation are shown in Fig. 3a for conditions of tensile loading and in Fig. 3b for conditions of compressive loading at times corresponding to the peak value of stress in Fig. 2. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination
262
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265 Table 1 Calculated values of flow (peak) stress during uniaxial deformation of nanocrystalline Cu in tension and compression at different average grain sizes at a constant strain rate of 109 s1. Grain size (nm)
2 3 4 6 8 10
Fig. 2. Plots of effective von Mises stress (re) as a function of effective strain (ee) for nanocrystalline Cu with an average grain size of 6 nm in tension (black) and compression (green) at strain rates of 109 s1. The nanocrystalline Cu shows greater strength in compression (C) as compared to tension (T). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of 12 other than fcc. It can be seen that the deformation under the conditions of tensile loading results in more dislocations than that under the conditions of compressive loading. This difference in plastic behavior gives rise to the strength asymmetry in tension vs. compression. The calculated values of the flow stress in tension and compression for nanocrystalline copper with grain sizes of 2–10 nm are tabulated in Table 1. It can be seen from the tabulated data that a decrease in grain size reduces the flow stress values as predicted by the inverse Hall–Petch effect. The calculated tension– compression strength asymmetry is also observed to decrease with a decrease in grain size down to a grain size of 4 nm, after which, a further decrease results in an increase in the strength asymmetry of the nanocrystalline metal. This high asymmetry below grain sizes of d = 4 nm is consistent with the mechanical behavior observed in metallic glasses where a strong strength asymmetry is observed and requires the use of a criterion that accounts for a normal stress dependence to predict plastic behavior. This confirms the result that the grain size of 2 nm is considered to be the lower
Tension
Compression
rTf (GPa)
rCf (GPa)
Strength asymmetry Calc. (%)
1.334 1.523 1.615 1.986 2.146 2.363
1.522 1.585 1.650 2.101 2.309 2.563
14.09 4.07 2.17 5.79 7.59 8.46
limit of nanocrystalline metals and closer to being amorphous. The variation of the flow stress values in compression and tension as a function of the nanocrystalline grain size is plotted in Fig. 4. It should be noted from the data in Table 1 that the MD simulations of the strength asymmetry predict an increase in strength asymmetry as the grain size decreases from 4 nm to 2 nm, whereas the MS simulations for nanocrystalline Ni predict an increase in the asymmetry as the grain size increases from 2 nm to 4 nm. The difference in behavior can be attributed to the difference in the material system (Ni for MS simulations as compared to Cu for MD simulations). As a result, MD simulations were carried out to calculate the flow stresses in tension and compression for nanocrystalline Ni with grain sizes from 2 nm to 4 nm. The calculated values of the flow stress in tension and compression for nanocrystalline Ni are tabulated in Table 2 in comparison with the flow stress values reported in the literature [9,10] for quasi-static loading at grain sizes of 2–4 nm. It should be noted from the data in Table 2 that the strength asymmetry at grain sizes less than 4 nm for nanocrystalline Ni is consistent with that obtained for nanocrystalline Cu. The increase in strength asymmetry as the grain size decreases below 4 nm at high strain rates, however, contradicts the behavior predicted using MS simulations at these grain sizes. The striking difference can be related to high strain rate behavior in MD simulations as compared to quasi-static loading conditions in MS simulations. However, the flow stress values calculated for nanocrystalline Ni from MD simulations are much lower than those calculated using during quasi-static loading using MS simulations. One reason for the difference in the flow stress values can be due to the added velocity component to the stress calculations in MD simulations as indicated in Eq. (1) as compared to the stress
Fig. 3. The nanocrystalline Cu system during MD simulations of deformation for (a) conditions of tensile loading, and (b) for conditions of compressive loading, at times corresponding to the peak value of stress in Fig. 2 at a constant strain rate of 108 s1. The atoms are colored according to CNA values. The contour scale is the same as in Fig. 1.
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
263
Fig. 4. (a) Plot of the calculated flow (peak) stress in tension and compression for nanocrystalline Cu with an average grain size ranging from 2 nm to 10 nm at a strain rates 109 s1. The flow stress values in compression (circles) are greater than those in tension. (b) Plot of the calculated tension–compression strength asymmetry for nanocrystalline Cu with an average grain size ranging from 2 nm to 10 nm at a strain rate of 109 s1.
Table 2 Calculated values of flow (peak) stress during uniaxial deformation of nanocrystalline Ni in tension and compression at different average grain sizes at a constant strain rate of 109 s1 in comparison with those obtained using MS simulations under quasi-static (Q.S.) loading conditions [9,10] reported in the literature. Grain size (nm)
2 3 4
Tension
Compression
Strength asymmetry
rTf (GPa)
Q.S.
rCf (GPa)
Q.S.
Calc. (%)
Q.S. (%)
2.77 3.33 3.89
3.30 4.50 4.90
2.98 3.46 3.92
4.30 6.10 7.10
7.58 3.90 0.77
30.3 35.5 43.9
calculations in MS simulations. This contribution to stresses, however, is found to be significantly small as compared to the total value of the stress calculated in MD simulations. This raises the question: ‘‘Are the conditions of deformation used in MS simulations really quasi-static”? To address this question, the details of the deformation conditions in the MS simulations are discussed below and compared with MD simulations. Quasi-static deformation in MS simulations is obtained by scaling the atomic coordinates (strain increments along the axis) followed by relaxation of the system using the conjugate gradient technique [9,10]. This relaxation only allows a local relaxation of the atoms and does not allow for complete adiabatic relaxation of the system during the deformation process. Given the system sizes used, the strain increments used (60.1%) at each step in the MS simulations are orders of magnitude larger than that used (60.0004% at 109 s1) in the MD simulations. The deformation strain in the MS simulations may therefore result in deformation at ultra-high strain rates (P1011 s1) which results in the large strength asymmetry (30%) in tension and compression for nanocrystalline Ni. To test this, the flow stress was calculated for nanocrystalline Ni for uniaxial stress loading in tension and compression using MD simulations at a strain rate of 1011 s1. The tension compression strength asymmetry is calculated to be 21%, 20% and 22% at 2 nm, 3 nm, and 4 nm, respectively at a strain rate of 1011 s1. The calculated tension–compression strength asymmetry using MD simulations for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm is shown in Fig. 5 at a strain rate of 109 s1 (solid line), 1011 s1 (dash-dot line) in comparison with the tension–compression strength values for MS simulations [9,10] under Q.S. loading conditions (dashed line). The values of
the strength asymmetry at 1011 s1 compare better with those observed for quasi-static loading of nanocrystalline Ni. As a result, the plastic behavior at higher strain rates (>109 s1) results in a higher asymmetry due to the fact that the atoms have less time to respond to the deformation. This high strain rate behavior is inherently included in MS simulations due to the conditions of the local relaxation during deformation, and therefore, the MS simulations suggests higher values of flow stresses as well as strength asymmetry. Thus, the yield behavior of materials predicted using computational techniques is largely dependent on the method used. MD simulations are typically carried out at strain rates P107 s1. As a result, a strain rate effect is inherently included in the material response and needs to be considered when comparing with the results obtained experimentally under quasi-static loading conditions. Molecular statics simulations, on the other hand do not allow for diffusion processes and thermal relaxation that occur during quasi-static loading conditions. As a result, the conditions of deformation used molecular static simulations (in spite of having stress states as ry = rz = 0, and rx = r) cannot be considered as quasi-static conditions of deformation. Unfortunately, there is no experimental evidence of the strength asymmetry at these grain sizes to compare the results obtained suing computational techniques. The strength asymmetry calculated for nanocrystalline copper using hardness and tensile testing [16,17] suggested an increase in asymmetry as the grain size reduced from 100 nm to 16 nm with the asymmetry calculated to be 3% at 16 nm. To compare this value with that obtained using MD simulations at high strain rates, the strength asymmetry values are calculated for nanocrystalline Cu with an average grain size of 6 nm at strain rates ranging from 109 s1 to 1011 s1 and tabulated in Table 3 and fitted to a quadratic curve as shown in Fig. 6. The fitting parameters of the quadratic curve predict a strength asymmetry value of 5% under quasi-static loading conditions. This value is in close agreement with the experimental value of 3% at a grain size of 16 nm. However, it should be noted that the data fitted is for strain rates P109 s1 wherein GB based processes are limited. At lower strain rates, it can be expected that GB sliding and diffusional processes occur which will result in the weakening of the nanocrystalline metal and also lower asymmetry values. Thus, more simulations of the plastic yielding of the nanocrystalline metals at lower strain rates (104 s1–109 s1)
264
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
Fig. 5. (a) Plot of the calculated flow (peak) stress in tension (green) and compression (black) for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm at a strain rates 109 s1 in comparison with the values reported from molecular static (MS) simulations [9,10] under quasi-static (Q.S.) loading conditions (dashed line). (b) Plot of the calculated tension–compression strength asymmetry for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm at a strain rate of 109 s1 (solid black line) and 1011 s1 (dash-dot line) in comparison with the tension–compression strength values for molecular static (MS) simulations [9,10] under quasi-static (Q.S.) loading conditions (dashed line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 3 Values of flow (peak) stress during uniaxial deformation of nanocrystalline Cu with an average grain size of 6 nm at various strain rates in tension and compression. Strain rate e_ (s1)
Tension (flow stress) rTf (GPa)
Compression (flow stress) rCf (GPa)
Strength asymmetry (%)
109 2 109 4 109 8 109 1011
1.986 2.164 2.405 2.794 6.399
2.101 2.303 2.577 3.010 8.712
5.79 6.42 7.15 7.73 36.15
4. Conclusions The MD simulations reported here suggest that strength asymmetry is sensitive to the grain size of the nanocrystalline metal as well as the strain rates used. MD simulations suggest that a decrease in grain size reduces the flow stress values as predicted by the inverse Hall–Petch effect. However, the tension–compression strength asymmetry is observed to decrease as the grain size decreases down to a size of 4 nm, after which, a further decrease in grain size results in an increase in the strength asymmetry values. The higher asymmetry at a size of 2 nm is consistent with the mechanical behavior observed in metallic glasses and can be attributed to the amorphous nature of the nanocrystalline metal. However, the flow stress values obtained using MD simulations are much less as compared to those obtained using MS simulations for quasi-static loading conditions. A comparison of the methods used to calculate the flow stresses suggests that the high flow stress values result from the very high effective deformation strain rates resulting from the MS simulation technique. As a result, the atomic scale yield behavior of metals with nanometer-scale grains under quasi-static loading conditions at these small grain sizes is still not clearly understood. This suggests the need to design computational tools that will be capable of modeling deformation behavior under quasi-static loading conditions so as to understand the contributions of GB based processes and dislocation based processes to plasticity under multi-axial loading conditions and in turn the yield criterion.
Acknowledgements
Fig. 6. Plot of the calculated tension–compression strength asymmetry for nanocrystalline Cu with an average grain size of 6 nm as a function of strain rate of deformation. The solid black line shows the quadratic fit to the data.
are needed to be able to accurately predict the strain rate dependence of the yield behavior of nanocrystalline metals.
AMD is supported by the US Army Research Office (ARO) through the National Research Council Research Associateship Program. DWB and MAZ are also supported by the National Science Foundation through Grants DMR-03,04,299 and DMR-08,06,323. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or other funding agencies.
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
References [1] V. Yamakov, D. Wolf, S.R. Phillpot, A.K. Mukherjee, H. Gleiter, Nat. Mater. 1 (2002) 45. [2] D.W. Brenner, Computer modeling of nanostructured materials, in: Carl Koch (Ed.), Nanostructured Materials, second ed., Noyes Publications, Norwich, NY, 2006 (Chapter 7). [3] H. Van Swygenhoven, Science 296 (2002) 66. [4] J. Schiotz, K.W. Jacobsen, Science 301 (2003) 1357. [5] A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Metall. Mater. Trans. A 41A (2010) 523. [6] A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Phys. Rev. B 80 (2009) 104108. [7] E.M. Bringa, A. Caro, Y. Wang, M. Victoria, J.M. McNaney, B.A. Remington, R.F. Smith, B.R. Torralva, H. Van Swygenhoven, Science 309 (2005) 1838.
265
[8] R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, fourth ed., Wiley, New York (NY), 1996. [9] A.C. Lund, C.A. Schuh, Acta Mater. 53 (2005) 3193–3205. [10] A.C. Lund, T.G. Nieh, C.A. Schuh, Phys. Rev. B 69 (2004) 012101. [11] P.M. Derlet, H. Van Swygenhoven, Phys. Rev. B 67 (2003) 014202. [12] S. Kumar, S.K. Kurtz, J.R. Banavar, M.G. Sharma, J. Stat. Phys. 67 (1992) 523. [13] D.J. Honneycutt, H.C. Andersen, J. Phys. Chem. 91 (1987) 4950. [14] A.F. Voter, The embedded atom method, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, vol. 77, Wiley, New York, 1994. [15] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393. [16] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019. [17] M. Legros, B.R. Elliott, M.N. Rittner, J.R. Weertman, K.J. Hemker, Philos. Mag. A 80 (2000) 1017.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Tension–compression asymmetry in nanocrystalline Cu: High strain rate vs. quasi-static deformation Avinash M. Dongare a,b,*, Arunachalam M. Rajendran c, Bruce LaMattina d, Mohammed A. Zikry b, Donald W. Brenner a a
Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC, USA Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, USA Department of Mechanical Engineering, University of Mississippi, University, MI, USA d US Army Research Office, Research Triangle Park, NC, USA b c
a r t i c l e
i n f o
Article history: Received 1 March 2010 Received in revised form 25 April 2010 Accepted 4 May 2010 Available online 8 June 2010 Keywords: Molecular dynamics Nanocrystalline metals Tension–compression asymmetry Plastic deformation
a b s t r a c t Large-scale molecular dynamics (MD) simulations are used to understand the yield behavior of nanocrystalline Ni and Cu with grain sizes 610 nm at high strain rates. The calculated flow stress values at a strain rate of 109 s1 suggest an asymmetry in the strength values in tension and compression with the nanocrystalline metal being stronger in compression than in tension. This tension–compression strength asymmetry is observed to decrease with a decrease in grain size of the nanocrystalline metal up to a grain size of 4 nm, after which, a further decrease in grain size results in an increase in the strength asymmetry. The effect of strain rate on the yield behavior of nanocrystalline metals as obtained from MD simulations is discussed and compared with that reported in the literature obtained by molecular statics simulations for quasi-static loading conditions. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Nanocrystalline metals have been extensively studied by computational methods [1–6] to understand the micromechanisms governing their macroscopic mechanical behavior. Deformation mechanisms in loaded nanocrystalline materials can be conveniently classified into intra-grain dynamics, such as dislocation motion and twinning, and inter-grain processes such grain boundary sliding and grain rotation. At sufficiently small grain sizes with diameters under about 20 nm (depending on the material) molecular dynamics simulations and recent experimental studies have shown that inter-grain processes typically dominate deformation. These processes can result in weakening of a metal with decreasing grain size, e.g. inverse Hall–Petch behavior. Nanocrystalline metals with grain sizes in the inverse Hall–Petch regime have gained considerable attention due to their increased strengths during deformation at high strain rates. For example, shock loading of these nanocrystalline metals (strain rates P108 s1) limits the GB sliding mechanism, which can result in ultra-high strengths [7]. As a result, these materials show significant promise for use in extreme conditions (impact/blast) and a fundamental understanding of the
* Corresponding author at: Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC, USA. E-mail address: [email protected] (A.M. Dongare). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.004
deformation and failure of their constituent materials is required to maximize the performance and reliability of these materials. While there has been significant progress in understanding mechanisms of plastic deformation in nanocrystalline metals, understanding the effect of grain size and strain rate on macroscopic deformation behavior at high strain rates is still in its infancy. Computer simulations, with their current capabilities, allow the study of these phenomena and can complement experiments in the design of new materials with superior strength. The performance and reliability of materials is typically analyzed using the commonly used phenomenological yield criteria [8], such as von Mises and Tresca. These conventional yield criteria are based on the assumption that the yielding of the metal is determined by the deviatoric stress and are independent of the pressure and the normal stress. While, these criteria have been found to be appropriate to study deformation behavior in polycrystalline metals, molecular statics (MS) simulations suggest these conventional yield criteria to be inappropriate at grain sizes in the inverse Hall– Petch regime (d 6 15 nm) [9]. For example, deformation studies on the of nanocrystalline Ni (2–4 nm) using molecular static (MS) simulations under quasi-static (Q.S.) loading conditions indicate a strong asymmetry in tensile and compressive strengths that cannot be predicted by the conventional von Mises and Tresca criteria [9,10]. This strength asymmetry is observed to decrease with a decrease in grain size and can be well described using the Mohr– Coulomb criterion typically used to predict deformation behavior
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
in metallic glasses. The MS simulation results suggest that the ultra-fine grains can be considered as amorphous materials at the lower limit of grain size [9]. However, recent results on the yield behavior in nanocrystalline copper with a grain size of 6 nm at a strain rate of 109 s1 suggest the applicability of the von-Mises/ Tresca criteria at these small grain sizes at high strain rates [5]. As a result, the focus of this paper is to understand the difference in the response of the nanocrystalline metal (Ni, Cu) to loading at high strain rates using MD simulations as compared to that observed during quasi-static loading conditions. The strain rates (P109 s1) used here are chosen to reproduce those observed during shock loading to create ultra-high strengths in nanocrystalline metals [7]. The computational details are presented in Section 2. The effect of grain size on tension–compression strength asymmetry at high strain rates is discussed in Section 3 in comparison with that reported in the literature for quasi-static loading, followed by conclusions in Section 4.
2. Computational methods The initial MD system of nanocrystalline Ni and Cu used in this study was created using a Voronoi construction as suggested by Derlet and Van Swygenhoven [11]. Periodic boundary conditions were used in all the three directions, and the system was equilibrated at 300 K for 200 ps to relax the initial structure. The density of the final configuration is 99% of the bulk density of the metal. This procedure constructs a system with random grain orientations containing no textures and a grain-size distribution close to a log-normal distribution [12]. The initial nanocrystalline Cu system with an average grain size of 6 nm is shown in Fig. 1 with the atoms colored according to common neighbor analysis [13]. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination of 12 other than fcc. To study the effect of grain size, nanocrystalline systems with were created
Fig. 1. The initial configuration of nanocrystalline Cu system with an average grain size of 6 nm. The system consists of approximately 1.2 million atoms arranged in 122 grains and each atom is colored according to the common neighbor analysis. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination of 12 other than fcc. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
261
with an average grain size of 2 nm, 3 nm, 4 nm, for Ni and Cu as well as with 6 nm, 8 nm and 10 nm for Cu. The smallest nanocrystalline sample created had at least 122 grains and had a size of at least 16 nm in X, Y and Z directions. This resulted in systems with atoms ranging from a few hundred thousand atoms to 4 million atoms. The MD simulations were carried out using the using the Voter–Chen formulation [14] of the embedded atom method (EAM) potential for copper and with Mishin formulation [15] of the EAM potential for nickel. Both these potentials provide a good description of the unstable and stable stacking fault energies as well as grain boundary energies and therefore is well suited to describe deformation behavior for nanocrystalline Cu and Ni. The elements of the tensor of atomic-level stresses were calculated as
rab ðiÞ ¼
" # 1X a b F ij rij þ M i v ai v bi X0 2 j 1
ð1Þ
where a and b label the Cartesian components, X0 is the atomic volume, Fij is the force on atom i due to atom j, Mi is the mass of atom i, and vi is the velocity of atom i. An effective stress (re) and an effective strain (ee) according to the von Mises criterion for yielding was calculated as 1=2 1 re ¼ ððrx ry Þ2 þ ðry rz Þ2 þ ðrz rx Þ2 þ 6ðr2xy þ r2yz þ r2xz ÞÞ ð2Þ 2 pffiffiffi h i1=2 2 2 2 2 ððex ey Þ þ ðey ez Þ þ ðez ex Þ Þ ee ¼ ð3Þ 3 where rx, ry, and rz are the stresses averaged over the entire system in the X, Y, and Z directions, respectively, and ex, ey, and ez are the engineering strains in the X, Y, and Z directions, respectively. The time step for all of the MD simulation was 2 fs. The temperature was allowed to evolve during the deformation process. 3. Tension–compression asymmetry MD simulations of uniaxial stress loading of the nanocrystalline Ni and Cu sample were carried out at constant strain by deforming the sample in the X direction while maintaining zero stress conditions in the Y and Z directions (ry = rz = 0, and rx = r). At each time step, deformation was achieved by scaling the x coordinate of all the atoms and the periodic box size by a parameter chosen to achieve the desired strain rate (e_ ). The atomic coordinates and periodic box sizes in the Y and Z directions were simultaneously scaled to have a zero stress in the Y and Z directions. The scaling parameter in MD simulations is chosen so as to achieve a strain rate of 109 s1. Based on Eqs. (2) and (3), for the conditions of uniaxial stress loading, the effective von Mises stress (re) reduces to the stress in the loading direction (rx), and the effective strain (ee) reduces to the strain in the loading direction (ex). The effective von Mises stress as a function of effective strain during uniaxial deformation of the nanocrystalline Cu system with a 6 nm grain size in tension and compression at strain rates of 109 s1 is plotted in Fig. 2. The curves are initially linear up to the yield point, after which they start to deviate from elastic behavior. The flow stress (rf) is defined as the peak value of the stress in the stress–strain curve [4,5,9,10]. It can be seen from Fig. 2 that the nanocrystalline Cu is stronger in compression than in tension. The snapshots of the system during MD simulations of deformation are shown in Fig. 3a for conditions of tensile loading and in Fig. 3b for conditions of compressive loading at times corresponding to the peak value of stress in Fig. 2. The contour for the atoms colored according to their CNA values is as follows: the white (online = red) colored atoms represent local hexagonal close-packed order (stacking faults), the gray (online = light green) atoms represent bulk fcc stacking, the dark gray atoms (online = light blue) represent a coordination greater than 12, and the black (online = blue) colored atoms represent a coordination
262
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265 Table 1 Calculated values of flow (peak) stress during uniaxial deformation of nanocrystalline Cu in tension and compression at different average grain sizes at a constant strain rate of 109 s1. Grain size (nm)
2 3 4 6 8 10
Fig. 2. Plots of effective von Mises stress (re) as a function of effective strain (ee) for nanocrystalline Cu with an average grain size of 6 nm in tension (black) and compression (green) at strain rates of 109 s1. The nanocrystalline Cu shows greater strength in compression (C) as compared to tension (T). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
of 12 other than fcc. It can be seen that the deformation under the conditions of tensile loading results in more dislocations than that under the conditions of compressive loading. This difference in plastic behavior gives rise to the strength asymmetry in tension vs. compression. The calculated values of the flow stress in tension and compression for nanocrystalline copper with grain sizes of 2–10 nm are tabulated in Table 1. It can be seen from the tabulated data that a decrease in grain size reduces the flow stress values as predicted by the inverse Hall–Petch effect. The calculated tension– compression strength asymmetry is also observed to decrease with a decrease in grain size down to a grain size of 4 nm, after which, a further decrease results in an increase in the strength asymmetry of the nanocrystalline metal. This high asymmetry below grain sizes of d = 4 nm is consistent with the mechanical behavior observed in metallic glasses where a strong strength asymmetry is observed and requires the use of a criterion that accounts for a normal stress dependence to predict plastic behavior. This confirms the result that the grain size of 2 nm is considered to be the lower
Tension
Compression
rTf (GPa)
rCf (GPa)
Strength asymmetry Calc. (%)
1.334 1.523 1.615 1.986 2.146 2.363
1.522 1.585 1.650 2.101 2.309 2.563
14.09 4.07 2.17 5.79 7.59 8.46
limit of nanocrystalline metals and closer to being amorphous. The variation of the flow stress values in compression and tension as a function of the nanocrystalline grain size is plotted in Fig. 4. It should be noted from the data in Table 1 that the MD simulations of the strength asymmetry predict an increase in strength asymmetry as the grain size decreases from 4 nm to 2 nm, whereas the MS simulations for nanocrystalline Ni predict an increase in the asymmetry as the grain size increases from 2 nm to 4 nm. The difference in behavior can be attributed to the difference in the material system (Ni for MS simulations as compared to Cu for MD simulations). As a result, MD simulations were carried out to calculate the flow stresses in tension and compression for nanocrystalline Ni with grain sizes from 2 nm to 4 nm. The calculated values of the flow stress in tension and compression for nanocrystalline Ni are tabulated in Table 2 in comparison with the flow stress values reported in the literature [9,10] for quasi-static loading at grain sizes of 2–4 nm. It should be noted from the data in Table 2 that the strength asymmetry at grain sizes less than 4 nm for nanocrystalline Ni is consistent with that obtained for nanocrystalline Cu. The increase in strength asymmetry as the grain size decreases below 4 nm at high strain rates, however, contradicts the behavior predicted using MS simulations at these grain sizes. The striking difference can be related to high strain rate behavior in MD simulations as compared to quasi-static loading conditions in MS simulations. However, the flow stress values calculated for nanocrystalline Ni from MD simulations are much lower than those calculated using during quasi-static loading using MS simulations. One reason for the difference in the flow stress values can be due to the added velocity component to the stress calculations in MD simulations as indicated in Eq. (1) as compared to the stress
Fig. 3. The nanocrystalline Cu system during MD simulations of deformation for (a) conditions of tensile loading, and (b) for conditions of compressive loading, at times corresponding to the peak value of stress in Fig. 2 at a constant strain rate of 108 s1. The atoms are colored according to CNA values. The contour scale is the same as in Fig. 1.
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
263
Fig. 4. (a) Plot of the calculated flow (peak) stress in tension and compression for nanocrystalline Cu with an average grain size ranging from 2 nm to 10 nm at a strain rates 109 s1. The flow stress values in compression (circles) are greater than those in tension. (b) Plot of the calculated tension–compression strength asymmetry for nanocrystalline Cu with an average grain size ranging from 2 nm to 10 nm at a strain rate of 109 s1.
Table 2 Calculated values of flow (peak) stress during uniaxial deformation of nanocrystalline Ni in tension and compression at different average grain sizes at a constant strain rate of 109 s1 in comparison with those obtained using MS simulations under quasi-static (Q.S.) loading conditions [9,10] reported in the literature. Grain size (nm)
2 3 4
Tension
Compression
Strength asymmetry
rTf (GPa)
Q.S.
rCf (GPa)
Q.S.
Calc. (%)
Q.S. (%)
2.77 3.33 3.89
3.30 4.50 4.90
2.98 3.46 3.92
4.30 6.10 7.10
7.58 3.90 0.77
30.3 35.5 43.9
calculations in MS simulations. This contribution to stresses, however, is found to be significantly small as compared to the total value of the stress calculated in MD simulations. This raises the question: ‘‘Are the conditions of deformation used in MS simulations really quasi-static”? To address this question, the details of the deformation conditions in the MS simulations are discussed below and compared with MD simulations. Quasi-static deformation in MS simulations is obtained by scaling the atomic coordinates (strain increments along the axis) followed by relaxation of the system using the conjugate gradient technique [9,10]. This relaxation only allows a local relaxation of the atoms and does not allow for complete adiabatic relaxation of the system during the deformation process. Given the system sizes used, the strain increments used (60.1%) at each step in the MS simulations are orders of magnitude larger than that used (60.0004% at 109 s1) in the MD simulations. The deformation strain in the MS simulations may therefore result in deformation at ultra-high strain rates (P1011 s1) which results in the large strength asymmetry (30%) in tension and compression for nanocrystalline Ni. To test this, the flow stress was calculated for nanocrystalline Ni for uniaxial stress loading in tension and compression using MD simulations at a strain rate of 1011 s1. The tension compression strength asymmetry is calculated to be 21%, 20% and 22% at 2 nm, 3 nm, and 4 nm, respectively at a strain rate of 1011 s1. The calculated tension–compression strength asymmetry using MD simulations for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm is shown in Fig. 5 at a strain rate of 109 s1 (solid line), 1011 s1 (dash-dot line) in comparison with the tension–compression strength values for MS simulations [9,10] under Q.S. loading conditions (dashed line). The values of
the strength asymmetry at 1011 s1 compare better with those observed for quasi-static loading of nanocrystalline Ni. As a result, the plastic behavior at higher strain rates (>109 s1) results in a higher asymmetry due to the fact that the atoms have less time to respond to the deformation. This high strain rate behavior is inherently included in MS simulations due to the conditions of the local relaxation during deformation, and therefore, the MS simulations suggests higher values of flow stresses as well as strength asymmetry. Thus, the yield behavior of materials predicted using computational techniques is largely dependent on the method used. MD simulations are typically carried out at strain rates P107 s1. As a result, a strain rate effect is inherently included in the material response and needs to be considered when comparing with the results obtained experimentally under quasi-static loading conditions. Molecular statics simulations, on the other hand do not allow for diffusion processes and thermal relaxation that occur during quasi-static loading conditions. As a result, the conditions of deformation used molecular static simulations (in spite of having stress states as ry = rz = 0, and rx = r) cannot be considered as quasi-static conditions of deformation. Unfortunately, there is no experimental evidence of the strength asymmetry at these grain sizes to compare the results obtained suing computational techniques. The strength asymmetry calculated for nanocrystalline copper using hardness and tensile testing [16,17] suggested an increase in asymmetry as the grain size reduced from 100 nm to 16 nm with the asymmetry calculated to be 3% at 16 nm. To compare this value with that obtained using MD simulations at high strain rates, the strength asymmetry values are calculated for nanocrystalline Cu with an average grain size of 6 nm at strain rates ranging from 109 s1 to 1011 s1 and tabulated in Table 3 and fitted to a quadratic curve as shown in Fig. 6. The fitting parameters of the quadratic curve predict a strength asymmetry value of 5% under quasi-static loading conditions. This value is in close agreement with the experimental value of 3% at a grain size of 16 nm. However, it should be noted that the data fitted is for strain rates P109 s1 wherein GB based processes are limited. At lower strain rates, it can be expected that GB sliding and diffusional processes occur which will result in the weakening of the nanocrystalline metal and also lower asymmetry values. Thus, more simulations of the plastic yielding of the nanocrystalline metals at lower strain rates (104 s1–109 s1)
264
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
Fig. 5. (a) Plot of the calculated flow (peak) stress in tension (green) and compression (black) for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm at a strain rates 109 s1 in comparison with the values reported from molecular static (MS) simulations [9,10] under quasi-static (Q.S.) loading conditions (dashed line). (b) Plot of the calculated tension–compression strength asymmetry for nanocrystalline Ni with an average grain size ranging from 2 nm to 4 nm at a strain rate of 109 s1 (solid black line) and 1011 s1 (dash-dot line) in comparison with the tension–compression strength values for molecular static (MS) simulations [9,10] under quasi-static (Q.S.) loading conditions (dashed line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 3 Values of flow (peak) stress during uniaxial deformation of nanocrystalline Cu with an average grain size of 6 nm at various strain rates in tension and compression. Strain rate e_ (s1)
Tension (flow stress) rTf (GPa)
Compression (flow stress) rCf (GPa)
Strength asymmetry (%)
109 2 109 4 109 8 109 1011
1.986 2.164 2.405 2.794 6.399
2.101 2.303 2.577 3.010 8.712
5.79 6.42 7.15 7.73 36.15
4. Conclusions The MD simulations reported here suggest that strength asymmetry is sensitive to the grain size of the nanocrystalline metal as well as the strain rates used. MD simulations suggest that a decrease in grain size reduces the flow stress values as predicted by the inverse Hall–Petch effect. However, the tension–compression strength asymmetry is observed to decrease as the grain size decreases down to a size of 4 nm, after which, a further decrease in grain size results in an increase in the strength asymmetry values. The higher asymmetry at a size of 2 nm is consistent with the mechanical behavior observed in metallic glasses and can be attributed to the amorphous nature of the nanocrystalline metal. However, the flow stress values obtained using MD simulations are much less as compared to those obtained using MS simulations for quasi-static loading conditions. A comparison of the methods used to calculate the flow stresses suggests that the high flow stress values result from the very high effective deformation strain rates resulting from the MS simulation technique. As a result, the atomic scale yield behavior of metals with nanometer-scale grains under quasi-static loading conditions at these small grain sizes is still not clearly understood. This suggests the need to design computational tools that will be capable of modeling deformation behavior under quasi-static loading conditions so as to understand the contributions of GB based processes and dislocation based processes to plasticity under multi-axial loading conditions and in turn the yield criterion.
Acknowledgements
Fig. 6. Plot of the calculated tension–compression strength asymmetry for nanocrystalline Cu with an average grain size of 6 nm as a function of strain rate of deformation. The solid black line shows the quadratic fit to the data.
are needed to be able to accurately predict the strain rate dependence of the yield behavior of nanocrystalline metals.
AMD is supported by the US Army Research Office (ARO) through the National Research Council Research Associateship Program. DWB and MAZ are also supported by the National Science Foundation through Grants DMR-03,04,299 and DMR-08,06,323. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or other funding agencies.
A.M. Dongare et al. / Computational Materials Science 49 (2010) 260–265
References [1] V. Yamakov, D. Wolf, S.R. Phillpot, A.K. Mukherjee, H. Gleiter, Nat. Mater. 1 (2002) 45. [2] D.W. Brenner, Computer modeling of nanostructured materials, in: Carl Koch (Ed.), Nanostructured Materials, second ed., Noyes Publications, Norwich, NY, 2006 (Chapter 7). [3] H. Van Swygenhoven, Science 296 (2002) 66. [4] J. Schiotz, K.W. Jacobsen, Science 301 (2003) 1357. [5] A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Metall. Mater. Trans. A 41A (2010) 523. [6] A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Phys. Rev. B 80 (2009) 104108. [7] E.M. Bringa, A. Caro, Y. Wang, M. Victoria, J.M. McNaney, B.A. Remington, R.F. Smith, B.R. Torralva, H. Van Swygenhoven, Science 309 (2005) 1838.
265
[8] R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, fourth ed., Wiley, New York (NY), 1996. [9] A.C. Lund, C.A. Schuh, Acta Mater. 53 (2005) 3193–3205. [10] A.C. Lund, T.G. Nieh, C.A. Schuh, Phys. Rev. B 69 (2004) 012101. [11] P.M. Derlet, H. Van Swygenhoven, Phys. Rev. B 67 (2003) 014202. [12] S. Kumar, S.K. Kurtz, J.R. Banavar, M.G. Sharma, J. Stat. Phys. 67 (1992) 523. [13] D.J. Honneycutt, H.C. Andersen, J. Phys. Chem. 91 (1987) 4950. [14] A.F. Voter, The embedded atom method, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, vol. 77, Wiley, New York, 1994. [15] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393. [16] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019. [17] M. Legros, B.R. Elliott, M.N. Rittner, J.R. Weertman, K.J. Hemker, Philos. Mag. A 80 (2000) 1017.