Microstructural investigation of the hardening mechanism in fcc crystals during high rate deformations

Computational Materials Science 138 (2017) 10–15

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Microstructural investigation of the hardening mechanism in fcc crystals during high rate deformations Mohammadreza Yaghoobi, George Z. Voyiadjis ⇑ Computational Solid Mechanics Laboratory, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, United States

a r t i c l e

i n f o

Article history: Received 7 April 2017 Received in revised form 31 May 2017 Accepted 2 June 2017

Keywords: Molecular dynamics Dislocation Hardening mechanism High rate deformations

a b s t r a c t The present paper studies the hardening mechanism in fcc metallic structures during high rate deformations by incorporating the dislocation network properties. The large scale atomistic simulation is used to study the variations of dislocation length distribution and its characteristic lengths as the strain rate changes. First, the dislocation length distributions at different strain rates are studied to qualitatively capture the relation between the material strength and applied strain rate. It is observed that increasing the strain rate decreases the dislocation network lengths. Accordingly, the required stress to activate the dislocation sources increases. Since dislocation movements sustain the imposed plastic flow, higher activation stress leads to material hardening. Furthermore, the results show that the properties of dislocation length distribution at high deformation rates are different from those of lower strain rates due to the activation of cross-slip mechanism at high strain rates. In order to quantitatively describe the relation between the material strength and dislocation network properties as the strain rate varies, the variations of average and maximum dislocation length are investigated in the cases of different applied strain rates. The relation between the material strength and average length of mobile dislocations is captured using an inverse linear equation which shows a good agreement with the atomistic simulation results. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Up to now, many strengthening mechanisms have been identified in bulk metallic samples including forest hardening mechanism (i.e. interaction of dislocations with each other), Hall-Petch effects (i.e. interaction of dislocation with the grain boundary), solid-solution strengthening, precipitation strengthening, and variation of lattice resistance [1]. When it comes to the high rate deformations, the experimental and theoretical studies have demonstrated intriguing deformation and strengthening mechanisms that are different from those of the slower rates. Several experimental techniques of gas guns, split Hopkinson bars, Zpinch, and high-energy pulsed lasers have been developed to study the material behavior during high rate deformations [2,3]. The experiments cover a wide range of strain rates up to 1010 s
1 . Recently, Crowhurst et al. [4] incorporated a laser pulse to accelerate the free surface of tantalum and studied its yielding at strain rates up to 109 s
1 . Remington et al. [3] reviewed the application of high-power pulsed to induce the strain rates of 106
1010 s
1 . They addressed the fundamental issues at high strain rates includ⇑ Corresponding author. E-mail address: [email protected] (G.Z. Voyiadjis). http://dx.doi.org/10.1016/j.commatsci.2017.06.003 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

ing dislocation velocity, the slip-twinning transition, and the transition between thermally-activated and phonon drag regimes [3]. The experimental results showed that the mechanisms governing the strengthening in the high rate deformations are different from those of the slower strain rates. As an example, Park et al. [2] showed that the plastic flow in metallic materials is independent of the grain size at ultrahigh pressures and strain rates and the conventional Hall-Petch effect is not available anymore. The governing deformation mechanisms in metallic samples depend on the applied strain rate. The Frank-Read source operation, which is a well-stablished deformation mechanism (see, e.g., Hirth and Lothe [5] and Hull and Bacon [6]), is the primary mechanism of deformation for strain rates up to 103
106 s
1 (see, e.g., Gurrutxaga-Lerma et al. [7]). Increasing the strain rates, the cross-slip [7–16] and homogenous dislocation nucleation [7,17–21] become the dominant mechanisms of deformation. The relation between the material strength and strain rate have been captured using various mechanisms. The primary relation between the material strength and strain rate can be developed using the Arrhenius form equation that leads to a linear relation between the strength and the natural logarithm of strain rate. The linear relation is applicable for strain rates up to 104 s
1 [22–24]. In this range of strain rates, the thermal-activation mechanism controls

M. Yaghoobi, G.Z. Voyiadjis / Computational Materials Science 138 (2017) 10–15

the variation of strength as the strain rate changes [23,24]. Gurrutxaga-Lerma et al. [7] captured the variation of strength versus the strain rate by relating the activation time and strength of Frank-Read sources to the applied strain rate that is applicable to strain rates up to 106 s
1 . The relation between the strength and strain rate for strain rates higher than 104 s
1 can be described using the interaction between the moving dislocations with phonon drag [23,24]. Furthermore, in the cases of strain rates larger than 104 s
1 , Fan et al. [22] incorporated the non-Arrhenius behavior in their model to capture the relation between the strength and the applied strain rate. Theoretical models which define the relation between the strain rate and strength of fcc metallic structures have been widely investigated (see, e.g., Hirth and Lothe [5] and Hull and Bacon [6]). The microstructural studies can be incorporated to evaluate and modify the developed models. In the case of fcc metallic structures, the hardening mechanism is controlled by the properties of dislocation network (see, e.g., Hirth and Lothe [5] and Hull and Bacon [6]). In order to study the evolution of dislocation network as the strain rate varies, the incorporated microstructural method should be able to capture dislocations as separate entities. Discrete dislocation dynamics (DDD) has been incorporated to study the hardening mechanisms in metallic materials using the dislocation network properties (see, e.g., El-Awady et al. [25]). In the case of high rate deformations, the cross-slip is one of the major governing deformation mechanisms [7–16]. The cross-slip has been incorporated in DDD using some ad-hoc assumptions (see, e.g., Wang et al. [11] and Hussein et al. [15]), and more investigations should be conducted to evaluate the accuracy of DDD to capture the crossslip at high strain rates. Furthermore, Gurrutxaga-Lerma et al. [26] showed that a dynamic DDD should be incorporated to capture the hardening mechanism at high strain rates. However, until now, an investigation of the hardening mechanism in bulk fcc materials at high rate deformations that relates the material strength to the dislocation network properties, such as dislocation length distribution, has not been conducted using a DDD which incorporates a well-established cross-slip model and appropriate dynamic framework. Another method to investigate the microstructural evolution of metallic structures as the strain rate varies and its relation with the hardening mechanism is to model the sample with full atomistic details using molecular dynamics (MD). After defining appropriate atomic interactions, MD is fully capable of capturing all the deformation mechanisms, such as cross-slip, without using any ad-hoc approximation. However, MD, due to its nature, has limitations on the simulation length scale and time scale. Nowadays, by developing efficient parallel algorithms and powerful supercomputers, samples consisting of one billion atoms can be modeled using atomistic simulation [27–30]. In the case of time scale, although the MD cannot capture the quasi-static experiments, it is an appropriate method to model the high rate deformations [31–34]. Unlike the DDD which directly reports the dislocation properties, MD only calculates the atomic trajectories and velocities. Accordingly, the dislocation properties should be obtained from the atomic trajectories using some post-processing methods. Stukowski [35] implemented and compared various methods of defect detection in crystalline materials including energy filtering, centrosymmetry parameter analysis, bond order analysis, Voronoi analysis, adaptive common neighbor analysis, and neighbor distance analysis. Stukowski and his co-workers [35,36] have developed the Crystal Analysis Tool which is able to extract the required dislocation information including the dislocation length, position, and Burgers vector from atomic trajectories. Until now, the dislocation density or the total length of dislocations are the only variables which have been considered to study the hardening mechanisms in fcc bulk

11

metallic samples [16,28,31,37–42]. Accordingly, deeper understanding of dislocation network properties, such as dislocation length distribution, average length of dislocations, and maximum dislocation length should be acquired to fully capture the hardening mechanisms. One of the most important properties of dislocation network is the dislocation length distribution. In the case of metallic bulk samples and quasi-static strain rates, El-Awady et al. [43] incorporated the experimental results reported by Mughrabi [44] and described the probability density function (PDF) of the dislocation length using a Weibull distribution. In the cases of metallic samples of confined volumes with free surfaces, such as pillars, the PDF of the dislocation length changes due to the interaction of dislocations with free surfaces [43,30]. Accordingly, the size effects in pillars can be captured using the modified dislocation length distribution [43,30]. In the cases of bulk metallic samples and high rate deformations, however, the evolution of dislocation length distribution has not been investigated as the strain rate varies. In this work, the large scale atomistic simulation is incorporated to investigate the hardening mechanism in fcc bulk materials during high rate deformations using the properties of dislocation length distribution. The variations of dislocation network and its characteristic lengths including the average and maximum dislocation length versus the strain rate are quantitatively studied using large scale atomistic simulation. As the strain rate increases, the dislocation density increases to sustain the imposed plastic flow. Accordingly, increasing the dislocation density leads to the reduction of the characteristic lengths of dislocation network. Some characteristic lengths of the dislocation network including the average and maximum dislocation length are investigated to capture the variation of strength as the strain rate changes. The smaller dislocation characteristic lengths lead to the larger strength. It is due to the fact that larger stresses are required to activate smaller dislocations. Accordingly, increasing the strain rate leads to the material hardening. 2. Simulation details and methodology Large scale atomistic simulation of a Ni single crystal cube, as an example of fcc material, during uniaxial compression test is incorporated to study the hardening mechanism in the region of high strain rates ðe_  108 s
1 Þ. The cube with the dimensions of  0, ½1 1 2,  and ½1 1 1 direc1500 Å, 1500 Å, and 1500 Å along ½1 1 tions, respectively, is generated (Fig. 1). The parallel code LAMMPS [45] is incorporated to perform the atomistic simulation. The sample consists of around 310,000,000 atoms. The embedded-atom method (EAM) potential developed by Mishin et al. [46] is used to model the Ni-Ni atomic interaction [47]. The periodic boundary conditions are incorporated for all three directions. Newton’s equations of motion are integrated using the velocity Verlet algorithm with a time step of 1.5 fs. The simulations are performed using NPT ensemble. The sample is relaxed for 100 ps with the temperature increasing from 1 K to 300 K and zero pressure in all directions. Next, it is relaxed for another 100 ps at 300 K and zero pressure in all directions. Finally, the uniaxial compression is conducted along ½1 1 1 direction from the sample top and bottom with three constant strain rates of e_ 1 ¼ 2:0e8 s
1 , e_ 2 ¼ 1:0e9 s
1 , and e_ 3 ¼ 5:0e9 s
1 at 300 K and zero pressure on the lateral boundaries (Fig. 1). The Crystal Analysis Tool developed by Stukowski and his co-workers [35,36] is incorporated to extract the dislocation structures from the atomic trajectories, and provide additional information such as dislocation length and Burgers vector. The Crystal Analysis Tool is developed based on the commonneighbor analysis method. The atomic arrangements and structures are initially identified. Next, a pattern matching algorithm

12

M. Yaghoobi, G.Z. Voyiadjis / Computational Materials Science 138 (2017) 10–15

Fig. 1. Geometry and orientation of the simulated sample.

groups atoms into the so-called clusters. A Delaunay mesh is generated that connects all atoms. The elastic deformation gradient tensor is computed for the constructed Delaunay mesh. The elastic deformation gradient is multi-valued if a dislocation intersects a tessellation element. Next, trial circuits are built on the interface mesh to define the dislocation lines. The code provides a onedimensional line representation of the dislocation segments. The dislocation network is visualized and analyzed using the software Paraview [48]. Finally, the dislocation density is obtained by dividing the total dislocation length by the sample volume. 3. Results and discussions The strain rate e_ induced by dislocation movements can be described as follows [49–51]:

e_ ¼ gbqm v þ gbq_ ml;

ð1Þ

 is the average dislocation velocity, qm is the mobile dislowhere v cation density, l is the distance associated with the displacement of nucleated dislocations, q_ m is the mobile dislocation nucleation rate, b is the Burgers vector magnitude, and g is a microstructure factor on the order of unity. Based on Eq. (1), high strain rates can be achieved through the movement of dislocation network with large qm at subsonic velocities, the movement of a dislocation network with small qm at very high (supersonic) velocities, or the large nucleation rate of mobile dislocations (i.e. large q_ m ) associated with l. Gurrutxaga-Lerma et al. [7] incorporated the three dimensional dislocation dynamics and showed that in the region of high strain rates e_  108 s
1 , the dislocation velocity changes very slightly by increasing the strain rate. In other words, according to Eq. (1), the increase in the strain rate is sustained either by increasing the mobile dislocation density or nucleation rate of mobile dislocations. Remington et al. [3] proposed an analytical dislocation velocity expression based on the atomistic simulation results reported by Tang et al. [52] and Deo et al. [53] as follows:

   m  r ; v ¼ v s 1
A exp
B

sp

ð2Þ

where v s is the shear wave velocity, A, B, and m are fitting parameters, and sp is the Peierls-Nabarro stress. Based on Eq. (2), the dislocation velocity cannot exceed the shear wave velocity. One should note that the results reported by Deo et al. [53] are obtained for movement of a single dislocation. In the case of a dislocation interacting with other dislocations, especially in samples with large dislocation density, the average dislocation velocity becomes much smaller. Fig. 2(a) shows the variation of dislocation density q versus the strain e for different strain rates of e_ 1 , e_ 2 , and e_ 3 . The dislocation density in all three samples starts from nearly a zero value because all samples are initially defect free. Fig. 3 shows the initial stages of dislocation nucleation in the case of e_ 1 ¼ 2:0e8 s
1 . The majority of dislocations are Shockley partial dislocations with the Burgers vector of 1=6h1 1 2i, which are illustrated with green color in Fig. 3. The results show that the cross-slip is the dominant deformation mechanism that increases the pinning point numbers. The dislocations pinned at their ends are then elongated and provide the required mobile dislocation length to sustain the applied strain rate. The output of atomistic simulation is the total dislocation density and not the mobile one, and it is nearly impossible to obtain the mobile dislocation density. Accordingly, in order to study the trend of hardening mechanisms, it is assumed that the mobile dislocation density is a constant share of the total one. In the initial stages of plasticity [Fig. 3(a) and (b)] since the value of the mobile dislocation density is small, large dislocation nucleation rate is required to sustain the applied strain rate that results in a large initial increase of the total dislocation density. As the applied strain rate increases, the required dislocation nucleation rate also increases that leads to the larger dislocation density. Fig. 2(a) shows that after the initial sharp increase in dislocation density, there is a drop in the dislocation content. It is due to the annihilation of dislocations nucleated at the large stresses prior to the softening. Since the samples do not experience those large stresses after softening, a portion of annihilated dislocations cannot be nucleated again. Furthermore, a higher strain rate leads to a higher drop in the dislocation density. It is due to the fact that the stress values experienced before the softening are larger for higher strain rates. Accordingly, a larger portion of the annihilated dislocations cannot be further nucleated after softening. The dislocation nucleation rate decreases as the mobile dislocation density increases,  to sustain the imposed strain rate which activates the term gbqm v [Fig. 3(c)]. The molecular dynamics results show that in the range of high strain rates ðe_  108 s
1 Þ, as the strain rate increases, the dislocation density increases. The range of dislocation densities are in the order of 1017 m
2 that is large compared to the quasistatic experiments due to the high strain rates. Fig. 2(b) investigates the variation of stress r versus the strain e for different strain rates of e_ 1 , e_ 2 , and e_ 3 . The initial response is elastic and the sample is defect free. After the initial dislocation nucleation, the stress drops since the movement of nucleated dislocations sustain the applied plastic flow. The results show that the material strength increases as the strain rate increases. In order to study the hardening due to strain rate increase, the dislocation length distribution should be investigated. El-Awady et al. [44] incorporated a Weibull distribution to capture the dislocation length distribution in a pillar with the diameter of D as follows:

f ðkÞ ¼

 b
1 b k k e
ðhÞ h h

for k P 0;

ð3Þ

where f ðkÞ is the probability density function of dislocation source length of k, b > 0 is the shape parameter, and 0 < h < D. El-Awady et al. [43] used the experimental data of Mughrabi [44], which is illustrated in Fig. 4, to obtain the parameters of the Weibull

13

M. Yaghoobi, G.Z. Voyiadjis / Computational Materials Science 138 (2017) 10–15

25 = . Slow

50

20

40

Stress (GPa)

Dislocation density (1016 m-2)

60

30

= . Medium = Fast

.

15

10

20

5 10

0

0 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Strain

Strain

(b) Stress-Strain

(a) Dislocation density-Strain

Fig. 2. The compressive response of Ni single crystal cube at different strain rates of e_ 1 ¼ 2:0e8 s
1 , e_ 2 ¼ 1:0e9 s
1 , and e_ 3 ¼ 5:0e9 s
1 .

Fig. 3. Initial dislocation nucleation and evolution in Ni single crystal cube subjected to the strain rate of e_ 1 ¼ 2:0e8 s
1 at different strains of: (a) e ¼ 0:012 (b) e ¼ 0:018 (c) e ¼ 0:036.

0.3

1.E+00 SlowE0.21 = 2.0e8 s , = 2.0e8 s , SlowE0.33 SlowE0.45 = 2.0e8 s , = 1.0e9 s , MediumE0.21 = 1.0e9 s , MediumE0.33 = 1.0e9 s , MediumE0.45 = 5.0e9 s , FastE0.21 = 5.0e9 s , FastE0.33 = 5.0e9 s , FastE0.45

0.25 1.E-01

Log ( PDF )

0.2 0.15 0.1 0.05 0

1.E-02

= 0.21 = 0.33 = 0.45 = 0.21 = 0.33 = 0.45 = 0.21 = 0.33 = 0.45

1.E-03

1.E-04

0

0.5

1

1.5

2

Increasing strain rate

= 1.E-05

0

100

200

300

400

500

600

700

Dislocation length (Å) Fig. 4. Probability distribution function of dislocation link-length. The data is experimentally measured for Cu (Mughrabi [44]) and used by El-Awady et al. [43] in their DDD calculations.

Fig. 5. Variation of probability distribution function of dislocation length as the strain rate changes for different strain rates of e_ 1 ¼ 2:0e8 s
1 , e_ 2 ¼ 1:0e9 s
1 , and e_ 3 ¼ 5:0e9 s
1 .

14

M. Yaghoobi, G.Z. Voyiadjis / Computational Materials Science 138 (2017) 10–15

Fig. 6. Variation of the normalized maximum dislocation length ðkmax Þe_ =ðkmax Þe_ 3 , normalized stress ðrÞe_ 3 =ðrÞe_ , and normalized average dislocation length ð kÞe_ =ð kÞe_ 3 versus the strain for the applied strain rates of: (a) e_ 1 ¼ 2:0e8 s
1 (b) e_ 2 ¼ 1:0e9 s
1 .

distribution. In the case of high rate deformations, Fig. 5 shows the PDFs of the dislocation length for the simulated strain rates of e_ 1 , e_ 2 , and e_ 3 and selected strains of 0.21, 0.33, and 0.45 obtained from atomistic simulation. The results show that the PDF of the dislocation length at high strain rates is different from that of the slower rates, and the maximum probability of dislocation length belongs to the very small dislocations. It is due to the activation of cross-slip that is the dominant deformation mechanism at high rate deformations (Fig. 3). During the high rate deformations, many small dislocations are activated via the cross-slip that their movement are immediately ceased because the stresses required to activate these sources are very large. Fig. 5 shows that as the strain rate increases, the PDF of the dislocation length moves toward the smaller dislocation lengths. To unravel the mechanism that controls the dislocation length distribution, one should investigate the variation of dislocation density versus the strain rate. Fig. 2(a) shows that the dislocation density increases as the strain rate increases which leads to the smaller dislocation lengths (Fig. 5). In other words, increasing the dislocation density increases the probability of dislocations colliding with each other that decreases the dislocation lengths. The material hardening as the strain rate increases [Fig. 2(b)] can be qualitatively captured using the results presented in Fig. 5 that show that the observed hardening is due to the dislocation length reduction. In order to quantitatively analyze the dislocation length distribution, some characteristic lengths should be introduced. The most appropriate choice is the average length of mobile dislocations  km . However, from the output of atomistic simulation, it is nearly impossible to obtain  km . On the other hand, two other characteristic lengths that can be introduced are the average dislocation length  k and the maximum dislocation length kmax . The stress required to activate a dislocation source has an inverse relation with the dislocation length. Accordingly, considering the same applied stress, the length of a mobile dislocation is larger than that of an immobile one. Since  k includes both mobile and immobile dislocations, therefore  k< km . The maximum dislocation length kmax , on the other hand, is the largest dislocation in the sample and kmax >  km >  k. Instead of studying the average length of mobile dislocations  km , one can study the variation of maximum dislocation length kmax and the average dislocation length  k as the upper and lower bounds of  km . A new term rk should be added to the gen-

eral definition of material strength to incorporate this hardening mechanism. In the case of pillars that the dislocation length distribution changes because of the dislocation interaction with free surfaces, Parthasarathy et al. [54] proposed that the resolved shear strength has an inverse linear relation with the mean value of the source length. Parthasarathy et al. [54] developed the model for the operation of Frank-Read sources in the case of pillars. Here, the dislocation length distribution changes due to the strain rate variation. However, the Frank-Read source operation is not the governing deformation mechanism anymore, and cross-slip controls the plastic deformation (Fig. 3). In the case of cross-slip, the smaller dislocations are still harder to be activated. Accordingly, an inverse linear relation between rk and  km is proposed here which can be described as follows:

g Gb rk ¼  ; km

ð4Þ

 is a geometrical constant, and G is the shear modulus. In where g the case of fcc metals, the majority of the dislocations are Shockley partial dislocations. Accordingly, the Burgers vector of the Shockley partial dislocations can be incorporated in Eq. (4). Since the samples are all from Ni, it can be assumed that the Burgers vector for different applied strain rates and strains are similar. Consequently, in the cases of simulated samples, Eq. (4) states that the stress has a linear relation with 1= km . In order to study Eq. (4), the applied stress r, maximum dislocation length kmax , and average dislocation length  k of the sample subjected to the applied strain rates of e_ 1 ¼ 2:0e8 s
1 and e_ 2 ¼ 1:0e9 s
1 are normalized using the results obtained from the sample subjected to the strain rate of e_ 3 ¼ 5:0e9 s
1 for different strain values. In the cases of e_ 1 , Fig. 6 shows that ðrÞe_ 3 =ðrÞe_ 1 is between 1.9 and 2.6, ð kÞe_ 1 =ð kÞe_ 3 is between 1.3 and 2.1, and ðkmax Þe_ 1 =ðkmax Þe_ 3 is between 2.3 and 3.5. The results  _ =ðkÞ  _ and smaller than show that ðrÞ _ =ðrÞ _ is larger than ðkÞ e3

e1

e1

e3

ðkmax Þe_ 1 =ðkmax Þe_ 3 for all strain values. In the case of e_ 2 , Fig. 6 shows that ðrÞe_ 3 =ðrÞe_ 2 is between 1.3 and 1.9, ð kÞe_ 2 =ð kÞe_ 3 is between 1.2 and 1.6, and ðkmax Þe_ 2 =ðkmax Þe_ 3 is between 1.5 and 2.1. Again, the kÞ _ =ð kÞ _ and smaller results show that ðrÞ _ =ðrÞ _ is larger than ð e3

e2

e2

e3

than ðkmax Þe_ 2 =ðkmax Þe_ 3 . The results show that the normalized values of r can be correlated with the normalized values of 1= k and

M. Yaghoobi, G.Z. Voyiadjis / Computational Materials Science 138 (2017) 10–15

1=kmax , and Eq. (4) can capture the results obtained from the atomistic simulation. One should notice that the oscillations in normalized kmax is due to the stochastic nature of kmax , while the k do not exhibit large oscillations. normalized r and  4. Conclusions The present work investigates the hardening mechanism in fcc metallic structures during high rate deformations using the large scale atomistic simulation both qualitatively and quantitatively. The dislocation network properties including the dislocation length distribution, average length of dislocations, and maximum dislocation length are incorporated to study the hardening mechanism. First, the dislocation length distribution variation as the strain rate changes is studied during high rate deformations. The results show that the distribution of dislocation length at high strain rates is different from that of the slower rates due to the activation of cross-slip mechanism which induces many small dislocations. As the strain rate increases, the dislocation lengths decreases which qualitatively explains the increase in material strength. In order to quantitatively describe the strength enhancement as the strain rate increases, the variations of two dislocation network properties including the average and maximum dislocation length are studied as the strain rate changes. An inverse linear equation is proposed to capture the relation between the material strength and average length of mobile dislocations. The inverse linear relation shows a good agreement with the atomistic simulation results. Acknowledgements This research was conducted with high performance computing resources provided by Louisiana State University (http://www.hpc. lsu.edu) and Louisiana Optical Network Initiative (http://www. loni.org). The current work is partially funded by the NSF EPSCoR CIMM project under award #OIA-1541079. References [1] A.S. Argon, Strengthening Mechanisms in Crystal Plasticity, Oxford University Press, Oxford, UK, 2008. [2] H.-S. Park, R.E. Rudd, R.M. Cavallo, N.R. Barton, A. Arsenlis, J.L. Belof, K.J.M. Blobaum, B.S. El-dasher, J.N. Florando, C.M. Huntington, B.R. Maddox, M.J. May, C. Plechaty, S.T. Prisbrey, B.A. Remington, R.J. Wallace, C.E. Wehrenberg, M.J. Wilson, A.J. Comley, E. Giraldez, A. Nikroo, M. Farrell, G. Randall, G.T. Gray, Phys. Rev. Lett. 114 (2015) 65502. [3] T.P. Remington, B.A. Remington, E.N. Hahn, M.A. Meyers, Mater. Sci. Eng. A 688 (2017) 429–458. [4] J.C. Crowhurst, M.R. Armstrong, S.D. Gates, J.M. Zaug, H.B. Radousky, N.E. Teslich, Appl. Phys. Lett. 109 (2016) 094102. [5] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., John Wiley & Sons, New York, 1982. [6] D. Hull, D.J. Bacon, Introduction to Dislocations, fifth ed., Elsevier Ltd, 2011. [7] B. Gurrutxaga-Lerma, D.S. Balint, D. Dini, A.P. Sutton, J. Mech. Phys. Solids 84 (2015) 273–292. [8] J. Bonneville, B. Escaig, Acta Metall. 27 (1979) 1477–1486. [9] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, S.G. Srinivasan, H. Jonsson, Phys. Rev. Lett. 79 (1997) 3676–3679.

15

[10] D. Caillard, J.L. Martin, Thermally Activated Mechanisms in Crystal Plasticity, Elsevier Science, Netherlands, 2003. [11] Z.Q. Wang, I. Beyerlein, R. LeSar, Modell. Simul. Mater. Sci. Eng. 15 (2007) 675– 690. [12] S. Rao, D.M. Dimiduk, J. El-Awady, T.A. Parthasarathy, M.D. Uchic, C. Woodward, Philos. Mag. 89 (2009) 3351–3369. [13] S.I. Rao, D.M. Dimiduk, J.A. El-Awady, T.A. Parthasarathy, M.D. Uchic, C. Woodward, Acta Mater. 58 (2010) 5547–5557. [14] C. Zhou, I.J. Beyerlein, R. LeSar, Acta Mater. 59 (2011) 7673–7682. [15] A. Hussein, S. Rao, M. Uchic, D. Dimiduk, J. El-Awady, Acta Mater. 85 (2015) 180–190. [16] M. Yaghoobi, G.Z. Voyiadjis, Comput. Mater. Sci. 111 (2016) 64–73. [17] M.A. Meyers, Scr. Metall. 12 (1978) 21–26. [18] J. Weertman, Mech. Mater. 5 (1986) 13–28. [19] J. Weertman, P.S. Follansbee, Mech. Mater. 7 (1988) 177–189. [20] M.Y. Gutkin, I.A. Ovidko, Appl. Phys. Lett. 88 (2006) 211901. [21] M.Y. Gutkin, I.A. Ovidko, Acta Mater. 56 (2008) 1642–1649. [22] Y. Fan, Y.N. Osetsky, S. Yip, B. Yildiz, Phys. Rev. Lett. 109 (2012) 135503. [23] N.R. Barton, J.V. Bernier, R. Becker, A. Arsenlis, R. Cavallo, J. Marian, M. Rhee, H. S. Park, B.A. Remington, R.T. Olson, J. Appl. Phys. 109 (2011) 073501. [24] N.R. Barton, M. Rhee, J. Appl. Phys. 114 (2013) 123507. [25] J.A. El-Awady, H. Fan, A.M. Hussein, in: C.R. Weinberger, G.J. Tucker (Eds.), Multiscale Mater. Model. Nanomechanics, Springer, 2016. [26] B. Gurrutxaga-Lerma, D.S. Balint, D. Dini, D.E. Eakins, A.P. Sutton, Proc. R. Soc. A 469 (2013) 20130141. [27] J. Roth, F. Gähler, H.-R. Trebin, Int. J. Mod. Phys. C 11 (2000) 317–322. [28] F.F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T.D. de La Rubia, M. Seager, Proc. Natl. Acad. Sci. 99 (2002) 5783–5787. [29] P. Vashishta, R.K. Kalia, A. Nakano, J. Nanoparticle Res. 5 (2003) 119–135. [30] G.Z. Voyiadjis, M. Yaghoobi, Scr. Mater. 130 (2017) 182–186. [31] E.M. Bringa, K. Rosolankova, R.E. Rudd, B.A. Remington, J.S. Wark, M. Duchaineau, D.H. Kalantar, J. Hawreliak, J. Belak, Nat. Mater. 5 (2006) 805–809. [32] Y.M. Wang, E.M. Bringa, J.M. McNaney, M. Victoria, A. Caro, A.M. Hodge, R. Smith, B. Torralva, B.A. Remington, C.A. Schuh, H. Jarmarkani, M.A. Meyers, Appl. Phys. Lett. 88 (2006) 061917. [33] W. Ma, W.J. Zhu, Y. Hou, J. Appl. Phys. 114 (2013) 163504. [34] M. Yaghoobi, G.Z. Voyiadjis, Acta Mater. 121 (2016) 190–201. [35] A. Stukowski, Model. Simul. Mater. Sci. Eng. 20 (2012) 045021. [36] A. Stukowski, V.V. Bulatov, A. Arsenlis, Model. Simul. Mater. Sci. Eng. 20 (2012) 085007. [37] M.J. Buehler, Alexander Hartmaier, Huajian Gao, Mark Duchaineau, F.F. Abraham, Comput. Methods Appl. Mech. Eng. 193 (2004) 5257–5282. [38] B.A. Remington, P. Allen, E.M. Bringa, J. Hawreliak, D. Ho, K.T. Lorenz, H. Lorenzana, J.M. McNaney, M.A. Meyers, S.W. Pollaine, K. Rosolankova, B. Sadik, M.S. Schneider, D. Swift, J. Wark, B. Yaakobi, Mater. Sci. Technol. 22 (2006) 474–488. [39] Z. Chen, S. Jiang, Y. Gan, Y.S. Oloriegbe, T.D. Sewell, D.L. Thompson, J. Appl. Phys. 111 (2012) 113512. [40] Y. Gao, C.J. Ruestes, D.R. Tramontina, H.M. Urbassek, J. Mech. Phys. Solids 75 (2015) 58–75. [41] G.Z. Voyiadjis, M. Yaghoobi, Mater. Sci. Eng. A 634 (2015) 20–31. [42] G.Z. Voyiadjis, M. Yaghoobi, Comput. Mater. Sci. 117 (2016) 315–329. [43] J.A. El-Awady, M. Wen, N.M. Ghoniem, J. Mech. Phys. Solids 57 (2009) 32–50. [44] H. Mughrabi, J. Microsc. Spectrosc. Electron 1 (1976) 571–584. [45] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [46] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393–3407. [47] C.A. Becker, F. Tavazza, Z.T. Trautt, R.A. Buarque de Macedo, Curr. Opin. Solid State Mater. Sci. 17 (2013) 277–283. http://www.ctcms.nist.gov/potentials. [48] A. Henderson, Paraview Guide, A Parallel Visualization Application, Kitware Inc, 2007. . [49] M.A. Meyers, K.K. Chawla, Mechanical Behavior of Materials, Cambridge University Press, Cambridge, UK, 2009. [50] R.W. Armstrong, W. Arnold, F.J. Zerilli, J. Appl. Phys. 105 (2009) 023511. [51] R.W. Armstrong, F.J. Zerilli, J. Phys. D: Appl. Phys. 43 (2010) 492002. [52] Y. Tang, E.M. Bringa, B.A. Remington, M.A. Meyers, Acta Mater. 59 (2011) 1354–1372. [53] C.S. Deo, D.J. Srolovitz, W. Cai, V.V. Bulatov, J. Mech. Phys. Solids 53 (2005) 1223–1247. [54] T.A. Parthasarathy, S.I. Rao, D.M. Dimiduk, M.D. Uchic, D.R. Trinkle, Scr. Mater. 56 (2007) 313–316.