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# Journal Club Theme of June 2014: Modeling Crystal Plasticity through Discrete Dislocation Dynamics Simulations

Molecular Dynamics (MD) simulations is a proven and powerful method in studying plasticity, deformation, and defect interactions in different material systems. However, some major challenges that face MD modeling include: being able to simulate reasonably large enough volumes that would capture enough statistics of the variations in the microstructure without worry about imposed boundary conditions; and the needed to overcome the severe-limitations in the temporal domain. These add to the difficulty of comparing traditional MD simulations to experimental results without making sometime outrageous assumptions.

A number of methods have been proposed over the years to overcome the severe limitations in time-scales (e.g. the parallel-replica dynamics method [1], the hyperdynamics method [2], and the temperature accelerated dynamics method [3]). In addition, to overcome the limitations in accessible length-scales a number of coarse graining methods have also been proposed. Such methods are strictly material and mechanisms based methods. Discrete dislocation dynamics is one such coarse-graining approach in the field of dislocation mediated plasticity.

**Background:**

In dislocation-mediated plasticity of crystalline materials, Discrete Dislocation Dynamics (DDD) methods provide a coarse graining approach in which both the time-scale and length-scale limitations in MD simulations are greatly reduced. DDD methods were developed to simulate plastic deformation at the mesoscale in crystalline structures by direct numerical simulations of the collective motion of dislocation ensembles with minimum ad hoc assumptions by direct numerical solution of their equations of motion. In this method dislocations are represented as discrete line-defects, from which their interactions among themselves as well as with other defects can be calculated, and in response their motion can be fully determined [4].

These computations can be performed in two dimensional (2D) or three-dimensional (3D) computational cells. In 2D DDD simulations dislocations are modeled as infinite long straight dislocations, and therefore can be represented by point defects in the 2D plane (e.g. [5, 6]). While such 2D methods provide a wealth of knowledge in a number of applications, they are limited in the sense that they do not fully capture the true 3D nature of dislocations. It is worth noting that a recent model has attempted to incorporate some 3D mechanisms into the 2D framework in what has been known as the 2.5D DDD method [7]. On the other hand, full 3D DDD methods are based on modeling the full dynamics of dislocation loops in space. Each loop can be discretized as connected straight having a pure edge or pure screw segments [8], or having mixed edge and screw segments [9]. In addition, a parametric approach was also used to represent dislocation loops can as curved splines [10].

Based on these methods a few widely used parallelized 3D DDD simulation codes have been developed over the past two decades. These codes include in no particular order: the “MDDP” code developed by Zbib et al [9, 11], the “ParaDiS” code developed at Lawrence Livermore National Laboratory (LLNL) [12], the “PDD” code developed by Ghoniem et al [13, 14], the “MicroMegas” developed by Devincre et al [8], the “PARANOID” code developed by Schwarz at IBM [15], the “TRIDIS” code developed by Fivel et al [], and the code developed by Weygand et al [9]. It is worth noting that while these codes vary in the approximations made, mechanisms, formulation, scalability, performance, and complexity, they in general produce qualitatively similar observations and predictions.

**Success Stories: **

Over the past decade both 2D and 3D DDD methods have been put to work to model a large set of technologically important problems. These include simulations on different materials (ranging from metals, semi-conductors, and reactive materials), crystallographic systems (FCC, BCC, and HCP crystals), loading conditions (mechanical, thermal, uniaxial, multiaxial, torsion, bending, cyclic, etc.), and applications (bulk response, thin films, micro-crystals, precipitate hardening, etc.). I would be happy to point to specific citations upon request since it is incomprehensible to list them all here

All in all, I personally believe that one of the greatest success stories of DDD simulations that demonstrated its great potential to the field of dislocation mediated plasticity, is the major insights DDD has provided over the past decade in the problem of size-scale effects in microcrystals. From the get-go DDD simulation results from multiple independent groups have shown that in the presence of a pre-existing dislocation networks that the crystal strength in microcrystals is fully governed by the weakest-dislocation sources available in the crystal (e.g. [17, 18]). This was subsequently validated through detailed in-situ TEM studies (e.g. [19]). Moreover, one of the first systematic computational studies to evaluate the effect of the initial dislocation density on the strength-scaling exponent was made utilizing 3D-DDD simulations [17], which was published before a similar experimental study on Mo micro-fibers [20]. Moreover, recent experiments and predictions from DDD simulations on the same materials and in the same range of crystal sizes and dislocation densities show excellent agreement (See Figure 1 below) [21]. Such simulations undoubtedly shows the strength of DDD methods in predicting dislocation mediated plasticity from submicron to tens of micron in length scale.

It is interesting to point out that in particular, the DDD simulation results published in literature on some materials (in particular nickel) span a much bigger range of initial dislocation densities than those performed experimentally, and further analysis of these simulations can lead to a wealth of knowledge not directly accessible with the available experiments in literature. In particular DDD simulation results include details of dislocation density evolution, variations in effective dislocation length, distributions of local strains and stresses, etc, all as a function of strain and time. Further analysis of these existing simulation results can thus potentially lead to identifying size-dependent constitutive laws that can be incorporated into crystal plasticity simulations to account for size-effects with minimal ad-hoc assumptions.

Figure 1: Log–log plots of: (a) Resolved shear strength vs. microcrystal diameter from 3-D DDD simulations (solid lines show the best power-law fit for each initial dislocation density); and (b) the magnitude of the power-law exponent, |n|, vs. initial dislocation density as computed from 3-D DDD simulations, alongside those computed from the current experiments and those published in the literature. Figure published in [21]

**Current Directions:**

DDD simulations are posed to address many of the still open questions during deformation of crystalline materials. As an example: resolving the details of grain-size, grain boundary character and overall evolution of plasticity in polycrystalles is of great important. While MD simulations can provide details of dislocation nucleation and interactions with a particular grain boundary or twin boundary, they are mostly focused on idealized nanocrystalline materials with grain sizes below a few tens of nanometers. In addition it is difficult to extrapolate from these results to provide quantitative assessment of the collective behavior of dislocations in ultrafine-grained of coarse-grained metals. On the other hand, while CP simulations can provide great wealth of knowledge on the evolution of plasticity in polycrystals, questions still remain on the applicability of some of the constitutive relations used to represent dislocation interactions near boundaries.

To address some of these gaps between MD and CP a number of efforts are underway to develop atomistically informed DDD simulations of polycrystalline systems [24-27], and twinned crystals [23]. In addition, a recent highly parallelized finite element method has been integrated within the highly parallel DDD code ParaDiS [28]. This recent model shows great potential in accurately resolving the complex fields near interfaces and free surfaces, and can prove importance in bridging the gap of accurately modeling polycrystalline materials with DDD simulations.

Other important and new directions include: developing large scale DDD simulations of anisotropic plasticity [29], dislocation evolutions and yield strength in single crystal superalloys with a high fraction of γ′ precipitates and at high temperatures [30, 31], dislocation climb and creep [32], and coupling between plastic deformation induced by discrete dislocations, vacancy diffusion, and heat propagation in solid crystals [33]. These are but a few citations/advances in the DDD community that can potentially enhance our understanding of many fundamental issues in the greater Mechanics and Materials community.

**Challenges:**

There are a number of challenges to DDD simulations that still need to be addressed. One of the shortcomings in DDD is reaching large strains for larger simulation volumes (on the order of 10s of microns). This is due to that dislocation densities typically multiply profusely with increasing strain, which subsequently translates as increasing number of degrees of freedom in the simulations (sometimes by a few orders of magnitude). While this issue might be overcome by increasing the number of processors during the simulations, most researchers do not have access to more than a few thousand processors for each simulation at a time, which greatly hinders their ability in modeling larger systems to higher strains and dislocation densities. Revisiting the architecture of DDD methods taking advantage of the advances in GPUs might help resolve such issues.

Another major challenge is validating some of the dislocation-mechanisms typically incorporated in the simulations. One such example is the cross-slip mechanisms in FCC crystals. Recent MD studies identified that cross-slip to preferentially occur at screw dislocation intersections with forest dislocations or at free surfaces (e.g. [34-36]). While there is no direct experimental evidence, the activation parameters computed from these studies MD agree well with experimental measurements (e.g. [37]). Our recent work incorporating these newly atomistically identified cross-slip mechanisms into DDD encouragingly shows that dislocation patter formation is strongly coupled with these mechanisms more than with the traditional bulk cross-slip mechanism. Nevertheless, detailed experimental characterization of hardening and dislocation microstructure formation in microcrystals is still necessary to validate these proposed mechanisms and simulations.

Dealing with dislocation climb mechanistically within the framework of DDD is also challenging due to the time-scale differences between dislocation glide and climb. Some DDD simulations do allow for a dislocation-climb mobility but that is mostly a fictitious model that does not account for the nonlinear vacancy-dislocation interactions inherent to climb [32]. Nevertheless, a recent 2D DDD simulations that fully accounts for matter transport due to vacancy diffusion and its coupling with dislocation motion [32]. Such formulation if incorporated into large scale 3D DDD simulations could potentially enhance our prediction of material response at elevated temperature, and when diffusion of species plays a major role in the deformation process (e.g. hydrogen embrittlement and stress assisted corrosion).

Finally, while many efforts have been performed to connect MD simulations with DDD simulations, either through hierarchically (e.g. [38]) or concurrently (e.g. in 2D [39, 40]), however, connecting DDD with CP is far more difficult since DDD is a discretized representation while CP is a continuum representation. Ongoing efforts in developing constitutive rules from DDD simulations (e.g. [41, 42]), tools to represent the discretized representation of dislocation in a continuum framework, efforts in developing continuum theories of dislocations, could serve as a tool towards achieving such linking (e.g. [44]).

In summary, recent advances in DDD methods provide a microstructurally based tool with minimum ad-hoc assumptions capable of modeling the plastic deformation in crystalline structures. These tools can provide insightful understanding and characterization of plasticity in a variety of crystals and for a variety of applications. Phase field (e.g. [45]) and level set (e.g. [46]) based dislocation dynamics method do provide over avenues of studying plasticity. However such methods are still a few year away before their potential is fully realized [47]. For further reading and citations regarding the DDD method and its applications the following two recent review articles are very useful: [48, 49].

References:

[1] A.F. Voter, Parallel replica method for dynamics of infrequent events, Phys. Rev. B, 57:R13985-R13988, (1998).

[2] A.F. Voter, Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events, Phys. Rev. Lett., 78:3908-3911, (1997).

[3] F. Montalenti, M.R. Sørensen, A.F Voter, Closing the gap between experiment and theory: crystal growth by temperature accelerated dynamics, Phys. Rev. B, 87:126101, (2001).

[4] R. Lesar, J.M Rickman, Coarse Graining of Dislocation Structure and Dynamics, in Continuum Scale Simulation of Engineering Materials: Fundamentals Microstructures Process Applications, (2005).

[5] J. Lepinoux, L.P. Kubin, The dynamic organization of dislocation structures - A simulation, Scripta Metall., 21:833-838, (1987).

[6] E. Van der Giessen, A.Needleman. Discrete dislocation plasticity: a simple planar model, Mod. Sim. Mater Sci. Eng., 3:689-735 (1995).

[7] A. Benzerga, Y. Brchet, A. Needleman, E. Van der Giessen, Incorporating three dimensional mechanisms into two-dimensional dislocation dynamics, Mod. Sim. Mater Sci. Eng., 12, 159–196 (2004).

[8] B. Devincre, L.P. Kubin, Simulation of forest interactions and strain-hardening in fcc crystals, Mod. Sim. Mater Sci. Eng., 2:559-570, (1994).

[9] H.M. Zbib, M. Rhee, J.P. Hirth, On plastic deformation and the dynamics of 3D dislocations, Int. J. Mech. Sci., 40:113-127, (1998).

[10] N.M Ghoniem, L.Z. Sun, Fast sum method for the elastic field of three-dimensional dislocation ensembles, Phys. Rev. B, 60:128-140, (1999).

[11] H.M. Zbib, T. Diaz de la Rubia, Int. J. Plasticity, A multiscale model of plasticity, 18 (2002) 1133-1163.

[12] A. Arsenlis, W. Cai, M. Tang, M. Rhee, T. Oppelstrup, G. Hommes, T.G. Pierce, V.V. Bulatov, Enabling strain hardening simulations with dislocation dynamics, Mod. Sim. Mater Sci. Eng., 15:553–595, (2007).

[13] Q.Z. Wang, N. Ghoniem, S. Swaminarayan, R. LeSar, A Parallel Algorithm for 3D Dislocation Dynamics, J. Comp. Phys. 219:608-621 (2006).

[14] J.A. El-Awady, S.B. Biner, N.M. Ghoniem, A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes, J. Mech. Phys. Solids, 56:2019–2035 (2008).

[15] K.W. Schwarz, Mod. Sim. Mater Sci. Eng., Local rules for approximating strong dislocation interactions in discrete dislocation dynamics, 11:609 (2003).

[16] D.Weygand, L.H. Friedman, E. Van der Giessen, A. Needleman, Aspects of boundary-value problem solutions with three dimensional dislocation dynamics, Mod. Sim. Mater Sci. Eng., 10:437–468, (2002).

[17] S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, M.D. Uchic, M. Tang, C. Woodward, Athermal mechanisms of size-dependent crystal flow gleaned from three-dimensional discrete dislocation simulations, Acta Mater., 56:3245, (2008).

[18] J.A. El-Awady, M. Wen, and N.M. Ghoniem, The role of the weakest-link mechanism in controlling the plasticity of micropillars, J. Mech. Phys. Solids, 57:32, (2009).

[19] F. Mompiou and M. Legros, Plasticity Mechanisms in Sub-Micron Al Fiber Investigated by In Situ TEM, Advanced Engineering Materials, 14:955–959, (2012).

[20] H. Bei, S. Shim, G.M. Pharr, E.P. George, Acta Mater., 56:4762, (2008).

[21] J.A. El-Awady, M.D. Uchic, P.A. Shade, S.-K. Kim, S.I. Rao, D.M. Dimiduk, C. Woodward, Pre-Straining Effects on the Power-Law Scaling of Size Dependent Strengthening in Ni Single Crystals, Scripta Mater., 68, (2013).

[22] J.A. El-Awady, in review, (2014).

[23] H. Fan, S. Aubry, A. Arsenlis, J.A. El-Awady, Unpublished work, (2014).

[24] C. Zhou, R. Lesar, Dislocation dynamics simulations of plasticity in polycrystalline thin films, Int. J. Plasticity, 30-31:185-201, (2012).

[25] A. T. Lim, M. Haataja, W. Cai, D.J. Srolovitz, Stress-driven migration of simple low-angle mixed grain boundaries, Acta Mater., 60:1395-1407, (2012).

[26] B. Liu, P. Eisenlohr, F. Roters, D. Raabe, Simulation of dislocation penetration through a general low-angle grain boundary, Acta Mater, 13-14:5380-5390, (2012).

[27] J. Wang, C. Zhou, I.J. Beyerlein, S. Shao, Modeling interface-dominated mechanical behavior of nanolayered crystalline composites, JOM, 66:102-113, (2014).

[28] J. Crone, P. Chung, K. Leiter, J. Knap, S. Aubry, G. Hommes, A. Arsenlis, A multiply-parallel implementation of finite element-based discrete dislocation dynamics for arbitrary geometries, Mod. Sim. Mater Sci. Eng., 22:035014, (2014).

[29] S. Aubry, S.P. Fitzgerald, A. Arsenlis, Methods to compute dislocation line tension energy and force in anisotropic elasticity, Mod. Sim. Mater Sci. Eng., 22:015001, (2014).

[30] A. Vattré, B. Fedelich, On the relationship between anisotropic yield strength and internal stresses in single crystal superalloys, Mech. Mater., 43:930-951, (2011).

[31] A. Vattréa, B. Devincre, F. Feyel, R. Gatti, S. Groh, O. Jamond, A. Roos, Modelling crystal plasticity by 3D dislocation dynamics and the finite element method: The Discrete-Continuous Model revisited, J. Mech. Phys. Solids, 63:491–505, (2014).

[32] S.M. Keralavarma, T. Cagin, A. Arsenlis, A.A. Benzerga, Power-law creep from discrete dislocation dynamics, Pys. Rev. Lett., 109:265504, (2012).

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[35] S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, J.A. El-Awady, C. Woodward M.D. Uchic, Calculations of Intersection Cross-Slip Activation Energies in FCC Metals Using Nudged Elastic Band Method, Acta Mater.,59:7135-7144, (2011).

[36] S.I. Rao, D. Dimiduk, T.A. Parthasarathy, J.A. El-Awady, M.D. Uchic, C. Woodward, The activated state of cross-slip at screw intersections in FCC crystals, Acta Mater., 58:5547-5557, (2010).

[37] O. Couteau, T. Kruml, J.-L. Martin, About the activation volume for cross-slip in Cu at high stresses, Acta Mater., 59:4207-4215, (2011).

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## Comments

## Handling boundaries in DD

Jaafar -

Thanks for initiating a very interesting topic. Certainly DD framework provides an excellent brigde between atomistic and crystal plasticity simulations. I was wondering if you could shed some light on the effort along the following directions:

1. Realistically representing interfaces, especially grain boundaries and twin boundaries.

2. Modeling nucleation and evolution voids, twins, and similar volumetric defects.

Thanks,

~Shailendra

## Boundaries in DD

Shailendra,

Thanks for your interest in the topin and your comment. Regarding your two questions:

1) Dealing with twin and grain boundaries in DDD is very similar to dealing

with free surfaces. They are represented as a 2D interface, and the image field

resulting from that interface can be computed through an FEM simulations that

is coupled with DDD. The main difference between dealing with free-surfaces and

twin or grain boundaries is the result of the dislocation-interaction with the specific

interface. With free surfaces the dislocation can only exit the crystal and

create a surface step. On the other hand with twin and grain boundaries

multiple scenarios may occur (e.g. dislocation pileup, absorption, or transmission).

The details of these interactions thus must be hierarchically informed into DDD

from MD simulations (see e.g. de Koning et al in Phil Mag A, 82, pp. 2511

(2002) and JNM, 323, pp. 281 (2003)). Such rules have already been implemented

in DDD simulations of FCC crystals (e.g. Fan et al Scripta Mater. 66, pp 813

(2012)). It should be mentioned that the surface of the grain and twin boundaries

can be quite complex atomistically. However DDD is a coarse-grained

representation and is not expected to give precise details of the grain

structure, but rather the influence of the grain boundary on the collective

behavior of dislocations in the grains.

2) Since the stress variation within the crystal is fully resolved in

any coupled DDD-FEM framework, nucleation of volumetric defects can be handed

if the appropriate nucleation model is hierarchically informed from MD

simulations. The challenge however is how to deal with an evolving volumetric

defect? With respect to the DDD framework, interactions between dislocations

and the interfaces of the evolving volumetric defect is no different than

dealing with a stationary surface, however the main complexity is related to

solving the boundary value problem with the evolving volumetric defect. I am

not aware of any published efforts to resolve these issue yet.

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