Discrete dislocation dynamics simulations of dislocation- θ ′ precipitate interaction in Al-Cu alloys
R. Santos-Güemes, G. Esteban-Manzanares, I. Papadimitriou, J. Segurado, L. Capolungo, J. LLorca
DDiscrete dislocation dynamics simulations ofdislocation- θ (cid:48) precipitate interaction in Al-Cu alloys R. Santos-G¨uemes , , G. Esteban-Manzanares , , I. Papadimitriou , J.Segurado , , L. Capolungo , J. LLorca , , ∗ IMDEA Materials Institute, C/ Eric Kandel 2, 28906, Getafe, Madrid, Spain. Department of Materials Science, Polytechnic University of Madrid/UniversidadPolit´ecnica de Madrid, E. T. S. de Ingenieros de Caminos. 28040 - Madrid, Spain. Material Science and Technology Division, MST-8, Los Alamos National Laboratory,Los Alamos 87545 NM, USA.
Abstract
The mechanisms of dislocation/precipitate interaction were studied bymeans of discrete dislocation dynamics within a multiscale approach. Sim-ulations were carried out using the discrete continuous method in combina-tion with a fast Fourier transform solver to compute the mechanical fields(Bertin et al., 2015). The original simulation strategy was modified to in-clude straight dislocation segments by means of the field dislocation mechan-ics method and was applied to simulate the interaction of an edge dislocationwith a θ (cid:48) precipitate in an Al-Cu alloy. It was found that the elastic mis-match has a negligible influence on the dislocation/precipitate interaction inthe Al-Cu system. Moreover, the influence of the precipitate aspect ratioand orientation was reasonably well captured by the simple Orowan modelin the absence of the stress-free transformation strain. Nevertheless, the in-troduction of the stress-free transformation strain led to dramatic changesin the dislocation/precipitate interaction and in the critical resolved shearstress to overcome the precipitate, particularly in the case of precipitateswith small aspect ratio. The new multiscale approach to study the dislo-cation/precipitate interactions opens the possibility to obtain quantitativeestimations of the strengthening provided by precipitates in metallic alloystaking into account the microstructural details. ∗ Corresponding author
Journal of the Mechanics and Physics of Solids June 7, 2018 a r X i v : . [ c ond - m a t . m t r l - s c i ] J un eywords: Dislocation dynamics, precipitate strengthening, multiscalemodeling.
1. Introduction
Plastic deformation in metallic alloys is carried by dislocation slip andstrengthening is achieved with obstacles that hinder the motion of disloca-tions. These obstacles can take the form of dislocations during deformation,solute atoms in the lattice, grain boundaries, etc. but precipitation hard-ening is well established as the most effective mechanism to increase theyield strength in metallic alloys (Ardell, 1985). Obviously, the strengtheningprovided by the dispersion of second phases depends on the chemical compo-sition, size, shape, orientation, spatial distribution, etc. of the precipitates,which have been optimized over the years through costly, experimental trial-and-error approaches. Nevertheless, these strategies are being overcome byrecent advances in multiscale modelling approaches based on the coupling of ab initio and atomistic simulations with computational thermodynamics andphase-field models that allow an accurate prediction of the precipitate fea-tures as a function of the alloy chemical composition and thermo-mechanicaltreatment (Liu et al., 2013, 2017; Ji et al., 2014).In addition to these tools, the design of metallic alloys in which precipitatestrengthening has been maximized requires the development of multiscalemodelling strategies that are able to account for the mechanisms of dislo-cation/precipitate interaction. In the case of very small precipitates ( < >
100 nm), which are overcome by the formation of dislocationloops, due to computational reasons. The analysis of this process was pio-neered by Orowan using a constant line tension model, which computed thecritical resolved shear stress (CRSS) necessary to overcome a periodic squarearray of spherical precipitates impenetrable for dislocations (Orowan, 1948).Later, Bacon et al. (1973) included the effect of the interaction stresses be-tween the dislocation segments, while other authors expanded the results ofOrowan to deal with random distributions of obstacles (Foreman and Makin,1966; Kocks, 1966). 2hile these approaches can provide qualitative trends, they cannot bequantitative because the precipitate geometry and orientation as well as thedetails of the dislocation/precipitate interaction are not taken into accountand numerical approaches have been used in recent years. Xiang et al. (2004);Xiang and Srolovitz (2006) used a level-set representation of the dislocationline to simulate the interaction of both edge and screw dislocations withspherical precipitates. Matrix and precipitate were elastic and isotropic withthe same elastic constants and it was assumed that the precipitates couldor could not be sheared by dislocations, the latter by including a strongrepulsive force on the dislocation within the precipitate. Moreover, the effectof a misfit dilatational strain between the matrix and the precipitate wasincluded. The simulations showed that the richness and complexity of thedislocation/precipitate interactions and postulated new by-pass mechanisms.However, it should be noted that these simulations did not take into accountthe crystallography of slip, leading to limitations in the precise modelling ofthe dislocation mobility (cross-slip, climb).The influence of crystallographic slip was taken into account by Mon-net (2006) and Monnet et al. (2011), who used discrete dislocation dynam-ics (DDD) simulations to obtain the CRSS necessary to overcome sphericalprecipitates, which were made impenetrable to the dislocation by adding afriction stress within the precipitate. Further DDD simulations included theeffect of the image stresses induced by the elastic modulus mismatch be-tween the matrix and the spherical precipitate on the CRSS (Takahashi andGhoniem, 2008; Takahashi and Terada, 2011; Shin et al., 2003) using the su-perposition technique developed by Van der Giessen and Needleman (1995).Takahashi and Ghoniem (2008) reported the influence of the shear modulusmismatch on the CRSS when the spherical precipitates were sheared whileShin et al. (2003) address the case of the formation dislocation loops. Theinfluence of the image stresses was higher in the former case but increased inthe latter as several dislocations loops were around the precipitate, increasingthe hardening rate. However, the superposition method requires to solve theelastic boundary value problem (using either the finite element or the bound-ary element method) in each time step of the DDD simulations to obtain theimage stresses. Further DDD simulations to study dislocation/precipitateinteractions (many of them focussed in the case of shearable γ (cid:48) precipitatesin Ni-based superalloys) have ignored the modulus mismatch (Yashiro et al.,2006; Vattr´e et al., 2009; Yang et al., 2013; Hafez Haghighat et al., 2013;Huang et al., 2012; Zlek et al., 2017; Monnet, 2015). Only more recently,3ao et al. (2015) carried out DDD simulations of dislocation/precipitate in-teraction that took into account the effect of modulus misfit as well as of themisfit stresses arising from the lattice mismatch between γ and and γ (cid:48) phasesin Ni-based superalloys. The Fast Fourier Transform (FFT) method – muchfaster than the traditional methods – was used in this case to compute theimage stresses.While these results have improved our understanding of precipitationstrengthening, they often ignore the actual details of the precipitate shapeand orientation, of the dislocation mobility as well as of the complex stressfield around the precipitates (misfit and stress-free transformation strains).More recent analyses are trying to overcome these limitations by obtainingthis critical information from simulations at lower length scales. For instance,Lehtinen et al. (2016) carried out DDD simulations of dislocation/precipitateinteraction in BCC Fe in which the parameters of the simulation (disloca-tion mobility, shear modulus, dislocation core energy) were obtained fromatomistic simulations. The precipitates were spherical and the dislocation-precipitate interaction was modelled by a Gaussian potential that was cali-brated from atomistic simulations.This investigation presents a comprehensive multiscale modelling strategybased on DDD to study the mechanisms of dislocation/precipitate interac-tion. The methodology is applied to Al-Cu alloys but it is general and canbe extended to any other metallic alloy. The details of the the θ (cid:48) (Al Cu)precipitates (size, shape and orientation) as well as the stress-free transforma-tion strains around the precipitate were obtained in a previous investigationby the coupling of ab initio and atomistic simulations with computationalthermodynamics and phase-field models (Liu et al., 2017) and were in goodagreement with the experimental data. In addition, the elastic constantsand the dislocation mobility laws were obtained from atomistic simulationsand this information was used to determine the actual mechanisms of dis-location/precipitate interaction in this system by means of DDD simula-tions in which all the relevant physical processes were accounted for. Inparticular, the influence of the precipitate shape, orientation, modulus mis-match and stress-free transformation strain on the dislocation/precipitateinteraction mechanisms was analyzed and their influence on the CRSS wasdetermined and compared with the predictions of the classical models fordislocation/precipitate interaction.The paper is organized as follows: the characteristics of the dislocations inthe Al matrix and of the θ (cid:48) precipitates (obtained using different simulations4trategies) are presented in section 2, while the DDD simulation strategy isdetailed in section 3. The results of the DDD simulations are shown anddiscussed in Section 4 while the conclusions are drawn in section 5.
2. Material system θ (cid:48) precipitates are the key strengthening phase in Al-Cu alloys aged athigh temperature (Polmear (1995)). θ (cid:48) is a stoichiometric phase with chem-ical composition Al Cu and tetragonal structure (space group I /mmm , a θ (cid:48) = 0.404 nm, c θ (cid:48) = 0.580 nm). The unit cells of α -Al (space group F m ¯3 m , a α = 0.404 nm) matrix and θ (cid:48) are shown in Figs. 1 a) and b), respectively.Previous studies of the transformation path of the θ (cid:48) precipitate from the α -Al lattice have shown three successive steps, which are shown in Fig. 2(Dahmen and Westmacott, 1983; Nie and Muddle, 1999; Nie, 2014). TheAl atoms in layers 2 and 3 of the α -Al lattice are first shifted in oppositedirections by a distance a α /6. This step is followed by a homogeneous sheardeformation of the cell by an angle arctan(1/3) and, finally, by the shufflingof one Cu atom to the center of the cell and diffusion of the other Cu atominto the Al matrix. According to this transformation path, the lattice corre-spondence between the parent phase ( α -Al) and the θ (cid:48) precipitates is givenby [013] α → [001] (cid:48) θ and [010] α → [010] (cid:48) θ and the transformation matrix, T ,that relates the lattice parameters in the α -Al, e α , and in the θ (cid:48) precipitate, e θ (cid:48) ( Te α = e θ (cid:48) ) is expressed as (Gao et al., 2012; Liu et al., 2017) T = a θ (cid:48) /a α a θ (cid:48) /a α − /
30 0 c θ (cid:48) / . a α . (1)The transformation matrix includes both strains and rigid body rotationsand the corresponding stress-free transformation strain (SFTS), (cid:15) , can becomputed as (cid:15) = 12 ( T T T − I ) (2)where I stands for the identity matrix.The nucleation and growth of θ (cid:48) precipitates during high temperatureageing has been recently modelled in 3D by means of a multiscale phase-field approach, that takes into account the contribution of the chemical freeenergy, the interface energy and the elastic energy due to the SFTS (Liu5 igure 1: Crystal structure of: (a) FCC α -Al. (b) BCT θ (cid:48) precipitates. (Nie, 2014). Redand blue spheres stand for Al and Cu atoms, respectively. From Liu et al. (2017). STEP 1 STEP 2 STEP 3
Figure 2: Transformation path from α -Al to θ (cid:48) precipitates during high temperature ageing(Nie, 2014). Red and blue spheres stand for Al and Cu atoms, respectively. From Liuet al. (2017). et al., 2017). The chemical free energy was given by computational ther-modynamics results, while the interface energy and the lattice parametersof both phases (which determine the elastic energy associated to the SFTS)were obtained from density functional theory simulations. The computedlattice parameters were a α = 0.405 nm, a θ (cid:48) = 0.408 nm, c θ (cid:48) = 0.5701 nm,very close to the experimental data reported above (Nie, 2014) and it wasassumed that a α (cid:39) a θ (cid:48) to compute the SFTS. The multiscale simulation pre-dicted that the θ (cid:48) precipitates grew with an orientation relationship (001) α (cid:107) (001) θ (cid:48) , [100] α (cid:107) [100] θ (cid:48) . The precipitates were circular disks and the broadface of the disk was coherent with the Al matrix and parallel to either the(100), (010) or (001) planes of the FCC Al lattice, leading to three differentorientation variants, while the edges of the circular plates were semi-coherent.6our different deformation variants were possible for each orientation variantof the precipitate due to the four-fold symmetry of the (100) planes in theFCC lattice, leading to a total of 12 deformation variants, in agreement withthe experimental observations (Dahmen and Westmacott, 1983; Nie, 2014),which were characterized for their corresponding SFTS matrix that can befound in Liu et al. (2017). The simulations predicted a precipitate diameterin the range 120-180 µ m and a thickness of 4-8 µ m, with an average aspectratio of ≈
26. These results were in close agreement with experimental datain the literature for peak-aged Al- 4 wt. % Cu alloys (Liu et al., 2017; Zhuet al., 2000; Biswas et al., 2011).The elastic constants of the α -Al matrix and of the θ (cid:48) precipitates were de-termined using Density Functional Theory (DFT) with the Quantum Espressoplane-wave pseudopotential code (Giannozzi et al., 2009). The exchange-correlation energy was evaluated with the help of the Perdew-Burke-Erzenhofapproach (Perdew et al., 1996) within the generalized gradient approxima-tion. Ultrasoft pseudopotentials were used to reduce the basis set of planewavefunctions used to describe the real electronic functions (Vanderbilt,1990). After careful convergence tests, a cutoff of 37 Ry was found to besufficient to reduce the error in the total energy below 1 meV/atom. Ak-point grid separation of 0.03 ˚A − was employed for the integration overthe Brillouin zone according to the Monkhorst-Pack scheme (Monkhorst andPack, 1976).The elastic constants were obtained by applying a given strain to theunit cell in the ground state and calculating the corresponding stress afterthe atom coordinates in the unit cell were relaxed. Taking into account thecrystal symmetries, the cubic α - Al unit cell was deformed in the directionnormal to the cube face and in shear to compute the three independentelastic constants. The BCT cell of the θ (cid:48) precipitate was deformed along twonormal directions perpendicular to two faces of the tetragonal lattice and intwo shear directions to compute the six independent elastic constants. Sixstrain levels (varying from -0.003 to 0.003) were used for each deformationpattern to obtain a reliable linear fit of the stress-strain relationship. Theelastic constants of α -Al and of θ (cid:48) precipitates obtained by DFT are depictedin Tables 1 and 2, respectively. The ones for α -Al were very close to theexperimental data in the literature (Vallin et al., 1964; Sinko and Smirnov,2002). To the best of the authors’ knowledge, no experimental data areavailable for θ (cid:48) . 7 C C DFT 110.4 60.0 31.6Experimental 114.3 61.9 31.6
Table 1: Elastic constants (in GPa) of α - Al at 0K obtained from DFT. The experimentalvalues extrapolated at 0K (Sinko and Smirnov, 2002) are included for comparison. C = C C C = C C C = C C Table 2: Elastic constants (in GPa) of θ (cid:48) - Al Cu precipitates at 0K obtained from DFT.
3. Discrete Dislocation Dynamics Strategy
The dislocation/precipitate interaction is analyzed by means of DDD sim-ulations following the discrete continuous model developed by Lemarchandet al. (2001). In this approximation, the dislocations are treated as plate-likeinclusions with an eigenstrain that corresponds to the plastic strain asso-ciated with the area sheared by the dislocation. The dislocation loops arediscretized in segments which move depending on the stresses acting on thesegments and the mobility rules and the plastic strain is computed from thedislocation glide. The DDD code was coupled in the original model with afinite element code that computed the displacement field that is solution ofthe boundary value problem taking into account the plastic strain providedby the DDD simulations (Lemarchand et al. (2001)). This framework neitherrequires the use of analytical expressions for the displacement fields of thedislocation segments (and, thus, can be easily extrapolated to anisotropicmaterials), nor the computational power increases with the square of thenumber of dislocations segments. However, computational efforts are lim-ited by the fine finite element discretizations necessary to achieve accurateresults, particularly in the case of precipitates with very large aspect ratio.These limitations were overcome recently by Bertin et al. (2015), who usedthe Fast Fourier Transform (FFT) method to compute the mechanical fieldsand solve the boundary value problem for periodic cases. Moreover, the het-erogeneous stress distribution that appear due the elastic modulus mismatchbetween the matrix and the precipitate and the stresses induced by SFTScan be easily incorporated to the simulations.Dislocations are discretized into segments. The dislocation mobility fol-lows a viscous linear law, where the velocity of node i , v i , of the dislocation8ine is given by F i = B v i (3)where B is viscous drag coefficient that depends on the dislocation characterand F i the force acting on node i , which is given by F i = (cid:88) j f ij (4)where f ij is the force acting on the segment ij , which is computed accordingto f ij = (cid:90) x j x i N i ( x ) f pkij ( x )d x (5)where N i is the interpolation function associated to node i and f pkij is thePeach-Koeler force given by f pkij ( x ) = (cid:16) σ ( x ) · b ij (cid:17) × ˆt ij (6)where b ij is the Burgers vector of the segment ij and ˆt ij the unit vectorparallel to the dislocation line. The stress field within the simulation domain is computed using the FFTalgorithm. The mechanical state of the system is determined by solving themechanical equilibrium equations in the domain V with periodic boundaryconditions according to σ ( x ) = C ( x ) : [ (cid:15) ( x ) − (cid:15) p ( x ) − δ ( x ) (cid:15) ] , ∀ x ∈ V div( σ ( x )) = 0 x ∈ V σ · n has opposite orientation on opposite sides of ∂V (cid:82) V (cid:15) ( x ) = E (7)where C denotes the fourth order elasticity tensor, (cid:15) the total strain, (cid:15) p the plastic strain, (cid:15) the SFTS, ∂V stand for the boundaries of domain V with normal n and E is the imposed macroscopic strain. The SFTS is anhomogeneous eigenstrain within the precipitate (that only depends on theprecipitate variant) and, thus, δ ( x ) is a Dirac delta function that is equal to1 when x is within the precipitate and 0 otherwise. This discontinuous strain9eld may lead to Gibbs fluctuations when using a FFT solver. They wereattenuated by the use of discrete gradient operators in Fourier space (in thisparticular case, the rotational discrete gradient operator proposed by Willot(2015)). The fluctuations in the simulations were not significant, as shownin the stress fields below.The plastic strain (cid:15) p ( x ) is computed directly from dislocation motion inthe DCM (Bertin et al., 2015), while (cid:15) is given in Section 2. The polariza-tion scheme proposed by Moulinec and Suquet (1998) is used to solve themechanical equilibrium problem in each step of the DDD simulation. This isachieved through the introduction of a reference medium with stiffness C σ ( x ) = C : (cid:15) ( x ) + τ ( x ) (8)where τ ( x ) is the polarization tensor, which is given by τ ( x ) = δ C ( x ) : (cid:15) ( x ) − C ( x ) : (cid:20) (cid:15) p ( x ) + (cid:15) ( x ) (cid:21) (9)where δ C ( x ) = C ( x ) − C . The SFTS tensor (cid:15) ( x ) takes the value for thecorresponding precipitate variant within the precipitate and it is equal to 0outside of the precipitate (Bertin and Capolungo, 2018). From the expressionfor the total stress in (8) and the mechanical equilibrium condition (7), themechanical fields can be obtained using the FFT algorithms detailed in Bertinet al. (2015); Bertin and Capolungo (2018). Only dislocation loops can be initially introduced in the DCM but thisconfiguration is not appropriate to analyze the interaction of a single dislo-cation line with the precipitate. This limitation was overcome in the cubicdomain (with periodic boundary conditions) by introducing within the do-main a rectangular prismatic loop parallel to one cube faces (Fig. 3). Twoopposite sides of the prismatic loop (shown as discontinuous lines in the fig-ure) were moved in opposite directions until they reached the boundaries ofthe domain and annihilate each other, because they have opposite directions,leading to two straight dislocations forming a dipole within the domain. Oneof the dislocations was fixed during the simulation (yellow line in the figure)while the other was free to move following the mobility rules established inthe following section. The Field Dislocation Mechanics (FDM) method was10hen used to cancel the stress field created in the domain by the fixed dis-location following the procedure reported in Berbenni et al. (2014); Brenneret al. (2014); Djaka et al. (2017). [110][112] [111] b Figure 3: Introduction of a prismatic dislocation loop parallel to one cube faces. Twoopposite sides of the prismatic loop (shown as discontinuous lines in the figure) were movedin opposite directions until they reached the boundaries of the domain and annihilate eachother, leading to two straight dislocations forming a dipole within the domain. One of thedislocations (yellow line) was fixed during the simulation while the other was free to movein the slip plane (shaded) and interacted with the precipitate.
The FDM method involves the Stokes-Helmholtz decomposition of theelastic distortion into incompatible and compatible parts. The existence ofa non-zero dislocation density in the material is accounted for by the incom-patible part, while the compatible part ensures that the boundary conditionsand the stress equilibrium conditions are fulfilled (Acharya, 2001). The in-compatible elastic distortion is included in the FDM method through Nye’sdislocation tensor α (Nye, 1953), which is defined as α ij = b i t j , where b i isthe net Burgers vector in direction e i per unit surface S and t j the dislocationline direction along e j .The incompatibility equation and the conservation law are expressed,respectively, by curl( U e ) = α (10)div( α ) = 0 (11)11here U e is the elastic distortion tensor. Applying the Stokes-Helmholtzdecomposition to the elastic distortion tensor, U e = U e, ⊥ + U e, (cid:107) (12)where U e, ⊥ and U e, (cid:107) stand for the incompatible and compatible parts of theelastic distorion respectively. Taking into account that α = curl( U e , ⊥ ) (13)div( U e , ⊥ ) = 0 (14)and applying again the operator curl to the expression (13), after some ma-nipulation, it yields a Poisson-type equationdiv(grad( U e , ⊥ )) = − curl( α ) (15)that can be expressed in component form as U e, ⊥ ij,kk = − e jkl α il,k (16)The Poisson equation (16) can be solved in the Fourier space. To thisend, the dislocation density α ( x ) is computed in the Fourier space, ˜ α ( ξ ), andthe incompatible elastic distortion is obtained in the Fourier space as:˜ U e, ⊥ ij ( ξ ) = i ξ k ξ e jkl ˜ α il ( ξ ) ∀ ξ (cid:54) = ˜ U e, ⊥ ij ( ) = . (17)Once ˜ U e, ⊥ ( ξ ) is known, the inverse Fourier transform is computed to getthe incompatible elastic distortion in the real space. Finally, the incompatibleelastic strain (cid:15) e, ⊥ is the symmetric part of U e, ⊥ . The incompatible elasticstrain is introduced as plastic strain in the Lippmann-Schwinger equation,which is solved using the FFT algorithm following the same procedure usedin section 3.1.In order to screen the stress field of a straight edge dislocation with Burg-ers vector b , the corresponding Nye tensor is given by α = − b/l (18)12here − b is the magnitude of the Burgers vector (opposite to the one of thedislocation to cancel the stress field) and l the voxel size of the discretization.This value of the Nye tensor is applied in the voxels where the immobiledislocation is located, being zero in the rest of the simulation domain. The drag coefficient vector, B , that characterizes the dislocation mobil-ity has been recently determined in Al by Cho et al. (2017). They carriedout molecular dynamics simulations of straight dislocation segments withdifferent character and determined B as a function of temperature in theregime in which the dislocation mobility is controlled by the viscous frictionforce arising from phonon damping. it was found that the drag coefficientof a mixed dislocation cannot be obtained by a linear interpolation betweenthose of edge and screw dislocations (Fig. 4). The maximum drag coeffi-cients were found for the screw dislocation and a mixed dislocation whoseBurgers vector forms an angle of 60 ◦ with the dislocation line. The dragcoefficients obtained from the molecular dynamics simulations were fitted totwo parabolic functions according to B ( θ ) = B (0) − (cid:18) B (0) − B (0)1 . (cid:19) π | θ | + (cid:18) B (0) − B (0)1 . (cid:19) π θ < θ < π B ( θ ) = B (0) + (cid:18) B ( π/ − B (0) (cid:19) π (cid:16) | θ | − π (cid:17) −− (cid:18) B ( π/ − B (0) (cid:19) π (cid:16) | θ | − π (cid:17) π < θ < π B (0) and B ( π/
2) stand for the drag coefficients of pure screw andedge dislocations, respectively. This drag coefficient was used in the DDDsimulations for the Al matrix. The drag coefficient in the θ (cid:48) precipitates wasassumed to be infinity and, therefore, the dislocations could not shear theprecipitates and were forced to by-pass them.
4. Results and discussion
DDD simulations were carried out using a cubic domain of 729 x 729 x729 nm with periodic boundary conditions, which was discretized with a gridof 128 x 128 x 128 voxels. The axes of the cubic domain were aligned with13
15 30 45 60 75 90
Dislocation character, D r a g c o e ff i c i e n t , B ( P a s ) x10 - Eq. (19)Molecular dynamics [Cho et al., 2017]
Figure 4: Drag coefficient B for the dislocation mobility in Al at 300K as a function ofthe dislocation character. θ = 0 ◦ stands for pure screw dislocation and θ = 90 ◦ for pureedge dislocation. The solid symbols stand for moecular dynamics simulation in Cho et al.(2017), while the solid line corresponds to eq. (19). the [1¯12], [110] and [¯111] directions of the Al lattice. A straight edge disloca-tion (¯111)[110] was introduced in the simulation box following the strategydescribed above. The precipitate was inserted at the center of the simulationbox as a circular disk parallel to either the (001) and (010) plane, whichstand for the respective habit planes. θ (cid:48) precipitates also grow along the(001) habit plane but this dislocation/precipitate configuration is equivalentto the that of the (010) precipitates. The slip plane of the dislocation inter-sects the center of the precipitate. The initial configuration is represented infigure 5 for both orientations of the precipitate. For the precipitate parallelto the (001) plane, the section of the precipitate along the glide plane wasparallel to the Burgers vector (Fig. 5a), whereas it formed an angle of 60 ◦ forthe precipitate in the (010) plane (Fig. 5b). A shear strain rate is applied tothe cubic domain along the [110] direction, as shown in Figure 5.The precipitate volume fraction was held constant and equal to 3.1 10 −
14n the simulations, so the interaction between precipitates can be neglected.The elastic constants of the Al matrix and of the θ (cid:48) precipitate in Tables 1and 2, respectively, were used, while the dislocation mobility in Al was givenby the drag coefficient B in eq. (19). It was assumed that the precipitatewas impenetrable to dislocations. All the simulations presented below werecarried out at an applied strain rate of 10 s − because the results obtained atthis strain rate are equivalent to those obtained under quasi-static conditions,as shown in the Appendix. [110][112] [111] τ [110][112] [111] τ (a) (b) b b X YZ X YZ b b Figure 5: Initial configuration of the edge dislocation and the θ (cid:48) precipitate for the DDDsimulations. (a) Precipitate habit plane (001). The angle between the Burgers vector andthe section of the precipitate along the glide plane is 0 ◦ . (b) Precipitate habit plane (010).The angle between the Burgers vector and the section of the precipitate along the glideplane is 60 ◦ . The orientation of the dislocation line and the precipitate in the slip planeare shown for both configurations below each figure. The mechanisms of dislocation precipitate interaction and the particularrole played by the SFTS around the precipitate can be understood fromthe shear stress-strain curves obtained from the DDD simulations. In thissection, the stress-strain curves and the path followed by the dislocation15s analyzed for each precipitate variant. The precipitate diameter in thesesimulations was 156 nm and the aspect ratio 26:1, in agreement with theresults of the phase-field simulations for θ ‘ in Al-Cu alloys (Liu et al., 2017).Simulations were carried with and without including the effect of the SFTS toassess the influence of this factor on the mechanics of dislocation/precipitateinteractions. The interaction of the dislocation with the 12 deformationvariants induced by the presence of the SFTS is reduced to 6 independentcases due to the symmetries of the FCC lattice, two corresponding to the0 ◦ configuration (Fig. 5a) and four to the 60 ◦ configuration (Fig. 5b). ◦ orientation The shear stress-strain curve of the simulation in the 0 ◦ orientation with-out SFTS is plotted in Fig. 6. The configuration of the dislocation linearound the cross-section of the precipitate in the glide plane is also includedin the figure for different values of the applied strain. In the initial stages ofdeformation, marked with (i) in the figure, dislocation glide takes place atvery low stress and the dislocation line remains straight, indicating that thereis no influence of the precipitate. Linear hardening is observed afterwardsin region (ii) as the dislocation starts to bow around the precipitate. Thedislocation overcomes the precipitate by the formation of an Orowan loop, asshown in (iii) and, as the dislocation leaves the domain, another dislocationenters the domain by the opposite boundary due to the periodic boundaryconditions (region iv), leading to hardening.The dislocation precipitate interaction for the 0 ◦ configuration is changedin the presence of the stress fields around the precipitate induced by theSFTS, which can be found in Table 1 of Liu et al. (2017). The magnitudeof the SFTS in this table is given in a reference system which follows theorientation relationship between the matrix and the precipitate. They haveto be rotated to the reference frame in Fig. 5 and the two SFTS consideredfor this precipitate configuration are (cid:15) and (cid:15) , which are given by (Liu et al.,2017). (cid:15) = R T − . − . − . R and (cid:15) = R T . . − . R (20)where R is the rotation matrix, which is expressed as16 Shear strain, S h ea r s t r ess , [ M P a ] (i) (ii) (iii) (iv) Figure 6: Shear stress-strain curve corresponding to the dislocation/precipitate interactionin the 0 ◦ configuration without SFTS. The evolution of the dislocation line around theprecipitate is showed in several points along the curve. R = . − .
408 0 . .
707 0 .
707 0 − .
577 0 .
577 0 . . (21)In the case of (cid:15) , (Fig. 7a), the stress field around the precipitate leadsto an initial repulsion between the dislocation and the precipitate, which isshown in the initial hardening in the stress-strain curve in region (i) and inthe shape of the dislocation line. After this initial barrier is overcome, onesmall segment of the dislocation line is attracted to the precipitate (region ii)and the dislocation starts to bow around the precipitate (region iii) but thedislocation loop is not symmetric due to the SFTS. The Orowan loop is finallycreated around the precipitate (regions iv and v) and the process is repeatedas a new dislocation enters the domain (region vi). The stress field createdby the SFTS (cid:15) leads to a different behavior, as shown in Fig. 7b). The dis-location line is initially attracted to the precipitate (region i) and a minimumin the stress-strain curve is found when the dislocation line gets in contactwith the precipitate (region ii). Linear hardening is observed afterwards asthe dislocation bows around the precipitate (region iii) and overcomes the17 i) (ii)(iii) (iv) (v)(vi) (a) (i) (ii)(iii) (iv) (v) (b) Figure 7: Shear stress-strain curve corresponding to the dislocation/precipitate interactionin the 0 ◦ configuration with SFTS. (a) (cid:15) = (cid:15) . (b) (cid:15) = (cid:15) .The evolution of the dislocationline around the precipitate is showed in several points along the curve. obstacle by the formation of an Orowan loop (region iv). However, the finalOrowan loop is not attached to the precipitate surface and the final shape ofthe Orowan loop is different from the ones found in Figs. 6 and 7a).The presence of the SFTS increased considerably the CRSS (i.e. the max-imum stress in the shear stress-strain curve) necessary to overcome the pre-cipitate. According to the line tension model, the CRSS is controlled by theminimum radius of curvature of the dislocation line during the Orowan pro-cess, which decreased in the presence of the SFTS because of the anisotropyintroduced in the development of the Orowan loops. In addition, the CRSSin the presence of (cid:15) was slightly higher than the one in the presence of (cid:15) . ◦ orientation Similar DDD simulations were carried out when the precipitate was in60 ◦ configuration. In the absence of the SFTS, the dislocation overcomes theprecipitate by the formation of an Orowan loop (Fig. 8) and the regions ofthe shear stress-strain curve are equivalent to those found in Fig. 6 in the0 ◦ orientation in the absence of the SFTS. In this case, the dislocation lineadvances toward the precipitate and rotates until is in full contact with thebroad face of the precipitate (region ii). Afterwards, the dislocation armsadvance until an Orowan loop is formed around the precipitate (region iii).In the orientation 60 ◦ , there are four independent SFTS that lead to18 Shear strain, S h ea r s t r ess , [ M P a ] (i)(ii) (iii) Figure 8: Shear stress-strain curve corresponding to the dislocation/precipitate interactionin the 60 ◦ configuration without SFTS.The evolution of the dislocation line around theprecipitate is showed in several points along the curve. changes in the dislocation-precipitate interaction mechanisms. The corre-sponding STFS are given by (cid:15) = R T − . . . R and (cid:15) = R T . . − . R (22) (cid:15) = R T − . − . − . R and (cid:15) = R T − . − . − . R (23)and the dislocation-precipitate interactions in the presence of the stress fieldscreated by the SFTS are plotted in Fig. 9, together with the correspondingshear stress-strain curves. In all cases, the dislocation line tends to rotate andto become parallel to the broad face of the precipitate, and an Orowan loopis formed afterwards as the dislocation arms propagate at both sides of the19 i)(iii) (iv) (v)(vi) (a) (ii) (b) (iii)(i) (ii) (iv)(v) (c) (iv)(i)(ii) (iii) (d) (v)(i)(ii)(iii) (iv) Figure 9: Shear stress-strain curve corresponding to the dislocation/precipitate interactionin the 60 ◦ configuration with SFTS. (a) (cid:15) = (cid:15) . (b) (cid:15) = (cid:15) . (c) (cid:15) = (cid:15) . (d) (cid:15) = (cid:15) .The evolution of the dislocation line around the precipitate is showed in several pointsalong the curve. precipitate. However, the approximation of the dislocation to the precipitateand the formation of the Orowan loop is modulated by the SFTS.In the case of (cid:15) (Fig. 9a), the stress field near the precipitate initiallyattracts the dislocation toward the precipitate (region i), but this is followedby a strong repulsion between the dislocation line and the broad face of theprecipitate (region ii), leading to the formation of a half loop whose extremesare in contact with precipitate (regions iii and iv). The interaction between20 igure 10: Contour plots of the shear stress, τ yz corresponding to the disloca-tion/precipitate interaction in the 60 ◦ configuration with SFTS (cid:15) = (cid:15) . (a) Initial config-uration. (b) Dislocation/precipitate configuration corresponding to (iii) in Fig. 9a. Thedislocation is shown as a black line and the cross-section of the precipitate is white. the stress field of the dislocation and the stress field created by the STFS inthis case is shown in the contour plots of τ yz in Fig. 10. The repulsion betweenthe dislocation and the precipitate due to the presence of the STFS leads tothe formation of the ellipsoidal Orowan loop which is only in contact withthe edges of the precipitate (regions v and vi). Interestingly, the minimumradius of curvature of the dislocation line during the Orowan process washigher than that in the case without SFTS (Fig. 8) and the CRSS for the60 ◦ configuration with SFTS (cid:15) was smaller than that obtained in the absenceof the SFTS (Fig. 8).In the case of (cid:15) (Fig. 9c), the dislocation line is initially repulsed bythe precipitate (region i) but it is strongly attracted afterwards towards thebroad face of the precipitate (region ii). The final Orowan loop is in contactwith the precipitate along the whole matrix/precipitate interface (regions iiiand iv). This leads to a very small radius of curvature of the dislocationand the CRSS in this case is much higher than the one in the absence of theSFTS (Fig. 8). The situations in the presence of the two other SFTS (Fig.9b and d) are in between those reported above and the CRSS in these caseswere equal ( (cid:15) ) or slightly higher ( (cid:15) ) than that in the absence of the SFTS.21 a) (i)(ii) (iii) (iv) (b) (i) (ii) (iii) (iv) (v) Figure 11: Shear stress-strain curve corresponding to the dislocation/precipitate interac-tion in the 0 ◦ configuration with SFTS. (a) (cid:15) = (cid:15) . (b) (cid:15) = (cid:15) . The evolution of thedislocation line around the precipitate is showed in several points along the curve. Although θ (cid:48) precipitates in Al-Cu alloy have a large aspect ratio, this ge-ometric feature may be changed by the addition of alloying elements whichmodify the interfacial energy between the Al matrix and the precipitate(Mitlin et al., 2000; Yang et al., 2016; Duan et al., 2017). Thus, it is in-teresting to analyze the influence of the precipitate aspect ratio on the mech-anisms of dislocation-precipitate interaction in the presence of the SFTS.To this end, DDD simulations in the 0 ◦ and 60 ◦ configuration were carriedout with precipitates with aspect ratios in the range 26:1 to 1:1 while theprecipitate volume fraction (3.1 10 − ) was held constant.In the absence of SFTS, the dislocations overcome the precipitate by theformation of an Orowan loop and the corresponding results are not plottedfor the sake of brevity. The shear stress-strain curves corresponding to the0 ◦ configuration with SFTS given by (cid:15) and (cid:15) are plotted in Figs. 11a) andb), respectively. The corresponding contour plots of the shear stress τ yz inthe initial configuration are shown in Figs. 12a) and b).In the first case (12a), the stress field induced by the SFTS repels thedislocation (region i) and impedes that the dislocation line gets in touch withthe precipitate (region ii). As a result, the effective precipitate diameter thatcontrols the radius of curvature of the dislocation arms to form an Orowanloop is increased (region iii). On the contrary, the stress field created by (cid:15) igure 12: Contour plots of the shear stress, τ yz corresponding to the disloca-tion/precipitate interaction in the 0 ◦ configuration. (a) SFTS (cid:15) = (cid:15) . (b) SFTS (cid:15) = (cid:15) . The dislocation is shown as a black line and the cross-section of the precipitate iswhite. (Fig. 11b) strongly attracts the dislocation line (regions i and ii) and thedislocation spontaneously overcomes the precipitate by the formation of avery tight Orowan loop (region ii). This process is repeated is further strainapplied to the simulation domain (regions iv and v).The mechanisms of dislocation-precipitate interaction in the case of 60 ◦ con-figuration with smaller aspect ratio are qualitatively similar to those reportedabove and are not included for the sake of brevity. It is, however, importantto assess the influence of the SFTS and of the precipitate aspect ratio on theCRSS for both precipitate orientations. These results are plotted in Figs.13a) and b) for the 0 ◦ and 60 ◦ orientations, respectively. The CRSS obtainedfrom the simulations with and without SFTS are included in each figure,together with the predictions of the Orowan model for the CRSS, τ O , whichis given by τ O = GbL (24)where G (= 29.9 GPa) is the shear modulus of the Al matrix in the slipplane parallel to the Burgers vector b (= 0.2863 nm) and L is the distancebetween precipitates along the dislocation line (Fig. 13). In the case of the0 ◦ orientation, the variation of the precipitate aspect ratio from 26:1 to 1:1(while the precipitate volume fraction was held constant) did not change L ε DDD + ε CR SS ( M P a ) Precipitate aspect ratio (a) ε DDD + ε DDD + ε DDD + ε CR SS ( M P a ) Precipitate aspect ratio (b)
Figure 13: CRSS as a function of the precipitate aspect ratio. (a) Precipitate in the0 ◦ configuration. (b) Precipitate in the 60 ◦ configuration. The results corresponding toDDD with and without SFTS as well as to the Orowan model are presented in each figurefor comparison. significantly (from L = 718 nm to 636 nm, respectively), while the differencesin L with the aspect ratio were slightly different in the 60 ◦ configuration(from L = 587 nm for 26:1 to L = 637 nm for 1:1 aspect ratio). Thus, theCRSS given by eq. (24) was almost constant for the 0 ◦ configuration anddecreased slightly with the aspect ratio in the 60 ◦ configuration (Fig. 13).The predictions of the Orowan model were in good agreement with the DDDsimulations in the absence of the SFTS in the 60 ◦ configuration but theyoverestimated by ≈
20% the CRSS for precipitates with large aspect ratiooriented at 0 ◦ . These results are in agreement with the main hypotheses ofthe Orowan model, which assumed that the precipitates were spherical (smallaspect ratio) and that the dislocation line formed a circular loop betweenprecipitates.The introduction of the SFTS led to large variations in the CRSS, whichwere more important if the precipitates had small aspect ratio. Dependingon whether the stress field associated with the SFTS attracted or repelledthe dislocation, the CRSS could increase or decrease dramatically in the24recipitates in the 0 ◦ orientation (Fig. 13a) when the aspect ratio was 1:1. Asshown in Fig. 11, the stress field associated to the SFTS controlled the shapeof the dislocation loop at the instability point and, thus, the magnitude of theCRSS. Similar results were obtained in the case of the precipitate oriented at60 ◦ with an aspect ratio of 1:1 (Fig. 13b) for (cid:15) and (cid:15) . It should be noted,however, that the SFTS is an important factor to determine the shape of theprecipitate, as shown by Liu et al. (2017). The SFTS used in this simulationsled to precipitates with large aspect ratio (¿ 10) and may not be realistic forequiaxed precipitates.The influence of the SFTS decreased as the precipitate aspect ratio in-creased because the dislocation loop configuration depended not only in theSFTS but also on the precipitate shape, leading always to an elongated loopparallel to the main axis of the precipitate (Figs. 7 and 9). Nevertheless,it should be noticed that the presence of the SFTS increased the CRSS ofprecipitates with an aspect ratio of 26:1 in all cases (with the exceptionof the SFTS (cid:15) in the 60 ◦ configuration) the CRSS with respect to the val-ues obtained by DDD simulations without the SFTS. These results indicatethat the stress fields around the precipitate (due to the SFTS or to thermalstresses generated upon cooling from the ageing temperature as a result of thethermal expansion mismatch between the matrix and the precipitates) haveto be taken into account to make quantitative predictions of the strength-ening provided by precipitates in metallic alloys. It should be finally notedthat the presence of the SFTS (cid:15) in the 60 ◦ configuration changed the shapeof the Orowan loop around the precipitate when the aspect ratio increasedfrom 1:1 to 2:1 and again for larger values of the aspect ratio, leading to acomplex variation of the CRSS with this parameter for this particular SFTS.Moreover, the stress fields around precipitates can interact with each otherfor large precipitate volume fractions (far away from the dilute conditions ofthis investigation), leading to complex interaction patterns between disloca-tions and precipitates. All the results presented above were obtained using different values ofthe elastic properties for the matrix and the precipitate, according to theDFT results in Tables 1 and 2. However, it is interesting to assess the in-fluence of the elastic heterogeneity on the stress-strain curves and on theCRSS. Thus, two simulations were carried out for the 0 ◦ and 60 ◦ orientations(without STFS) for precipitates with an aspect ratio 26:1 in which the elastic25 s hea r s t r e ss , τ ( M P a ) shear strain, γ (a)0º orientation s hea r s t r e ss , τ ( M P a ) shear strain, γ (b)60º orientation Figure 14: Shear stress-strain curves corresponding to the dislocation/precipitate interac-tion with homogeneous and heterogeneous elastic constants for the matrix and the precip-itate. (a) 0 ◦ configuration without SFTS. (b) 60 ◦ configuration without SFTS. properties of the precipitate were identical to those of the matrix. The corre-sponding stress-strain curves for these homogeneous simulations are plottedin Figs. 14a) and b) for the precipitates oriented at 0 ◦ and 60degree, togetherwith the results obtained with the actual elastic constants of the matrix andthe precipitate. The differences in the mechanisms and in the CRSS werenegligible, in agreement with previous investigations (Shin et al., 2003).
5. Concluding Remarks
The mechanisms of dislocation/precipitate interaction were analyzed bymeans of a novel DDD strategy based in the DCM (Lemarchand et al., 2001)in combination with a FFT solver to compute the mechanical fields (Bertinet al., 2015). This framework neither requires the use of analytical expressionsfor the displacement fields of the dislocation segments (and, thus, can beeasily extrapolated to anisotropic materials), nor the computational powerincreases with the square of the number of dislocations segments. Moreover,very fine discretizations (necessary to model precipitates with large aspectratio) can be used owing to the efficiency of the FFT solver, and the influenceof the image stresses (induced by the elastic modulus mismatch between thematrix and the precipitate) and of the SFTS can be easily incorporated to thesimulations. The original DDD strategy (Bertin et al., 2015) was modified26o include straight dislocation segments by means of the FDM model, theappropriate configuration to analyze the interaction of a single dislocationline with a precipitate.The novel DDD model was used to analyze the mechanisms of disloca-tion/precipitate interaction and the corresponding CRSS in Al-Cu alloys.The orientation, size and shape of the θ (cid:48) precipitates as well as the SFTSassociated to the different precipitate variants were obtained from recentmultiscale modelling simulations based on the phase-field model (Liu et al.,2017), while the elastic constants of the Al matrix and of the precipitates werecalculated by DFT and the dislocation mobility as a function of the dislo-cation character was obtained from molecular dynamics simulations (Choet al., 2017). This leads to a multiscale modeling strategy, in which all theparameters in the DDD simulations are obtained from calculations at lowerlength scales.The DDD simulations provided for the first time a detailed account ofthe influence of the precipitate aspect ratio, orientation, SFTS and elasticmismatch between the matrix and the precipitate on the dislocation path toform an Orowan loop and on the CRSS to overcome the precipitate. It wasfound than the elastic mismatch have a negligible influence on the disloca-tion/precipitate interaction in the Al-Cu system while the influence of theprecipitate aspect ratio and orientation was reasonably captured by the sim-ple Orowan model in the absence of the SFTS. Nevertheless, the introductionof the SFTS led to dramatic changes in the dislocation/precipitate interac-tion and in the CRSS. This effect decreased as the precipitate aspect ratioincreased but it was still very important (above 50% in the CRSS for someprecipitate variants) for θ ’ precipitates with the typical aspect ratio foundin Al-Cu alloys. Thus, this investigation reveals the large influence of theSFTS on the mechanics of dislocation/precipitate interaction, an importantfactor that has not been previously considered in the analysis of precipitationhardening.Finally, The methodology presented in this paper opens the possibility toexplore in more detail the mechanisms of dislocation/precipitate interactionin metallic alloys with realistic values of the precipitate size, shape and aspectratio as well as of the elastic mismatch and of the dislocation mobility. Theywill be able to provide quantitative assessments of the strengthening providedby the precipitates, taking into account the influence of the SFTS and of thethermal stresses that develop upon cooling from the ageing treatments athigh temperature. Finally, they can be extended to deal with larger volume27raction of precipitates to account for the interaction between the SFTS ofdifferent precipitates and to model the propagation of a dislocation througha forest of precipitates including the effect of cross-slip. These topics will bethe subject of future investigations.
6. Acknowledgments
This investigation was supported by the European Research Council un-der the European Union’s Horizon 2020 research and innovation programme(Advanced Grant VIRMETAL, grant agreement No. 669141). LC wasfunded by the US Department of Energys Nuclear Energy Advanced Model-ing and Simulation (NEAMS).
Appendix A. Influence of strain rate
Dislocation dynamics simulations are normally carried out at high strainrates (10 to 10 s − ) for computational reasons and this limitation oftenleads to question whether the results obtained are applicable under quasi-static conditions. In order to analyze this effect, several simulations werecarried out using a relaxation strategy that allows to study the dislocationdynamics under quasi-static conditions. In this approach, a strain incrementis applied to the simulation box at at a high strain rate, in this case 4.0 10 s − , and the energy of the system is relaxed afterwards during several stepsat a constant applied strain. The shear stress is reduced during relaxationand the process is finished when the difference in the shear stress between toconsecutive relaxation steps is lower than a certain tolerance, and the sys-tem can be considered to be in equilibrium. Then, a new strain increment isapplied and the whole relaxation process is repeated. The shear stress-straincurve obtained following this process is plotted in Fig. A.15a) in the caseof the interaction of an edge dislocation with a precipitate with a diameterof 156 nm and an aspect ratio of 26:1 in the 0 ◦ configuration. The blue linewith open symbols shows the successive strain increments followed by therelaxation of the shear stress and the red line with solid symbols stands forthe quasi-static shear stress-strain curve. The shear stress-strain curves ob-tained at different strain rates (10 and 10 s − ) for this case are plotted inFig. A.15b), together with the quasi-static curve in Fig. A.15a). The com-parison between these curves shows that the results obtained at an appliedstrain rate of 10 s − were very close to the quasi-static simulations and,28hus, the DDD presented in this paper were carried out at an applied strainrate of 10 s − . s hea r s t r e ss , τ ( M P a ) shear strain, γ (a) quasi-static s hea r s t r e ss , τ ( M P a ) shear strain, γ (b)strain rate (s -1 ) Figure A.15: (a) Shear stress-strain curve of the dislocation precipitate interaction ob-tained using the relaxation process. (b) Comparison between shear stress-strain curvesof the dislocation/precipitate interaction as a function of the applied strain rate. Thequasi-static results correspond to the red curve in (a). See text for details.
References
Acharya, A., 2001. A model of crystal plasticity based on the theory of con-tinuously distributed dislocations. Journal of the Mechanics and Physicsof Solids 49 (4), 761 – 784.Ardell, A., 1985. Precipitation hardening. Metallurgical Transactions A16 (12), 2131–2165.Bacon, D. J., Kocks, U. F., Scattergood, R. O., 1973. The effect of dislocationself-interaction on the orowan stress. Philosophical Magazine 28, 1241 –1263.Berbenni, S., Taupin, V., Djaka, K. S., Fressengeas, C., 2014. A numer-ical spectral approach for solving elasto-static field dislocation and g-disclination mechanics. International Journal of Solids and Structures51 (23-24), 4157–4175. 29ertin, N., Capolungo, L., 2018. A fft-based formulation for discrete disloca-tion dynamics in heterogeneous media. Journal of Computational Physics355, 366 – 384.Bertin, N., Upadhyay, M. V., Pradalier, C., Capolungo, L., 2015. A fft-basedformulation for efficient mechanical fields computation in isotropic andanisotropic periodic discrete dislocation dynamics. Modelling and Simula-tion in Materials Science and Engineering 23 (6), 065009.Biswas, A., Siegel, D. J., Wolverton, C., Seidman, D. N., 2011. Precipi-tates in AlCu alloys revisited: Atom-probe tomographic experiments andfirst-principles calculations of compositional evolution and interfacial seg-regation. Acta Materialia 59, 6187 – 6204.Bonny, G., Terentyev, D., Malerba, L., 2011. Interaction of screw and edgedislocations with cromium precipitates in ferritic iron: an atomistic study.Journal of Nuclear Materials 416, 70–74.Brenner, R., Beaudoin, A. J., Suquet, P., Acharya, A., 2014. Numericalimplementation of static field dislocation mechanics theory for periodicmedia. Philosophical Magazine 94, 1764 – 1787.Cho, J., Molinari, J.-F., Anciaux, G., 2017. Mobility law of dislocations withseveral character angles and temperatures in FCC aluminum. InternationalJournal of Plasticity 90, 66 – 75.Dahmen, U., Westmacott, K. H., 1983. Ledge structure and mechanism of θ (cid:48) precipitate growth in Al-Cu. Physica Status Solidi A 80, 249 – 252.Djaka, K. S., Villani, A., Taupin, V., Capolungo, L., Berbenni, S., 2017.Field dislocation mechanics for heterogeneous elastic materials: A numer-ical spectral approach. Computer Methods in Applied Mechanics and En-gineering 315, 921 – 942.Duan, S. Y., Wu, C. L., Gao, Z., Cha, L. M., Fan, T. W., Chen, J. H.,2017. Interfacial structure evolution of the growing composite precipitatesin al-cu-li alloys. Acta Materialia 129, 352 – 360.Foreman, A. J. E., Makin, M. J., 1966. Dislocation movement through ran-dom arrays of obstacles. Philosophical Magazine 14 (131), 911–924.30ao, S., Fivel, M., Ma, A., Hartmaier, A., 2015. Influence of misfit stresses ondislocation glide in single crystal superalloys: A three-dimensional discretedislocation dynamics study. Journal of the Mechanics and Physics of Solids76, 276 – 290.Gao, Y., Liu, H., Shi, R., Zhou, N., Xu, Z., Zhu, Y., Nie, J., Wang, Y., 2012.Simulation study of precipitation in an MgYNd alloy. Acta Materialia 60,4819 – 4832.Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni,C., Ceresoli, D., Chiarotti, G. L., Cococcioni, M., Dabo, I., Dal Corso,A., de Gironcoli, S., Fabris, S., Fratesi, G., Gebauer, R., Gerstmann, U.,Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N.,Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbrac-cia, C., Scandolo, S., Sclauzero, G., Seitsonen, A. P., Smogunov, A., Umari,P., Wentzcovitch, R. M., 2009. Quantum espresso: a modular and open-source software project for quantum simulations of materials. Journal ofPhysics: Condensed Matter 21 (39), 395502.Hafez Haghighat, S. M., Eggeler, G., Raabe, D., 2013. Effect of climb ondislocation mechanisms and creep rates in -strengthened ni base superalloysingle crystals: A discrete dislocation dynamics study. Acta Materialia 61,3709 – 3723.Huang, M., Zhao, L., Tong, J., 2012. Discrete dislocation dynamics mod-elling of mechanical deformation of nickel-based single crystal superalloys.International Journal of Plasticity 28, 141 – 158.Ji, Y. Z., Issa, A., Heo, T. W., Saal, J. E., Wolverton, C., Chen, L.-Q., 2014.Predicting β (cid:48) precipitate morphology and evolution in Mg-RE alloys us-ing a combination of first-principles calculations and phase-field modeling.Acta Materialia 76, 259 – 271.Kocks, U. F., 1966. A statistical theory of flow stress and work-hardening.Philosophical Magazine 13 (123), 541–566.Lehtinen, A., Granberg, F., Laurson, L., Nordlund, K., Alava, M. J., 2016.Multiscale modeling of dislocation-precipitate interactions in fe: Frommolecular dynamics to discrete dislocations. Physical Review E 93 (1),013309. 31emarchand, C., Devincre, B., Kubin, L., 2001. Homogenization method fora discrete-continuum simulation of dislocation dynamics. Journal of theMechanics and Physics of Solids 49, 1969 – 1982.Liu, H., Bell´on, B., LLorca, J., 2017. Multiscale modelling of the morphologyand spatial distribution of θ (cid:48) precipitates in al-cu alloys. Acta Materialia132, 611–626.Liu, H., Gao, Y., Liu, J. Z., Zhu, Y. M., Wang, Y., Nie, J. F., 2013. Asimulation study of the shape of β (cid:48)(cid:48)