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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 63 (2014) 491–505

0022-50http://d

n CorrE-m1 Pr2 Pr

Freiberg

journal homepage: www.elsevier.com/locate/jmps

Modelling crystal plasticity by 3D dislocation dynamicsand the finite element method: The Discrete-ContinuousModel revisited

A. Vattré a,b,1, B. Devincre b,n, F. Feyel a, R. Gatti b, S. Groh b,2,O. Jamond a,b, A. Roos a

a ONERA – The French Aerospace Lab, F-92322 Châtillon, Franceb LEM, CNRS–ONERA, 29 Avenue de la Division Leclerc, BP 72, 92322 Châtillon Cedex, France

a r t i c l e i n f o

Article history:Received 13 April 2012Received in revised form22 March 2013Accepted 9 July 2013Available online 20 September 2013

Keywords:Dislocation dynamicsFinite elementDislocation theoryCrystal plasticityDiscrete-Continuous Model

96/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jmps.2013.07.003

esponding author. Tel.: þ33 1 46734449; faail address: benoit.devincre@onera.fr (B. Devesent address: CEA, DAM, DIF, F-91297 Arpaesent address: Institute for Mechanics and, Germany.

a b s t r a c t

A unified model coupling 3D dislocation dynamics (DD) simulations with the finiteelement (FE) method is revisited. The so-called Discrete-Continuous Model (DCM) aims topredict plastic flow at the (sub-)micron length scale of materials with complex boundaryconditions. The evolution of the dislocation microstructure and the short-range disloca-tion–dislocation interactions are calculated with a DD code. The long-range mechanicalfields due to the dislocations are calculated by a FE code, taking into account the boundaryconditions. The coupling procedure is based on eigenstrain theory, and the precisemanner in which the plastic slip, i.e. the dislocation glide as calculated by the DD code,is transferred to the integration points of the FE mesh is described in full detail. Severaltest cases are presented, and the DCM is applied to plastic flow in a single-crystal Nickel-based superalloy.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Plastic deformation of crystalline materials results from the collective movement of dislocations under the influence ofthe interactions between the dislocation cores and the crystal lattice, their mutual interactions and the externally appliedloads. The resulting spatial distribution of dislocations is intrinsically heterogeneous and the manner in which this affectsthe mechanical response of material samples is a complex problem. Today, predicting these phenomena quantitatively withmathematical and numerical models remains one of the most challenging problems in materials science.

Outside their core region of a few Angstroms, dislocations are mostly linear defects with long-range elastic fields. Thesefields can be calculated with a classical procedure proposed long ago by Volterra (1907). The elastic strain field of adislocation loop is also expressed in the form of a line integral along the loop with a kernel involving the elastic Greenfunctions in an infinite domain (Mura, 1987). These solutions and other derived expressions are the basis of many modelsof plastic deformation.

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x: þ33 1 46734155.incre).jon, France.Fluid Dynamics, University of Technology Bergakademie Freiberg, Lampadiusstrasse 4, D-09599

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505492

3D dislocations dynamics (DD) simulations are among the most advanced techniques to model the collective propertiesof large dislocation ensembles (Kubin et al., 1992; Verdier et al., 1998; Zbib et al., 1998; Schwarz, 1999; Ghoniem and Sun,1999; Cai et al., 2004; Devincre et al., 2011). They are reasonably mature now and the computation of up to a few percent ofplastic strain in a volume with several hundreds of dislocations has become routine. Nevertheless, in DD simulationsboundary value problems are usually much simplified (Van der Giessen and Needleman, 1995; Devincre et al., 2003; Denget al., 2008). For instance, it is often assumed that external loads induce a uniform stress field inside the simulation volumeor in the vicinity of surfaces and interfaces. Subsequently, approximations are commonly used to calculate image forces andmany of the existing DD simulations codes are limited to isotropic elasticity, where analytical expressions for the stressfields for dislocation segments exist. In some 3D DD codes solutions for anisotropic media are used (Han et al., 2003; Chenand Biner, 2007), but they are computationally expensive due to the lack of closed-form solution of Green's functions forthat case. A significant step forward concerns the analytical expressions derived by Yin et al. (2010, 2012), based on thematrix formalism with the Willis–Steels–Lothe formula, but to the best of our knowledge these expressions have not yetbeen used in large-scale DD simulations.

Hybrid approaches have been proposed to solve the mechanical boundary problem of a finite volume containingdislocations. The first of these approaches superposes the mechanical fields associated to dislocations in an infinite mediumto the elastic solution of a modified boundary condition problem in a dislocation-free finite body. Initially applied to 2D DDsimulations by Van der Giessen and Needleman (1995), the superposition principle is now routinely used in 3D DDsimulations for different problems using finite element (FE) (Fivel and Canova, 1999; Weygand et al., 2001) or boundaryelement (BE) (Takahashi and Ghoniem, 2008; El-Awady et al., 2008) methods to solve the modified boundary problem.Recently, anisotropic elasticity has been taken into account in 2D DD (Shishvan and Mohammadi, 2011) to deal with theboundary value problem of a polycrystalline medium. The advantage of the superposition framework is the accuratedescription of the short-range dislocation stress field when dislocations are not too close to surfaces and interfaces.Unfortunately, numerical difficulties arise when the dislocation segments approach the boundaries (Weygand et al., 2001;Tang et al., 2006). Other methods include the extended finite element method (X-FEM) (Belytschko and Gracie, 2007; Gracieet al., 2007, 2008), and the level set method (Xiang et al., 2003). In the following we will refer to conventional codes whenthe boundary conditions are not accounted for, and to hybrid codes otherwise.

Another hybrid approach, which addresses some of these issues, is revisited in the present paper. Here, dislocation loopsare represented as thin plate-like inclusions causing an eigenstrain field in an isotropic or anisotropic elastic medium.The slipped areas calculated with a DD simulation are regularised as plastic strains at the integration points (IPs) of a FEmesh (Lemarchand et al., 2001; Devincre et al., 2003). The FE code can then calculate the associated equilibrium stress fieldinside a simulation volume of any shape, which can then be used to further drive the dislocation dynamics in a subsequenttime increment, and so on. This coupling between a DD simulation and a FE calculation, the so-called Discrete-ContinuousModel (DCM), is numerically straightforward: the DD simulation simply replaces the usual crystallographic constitutive lawin the FE calculation. The main difference with the superposition method is that the FE code not only takes into account theboundary conditions of the finite volume, but also the long-range dislocation–dislocation interactions. Problems withperiodic boundary conditions can also be treated without having to take into account explicitly the contributions of periodicreplicas, so at almost no additional computational cost. It was first developed by Lemarchand et al. (1999a, 1999b, 2000,2001) and subsequently by Groh et al. (2003a, 2003b). A comparable approach is sometimes used in phase-field simulationsto model the influence of dislocations on phase transformations (Rodney and Finel, 2001; Wang et al., 2001; Rodney et al.,2003). Another approach has been recently used in the framework of finite deformation theory (Liu et al., 2009; Gao et al.,2010).

The DCM has already been used to study plastic relaxation in anisotropic heteroepitaxial thin films (Groh et al., 2003b),plastic deformation in metal matrix composites (Groh et al., 2003a), compression tests in a micro-pillar of copper (Liu et al.,2009), orientation dependence of the yield strength (Vattré et al., 2010a) and channel size effects (Vattré et al., 2010b) insingle crystal superalloys. However, the algorithmic details of the regularisation procedure from slipped areas to plasticstrains, which constitutes the core of the DCM, have never been published, and they have also been improved in recent yearsin terms of precision and efficiency. The purpose of the present paper is thus to present these details (Section 2) togetherwith validation tests and a more complex application to a single crystal Nickel-based superalloy (Section 3). The last sectiongives a brief discussion and conclusions.

2. The Discrete-Continuous Model

This section discusses the DCM in detail. The manner in which the displacement jumps generated by the dislocationdynamics are calculated and the coupling algorithm between the DD code microMegas (2011) and the FE code Z-set (2012)is presented. Computational aspects of the treatment of surfaces and interfaces, and short-range interactions are alsodiscussed.

2.1. The boundary value problem

Material defects can be represented by equivalent incompatible stress-free strain distributions called eigenstrains (Mura,1987). Specifically, as shown in Fig. 1, a discrete dislocation loop of any shape can always be represented by a thin coherent

Fig. 2. The mechanical problem of a dislocated polycrystal using the eigenstrain formalism. A description of all quantities can be found in the text.At mechanical equilibrium, all fields must satisfy Eqs. (2) and (3).

Fig. 1. The eigenstrain formalism. A dislocation loop with a squared shape can be represented by a thin coherent inclusion Φ of thickness h and itsassociated stress-free strain tensor εp as defined by Eq. (1). The vector n is the normal to the habit plane of the inclusion, coinciding with the slip planenormal, the vector b is the Burgers vector of the loop.

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505 493

plate-like inclusion Φ with the same contour and thickness h (Nabarro, 1951). The associated eigenstrain results from theplastic slip generated during the expansion of the loop and the eigenstrain tensor εp (called plastic strain tensor hereafter) isdefined as the symmetric form of a dyadic product of the Burgers vector b with the normal n to its slip plane, according to

εp � 12h

b � nþn � bð Þ: ð1Þ

The elastic fields due to complex 3D configurations of dislocation loops can then be calculated from the eigenstraindistribution with stress-free strains as given by Eq. (1). Far from the dislocation core regions, such a procedure gives anaccurate description of the elastic fields (Saada and Shi, 1995). Close to the dislocation line (but outside the core region),smaller values of h give more accurate solutions with respect to the exact discrete dislocation solution. In linear elasticity,when a solid is sheared by an ensemble of dislocations, the corresponding total plastic strain εp can therefore be computedas a sum of the plastic strains εp;i due to each dislocation segment i moving in the volume.

The general boundary value problem of a dislocated finite crystal then becomes equivalent to finding the displacement u,the elastic strain εe and the stress r in mechanical equilibriumwith a given distribution of plastic strain εp and the boundaryconditions. As an example, Fig. 2 shows a polycrystalline volume Ω with boundary ∂Ω and outward normal nΩ, consisting ofseveral phases such as grains or precipitates Ωk. One part ST of ∂Ω can be subjected to imposed tractions T and another partSU to imposed displacements U . In the figure, each phase Ωk represents a different case:

1.

A dislocation-free phase Ω1 with εp ¼ 0 everywhere. 2. A dislocated phase Ω2 where each dislocation loop i is represented by a coherent inclusion Φi and the corresponding

plastic strain εp;i within. Outside Φi and at their boundaries ∂Φi, εp ¼ 0. Φ1 illustrates the simplest case of a planardislocation loop entirely inside Ω2. Φ2 represents a dislocation loop extended in two different glide planes, for instance asa result of cross-slip, and Φ3 represents a section of a dislocation loop emerging at a free surface. The last two casesrequire special numerical treatment, as discussed in Section 2.5.

3.

A phase Ω3 containing other material defects such as precipitate Ω4. Here, two cases must be distinguished: either Ω4

remains elastic and dislocations like Φ4 are blocked at the boundary ∂Ω4, or Ω4 can be plastically deformed. In the lastcase, either the boundary ∂Ω4 is penetrable to dislocations and the eigenstrains are continuously passing the interfacelike Φ5, or the boundary ∂Ω4 is only partially penetrable to dislocations and discontinuities in the eigenstrains at ∂Ω4

represent interface dislocations.

In the small-strain regime considered here, the total strain ε is the sum of elastic strain εe and total plastic strain εp.For a given plastic strain distribution, the dislocated solid is in mechanical equilibrium when the fields satisfy

ε¼ 12

∇uþ∇uT� �

r¼C : εe ¼C : ðε�εpÞ∇ � r¼ 0; ð2Þ

with boundary conditions

r � n¼ T at STu¼U at SU : ð3Þ

An initial plastic strain field εpðt ¼ 0Þ, corresponding to a pre-existing dislocation microstructure, also has to be takeninto account. The manner in which it is set up is discussed in Section 2.4. The first equation of the system (2) gives the

Fig. 3. The coupling between DD and FE consists of (i) a regularisation (DD-FE defines the plastic strain field within the solid) and (ii) an interpolation(FE-DD defines the stress field on dislocations) procedures.

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505494

strain–displacement relationship, with the subscript T denoting the transposition operator. The second equation is Hooke'slaw, with C the fourth-order elastic tensor and the third one gives the stress equilibrium equation. Eqs. (3) correspond to theNeumann and the Dirichlet boundary conditions. In the DCM simulation, Eqs. (2) and (3) are numerically solved using a FEcode. However, other methods such as the BE method might also be used.

2.2. Coupling procedure

As illustrated in Fig. 3, the two codes, microMegas ðDDÞ and Z-set ðFEÞ, are coupled into a hybrid system to form the DCM.The DCM can be considered as a standard FE computation, except that the classical constitutive law is replaced by the DDcode. However, because the stress fields due to the dislocations are long-range and because the computational domains ofboth codes are partly coincident and at the same length scale, the mechanical state of each material point is no longerindependent of its neighbours. In other words, the DD simulation provides the FE computation with a non-local constitutivelaw. This means that the DD code can only give the mechanical response at all material points simultaneously, contrarily to alocal constitutive law.

Both codes are strongly coupled in the sense that a mechanical equilibrium is sought for the coupled system at eachcoupled instant tn, i.e. at each coupled instant tn, Eqs. (2) and (3) should be verified (here in the weak sense of the FEapproximation). The coupling consists of two main procedures termed interpolation ðFE-DDÞ and regularisation ðDD-FEÞ.Between these procedures, both codes proceed as in their uncoupled counterparts in order to reach the next coupled instanttnþ1 ¼ tnþΔtn. The time increments used in each code are not necessarily equal to the coupled time increment Δtn, and theycan be subdivided into smaller ones. This is hardly ever done in the FE code, but in the DD code the coupled time incrementis often divided into em smaller time increments of duration δtm ¼Δtn= em. The time increment in the DD code is directlyrelated to the intensity of the short-range interactions (imposed by the minimum distance of approach between twosegments), and so this time step is limited by a maximum value. It is not strictly necessary that the δtm are all equal, but inthat case some of the equations below should be modified accordingly. For instance one could imagine simulations such asin Mordehai et al. (2008), Keralavarma et al. (2012), and Davoudi et al. (2012) with a large coupled time increment, duringwhich the DD codes switch between a mode treating dislocation glide and another mode treating dislocation climb, whichdo not take place at the same time scales. The manner in which the coupled system proceeds from one equilibrated coupledinstant to the next is now described.

Starting at coupled instant tn, all mechanical fields uðtnÞ, εðtnÞ, εpðtnÞ and rðtnÞ are supposed to fulfil Eqs. (2) and (3), in theweak sense proper to the FE approximation. For n¼ 0 a specific procedure is carried out first, as described in Section 2.4.A first prediction of the increment ΔuðtnÞ between tn and tnþ1, consistent with the imposed boundary conditions, ispredicted by the FE code using a quasi-Newton algorithm in which the tangent matrix is approximated by the local tensor ofelasticity. From this prediction the total strain increment ΔεðtnÞ is calculated at the IPs of the FE mesh.

Because of the smaller time increments δtm used in the DD code, the strain increment ΔεðtnÞ is divided into em incrementsδεðtmÞ ¼ΔεðtnÞ= em, with tm ¼ tnþðm�1Þδtm and m¼ 1;…; em. Between tm and tmþ1, i.e. during increment m, the stress rðtmÞincreases to rðtmþ1Þ according to

rðtmþ1Þ ¼ rðtmÞþδrðtmÞ ð4Þ

with

δrðtmÞ ¼C : ½δεðtmÞ�δεpðtmÞ�; ð5Þ

where δεpðtmÞ corresponds to the plastic strain increment due to the dislocation dynamics during the increment m(see Section 2.3). For m¼ 1, rðt1Þ is set to rðtnÞ.

However, before evaluating the dislocation dynamics, the stress fields at the IPs of the FE mesh (Eq. (4)) are firstextrapolated to the nodes using the inverse shape functions of the elements, and the different contributions of each elementare averaged at the given common node. Although conventional FE calculations allow for discontinuities in the stress fieldsbetween neighbouring elements, this averaging step is necessary here because the numerical integration of dislocationdynamics requires a continuous description of the stress throughout each phase. This stress is then interpolated to the

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505 495

midpoint of each segment by using the standard shape functions of the elements. There, the Peach–Köhler forces controllingthe mobility of the dislocation segments can then be calculated, just as in a classical uncoupled DD simulation.

The entire procedure between the calculation of the total strain increments ΔεðtnÞ at the IPs of the FE mesh to thecomputation of rðtmÞ at the midpoint of each segment is called the interpolation procedure, and is indicated by the arrowfrom the FE code to the DD code in Fig. 3.

The DD code now proceeds in a standard manner: the velocity υ of each dislocation segment can be calculated accordingto a mobility law. In the simplest case, a linear velocity equation is used: υ¼ τeffb=B with τeff the resolved shear stresscontaining contributions due to the Peach–Köhler force, line tension and other local interactions, B a drag constant and b themagnitude of the Burgers vector b. However, the exact form of the mobility law and the parameters on which it depends is acomplex issue outside the scope of the present paper. A detailed discussion of this aspect of DD simulations and the way it istaken into account in microMegas can be found in Devincre et al. (2011). From the velocities the dislocation motion of allsegments i is predicted up to tmþ1, after which the positions at location yiðtmþ1Þ ¼ yiðtmÞþυiΔtm are computed (taking intoaccount the possible contact reactions between dislocations which may annihilate or create segments, and events such ascross-slip).

Then, the resulting areas slipped by each mobile segment during time increment δtm are regularised into plastic strainincrements δεpðtmÞ at the IPs. At tnþ1, the plastic strain increment ΔεpðtnÞ at each IP accumulated during the em DDincrements is given by

ΔεpðtnÞ ¼ ∑em

m ¼ 1δεpðtmÞ: ð6Þ

These plastic strain increments ΔεpðtnÞ are then sent simultaneously to the FE code, where εpðtnþ1Þ ¼ εpðtnÞþΔεpðtnÞ and theassociated reactions rðtnþ1Þ (given by Hooke's law) are computed at each IP. If the mechanical equilibrium (Eqs. (2) and (3))is satisfied in the weak sense of the FE approximation, according to some standard convergence criterion (for instance basedon the absolute or relative norm of the residual forces), a new increment Δtnþ1 is started, if not, a new Newton–Raphsoniteration is started. Note that, as in all DD codes, the dynamics of the dislocation segments is integrated explicitly, so there isno corresponding criterion in the DD code for the dislocation configuration at the end of the DD increments.

The procedure in which Δεpðtnþ1Þ is calculated from the slipped areas is called the regularisation procedure, and isindicated by the arrow from the DD code to the FE code in Fig. 3. It is the most computationally expensive part of the DCM.The regularisation procedure is described in the next section.

In practice, the DCM time increment is mainly determined by the DD simulation code which takes place at small time-scales, and the FE computation convergence is usually reached at the first iteration. Hence, the implicit FE computationscheme presented here can be replaced if needed by a simpler explicit scheme.

For domains that remain elastic, for instance Ω1 in Fig. 2, the FE computation proceeds without inputs from the DD code.Also, for multiphase structures, the material parameters of the DD simulations can be different in each phase, e.g. differentdislocation mobility laws might be used in the phases Ω3 and Ω4 from Fig. 2. In the case of large and complex structures, theDCM methodology allows using one FE mesh, but several DD simulations can run in parallel, each applied to differentregions of the structure. It is also possible to use DD simulations in certain zones only and classical constitutive laws inothers.

2.3. Regularisation of dislocation slip

This section presents the algorithm used to calculate the plastic strain increment associated with the motion of onesingle straight segment AB, part of a discretised dislocation line in the DD simulation code. The regularisation proceeds intwo steps, which are now described. In the following, unless specified otherwise, positions and elastic fields are expressed ina common crystal reference frame ð0;X;Y ;ZÞ with X ¼ ½100�, Y ¼ ½010�, and Z ¼ ½001�.

In the first step, at each time increment δtm, each mobile straight segment AB of initial length ℓ0 gliding through adistance δy¼ υδtm generates plastic shear in a plate-like inclusion δΦAB surrounding its swept surface SAB (Fig. 4). This SABcan be thought of as the sum of elementary areas dS¼ dℓ� dy. Similarly, δΦAB can be thought of as the union of overlapping

Fig. 4. The surface area SAB swept by a segment AB during a glide δy¼ υδtm. The swept surface SAB and the distance δy are discretised into elementary areasdS and glide increments dy, respectively, as defined in Fig. 5.

Fig. 5. Regularisation of the plastic deformation associated to an elementary segment. (a) Area dS located at x swept by the glide dy of an elementarysection of length dℓ, and (b) its corresponding elementary homogeneous plastic shear dγ within a volume dΦðxÞ.

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505496

elementary spheres dΦðxÞ of radius h=2 and volume πh3=6, each one associated and centred on an elementary area dSlocated at xASAB (Fig. 5).

With these definitions, it becomes clear that the displacement jump across each elementary surface dS at x can beregularised to an elementary plastic shear increment dγ ¼ 6bdS=πh3, homogeneously distributed into its correspondingdΦðxÞ.

Now, as a second step, consider an IP of the FE mesh at position r inside the simulation volume. The shear δγrAB induced atr by the glide of AB during time increment δtm can only be non-zero if the IP is located within one or more elementaryvolumes dΦðxÞ. In that case, each elementary volume dΦðxÞ containing the IP contributes an elementary shear dγ to it, i.e.

δγrAB ¼6b

πh3

ZSAB

χ r; xð Þ dS� 6b

πh3SrAB; ð7Þ

with χðr; xÞ ¼ 1 if the IP at r lies within dΦðxÞ and 0 otherwise, so that the surface SrAB is that part of SAB whose associated

elementary volumes dΦðxÞ contain the IP.During a time increment δtm, the segment AB glides without rotating (but it can extend or shrink)3, so the surface SAB is a

trapezoid. A local frame ðA; xAB; yABÞ is considered, with xAB the line direction of the segment AB and yAB its normal glidedirection (Fig. 4). The surface Sr

AB is integrated continuously in the xAB direction but in a discrete way in the glide directionyAB: the total glide distance δy is decomposed into ej elementary glides dy¼ υδtm=ej. Then Sr

AB can be approximated as

SrABC ∑

ejj ¼ 1

dyZABj

χðr; xÞ dx� dy ∑ej

j ¼ 1LrABj ; ð8Þ

with ABj the segment AB at position yj � j dy, of length ℓj.For the sake of numerical efficiency, the length Lr

ABj is computed considering yet another local frame ðC j; xAB; yrABjÞ, with C j

the centre of ABj and yrABjthe unit vector normal to xAB in the plane passing through ABj and the IP at r. Note that because the

IPs can be located anywhere within a radius h=2 from ABj, yrABj is not necessarily normal to the glide plane. In this frame, thefollowing surfaces can be defined (Fig. 6):

Sþj ¼ fðx; yÞjðx�ℓj=2Þ2þy2oh2=4g;

S�j ¼ fðx; yÞjðxþℓj=2Þ2þy2oh2=4g; and

S�j ¼ fðx; yÞjy2oh2=4; x2oℓ2

j =4g [ S�j [ Sþ

j : ð9Þ

The surfaces S�j and Sþ

j partially overlap when ℓjoh. Denoting ðxr ; yrÞ the coordinates r in this frame, LrABj

is given by

LrABj

¼

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2

4�y2r

sif rASj\fSþ

j [ S�j gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h2

4�y2r

sþℓj

2�jxrj if rASþ

j [ S�j ; r=2Sþ

j \ S�j

ℓj if rASþj \ S�

j ;

8>>>>>>>><>>>>>>>>:ð10Þ

and Eq. (7) becomes

δγrAB ¼6bdy

πh3 ∑ej

j ¼ 1LrABj : ð11Þ

3 In the microMegas code, this condition is automatically satisfied because of the way in which the dislocation line is discretised into segments, but thisis not necessarily true in other DD codes.

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505 497

Finally, the plastic strain increment δεpAB at the IP located at r due to the glide of the segment AB during the time incrementδtm can be written as

δεpAB ¼6dy

πh3b � nþn � b

2∑ej

j ¼ 1LrABj : ð12Þ

The IPs outside the plate-like inclusion δΦAB do not have their plastic strain increased. In this work, only regular structuredmeshes are considered. Then the elements with IPs lying inside δΦAB can be determined very efficiently using spatialindexing (i.e. there is an analytical expression which gives the element number as a function of the location of eachsegment) and a near-neighbour table. Also, h must be at least of the order of the size of the element in a regular mesh toavoid that δΦAB does not contain any IPs, as was already discussed by Lemarchand et al. (2001). Typically, for quadratic brickelements of size L, h¼ 3L=2 was recommended. It is worth noting that h is not determined by dislocation spacing or linecurvature, because in that case the calculation of the short-range interactions of micro-sized features would rapidly becomean inextricable problem when large numbers of dislocations are involved. Rather, h is a numerical model parameter fixing aboundary between long-range and short-range contributions to the Peach–Köhler force. This issue will be addressed inSection 2.6.

This treatment is repeated for every segment in the simulation volume that moves during the current time incrementδtm. The regularisation procedure defined here is numerically efficient in handling dislocation glide, cross-slip and climb incrystals of any type of symmetry.

2.4. Initial configuration

As dislocations in a DCM simulation are represented by an eigenstrain field, the definition of the initial state uðt ¼ 0Þ is acritical step of calculations. Following the previous work in classical DD simulations (Devincre et al., 2011), it is relativelysimple to construct realistic 3D dislocation networks by relaxing a random distribution of dislocation loops. For reason ofsimplicity, in the DCM the initial configuration is usually a distribution of prismatic loops, i.e. dipolar dislocation loops madeof four edge dislocations with the same Burgers vector, which are then left to relax without any external loading. Theseprismatic loops are introduced in the simulation volume with a Volterra-like process. Two edge segments of the same initiallength ℓ0 and with the same Burgers vectors b but opposite line directions are first placed at the same position. Then, onesegment is held at its original position while the other one is moved along the complementary edge direction needed toform the contour of the prismatic loop. Alternatively, the same procedure, but moving the edge segment in the direction ofthe Burgers vector, might be used to create a loop fully contained in its glide plane, i.e. a glissile loop. The initial plastic strainεpðt ¼ 0Þ associated to this process is then distributed to the IPs of the FE mesh using the regularisation procedure aspresented in the previous section.

Fig. 6. Surfaces Sj , Sþj and S�

j appearing in the calculation of LrABj . All quantities are explained in the text.

Fig. 7. Plastic strain regularisation in the vicinity of phase boundaries (a) at penetrable interfaces and (b) at impenetrable interfaces or free surfaces. b1, n1

and b2, n2 denote the Burgers vectors and the normal slip planes of the two phases Ω1 and Ω2, respectively.

0 1 2 3 4 5 60

1

2

3

4

5

6

7

distance to surface (µm)

forc

e pe

r uni

t len

gth

(10-4

Nm

−1)

h=2.55µmh=1.28 µmreference solution

Fig. 8. Calculation of the image force at the centre of an edge dislocation segment parallel to a flat free surface for two different values of h. Goodagreement is found with the reference solution outside the regularised core region.

A. Vattré et al. / J. Mech. Phys. Solids 63 (2014) 491–505498

Then, the distribution of square loops is left to relax without any external loading. As a result, a 3D dislocation network isformed which may include strong junctions. On subsequent loading, some sections of the dislocation network will then actas dislocation sources.

2.5. Regularisation at domain boundaries

The regularisation procedure is now extended to dislocations in heterogeneous materials with internal interfaces(for instance Φ4 and Φ5 in Fig. 2) or external free surfaces (Φ3 in Fig. 2). As shown in Fig. 7, two cases should be considered.

In the case of penetrable interfaces (Fig. 7a), the transmission of the forces between Ω1 and Ω2 ensures the continuity ofthe traction across the interface, i.e. rΩ1 � nΩ1 ¼ �rΩ2 � nΩ2 . However, the eigenstrain keeping the trace of the dislocationglide from one phase to the other must be continuous. Because of Eq. (1), the amplitude of the eigenstrain directly reflectsthe type of defect (i.e. its orientation and magnitude) and the number of defects present. Modification of this amplitudewould therefore mean creation or destruction of dislocations, and this should only happen because of discrete eventsallowed by the local constitutive rules of the DD code. Thus, a local penetration rule has to be defined in the DD simulationcode to model the physical properties of the interface. However, no special treatment is required in the regularisationprocedure described in the previous section. This is an advantage of the spherical shape of dΦðxÞ, because of which thedisplacement jump generated by a dislocation crossing an interface or changing its glide plane direction is automaticallyregularised into a homogeneous plastic strain without discontinuities. This new feature of the DCM constitutes a significantimprovement with respect to earlier versions (Lemarchand et al., 2001).

In the case of non-penetrable interfaces (Fig. 7b), another problem appears: for elementary volumes dΦðxÞwhose centresare very close to the interface (i.e. at distances smaller than h=2), part of their volume would lie outside Ω1, but none of theassociated plastic strain can be affected there, and would simply be lost. The regularisation is therefore modified in thosecases: increments of plastic strain expected at the IPs within Ω2 from Eq. (12) are associated to a mirror volume with respectto the interface, and the IPs within that mirror volume, as for instance the open square in Fig. 7b. For regular meshes (as isthe case here), this mirror correction can be approximated by tabulating all IPs within Ω1 at a distance h=2 from the interfaceand simply multiplying the increments of plastic strain at those IPs by a linear interpolating function of value between 1 and2 (depending on the distance of the IP from the interface). This approximation, which is justified in the small strain limit, hasthe advantage of being simple and very cheap in terms of computing time. The same procedure applies to a free surface. Forthis last case, as illustrated in Fig. 8, the condition of zero traction at the free surface, rΩk � nΩk ¼ 0, is a numerically efficientapproximation to calculate the long-range effects of the image forces on dislocation segments. Nevertheless, as explained inSchwarz (1999), Devincre et al. (2011) a local line tension correction must be applied in the DD simulation code to fullyaccount for the influence of the surface on the line energy of dislocation segments touching a surface.

Finally, when a segment moves within a distance of h=2 of a periodic boundary, the eigenstrain increment associated toits periodic replica (i.e. the segment that is thought to move in the neighbouring computational cell by virtue of the periodicboundary conditions) must also be affected to the IPs inside the volume, in exactly the same way as described in Section 2.3.

2.6. Short-range interactions

As already discussed by Groh et al. (2003a), short-range interactions between dislocation segments cannot berepresented properly by considering only the stress fields calculated during a DCM step. This is because at very short

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distances, the quadratic shape functions of the standard finite elements cannot handle the dislocation interaction, which isinversely proportional to the distance between the segments. Moreover, any interaction at distances smaller than theregularisation length h cannot be calculated by the FE code. This problem is the major limitation of the DCM, and initiallythis limited applications to problems involving only a few dislocations in the simulation volume (Groh et al., 2003b).

To solve this problem, in the DD part of the DCM, analytic expressions for the stress field rloc of straight dislocationsegments (Devincre, 1995; Cai et al., 2006) can be added to the long-range stress fields given by the FE calculation each timetwo dislocation segments approach at short distance (Groh et al., 2003a). This procedure concerns only those segmentswhose points used to calculate the Peach–Köhler force approach at distance closer than h=2. As illustrated in Section 3.3, thiscorrection is necessary if one wishes to model dislocation interactions and reactions at short distances. This part of the DCMcalculation is easy to implement for isotropic elasticity calculations, but it significantly increases the computational burden.The expressions for anisotropic elasticity (Yin et al., 2010, 2012) might also be used, but this has not yet been tested in DCMsimulations. The implementation of these expressions in the DD part of the code would be of the same level of complexity asfor any other DD code, however the role of the Fast Multipole Method (FFM) in Yin et al. (2012) for the long-rangeinteractions is taken over here by the FE code, so nothing would need to be modified there. It is indeed interesting to notethat such a decomposition resembles the one used in the FFM (LeSar and Rickman, 2002) or alternative mean-fieldtechniques included in modern DD simulation codes. The main benefit of the DCM solution comes from the fact that thecomputational burden of the long-range stress field calculation is, by construction, only proportional to the number ofmobile dislocation segments in the simulation volume.

To summarise, when dislocation segments are closer to each other than h=2, two stress field contributions are used: (i) along-range contribution which is computed numerically by the FE code, and (ii) a short-range contribution (only within theregularised plastic strain volume of each segment), provided by analytic expressions for the elastic field of dislocationsegments (Devincre, 1995).

3. DCM simulations

In this section, simple test simulations are presented in order to illustrate the capabilities of the DCM. First, the stressfield due to a static prismatic loop is compared to an analytical solution. A classical test with a Frank-Read source is thencarried out, and a simulation of the zipping of a Lomer junction is presented in order to demonstrate that the DCM canhandle short-range interactions correctly. Finally, several aspects of plasticity in a single crystal superalloy are simulated asapplication to a multiphase system with either only a few or many dislocations.

It should be pointed out that the DCM is not meant as a speedup with respect to the conventional DD method, rather it ismeant for treating problems which cannot be treated by the conventional DD method alone. The calculation of theeigenstrain in the DCM adds a significant computational burden with respect to the conventional DD method, but on theother hand, the FE code takes care of the long-range interactions, which do not have to be calculated analytically anymore.A comparison of computational speeds is therefore not given here because such a comparison could hardly be very general:depending on the number of segments, the net loss or gain in computational speed is difficult to predict. As was alreadysuggested in Devincre et al. (2003), if only one dislocation loop is present, clearly the conventional DD is much faster, butwhen many dislocation loops are present, the DCM is faster.

3.1. Prismatic loop

Fig. 9 shows a square prismatic dislocation loop, generated in the manner described in Section 2.4. The prismatic loop isdefined with four ⟨100⟩ edge segments of length ℓ0 ¼ 0:25 μm in a ð010Þ slip plane of a simple cubic crystal lattice, so thatb¼ b½010� with b¼0.25 nm. It is placed in the middle of a volume of dimension L3 ¼ 0:48� 0:50� 0:52 μm3. The FE meshconsists of 16�16�16 quadratic elements to which periodic boundary conditions are applied, without external loading.It is assumed that the crystal is elastically isotropic with shear modulus μ¼ 51 GPa and Poisson ratio ν¼ 0:37. Fig. 9a showsthe initial displacement field u2ðt ¼ 0Þ caused during the initial setup of the prismatic loop, where the greyscale goes fromwhite for þju2ðt ¼ 0Þj to black for �ju2ðt ¼ 0Þj. Fig. 9b shows the initial associated plastic strain component ɛp21ðt ¼ 0Þ.

To validate the interpolation and regularisation of the DCM, the stress component s32 due to the dislocation loop in across-section of normal ½100� centred on the prismatic loop inside the simulation volume is compared to an exact analyticsolution. Fig. 10(a) and (b) shows the analytic solution and the FE solution, respectively, and (c) shows the comparisonbetween DCM computations with different values of h with the analytic solution. At distances larger than the plastic strainregularisation dimension h, both solutions predict the same stress state with a relative error r6%. The elastic singularity atthe dislocation line appears smeared out by the regularisation procedure in the DCM calculation. At distances smaller thanh, the interpolation procedure fails to yield the correct stress values which justifies the analytic correction introduced in theDCM to calculate the short-range elastic interactions (Section 2.6).

3.2. Frank-read source

The second test case is the computation of the critical stress of the activation of a Frank-Read source. One way of creatinga Frank-Read source in the DCM is to use a prismatic loop as initial configuration and to immobilise three of the four

Fig. 9. DCM computation of the mechanical fields associated to a prismatic dislocation loop. (a) A prismatic dislocation loop in a ð010Þ slip plane at thecentre of a periodic volume, discretised by 16� 16� 16 quadratic brick elements. The Burgers vector is b¼ b½010�. The displacement field is shown ingreyscale, where the scale goes from white þjuðt ¼ 0Þj to black �juðt ¼ 0Þj. (b) The initial plastic strain component ɛp21ðt ¼ 0Þ associated to the prismaticloop at the IPs in the FE mesh. The regularisation procedure delivers a continuous plastic strain field of the loop.

Fig. 10. Stress component s32 due to the prismatic loop of Fig. 9. (a) Analytic solution provided by continuum theory and (b) solution calculated by the DCMwith h equal to 63 nm and 16�16�16 FE mesh. The magnitude of the stresses is represented using the same scale. (c) Comparison among the analytic andDCM s32 stress components, plotted along the dashed line drawn in (a) and (b). The three different DCM solutions are obtained using the same prismaticloop of Fig. 9, but varying h and the mesh size. We can clearly see that at a distance larger than 2h, the computed stress fields (solid lines) perfectly matchthe theoretical calculation (dashed line). In the region included between h and 2h, the maximum relative error between theoretical and DCM stresssolutions is about 6%.

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segments of the loop (the dotted segments in Fig. 11. Under loading, the remaining mobile segment will act as a source.The determination of the critical shear stress for which a bowing segment can operate as a source is a classical bench test of3D DD simulations (Devincre and Condat, 1992; Schwarz and Tersov, 1996; Gómez-García et al., 1999). Here the critical stressof a Frank-Read source of initial length ℓ0 ¼ 0:1 μm is computed by the DCM with the same parameters as the previoussection. The DCM simulation yields τFR ¼ 116 MPa, whereas the classical DD simulation yields 113 MPa.

Fig. 12 shows snapshots at successive instants of the Frank-Read source operating in its glide plane. A uniform plasticstrain ɛp23 ¼ n⟨ɛp23⟩ is found inside the generated dislocation loops, corresponding to n eigenstrain inclusions Φ, with n thenumber of loops emitted from the Frank-Read source. Under periodic boundary conditions, the displacement jump increasesby one Burgers vector at each passage of a novel dislocation, as expected.

3.3. Contact reactions between dislocations

This section shows the role in the DCM of the additional stress field rloc discussed in Section 2.6. The aim is to show thatthe DCM can correctly handle contact reactions, such as zipping or unzipping of a dislocation junction. A simulation of aLomer junction formation is presented and compared with a conventional DD simulation. The capability of the DCM to

Fig. 11. Bow-out of an edge segment of initial length ℓ0 acting as a Frank-Read source in the DCM. The plastic strain slab ɛp23 of thickness h used for the DCMcomputation is shown in a darker shade of grey. n is the glide plane normal and b is the Burgers vector. A displacement jump of b is observed in the regionsswept by the dislocation line.

Fig. 12. 3D simulation of Frank-Read source on a cubic plane under periodic boundary conditions: successive snapshots of the cross sections of theoperating slip plane show the evolution of the plate-like inclusion. At each passage of a dislocation (represented in greyscale) the displacement jump acrossthe slip plane increases by b. The displacements are magnified by a factor of 100 for better visualisation.

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precisely reproduce such a type of contact reaction is important because contact reactions and short-range interactions areknown to control important properties during plastic flow (Gil Sevillano, 1993; Madec et al., 2002).

Fig. 13a presents the simulation of two attracting dislocation lines with lengths l1 ¼ l2 ¼ 5 μm and with Burgers vectors12½101� and 1

2 011�½ , respectively, on intersecting octahedral slip planes, n1 ¼ f111g and n2 ¼ f111g. Three different calculationswith the same periodic boundary conditions are compared in Fig. 13b, using the standard DD simulation, the DCM with theshort-range correction rloc , and the DCM without the correction. Snapshots are shown at two instants, at the equilibriumstate of the junction (without applied stress, i.e. rapp ¼ 0), and when a shear stress, close to the critical value rc needed tounzip the junction, is applied (the difference is about 5%). The first column of panel (b) shows that in the case of standard DD

Fig. 13. (a) Dislocation geometry used in the simulation of a junction zipping and unzipping processes. (b) Final state (first column, rapp ¼ 0) of the zippingprocess and initial stage (second column, rapp � rc) of the unzipping process for a standard DD simulation and two DCM simulations with and without theshort-range correction, as discussed in Section 2.6.

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and DCM with local correction the two attractive dislocations approach each other and form a junction segment of ½110�type, with comparable length (less than 4% difference). It also shows that only a very short junction is formed in the DCMsimulation without the rloc correction. This is because the short-range interactions at the origin of junction zipping cannotbe correctly interpolated from the stress calculated at the FE nodes. In the second column of panel (b) the resolved shearstress in the n1 slip plane needed to unzip the three different junctions is also shown. This demonstrates that the DCMsimulations compared with classical DD simulations correctly handle the unzipping process of a junction, while the DCMwithout local correction fails in describing the phenomenon. Other tests (not discussed here) have also been carried out in asimilar manner to verify that junction formation, dipole formation, coplanar dislocation annihilation and cross-slip whichare correctly reproduced with the DCM when including the short-range correction rloc.

3.4. Plasticity in multiphase materials

The purpose of this section is to demonstrate the capability of the DCM in capturing plasticity mechanisms in complexmultiphase materials. To illustrate this point, results for Nickel-based single crystal γ=γ′ superalloys are presented. Contraryto the previous examples with only a few segments, such simulations are massive computations in the sense that a largenumber of segments is present (more than 40,000 segments at the end of the simulation). Full details of similar mesoscopicsimulations have already been presented in Vattré et al. (2009, 2010a, 2010b), and have served as a basis for a physicallybased micromechanical model for structural analysis (Vattré and Fedelich, 2011).

These superalloys are extensively used in applications requiring high strength and fatigue resistance up to elevatedtemperatures, for instance in turbine blades. The difference in lattice parameter between the γ and γ′ phases creates acoherency stress field. In the DCM simulation, the latter can be computed with a preliminary thermo-elastic FE calculation,which would not be possible in classical DD simulations (see Vattré et al., 2010a for full details on the simulation procedure).In the following calculations we consider T ¼ 850 1C and linear elasticity is used with μ¼ 51 GPa and ν¼ 0:37.

In order to show that the DCM can easily capture size effects in plasticity, the results for three different simulationvolumes, increasing in size, are presented here. Each simulation volume contains the same initial configurations (dislocation

Fig. 14. The simulated resolved stress τ–plastic strain γ curves for uniaxial loading along the ½001� axis for the three channel widths. The dislocationmultiplication with increasing strain is illustrated by the 3D dislocation structures corresponding to different strain stages. (a) The dislocation configurationafter relaxation and (b) after 0.2% plastic strain.

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densities, source distributions and entanglements) and the same number of finite elements, but scaled up with a factor x¼1,2 or 3. The precipitates have orthorhombic dimensions 0:48x� 0:50x� 0:52x μm3, and their volume fraction is thereforeconstant at 61%, corresponding to a ratio between channel width w and precipitate size of 0.16, and the channel width wincreases from 0.08 to 0.16 and to 0:24 μm. The FE meshes contain 4096 quadratic brick elements and 56,361 degrees offreedom and periodic boundary conditions are applied at all external boundaries.

First, the three initial dislocation distributions made of prismatic loops are generated. Dislocation loops are randomlydistributed between the 12 octahedral slip systems with dislocation density of 2� 1012 m�2 and b¼0.25 nm. All detailsof the physical parameters of the simulations can be found in Vattré et al. (2009, 2010a, 2010b). The initial edge segments ofeach loop are assigned a random length within a pre-set interval. As explained in Section 2.4, initial plastic strains εpðt ¼ 0Þcorresponding to the prismatic loop expansion are assigned to the IPs of the FE mesh. A snapshot of the initial dislocationconfiguration of the smallest volume (i.e. x¼1 with w¼ 0:08 μm) and the von Mises stress sMis due to the coherency stresson a cross-section through the matrix is shown in the inset (a) of Fig. 14. Inside the precipitate, sMis is roughly uniform atabout 30 MPa.

The microstructures are then left to relax towards an equilibrium configuration, during which a network of interfacedislocations is formed. Equilibrium of the relaxed dislocation configuration is attained when all segments have left theprecipitate and the volume does not deform plastically any more, i.e. when all dislocation segments are immobile. The loopsections lying initially inside the precipitate are, after this relaxation, located at the interfaces and the precipitate becomesdislocation-free. During this relaxation, the elementary simulation time step is 5� 10�2 ns and the ratio em between the FEtime increment and the DD increment is 10. Complete relaxation takes about 3000 steps. The total dislocation density at theend of the relaxation is 6:2� 1013 m�2. The relaxed dislocation configurations mainly consist of misfit dislocations whichlocally reduce the coherency stress. Misfit dislocations are either screw or 7601 character dislocations and they aregenerated during the relaxation dynamics by gliding curved dislocations in the matrix.

In a second step, the relaxed dislocation microstructures in the three simulation volumes are then subjected to anexternal pure tensile loading along the ½001� crystallographic axis. Also, a lattice friction stress τf ¼ 107 MPa is considered inthe γ phase to account for solid-solution strengthening. In order to reduce computation time, a high resolved strain rate of_γ ¼ 20 s�1 is imposed.

Fig. 14 shows the resulting resolved stress–plastic strain curves. For all three cases, two different stages can bedistinguished. A first stage corresponds to a transient regime from zero to 0.01% plastic strain (indicated by the verticaldashed line in Fig. 14), during which plastic deformation mainly results from the bow-out of the dislocation loops initiallypresent inside the channels. A second stage corresponds to the irreversible plastic deformation from 0.01% plastic strain upto the end of simulations. During this second stage, the strain hardening rate is constant and plastic deformation arises fromdislocations passing around the precipitates.

For comparison with the relaxed dislocation network, the inset (b) of Fig. 14 shows the dense interfacial dislocationnetwork obtained at 0.2% plastic strain for the same smallest volume. The network of interfacial dislocations formed duringplastic deformation is polarised and creates a stress field in the channels opposing dislocation motion. In addition, nodislocation pile-up is observed. Irrespective of the strain value, the flow stress increases significantly with decreasingchannel width. To quantify the size effect, the simulated values of the 0.01% yield stress, called Orowan stress τOrw here, are

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fitted as

τOrw ¼ 1:02 μb=w: ð13Þ

This result corresponds quantitatively to the theoretical macroscopic Orowan stress, below which dislocations cannot curve.Hence, the onset of the irreversible plastic regime in superalloys is well reproduced with the DCM simulations.

4. Conclusion

A hybrid computational model, called the Discrete-Continuous Model (DCM), which couples 3D dislocation dynamics(DD) and finite element (FE) codes, is revisited. Crystal plasticity is simulated within a unified framework where (i) thedisplacement jumps across the slip planes of dislocations are represented by eigenstrains, obtained directly from the DDcode instead of the phenomenological evolution equations used in conventional crystal plasticity, and (ii) the boundaryconditions and the long-range dislocation–dislocation interactions are handled by the FE method. The precise manner inwhich the dislocation displacement jumps are regularised to the integration points of the FE mesh is presented for thefirst time.

Several standard tests and verifications such as the long-range stress field of a dislocation loop, the evolution of a Frank-Read source, and dislocation junction formation are presented. In order to demonstrate that the method is capable oftreating much more complex problems, the mechanical response of a single crystal superalloy is simulated. This problemcannot be treated by conventional DD simulation codes (although they might be treated with other hybrid methods such asthe superposition method) because of the misfit between γ and γ′ phases, which generates an initial dislocation network atinterfaces. Stress–strain curves are analysed and the onset of plasticity is found to scale as the inverse of the channel width,as expected from experience.

Work is currently in progress to further improve and optimise the DCM, such as using other finite elements than thebrick elements used here, and to adapt the coupled algorithm to the parallelised algorithms already present in each codeseparately.

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