Materials Science and Engineering A309–310 (2001) 294–299

Size and boundary effects in discrete dislocation dynamics:coupling with continuum finite element

Hasan Yasina, Hussein M. Zbiba, Moe A. Khaleelb,∗a School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA

b Pacific Northwest National Laboratory, Virtual Prototyping and Engineering Simulations Laboratory, MS# K2-18, Richland, WA 99352, USA

Abstract

In this work, we develop a framework coupling continuum elasto-viscoplasticity with three-dimensional discrete dislocation dynamics(micro3d). The main problem is to carry out rigorous analyses to simulate the deformation of single crystal metals (fcc and bcc) of finitedomains. While the overall macroscopic response of the crystal is based on the continuum theory, the constitutive response is determinedby discrete dislocation dynamics analyses using micro3d. Size effects are investigated by considering two boundary value problems: (1)uniaxial loading of a single crystal cube, and (2) bending of a single crystal micro-beam. It is shown that boundary conditions and the sizeof the computational cell have significant effect on the results due to image stresses from free-boundaries. The investigation shows thatsurface effects cannot be ignored regardless of the cell size, and may result in errors as much as 10%. Preliminary results pertaining todislocation structures under bending conditions are also given. Published by Elsevier Science B.V.

Keywords: Elasto-viscoplasticity; Discrete dislocation dynamics; Finite element analysis

1. Introduction

Recently, there have been significant advances inmodeling plasticity phenomena using discrete dislocationsdynamics models [1–3]. These models are based on basicdeformation mechanisms and aim at predicting strengthproperties of metals at small length scales. However, themain difficulty in these models, among other things, liesin connecting the observed macroscopic parameters (usedin continuum models), such as stress and strain, to thecollective behavior of dislocation groups, and their in-teraction with particles and interfaces. While, within thecontinuum mechanics framework, the governing equa-tions of the material response are developed based ona representative volume element (RVE) over which thedeformation field is assumed to be homogeneous, thedislocation dynamics models are presumably capable ofdescribing the heterogeneous nature of the deformationfield within the RVE. It is suggested that with a properhomogenization theory, or field averaging, one can couplethe discrete dislocation dynamics models with continuumapproaches, and provide a rigorous framework for ana-lyzing deformations at small scales, where surfaces and

∗ Corresponding author: Tel.:+1-509-375-2438; fax:+1-509-375-6605.E-mail address: moe.khaleel@pnl.gov (M.A. Khaleel).

interfaces are of great importance in determining materialsresponse. The idea was advocated by Van der Giessenand Needleman [6] by coupling two-dimensional (2D) dis-crete dislocations with a continuum finite element model.In the present work, we develop a computational modelcoupling our (3D) three-dimensional discrete dislocationsmodel (micro3d) with a continuum finite element modelbased on an elastic–plastic model. The main motivation isto develop a multi-scale model for investigating small-scaleplasticity phenomena and deformation of small-scalestructure.

2. The multi-scale elastic–plastic model

2.1. The continuum elastic–plastic model

Generally, the material response is measured in terms ofthe macroscopic strain rate tensorε and its relation to thestress tensorS. We consider a computational cell of a sizein the order of a few tens of micrometers containing manydislocations and point defects (microcracks, stacking faulttetrahedral, Frank sessile loops, rigid particles, etc.). On themacroscale level, we assume that the material obeys the basiclaw of continuum mechanics, i.e equations of equilibrium

divSSS = 0 (1)

0921-5093/01/$ – see front matter Published by Elsevier Science B.V.PII: S0921-5093(00)01731-7

H. Yasin et al. / Materials Science and Engineering A309–310 (2001) 294–299 295

For elasto-viscoplastic behavior, the strain rate tensorε isdecomposed into an elastic partεe and a plastic partεp suchthat

ε = εe+εp, ε = 12[∇vvv + ∇vvvT] (2)

For most metals the elastic response is linear and can beexpressed by the incremental form of Hooke’s law, i.e.

SSS = [CCCe][ ε − εp] (3)

whereCCCe is a fourth order tensor. The main difficulty inthe plasticity theory is the development of proper consti-tutive law for εp. More importantly, this law should bebased on the underlying microstructure, mainly disloca-tions. Nonetheless, this task is perhaps formidable, espe-cially when bridging two scales orders of magnitude apart,i.e. the continuum scale and the discrete dislocation scale.Here, we emphasis that the “assumed” constitutive natureof εp and flow stress, and their dependence upon internalvariables and gradients of internal variables is very critical,since they dictate, among other things, the length scale ofthe problem and the phenomena that can be predicted bythe model. In this respect, it goes without question that themost rigorous and physically based approach of comput-ing the plastic strain and strain hardening in metals, withall relevant length scales, is through explicit evaluation ofinteraction, motion and evolution of all individual discretedislocations and all relevant other defects in the crystal.

2.2. The 3D discrete dislocation dynamics model (DD)

The 3D discrete dislocation model (micro3d) developedat WSU is capable of simulating the dynamical behaviorof a large number of dislocations of arbitrary shapes andinteraction among groups of 3D dislocations. In the model,a crystal plasticity representation of a three-dimensionalcrystal is considered containing a number of dislocationloops and lines of arbitrary shapes lying on any or all slipsystems (for both fcc and bcc single crystals). The analysisis based on explicit evaluation of dislocation motion andtheir interaction with other defects and particles. A com-plete description of micro3d is given in many articles byZbib and co-workers [4,5]. The result is a set of nonlineardifferential equations governing the motion of the disloca-tion segments. The governing equation of motion for eachdislocation segment is given by equation of the form [7]

F(v) + v

M(T , p)= Fi, F = 1

v

(dW

dv

)∂v

∂t,

m∗ = 1

v

(dW

dv

)(4)

Here F = F(v) is the inertial force,M the mobility, Fi

the force arising from interactions with other defects, fromthe Peierls barrier if present, and that produced by appliedstresses, andW the total energy per unit length of a moving

dislocation (elastic energy and kinetic energy). The driv-ing forceFFF i is determined by first evaluating the total PKforce which arises from all dislocation stress fields and theapplied stress. The motion of each dislocation segment con-tributes to the overall macroscopic plastic strain rate tensorεp and plastic spinWWWp via the relations

εp =N∑i=1

−livgi

2V(nnni ⊗ bbbi + bbbi⊗nnni),

Wp =N∑i=1

−livgi

2V(nnni ⊗ bbbi − bbbi ⊗ nnni), nnni = vi × ξi

(5)

whereli is the dislocation segment length,nnni a unit normalto the slip plane,V the volume of the simulated crystal orthe RVE,bbbi the Burgers vector, andN the total number ofsegments in the RVE.

2.3. The superposition principle — homogenous materials

The solution for the stress field of a dislocation segmentis known for the case of infinite domain and homogeneousmaterials, which is used in DD codes. Therefore, the prin-ciple of superposition (developed by Van der Geissein andNeedleman [6] for the 2D case) is imposed to correct for theactual boundary conditions. Assuming that dislocation seg-ments, dislocations loops and any other internal defects withself-induced stress, are situated in a finite domainV boundedby ∂V, and subjected to arbitrary external traction andconstraints. Then the stress, strain and displacement fieldsare given by the sum of two solutions, i.e.SSS = SSS∞ + SSS∗,uuu = uuu∞ +uuu∗, ε = ε∞ + ε∗, whereSSS∞, ε∞ anduuu∞ are thestress, strain and displacement fields, respectively, causedby the internal defects as if they were in an infinite domain,whereasSSS∗, ε∗ anduuu∗ are the fields corresponding to theauxiliary problem satisfying certain displacement boundaryconditions along withttt = ttta − ttt∞ on ∂V, wherettta is theexternally applied traction, andttt∞ the traction inducedon ∂V by the defects (dislocations) in the infinite domainproblem. The tractionttta on ∂V results into animage stresswhich is superimposed onto the dislocations segments and,thus, accounting for surface-dislocation interaction. In pass-ing, we note that the same method is extended to the caseof dislocations in heterogeneous media [8].

2.4. Treatment of long-range stresses

Evaluation of the dislocationslong-range stress field iscomputationally expensive (N2 problem). A numerical tech-nique termed as thesuperdislocation method, and based onthe multipolar expansion method has been outlined by Zbibet al. (1998) which reduces the order of the problem toN logN. Here, we present another approach consistent withthe finite element framework. The stress field induced bythe dislocations contained within the RVE can be treated as

296 H. Yasin et al. / Materials Science and Engineering A309–310 (2001) 294–299

an internal stressSSSD (homogenized over the element withenough gauss points!). Then, the effective total stress withinthe RVE is the sum of the stress by all external agenciesSSS

and the internal stressSSSD. Hence, the constitutive equationin total form becomes

SSS + SSSD = [CCCe][ε − εp] (6)

Since, the dislocation stress field varies as 1/r, carefulapproximation ofSSSD over the RVE is most important. Oneneeds to consider high number of integration points to in-sure high accuracy. The treatment of long-range stress usingthe internal stress concept results into a body-force vector inthe finite element analysis (FEA) described in Section 2.5.This approximation works well for interaction among dislo-cations that do not reside in the same elements (in the FEA).The interaction of dislocations belonging to these elementsmust be computed one-to-one (M2, M = number of disloca-tion segments in a given element) as explained in this work.

2.5. The finite element formulation

The equations described before are re-written in total formas in Eq. (6), and the system of equations is then cast into afinite element framework leading to the standard form

[KKK]{UUU} = {fff a} + {fff B} + {fff∞} + {fff p} (7)

where [KKK] is the stiffness matrix,{fff a} the applied forcevector,{fff∞} = ∫

sttt∞[NNN ] ds a force vector arising from dis-

location image stresses,{fff∞} = ∫vSSS∞[BBB] dv a body force

vector from dislocations long-range interaction,{fff∞} =∫v[CCCe]εp[BBB] dv a body-force like vector from plastic strain

caused by dislocations, [NNN ] a vector containing the shapefunctions, and the matrix [BBB] the gradient of [NNN ]. Disloca-tions are sorted out in each element, and they contribute tothe plastic strain in Eq. (5).

The solution of Eq. (7) yields, among other things, thetotal stress fieldSSS which includes the effect of applied loads,image stresses and the “homogenized” dislocation stresses.However, as mentioned above, dislocation–dislocation in-teraction is very strong for dislocations that are close toeach other, and the homogeneous stressSSSD in this caseis not accurate. Therefore, for a dislocation segment “i”residing in an element “k” which containsM dislocationsegments, the PK force is computed as follows:

FFF i =

M∑j=1j =i

j =i+1j =i−1

(σDj ) · bbbi

× ξi + (SSSk − SSSDk )

×ξi +FFF i,i+1 +FFF i,i−1 (8)

whereσDj is the stress field of dislocation “j”, SSSk the stress

state in elementk as obtained from the FE analysis,ξ i the

line sense vector, andFFF i,i+1 and FFF i,i−1 the interactionforces between segmentsi and i + 1, andi and i − 1, re-spectively. This way, the interaction among dislocations inthe same element is treated explicitly (note that the homo-geneous stressSSSD

k must be subtracted, otherwise there willbe a double counting of the PK force!).

Fig. 1. Effect of free surfaces on dislocations: (a) initial configuration(dislocation loop); (b) stress contours of image stresses; (c) Peach–Kohlerforce on dislocation loop from self-force and image stresses.

H. Yasin et al. / Materials Science and Engineering A309–310 (2001) 294–299 297

2.6. The multi-scale plasticity model

The two models, DD model and the solid mechanics prob-lem are coupled and the result is a multi-scale model whichhas the following features: (a) couples the continuum prob-lem with the dislocation dynamics problem; (b) deals di-rectly with boundary conditions, especially image stresses;(c) incorporates shape changes associated with dislocationmotion and distortions; (d) incorporates directly dislocationlong-range stresses, and (e) incorporates lattice distortion(elastic) that arise from dislocations.

3. Results and discussion

The effect of the free surface, i.e. the image stresses,on the behavior of a dislocation line, a square dislocationloop, a circular dislocation loop with different dimensions

Fig. 2. (a) Dislocation sources in the initial configuration; (b) dislocations start to move; (c) the difference in the yield strength when applying thefree-boundaries effect and when ignoring it; (d) effect of box size.

and parameters, and a random distribution of Frank–Readsources was thoroughly investigated [9]. Here, we givesome representative results. We consider aluminum sin-gle crystal with the following properties (relevant to boththe continuum model and the DD model):µ = 26.6 MPa;v = 0.334; ρm = 2800 kg m−3; b = 2.862 e−10 m, andM = 100.0 Pa−1 s−1 (dislocation mobility). The 3D box(various sizes were simulated ranging from 1000b to10,000b) was divided into a finite element mesh usingeight-node elements (mesh size ranged from 5× 5 × 5 to20× 20× 20), and was fixed at its lower four corners andconstant extension rate was imposed on the upper surface.

The results given in Fig. 1 are for the interaction of a dis-location loop with free surfaces. Fig. 1(a) shows a contourplot of image stresses. The self-force of the loop is plottedin Fig. 1(c). Also, PK force caused by the image stresseson the dislocation loop is given in Fig. 1. These calcula-tions show that the loop in the(11 1) plane in a 1000b

298 H. Yasin et al. / Materials Science and Engineering A309–310 (2001) 294–299

Fig. 3. (a) Bending of a thin sheet (all dimensions in unit ofb): contours of shear stress. Dislocation distribution in bending of a thin sheet (all dimensionsare in unit ofb): (b) dislocation pile-up (side view); (c) dislocation distribution (top view).

(2.86�m) aluminum single crystal with a Burgers vectorof [1 1 0] propagates, if the initial configuration diameterof the loop is greater than 90.7% of the computational cellplaner size. This ratio, of course, depends on the crystalstructure, loop plane direction and Burgers vector direction.In all the simulations we performed, dislocation segments

in the simulation cell near the free boundaries tend to movetowards the boundary and annihilate. For example, a squaredislocation loop on a(11 1) plane 100b (28.6 nm) abovethe center of a 20,000b (5.72�m) aluminum single crys-tal with a Burgers vector of [1 1 0], the numerical resultsshow that when the loop size is greater than 86.5% of the

H. Yasin et al. / Materials Science and Engineering A309–310 (2001) 294–299 299

computational cell (crystal) size, the surface effect is greaterthan the self-force of the loop. Otherwise, the self-forceeffect is greater and the loop would collapse on itself asexpected.

Next, we examine the effect of free surfaces on the pre-dicted stress–strain curve. Frank–Read sources were dis-tributed randomly with random length ranging from 4000b(1.144�m) to 6000b (1.716�m) in a cubic aluminum singlecrystal (Fig. 2(a)) with side dimension of 20,000b (5.72�m).All the dislocation sources were placed on parallel (1 1 1)planes for a random distribution of three different Burgers di-rections of [11 0], [1 01] and [01 1]. The crystal was loadedunder constant strain rate of 10 s−1 in the [0 0 1] direction,and the base was totally fixed in all directions.

Fig. 2(b) illustrates the behavior of the dislocations withtime; the dislocation sources start operating when the re-solved shear stress on the dislocation reaches the criticalshear stress. Fig. 2(c) shows the difference between thestrength of the crystal when the free-boundary effect is ap-plied and when it is neglected. The percentage differenceis 8% calculated at 0.05% yield strength. For the samedislocation source distribution, inside a 20,000b (5.72�m)sub-domain cubic space, the size of the crystal was increasedto 22,000b (6.29�m), 30,000b (8.58�m) and 40,000b(11.44�m) to investigate the variation of the free-boundaryeffect with the crystal size. It is useful to define an aspectratio ζ = B/A, whereB is the size of the crystal andA thesize of the sub-domain where all the dislocation sourcesare initially situated. In this study,A is taken to be fixedand equal to 20,000b (5.72�m) while B is increased. Twomethods were followed to investigate the size effect: (1) byevaluating the strain–strain curve based on the whole crys-tal size, and (2) by taking the properties based on the sizeof A. No significant difference in the result was found. Theresult summarized in Fig. 2(d) shows that the percentagedifference in the 0.05% yield strength when the free sur-faces are taken into account and when they are neglected.The difference decreases as the aspect ratio increases, im-plying that as far as the initial macroscopic yield stress isconcerned, the effect of the free surface can be neglectedwhen the initial sources are far enough from the free sur-faces. But this is not true at large strain as can be deducedfrom Fig. 2(c). This leads to the conclusion that it does notmatter how large the simulation cellB is with respect to thesub-domainA, one cannot simply neglect the effect of theboundaries of cellB on the dislocations in sub-domainA.

Finally, we give preliminary results pertaining to the dis-location behavior during the bending of a thin sheet singlecrystal Al oriented as shown in Fig. 3. The sheet is simplysupported and loaded as shown in the figure (with load

distribution of 120 MPa) (dimensions are given in the fig-ure). Dislocations are initially situated on parallel (1 1 1)planes which are parallel to the loading plane as shown inthe figure. Fig. 3(b) shows the distribution of shear stress.The shear stress is maximum at the edges of the sheet, andreduces towards zero at the center of the sheet. This resultsinto a zone where the shear stress is not high enough tokeep the dislocations moving, resulting in a zone where thedislocations pile-up.

In conclusion, a numerical model coupling 3D discretedislocation dynamics with continuum finite element modelbased on elasto-plasticity has been developed. By usingthe super-position principle, dislocation-surface interactionis computed numerically. It is shown that one can’t sim-ply neglect these surface effects regardless of the simulationcell size. Although, the interaction between dislocations andsurfaces can be small when they are very far apart, thereis a progressive surface effect that is transmitted to thosedislocations through the dynamics of near-surface disloca-tions. Also, by using the finite element method, one canreduce the computation of dislocation long-range stress tothat of an internal stress, provided an extreme care is takenwith the homogenization of the dislocation stress field overthe RVE.

Acknowledgements

The support of the National Science Foundation undergrant number CMS-9634726 and the partial support ofthe Pacific Northwest National Laboratory is gratefullyacknowledged.

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