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REVISED 2/11/06

International Journal for Multiscale Computational Engineering, 4(1)x–x(2006)

Multiscale Total Lagrangian Formulation forModeling Dislocation-Induced Plastic

Deformation inPolycrystalline Materials

Xinwei Zhang, Shafigh Mehraeen, Jiun-Shyan Chen*Civil & Environmental Engineering Department, University of California,Los Angeles (UCLA), 5731G Boelter Hall, Los Angeles, CA 90095, USA

Nasr GhoniemMechanical and Aerospace Engineering, University of California,

Los Angeles, Los Angeles, CA 90095, USA

ABSTRACT

Multiscale mathematical and computational formulation for coupling mesoscale dis-location mechanics and macroscale continuum mechanics for prediction of plasticdeformation in polycrystalline materials is presented. In this development a totalLagrangian multiscale variational formulation for materials subjected to geometricand material nonlinearities is first introduced. By performing scale decompositionof kinematic variables and the corresponding dislocation kinematic variables, severalleading-order equations, including a scale-coupling equation, a mesoscale disloca-tion evolution equation, and a homogenized macroscale equilibrium equation, are ob-tained. By further employing the Orowan relation, a mesoscopic plastic strain is ob-tained from dislocation velocity and its distribution, and a homogenized elastoplas-tic stress-strain relation for macroscale is constructed. The macroscale, mesoscale,and scale-coupling equations are solved interactively at each macroscopic load incre-ment, and information on the two scales is passed through the macroscale integrationpoints. In this multiscale approach the phenomenological hardening rule and flowrule in the classical plasticity theory are avoided, and they are replaced by a homog-enized mesoscale material response characterized by dislocation evolution and theirinteractions.

KEY WORDS

*Address all correspondence to [email protected]

0731-8898/05/$35.00 c© 2006 by Begell House, Inc. 1

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AUTHOR, Please provide a few keywords for your paper

2 ZHANG ET AL

1. INTRODUCTION

Plastic deformation in metals occurs mainlydue to the breaking and reforming of atomicbonds, which allows dislocations to glidethrough crystalline materials. This process, to-gether with dislocation mutual interactions aswell as interactions with other mesoscopic de-fects, results in a heterogeneous plastic defor-mation in crystalline grains. The macroscalephenomenological description of material plas-tic behavior has intrinsic limitations in incorpo-rating mesoscale information and local defectsin the material law.

The classical theory of dislocations has led tosignificant understanding of the behavior of in-dividual crystal defects and elementary interac-tions among them. However, it does not seemto be well suited to the description of the highlyorganized and complex behavior of dislocationpopulations under dynamic conditions. Forthat reason, two new approaches have emergedover the last decade: direct computer simula-tion of dislocation dynamics [1–3] of the col-lective dynamics of dislocations and the self-organization approach [4–6] for describing dis-location structure formation. The computersimulation approach incorporates the behav-ior of individual dislocations, as known fromclassical theory, and the complexity of their in-teractions is explicitly handled by numericalmethods. For example, Kubin and Canova [7]and Kubin et al. [8] divided dislocation linesinto pure edge and screw segments withoutany mixed dislocations, which requires smallsegmentation as well as a large number ofsubsegments during the dislocation evolution.H. M. Zbib and others [9,10] proposed anothermodel based on a straight segment with mixedcharacter. A superdislocation model was alsointroduced to deal with the long-range dislo-cation interactions. Ghoniem et al. [3] andGhoniem and Sun [11] developed the paramet-ric dislocation dynamics approach by utilizing

parametrized curves to describe the dislocationlines that yield enhanced computational accu-racy and efficiency.

The mathematical theory of homogenizationhas been introduced by Bensoussan et al. [12].Multiscale methods have received significantattention in recent years, and many of thesedevelopments have been applied to the mod-eling of heterogeneous materials [13–17]. Thehomogenized Dirichlet projection method byOden et al. [18] and Oden and Zohdi [19] wasdeveloped based on the concept of hierarchi-cal modeling. In this method the mathemat-ical model at the coarsest level is representedby homogenized material properties. This isreferred to as the homogenized problem, andthe exclusion of heterogeneity generally makesthe homogenized problem computationally in-expensive compared to models of finer scale.An error estimate was developed for identi-fying the error between the coarse-scale so-lution using homogenized properties and thefine-scale solution of the heterogeneous mate-rial with microstructures [19]. This approachwas further extended to a hierarchical model-ing of heterogeneous materials where the mostessential scales of the problem can be adap-tively selected in the discretization [18]. Inthe asymptotic expansion approach introducedby Bensoussan et al. [12] the relationship be-tween the coarse-scale and fine-scale solutioncan be derived, and the corresponding homog-enized differential operator and homogenizedcoefficients for solving the macroscopic solu-tion can be obtained. This method has beenapplied to multiscale modeling of microstruc-tural evolution in the process of grain growthby Chen and Mehraeen [13,14]. Alternatively, amultigrid method for a periodic heterogeneousmedium has been introduced by Fish and Bel-sky [20] and Fish and Shek [21]. In this ap-proach a multigrid method was employed todevelop a fast iterative solver for differentialequations with oscillatory coefficients. An in-

International Journal for Multiscale Computational Engineering

Kate

Note

as meant?

MULTISCALE TOTAL LAGRANGIAN FORUMLATION 3

tergrid transfer operator was constructed fol-lowing the asymptotic expansion so that theproblem on the auxiliary grid gives rise to ahomogenization problem. Wavelet-based mul-tiscale homogenization has been introducedby Dorobantu and Engquist [22], Chen andMehraeen [13], and Mehraeen and Chen [23] forproblems with fixed microstructures.

For inelastic materials, Fish et al. [24] in-troduced a multiscale analysis of composites,where continuum plasticity in the matrix istaken into consideration. On the basis ofthe two-scale asymptotic expansion of the dis-placement field, a close form expression re-lating eigenstrains in the inclusion to the me-chanical fields was derived, and an averag-ing theorem was employed for solving theoverall structural response. This work con-sidered small strain in the macroscopic struc-ture. A multiscale approach presented by Smitet al. [25] considered a large deformation ef-fect, in which the micro-macro structural re-lations in large deformations were obtained,and the method was implemented in a mul-tilevel finite element framework. In the con-text of nonlinear analysis a homogenization-based multiscale method was also proposed byCricri and Luciano [26], where a macroscopicfailure surface is defined with microstructuretaken into consideration. Recently, a homog-enization method was proposed by Takano etal. [27,28], in which the scale-coupling func-tion between macroscopic and mesoscopic vari-ables was defined in a general, nonlinear form.Although the aforementioned approaches in-troduced multiscale formulation for model-ing inelastic behavior of heterogeneous materi-als, the phenomenological laws have been em-ployed.

This work aims to develop a multiscale vari-ational and computational formulation, whichallows the modeling of plastic deformation ina continuum by incorporating mesoscopic dis-location evolution and the corresponding ho-

mogenized stress-strain relation in a multi-scale framework. In this approach the aver-aged plastic strain in the microstructure is ob-tained by the mesoscopic simulation of dislo-cation evolution, and this information is uti-lized to form the homogenized elastoplastictangent modulus for macroscopic computation.The mesoscopic deformation and dislocationevolution are linked to the macroscopic defor-mation through the integration points in themacroscopic continuum. The outline of thispaper is as follows. Section 2 reviews dis-location mechanics and describes how plas-tic strain and elastoplastic tangent moduluscan be constructed in grain structures. InSection 3, the variational equation for cou-pled continuum mechanics and dislocation me-chanics is presented, and scale decompositionbased on asymptotic expansion is introduced.The multiscale variational equations, includ-ing a scale-coupling equation, the mesoscaledislocation evolution equation, and a homoge-nized macroscale equilibrium equation, are de-rived. The linearization of these multiscaleequations for general nonlinear problems aswell as numerical procedures for multiscalecomputation are also presented in this section.Numerical procedures of the proposed multi-scale formulation are given in Section 4. InSection 5, a numerical example demonstratingmultiscale modeling of the elastoplastic behav-ior of a CS.016 cold-worked carbon steel is pre-sented. Conclusion remarks are outlined in Sec-tion 6.

2. OVERVIEW OF DISLOCATIONMECHANICS

Plastic deformation in polycrystalline materialscan be described microscopically by consider-ing point defects (vacancies interstitials and im-purities), dislocations, grain boundaries, micro-cracks, and voids in grain structures. In thiswork, discrete dislocation evolution is modeled

Volume 4, Number 1, 2006

4 ZHANG ET AL

to characterize the plastic yielding and harden-ing process. The internal length scales withingrain structure, such as Burgers vector and dis-location density, are embedded in the meso-scopic model. In what follows, dislocation me-chanics is introduced as the basic mechanismof plastic deformation via a multiscale homog-enization.

2.1 Discrete Description of DislocationMechanics

The primary mechanisms at the mesoscalelevel controlling plastic deformation in contin-uum are the dislocation formation, evolution,and interactions between dislocation loops[3,10,16,22]. In this work, two-dimensional ge-ometry and the double slip systems with dis-crete edge dislocations are considered in eachgrain as shown in Fig. 1. Frank-Read sources,shown as solid dots in Fig. 1(b), are introducedat discrete nucleation points. Periodic bound-ary conditions are imposed for both dislocationmotions and grain boundary geometry.

Following Ghoniem et al. [3], the virtualwork of a dislocation motion can be expressedas

δEId =

∮

SId

δr · fPKds−∮

SId

δr · B · vds (2.1)

where B is the inverse mobility matrix, v is thedislocation velocity, SI

d is the Ith dislocationloop, and fPK is the Peach-Koehler force, whichis defined as

fPK = [(σ + σd) · b]× ξ (2.2)

where σ is the external applied Cauchy stressfield, σd is the Cauchy stress field induced byother dislocations, b is the Burger’s vector, andξ is the dislocation line vector as shown inFig. 2. In two dimensions the equation of mo-tion (Eq. (2.1)) can be simplified in the follow-ing linear drag relationship:

v = MgfPK (2.3)

where Mg is the mobility of dislocation glide.Here we assume that the motion of disloca-tion is limited to glide along the slip plane, andthe dislocation climbing process related to hightemperature is not considered.

Following the general coordinate convectionfor two interacting dislocations as shown inFig. 2, the stress tensor σd of an arbitrary testdislocation is given by

(a) (b)

FIGURE 1. (a) Microstructure of AL-6XN stainless steel; (b) unit cell/microstructure with grain bound-aries and slip planes utilized in multiscale analysis



FIGURE 2. General coordinate convection for twointeracting dislocations

σd=µb

2π(1−υ)

−rx(3r2x+ r2

y)(r2

x+ r2y)2

rx(r2x− r2

y)(r2

x+ r2y)2

0

rx(r2x− r2

y)(r2

x+ r2y)2

ry(r2x− r2

y)(r2

x+ r2y)2

0

0 0 − 2ry

r2x+ r2

y

(2.4)

where rx and ry are the components of posi-tion vector measured in the coordinate systemshown in Fig. 2, b is the magnitude of Burgersvector, µ is shear modulus, and υ is Poisson’sratio.

Dislocation multiplication is introduced byFrank-Read source points, which generate newdislocations when the resolved shear stress isbeyond the critical nucleation stress over agiven nucleation time period. Annihilation oftwo dislocations on the same slip plane withopposite Burgers vectors occurs when they arewithin critical annihilation distance. In addi-tion, if the resolved shear stress on a disloca-tion is below the friction stress, the dislocationbecomes immobilized. It can be remobilized ifthe resolved shear stress is higher than the fric-tion stress. The plastic deformation of a crys-talline material is largely dependent on the abil-

ity for dislocation to move within a material.Therefore grain structural parameters that im-pede the movement of dislocations, such as theexistence of grain boundaries, results in reduc-ing plastic deformation of the material. In thiswork we consider grain boundaries as the en-ergy barrier of dislocation motions, and trans-mission and absorption of dislocations acrossgrain boundaries are ignored.

2.2 Dislocation and Plastic Deformation

On the basis of Orowan [29], the plastic strainrate produced by gliding dislocations can be ex-pressed as

εp = ρm v bM (2.5)

where ρm is the mobile dislocation density, v isthe average dislocation velocity, b is magnitudeof Burgers vector, and M is the Schmid tensorof slip system, defined as

M = 12(m⊗ n + n⊗m) (2.6)

where m is slip direction and n is slip plane nor-mal direction. On the basis of the discrete dis-location mechanics and Orowan equation, theplastic strain rate in a double slip system, asshown in Fig. 1(b), can be expressed as

εp = 12

∑

i

ρ(i)m v(i)b(m(i)⊗ n(i) + n(i)⊗m(i)) (2.7)

where i is a given slip system. Further consid-ering strain rate ε, decomposition into elasticstrain rate εe and plastic strain rate εp:

ε = εe + εp (2.8)

A general stress-strain rate relation can be ex-pressed by

σ = C : εe (2.9)


6 ZHANG ET AL

where ε is the Cauchy stress rate and C is theelastic modulus tensor. Substituting Eqs. (2.7)and (2.8) into Eq. (2.9), we have

σ=C :[ε− 1

2ρm vb(m⊗ n + n⊗m)]

(2.10)

3. COUPLED DISLOCATION ANDCONTINUUM MECHANICS

3.1 Variational Equation

Define macroscopic and mesoscopic coordi-nates as shown in Fig. 3 in the undeformed con-figuration as X and Y, respectively, and macro-scopic and mesoscopic coordinates in the de-formed configuration as x and y, respectively.To start, we consider a continuum with a do-main ΩX and boundary ΓX = Γh

X ∪ ΓgX , sub-

jected to an external traction h and body forceb. At any point X in the macroscale domain,there is an associated representative mesoscaleunit cell describing the fine-scale material het-erogeneity and deformation expressed in bothmesoscale coordinate Y and macroscale coor-dinate X. To set up the multiscale variationalequation, we first account for all the macro- andmesoscale features and mechanisms in the con-tinuum domain ΩX . If a virtual material dis-

placement δu and a virtual dislocation motionδr are imposed on this system, we have the fol-lowing variational equation:

∫

ΩX

δui,JPjidΩ

+NUC×ND∑

I=1

∮

SId

δrifPKids−∮

SId

δriBij vjds

−∫

ΩX

δuibidΩ−∫

ΓhX

δuihidΓ = 0 (3.1)

with boundary conditions

PjiNj = hi on ΓhX

ui = gi on ΓgX (3.2)

where ΓhX and Γg

X are natural and essentialboundaries in the undeformed configuration,respectively, (.),J = ∂ (.)/∂Xj , Pji is first Piola-Kirchhoff stress, bi is the body force defined inΩX , hi is the surface force divided by the un-deformed surface area, Nj is the surface nor-mal defined on Γh

X , B is the inverse mobilitymatrix, v is the dislocation velocity, fPK is the

FIGURE 3. Micro- and mesocoordinate systems


Kate

Note

as meant?


Peach-Koehler force, SId is the Ith dislocation

loop, NUC is the total number of mesoscaleunit cells in ΩX , and ND is the total num-ber of dislocations within each mesoscale unitcell. It is understood immediately that sincethis equation is expressed in the macroscale do-main with all macroscale and mesoscale fea-tures considered, incorporating all dislocationloops on this scale is computationally impos-sible. In Section 3.2, a multiscale decompo-sition of macroscale and mesoscale kinematicvariables will be introduced so that certain vari-ables are only solved at their appropriate scalelevel, and a scale-coupling relation will be in-troduced to bridge variables at different scales.

3.2 Multiscale Decomposition

Define a representative mesoscale unit cell inthe mesoscale coordinate with domain ΩY asshown in Fig. 4, and periodicity in the unit cellat any deformation state is assumed. The inter-relation of macroscopic and mesoscopic coordi-nates in the undeformed configuration is iden-tified by

Yi =Xi

λ(3.3)

where the scale ratio λ is a very small number.Similarly, Eq. (3.3) in deformed configurationreads

FIGURE 4. Periodic grain structure with domainΩY and ΓY boundary

yi =xi

λ(3.4)

For notational convenience in this work, anymultiscale variable t is defined in the followingasymptotic form

t = t[0] + λt[1] + O(λ2

)(3.5)

where t is the mesoscopic variable, t[0] is re-ferred to as the macroscopic variable (coarse-scale component of t), and t[1] is defined as thefine-scale component of t.

By employing the asymptotic theory, themesoscopic displacement in the unit cell (withfixed X) measured using mesoscopic coordi-nates can be expressed by the asymptotic ex-pansion in Eq. (3.5) (scaled by 1/λ) to yield

ui (Y) =1λ

ui[0] (X) + ui

[1] (X,Y)

(holding X fixed) (3.6)

where ui (Y) is the mesoscopic displacementand ui

[0] (X) and ui[1] (X,Y) are coarse- and fine-

scale components of displacement, respectively.Here, the coarse-scale component ui

[0] (X) is in-dependent of mesoscale coordinate Y, whereasui

[1] (X,Y) is a perturbed term caused by the het-erogeneity of the grain structure.

Furthermore, we adopt the following coarse-fine scale interrelation

ui[1](X,Y)= αkli(Y)

(∂uk

[0]

∂Xl

)= αkli(Y)F [0]

kl (3.7)

where the coupling function αkli (Y) is inde-pendent of macroscopic coordinates X. Fromthe stress-strain relation, and considering thatthe coarse-scale component of displacementis independent of mesoscopic coordinates Y,Cauchy stress can be expressed as

σij = σ[0]ij + λσ

[1]ij + O

(λ2

)(3.8)


8 ZHANG ET AL

Consequently, the dislocation velocity v anddislocation stress σd induced by the multi-scale stress field can be expressed by the sameasymptotic form:

v = v[0] + λv[1] + O(λ2

)(3.9)

σd = σd[0] + λσd[1] + O(λ2

)(3.10)

where v[0] and v[1] are the dislocation veloci-ties induced by the coarse-scale and fine-scalestress fields, respectively, and σd[0] and σd[1]

are the corresponding coarse- and fine-scaledislocation-induced stresses, respectively. In-troducing Eq. (3.8) into Eq. (2.2) gives rise to

fPK(σ,σd) = f[0]PK(σ[0], σd[0])

+λf[1]PK(σ[1], σd[1]) + O

(λ2

)(3.11)

σ[0]PK(σ[0], σd[0]) = [(σ[0] + σd[0]) · b]× ξ (3.12)

f[1]PK(σ[1],σd[1]) = [(σ[1] + σd[1]) · b]× ξ (3.13)

Subsequently, the energy due to an evolvingdislocation follows the same asymptotic form:

δEId =

∮

SId

δr · fPKds−∮

SId

δr · B · vds

=∮

SId

δr · f[0]PKds−

∮

SId

δr · B · v[0]ds

+λ

∮

SId

δr·f[1]PKds−

∮

SId

δr·B·v[1]ds

+ O

(λ2

)

= δEI[0]d +λδE

I[1]d +O

(λ2

)(3.14)

where

δEI[0]d =

∮

SId

δr·f[0]PKds−

∮

SId

δr·B·v[0]ds (3.15)

δEI[1]d =

∮

SId

δr·f[1]PKds−

∮

SId

δr·B·v[1]ds (3.16)

To obtain multiscale variational equations,Eqs. (3.6) and (3.14) are introduced to the varia-tional Eq. (3.1) to yield

∫

ΩX

(∂δu

[0]i

∂Xj+

∂δu[1]i

∂Yj

)PjidΩ

+NUC×ND∑

I

(δEI[0]d + λδE

I[1]d )

=∫

ΩX

(δu

[0]i + λδu

[1]i

)bidΩ

+∫

ΓhX

(δu

[0]i + λδu

[1]i

)hidΓ (3.17)

in which we used the scaling relation inEq. (3.3) and employed the following chainrule:

∂ (.)∂Xi

=∂ (.)∂Yl

∂Yl

∂Xi=

1λ

∂ (.)∂Yi

(3.18)

Treating δu[0]i , δu

[1]i , and δri as independent

variations and considering the limit of λ → 0,Eq. (3.17) gives rise to three multiscale varia-tional equations:

3.2.1 Scale-Coupling Equation

∫

ΩX

∂δu[1]i

∂YjPji(σd, v, u)dΩ = 0 (3.19)

where the fine-scale variable u[1]i is related to

the coarse-scale variable ∂u[0]k

∂Xlvia the scale-

coupling function αkli (Y), as given in Eq. (3.7).



Note that the first Piola-Kirchhoff stress is afunction of dislocation evolution. This equationis to be used to solve for the coupling functionαkli (Y), where the details will be discussed inthe next section.

3.2.2 Macroscale Equilibrium Equation

∫

ΩX

∂δu[0]i

∂XjPji(σd, v, u)dΩ

=∫

ΩX

δu[0]i bidΩ +

∫

ΓhX

δu[0]i hidΩ (3.20)

where the homogenized stress-strain relation-ship required in this macroscale equation is tobe obtained from the scale-coupling functionsolved in Eq. (3.19) and the mesoscale disloca-tion evolution equation given below.

3.2.3 Mesoscale Dislocation EvolutionEquation

Utilizing the coarse-scale component of thestrain obtained by Eq. (3.20), the third decou-pled equation gives rise to

NUC×ND∑

I=1

δEI[i]d = 0, i = 0, 1 (3.21)

from which dislocation velocity v[i] can besolved. Separating the summation operator inEq. (3.21) into two levels, one over each unitcell, and the other over all unit cells in the do-main ΩX reads

NUC∑

m=1

ND∑

l=1

∮

Sl,md

δr·f[i]PK(σ[i], σd[i])ds−

∮

Sl,md

δr·B·v[i]ds

= 0, i = 0, 1 (3.22)

where NUC is the total number of unit cells inΩX and ND is the total number of dislocationswithin each unit cell. A sufficient condition tosatisfy Eq. (3.22) reads

ND∑

l=1

∮

Sld

δr·f[i]PK(σ[i],σd[i])ds−

∮

Sld

δr·B·v[i]ds

= 0, i = 0, 1 (3.23)

Given coarse-scale components of the strainat quadrature points of global structure,Cauchy stress is computed through an elas-tic trial predictor at each unit cell. If elastictrial stress is greater than the nucleation stress,Eq. (3.23) is then solved for dislocation veloc-ity v, which leads to the incremental plasticstrain. Consequently, an elastic trial stress iscorrected by means of calculated plastic strain.It is important to note that since the drivingforce of dislocation evolution is a linear func-tion of stresses σ and σd, we have f[0]

PK+λf[1]PK =

[(σ[0]+λσ[0])+(σd[0]+λσd[0])·b]×ξ = fPK . Thusit is straightforward to show that the disloca-tion velocity can be solved from the combined

equationND∑l=1

δEl[0]d + λδE

l[1]d =

ND∑l=1

δEld = 0, i.e.,

ND∑

l=1

∮

Sld

δr·fPK(σ, σd)ds−∮

Sld

δr·B·vds

= 0 (3.24)

This implies that dislocation velocity can betreated strictly as a mesoscale variable. Al-though the dislocation evolution equation nowcan be expressed at the mesoscale level, thestress fields involved in the evolution equationneed to be calculated from coarse- and fine-scale displacements that are to be solved fromthe macroscale equilibrium equation and scale-coupling equations, respectively.


Kate

Note

correct?

10 ZHANG ET AL

3.3 Linearization of Multiscale Equations

The multiscale Eqs. (3.19) and (3.20) are, ingeneral, nonlinear with respect to ui, and lin-earization of these two equations is needed.We first consider that coarse-scale componentu

[0]i (X) varies linearly with respect to X in the

mesoscale unit cell:

u[0]i (X)≈

(∂u

[0]i

/∂Xj

)Xj =λ

(∂u

[0]i

/∂Xj

)Yj (3.25)

in which ui (X,Y) is measured in macroscopicscale, and we ignore higher-order terms O

(λ2

).

Substituting Eq. (3.25) into the incrementalform of Eq. (3.6) yields

∆ui (Y) = ∆F[0]ij Yj + ∆u

[1]i (X,Y) (3.26)

where ∆F[0]ij = ∂∆ui

[0]/

∂Xj is the incrementalmacroscopic deformation gradient, and incre-mental fine-scale component of displacement∆u

[1]i (X,Y) is obtained from Eq. (3.7) as

∆ui[1] (X,Y) = αkli(Y)∆F

[0]kl (3.27)

3.3.1 Incremental Scale-Coupling Equation

Introducing Eqs. (3.26) and (3.27) into the incre-mental form of Eq. (3.19) yields

∫

ΩX

∂δui[1]

∂Yj(Dijkl+Tijkl)

(∂∆uk

[0]

∂Xl+

∂∆uk[1]

∂Yl

)dΩ

= −∫

ΩX

∂δui[1]

∂YjPjidΩ (3.28)

where

Dijkl = FimC2jmlnFkn, Tijkl = Sjlδik (3.29)

Here C2jlmn is the second elasticity tensor and

Sij is the second Piola-Kirchhoff stress. Further,substituting Eq. (3.27) into Eq. (3.28) and takingthe average over the unit cell gives rise to

∫

ΩX

1AY

[ ∫

ΩY

∂δui[1]

∂Yj(Dijkl+Tijkl)

×(δkmδnl+

∂αmnk(Y)∂Yl

)dΩ

]∂δum

[0]∂Xn

dΩ

= −∫

ΩX

1AY

∫

ΩY

(∂δui

[1]

∂Yj

)PjidΩ

dΩ (3.30)

where AY is the area of the unit cell in theundeformed configuration and ΩY is the do-main of the unit cell. Since the length scale ofthe mesoscale unit cell is considerably smallerthan the macroscopic length scale, the residual(R.H.S.) in the mesoscopic Eq. (3.30) can be ig-nored, and it yields

∫

ΩY

∂δui[1]

∂Yj(Dijkl + Tijkl)

∂αmnk(Y)∂Yl

dY

= −∫

ΩY

∂δui[1]

∂Yj(Dijmn + Tijmn) dY (3.31)

Note that mesoscale displacement ui must beused to calculate Dijkl and Tijkl in Eq. (3.31).Once the scale-coupling function αmnk is ob-tained from Eq. (3.31), the fine-scale incre-mental displacement can be calculated usingEq. (3.27).

3.3.2 Incremental Macroscale EquilibriumEquation

Substituting Eqs. (3.26) and (3.27) into the in-cremental equation of Eq. (3.20) and perform-ing an average process over the unit cell, the



following incremental equilibrium equation isobtained:

∫

ΩX

∂δui[0]

∂Xj

1

AY

∫

ΩY

(Dijkl+Tijkl)

×(δkmδnl+

∂αmnk(Y)∂Yl

)dY

∂∆um

[0]

∂XndΩ

=∫

ΩX

δui[0]bidΩ+

∫

ΓhX

δui[0]hidΓ

−∫

ΩX

(∂δui

[0]

∂Xj

)P

[0]ji dΩ (3.32)

Recall that the inner integral in Eq. (3.32) iscalculated over the unit cell, which leads tohomogenized geometric and material responsetensors in each unit cell. Recasting the inner in-tegral in Eq. (3.32) leads to homogenized mate-rial and geometric response tensors as follows:

Dijmn=1

AY

∫

ΩY

Dijkl

×(

δkmδnl +∂αmnk (Y)

∂Yl

)dΩ (3.33)

Tijmn=1

AY

∫

ΩY

Tijkl

×(

δkmδnl +∂αmnk (Y)

∂Yl

)dΩ (3.34)

where Dijmn is the homogenized material re-sponse tensor and Tijmn is the homogenizedgeometric response tensor. Equation (3.32) isrewritten as

∫

ΩX

∂δu[0]i

∂XjA

[0]ijmn

∂∆um[0]

∂XndΩ =

∫

ΩX

δui[0]bidΩ

+∫

ΓhX

δu[0]i hidΓ−

∫

ΩX

(∂δui

[0]

∂Xj

)P

[0]ji dΩ (3.35)

A[0]ijmn = Dijmn + Tijmn (3.36)

At each macroscopic iteration, homogenizedtensor A

[0]ijmn is calculated from the unit cell

that corresponds to each integration point ofthe macroscopic structure using the macro-scopic information F

[0]ij . Obtaining the homog-

enized material property at integration pointsrequires solving the scale-coupling function inEq. (3.31). With the homogenized materialproperty, the macroscopic displacement incre-ment is computed using Eq. (3.35), and themacroscopic deformation of structure is up-dated.

It can be easily identified that both the ho-mogenized material response tensor Dijmn andthe homogenized geometric response tensorTijmn do not possess major symmetry. Thesymmetry property in the material and homog-enized geometric response tensors can be re-covered by adopting a consistent homogeniza-tion procedure [30], as discussed in AppendixA, where the macroscopic strain energy den-sity W [0] is first calculated, and the homoge-nized incremental stress-strain relation is thenobtained consistently.

The dislocation evolution Eq. (3.23) is a lin-ear function of v, and thus the linearized equa-tion takes the same form as the total form equa-tion. The multiscale Eqs. (3.24), (3.31), and(3.35) are solved interactively to yield the com-plete macroscopic and mesoscopic solutions.

4. NUMERICAL PROCEDURES

In this multiscale analysis the macroscale equi-librium equation is discretized by a coarsemesh for computational efficiency. At eachintegration point of the macroscopic mesh,the scale-coupling function is solved on theunit cell using the scale-coupling equation andmacroscopic stress and strain information. Themesoscopic deformation and stress are then ob-tained using the scale-coupling function, and


12 ZHANG ET AL

this information is employed to calculate dislo-cation motion using the mesoscopic dislocationevolution equation. The computed dislocationvelocities and their distributions are used to ob-tain plastic deformation of the unit cell. Finally,the homogenized elastoplastic stress-strain re-lation is returned to the macroscopic equilib-rium equation for solving macroscopic defor-mation at the next load step. These computa-tional procedures are illustrated in Fig. 5.

5. NUMERICAL EXAMPLE

The elastoplastic behavior of a CS.016 cold-worked carbon (0.2% C) steel rod subjectedto uniaxial tension is analyzed by the pro-posed multiscale method, and the results arecompared with experimental data reported byBrooks [17]. This model problem is shown inFig. 6, in which the rod is subjected to an uniax-ial tension of 400 MPa. The mesoscopic para-

FIGURE 5. Computational algorithm of multiscale analysis

FIGURE 6. CS.016 cold-worked carbon steel rod subjected to an uniaxial tension



meters are given in Table 1. The grain structuresof the unit cell (size = 100 µm × 100 µm) aretaken from Mehraeen and Chen [23], as shownFig. 3, and the Young’s modulus of each grain isassigned randomly in the range of E0±(20%)E0

such that the averaged modulus is consistentwith the macroscopic Young’s modulus E0 givein Table 1.

The macroscopic tension increment of20 MPa is used. At each macroscopic loadstep, macroscopic strains and stresses based onelastic trial at the integration points are com-puted. For a given macroscopic incrementalstrain, mesoscopic incremental displacement isobtained through the scale-coupling functionby Eq. (3.31), and consequently, the mesoscopicstrain increment is computed. The dislocationcomputations in the grains are invoked if thetrial stress is greater than the dislocation nu-cleation stress. On the basis of the dislocationcalculations given in Section 4, the mesoscopicplastic strain increment is computed, and theelastoplastic tangent operator is returned tothe macroscopic integration points for macro-scopic equilibrium calculation. Figure 7 showsthe evolution of dislocations with increasingstrain in the grain structure. The dislocationpileups increase with increasing deformation.The predicted axial stress-strain response, asshown in Fig. 8, agrees well with experimentalobservations [17].

6. CONCLUSIONS

This work aims to develop a dislocation-based,multiscale formulation for modeling disloca-tion formation and evolution in grain structuresand for characterizing how mesoscale disloca-tions are related to plastic deformation in thecontinuum. This paper addresses the follow-ing issues: (i) multiscale variational equationsbetween continuum scale and mesoscale forgeneral, large deformation conditions; (ii) cou-

pling of mesoscale dislocations and continuumplastic deformation; (iii) numerical proceduresfor multiscale modeling of plastic deformation;and (iv) dislocation-based, multiscale analysisof plastic deformation of a CS.016 cold-workedcarbon steel without using phenomenologicalplasticity flow and hardening rules.

An asymptotic, expansion-based, homoge-nization method incorporated with mesoscaledislocation mechanics has been proposed toprovide a systematic approach for multiscalemodeling of plastic deformation in polycrys-talline materials. To yield the multiscale vari-ational equations and scale-coupling relation,a total Lagrangian formation and its lineariza-tion have been adopted. By introducing theasymptotic expansion of the material displace-ment in the test and trial functions in thevariational equation and by averaging strainenergy density of the unit cell, the result-ing leading-order equations yield the scale-coupling relation, coarse-scale homogenizedequilibrium equation, and fine-scale disloca-tion evolution equation. At every integrationpoint in the macroscopic structure, the point-wise macroscale strain is passed onto the cor-responding mesoscale unit cell. Within themesoscale unit cell, the scale-coupling functionwas first solved numerically, and mesoscaleincremental displacement, strain, and trialstress were computed using the scale-couplingfunction and macroscale strain information.The mesoscale dislocation evolution in thegrain structures was then simulated, and themesoscale incremental plastic strain was ob-tained from the dislocation velocity and dis-tribution. By incorporating the scale-couplingfunction and the mesoscale plastic strain, ahomogenized elastoplastic stress-strain relationwas obtained and returned to the macroscaleintegration points. Using the elastoplastic tan-gent moduli at integration points in the contin-uum domain, the macroscopic discrete equilib-rium equation was constructed and solved. In


14 ZHANG ET AL

TABLE 1. Material properties of AL-6XN stainless-steel and microstructure parameters

Young’s modules (E0) 207 GPaPoisson’s ratio 0.30Dislocation mobility 1.1× 103

(Pa−1s−1

)Dislocation nucleation time 3.0× 10−7sDislocation nucleation stress 150 MPa

FIGURE 7. Evolution of dislocations at different stain stages



FIGURE 8. Comparison of macroscopic stress-strain curve obtained from proposed multiscale ap-proach with the experimental results of Brooks [17]

this multiscale approach, the phenomenologi-cal hardening rule and the flow rule in classi-cal plasticity theory are avoided, and they arereplaced by a homogenized mesoscale materialresponse characterized by dislocation evolutionand their interactions.

In the numerical example, the elastoplas-tic deformation of a CS.016 cold-worked car-bon steel has been demonstrated. The macro-scopic stress-strain response has been obtainedby means of the proposed multiscale formu-lation and numerical algorithms. This stress-strain relation depicted a good agreement withthe experimental data.

REFERENCES

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3. Ghoniem, N. M., Tong, S.-H., and Sun, L. Z.,

Parametric dislocation dynamics: A thermody-namics-based approach to investigations ofmesoscopic plastic deformation, Phys. Rev. B61:913–927, 2000.

4. Gregor, V., and Kratochvl, J., Self-organizationapproach to cyclic microplasticity; a model of apersistent slip band, Int. J. Plasticity 14:159–172,1998.

5. Malygin, G. A., Self organization of disloca-tions and localization of strain in plastically de-formed crystals, Solid State Phys. 37:3–42, 1995.

6. Zaiser, M., Avlonitis, M., and Aifantis, E. C.,Stochastic and deterministic aspects of strainlocalization during cyclic plastic deformation.Acta Mater. 46:4143–4151, 1998.

7. Kubin, L. P., and Canova, G., The modelingof dislocation patterns, Scr. Metall. 27:957–962,1992.

8. Kubin, L. P., Canova, G., Condat, M., Devin-cre, B., Pontikis, V., and Brechet, Y., Dislocationmicrostructures and plastic flow a 3D simula-tion, Solid State Phenomena 23/24:455–472, 1992.

9. Rhee, M., Zbib, H. M., and Hirth J. P., Modelsfor long/short range interactions in 3D dislo-cation simulation. Model. Simul. Mater. Sci. Eng.6:467–492, 1998.

10. Zbib, H. M., Rhee, M., and Hirth, J. P., On plas-tic deformation and the dynamics of 3D dislo-cations, Int. J. Mech. Sci. 40:113–127, 1998.

11. Ghoniem, N. M., and Sun, L. Z., Fast summethod for the elastic field of 3-D dislocationensembles, Phys. Rev. B 60:128–140, 1999.

12. Bensoussan, A., Lions, J. L., and Papanicolau,G., Asymptotic Analysis for Periodic Structures,North Holland, Amsterdam, 1978.

13. Chen, J. S., and Mehraeen, S., Multi-scale mod-eling of heterogeneous materials with fixed andevolving microstructures, Model. Simul. Mater.Sci. Eng. 13:95–121, 2005.

14. Chen, J. S., and Mehraeen, S., Variationally con-sistent multi-scale modeling and homogeniza-tion of stressed grain growth, Comput. MethodsAppl. Mech. Eng. 193:1825–1848, 2004.

15. Chen, J. S., Kotta, V., Lu, H., Wang, D.,Moldovan, D., and Wolf, D., A variationalformulation and a double-grid method formeso-scale modeling of stressed grain growth


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16 ZHANG ET AL

in polycrystalline materials, Comput. MethodsAppl. Mech. Eng. 193:1277–1303, 2004.

16. Amodeo, R. J., and Ghoniem, N. M., Dislo-cation dynamics I: a proposed methodologyfor deformation micromechanics, Phys. Rev.41:6958–6967, 1990.

17. Brooks, C. R., Heat Treatment, Structure, andProperties of Nonferrous Alloys, American Soci-ety for Metals, 1982.

18. Oden, J. T., Vemaganti, K., and Moes, N., Hi-erarchical modeling of heterogeneous solids,Comput. Methods Appl. Mech. Eng. 172:3–25,1999.

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20. Fish, J., and Belsky, V., Multigrid methodfor periodic heterogeneous media, 1—Convergence studies for one-dimensionalmedia, Comput. Methods Appl. Mech. Eng.126:1–16, 1995.

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23. Mehraeen, S., and Chen, J. S., Wavelet-basedmulti-scale projection method in homogeniza-tion of heterogeneous media, finalist of MeloshMedal Competition, FE Anal. Des. 40:1665–1679, 2004.

24. Fish, J., Shek, K., Pandheeradi, M., and Shep-herd, M. S., Computational plasticity for com-posite structures based on mathematical ho-mogenization: theory and practice, Comput.Methods Appl. Mech. Eng. 148:53–73, 1997.

25. Smit, R. J. M., Brekelmans, W. A. M., and Mei-jer, H. E. H., Prediction of the mechanical be-havior of nonlinear heterogeneous systems bymulti-level finite element modeling, Comput.Methods Appl. Mech. Eng. 155:181–192, 1998.

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Nishiyabu, K., Microstructure-based deep-drawing simulation of knitted fabric reinforcedthermoplastics by homogenization theory, Int.J. Solids Sruct. 38:6333–6356, 2001.

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APPENDIX A: CONSISTENTHOMOGENIZATION OF THEMACROSCALE EQUILIBRIUM EQUATION

In this section we follow the consistent homog-enization method proposed by Bensoussan etal. [12]. First, the macroscale strain energy den-sity function is defined by

W [0] =1

AY

∫

ΩY

W (F) dΩ (A1)

Fij = ∂ui/∂Xj (A2)

where f and W are the mesoscopic deformationgradient and strain energy density in the unitcell, respectively, and W [0] is the pointwisemacroscopic strain energy density. Conse-


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quently, it is demonstrated that [12] the macro-scopic first PK stress tensor is obtained by

P[0]ij =

∂W [0]

∂F[0]ji

=1

AY

∫

Y

∂W

∂F[0]ji

dΩ

=1

AY

∫

Y

PnmKnmijdΩ (A3)

Pnm =∂W

∂Fmn(A4)

where

Knmij = deltajmδin + ηjimn,

ηklin =∂αkli (Y)

∂Yn(A5)

Recasting Eq. (A3) in the incremental formyields

∆P[0]ij = A

[0]ijkl∆F

[0]lk (A6)

A[0]ijkl = 1

AY

∫ΩY

ApqrsKpqijKrskldY ,

Apqrs =∂2W

∂Fqp∂Fsr= C2

pntrFqnFst + Sprδsq (A7)

Recall that Eq. (A6) demonstrates the ma-jor symmetry property of the homogenizedfirst elasticity tensor A

[0]ijkl. Furthermore, un-

like other asymptotic, expansion-based meth-ods [12, 20, 24, 25, 28], the second elasticity ten-sor in above proposed method also possessesa major symmetry property [12]. On the basisof the incremental form of macroscopic equilib-rium Eq. (3.35) and the above consistent ho-mogenization procedures, the macroscopic in-cremental equation is obtained:

∫

ΩX

∂du[0]i

∂XjA

[0]ijmn

∂∆u[0]m

∂XndΩ =

∫

ΩX

du[0]i bidΩ

+∫

ΓhX

du[0]i hidΓ−

∫

ΩX

∂du[0]i

∂YjP

[0]ij dΩ (A8)


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