Materials Science and Engineering A 387–389 (2004) 67–71

Energetic approach to subgrain formation

Jan Kratochvıl a,∗, Radan Sedlácekb

a Department of Physics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Prague, Czech Republicb Technische Universität München, Fakultät, für Maschinenwesen, Lehrstuhl für Werkstoffkunde und Werkstoffmechanik,

Boltzmannstr. 15, 85747 Garching, Germany

Received 26 August 2003; received in revised form 15 December 2003

Abstract

A variational formulation of the incrementally linear rigid-plastic approximation to crystal plasticity is set up. It is shown that the subgrainformation is a consequence of a tendency to minimize the sum of the incremental dissipated energy and the incremental energy that is relatedto the increase of flow stress. A physically motivated nonlocal formulation of the variational problem leads to a finite size of the subgrainsformed. A crystal deformed by symmetric double slip is considered as an example.© 2004 Elsevier B.V. All rights reserved.

Keywords:Subgrain structure; Geometrically necessary boundaries; Nonlocal plasticity; Variational formulation

1. Introduction

Dislocations stored in crystalline materials in the courseof plastic deformation are distributed into fairly regular mi-crostructures with characteristics that depend on the defor-mation process and material parameters[1]. Transmissionelectron microscopy (TEM) observations have shown thata structural description based on individual dislocations ordislocations arranged in tangles and cell walls is relevant atlow strains and lower temperatures. With increasing strainand/or temperature the characteristic structural features aredislocations assembled into boundaries with a net misorien-tations across them.

We will use the term subgrain structures to address themisoriented microstructures. This term has been establishedin the field of creep where stresses necessary for plastic de-formation are relatively low due to high temperature. Thesubgrain boundaries that separate the misoriented regionscalled subgrains come close to ideal small angle dislocationboundaries. The deviation from the ideal structure increaseswith decreasing temperature. In cold working, the structureis so disturbed that one usually does not use the term sub-grain boundary but speaks of cell boundaries, distinguish-ing between various kinds, such as geometrically necessary

∗ Corresponding author. Tel.:+420 2435 4701; fax:+420 3333 3226.E-mail address:krato@fsv.cvut.cz (J. Kratochvıl).

and incidental boundaries[2]. Anyway, the geometricallynecessary and subgrain boundaries are similar in that theyare associated with misorientations and that they constitutedislocation networks, that evolve during deformation fromthick walls with high dipole content to thin sheets consistingmostly of geometrically necessary dislocations.

Based on the earlier proposals[3,4], the subgrain forma-tion has been explained in the continuum-mechanics frame-work as an instability of a homogeneous deformation in[5,6]as follows. Trying to keep work hardening at the lowest pos-sible level, a plastically deformed crystal tends to locallydecrease the number of active slip systems compared to thenumber required by the Taylor criterion for compatible de-formation. To satisfy the imposed loading conditions andthe compatibility of the overall deformation anyway, the in-compatible plastic strain is accompanied by lattice rotations.The tendency to keep locally the lowest possible number ofactive slip systems will decrease or increase depending onwhether the Schmid factors of those slip systems increase ordecrease (geometrical softening and hardening). A detailedanalysis of a rate-independent, rigid-plastic crystal orientedfor symmetric double slip and deformed in plane strain hasbeen performed in the framework of classical (length-scaleindependent) continuum crystal plasticity[5,6]. Here we fo-cus on the variational formulation and solution of the sameproblem. The variational formulation provides a comple-mentary insight into the problem. As distinct form the classi-cal stability analysis[5,6] where only continuously altering

0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.msea.2004.01.069

68 J. Kratochv´ıl, R. Sedl´acek / Materials Science and Engineering A 387–389 (2004) 67–71

misorientations could be treated, the variational formulationenables to consider volume elements of common misorienta-tion surrounded by well-defined dislocation boundaries[7].Moreover, the variational approach provides a possibility toincorporate additional mechanisms that lead to a finite sizeof the subgrains formed. This is achieved by adding gradi-ent terms to the standard form of hardening, which leads toa finite subgrain size. Motivated by the theory of disloca-tions, the plastic strain gradients were introduced in[8] andused in the shear band theory[9,10] to arrange for a finitewidth of the shear bands, cf. also the recent review[11]. Analternative approach to explain the finite subgrain size wasput forward in[12].

2. The model

In order to substantiate the proposed approach we look atthe simple case of a crystal oriented for symmetric doubleslip. Here, the crystal can choose between two systems ofthe same Schmid factor. The preference of one of the sys-tems leads to crystal rotation around a common axis. It hasbeen reported by Pantleon and Hansen[13] that the exper-imentally observed misorientations can be explained on thebasis of rotations around two perpendicular axes. This wouldmean that the formation of real misoriented regions can beunderstood as a superposition of two compatible symmetricdouble slip subgranular deformations of the type consideredhere.

2.1. Rigid-plastic model

An incremental deformation of a pre-deformed metalcrystal in a pre-stressed state (i.e. in the presence of an ap-plied stress) is considered. This allows to use the linear the-ory of incremental deformation[14] even for the analysis ofadvanced stages of deformation. We adopt the rigid-plasticapproximation, which turns out to be the optimal viewpointrevealing the essential features of the subgrain formation.

The incremental deformation is represented by an in-cremental material displacementu(x). The displacementgradient∇u can be decomposed into a plastic part, which iscaused by the flow of dislocations, and an elastic part. In therigid-plastic approximation the elastic part of∇u reducesto a rigid rotation of the crystal latticeω∗. The small-straindecomposition reads∇u = β + ω∗, where the plastic dis-tortion β results from incremental slip along the consideredtwo slip systems. In the pre-stressed state the slip systemsare characterized by the slip planes with unit normalsn1 =(− cosφ, sinφ), n2 = ( cosφ, sinφ) and slip directionss1 = ( sinφ, cosφ), s2 = (− sinφ, cosφ); φ is the orienta-tion angle of the slip planes with respect to the symmetryaxis that coincides with the coordinate axisy. Then, theplastic distortionβ caused by the incremental slipsγ1, γ2can be expressed asβ = γ1s1 ⊗ n1 +γ2s2 ⊗ n2. The compo-nents of plastic strain incrementε = (∇u+ (∇u)rmT)/2 are

εxx = −εyy = −γ1 + γ2

2sin 2φ, εxy = γ2 − γ1

2cos 2φ.

(1)

The lattice rotationω∗ is related to the material,ω = ωxy =(∂yux−∂xuy)/2, and plastic,ωp = (γ1+γ2)/2, rotations by

ω∗ = ω − ωp. (2)

In general, the elastic and plastic parts of the incremen-tal material deformation∇u may be individually incompat-ible. The density of ‘geometrically necessary dislocations’(GNDs) required for the material deformation to be compat-ible is characterized by the tensorα = curlβ = −curlω∗that reveals the relation between the misorientation of theregions of crystal lattice measured byω∗ and the density ofGNDs represented byα. The dislocation density tensorαcan be resolved in the densitiesαi, i = 1,2, of the disloca-tions of the individual slip systems,

α = α1 + α2 =2∑i=1

curlβi =2∑i=1

curl(γisi ⊗ ni). (3)

If GNBs are present in the crystal, the tensorsαi specifiestheir dislocation composition.

2.2. Yield condition

We suppose that in the laboratory reference frame thepre-stress tensor (the applied stress) has only diagonalcomponentsσxx and σyy. We consider an incrementalrate-independent hardening model represented by the yieldcondition

τ1 = 12σ sin 2φ + hγ1 + qγ2 + hs1 · ∇α1 + qs1 · ∇α2,

(4)

τ2 = 12σ sin 2φ + qγ1 + hγ2 + qs2 · ∇α1 + hs2 · ∇α2,

where the shear stress resolved in thei-th slip systemτi isrelated to the stress tensorσ throughτi = si · σni, i = 1,2;σ = σyy − σxx, h andq are the active and latent harden-ing coefficients, respectively;si · ∇ represents the gradientin the direction of the slip vectorsi; αi is the density ofgeometrically necessary dislocations of the systemi; h andq are the non-local active and latent hardening coefficients,respectively.

In general, work hardening is controlled by the interac-tion among dislocations. On the mesoscale, the interactionis non-local and it should be expressed in an integral form.In the present brief analysis only two types of interactionterms of the expansion of such an integral are retained. Theterms inEq. (4) led byh andq represent the standard localhardening that depends on the densities of the so-called sta-tistically distributed dislocations. These densities increasewith increasing slipsγ1 andγ2. On the other hand, the termsled by h andq express that the resistance to dislocation mo-tion depends also on the arrangement of the neighboring

J. Kratochv´ıl, R. Sedl´acek / Materials Science and Engineering A 387–389 (2004) 67–71 69

dislocations. As GNDs clustered in subgrain boundaries areof prime interest here, the gradients of GND densities aretaken as simplified characteristics of that arrangement. Inthe present context, the gradient terms inEq. (4)can be in-terpreted as shear stresses needed to penetrate a cluster ofGNDs of the same (h) or of the other slip system (q). Thelength scale contained inh and q (in contrast toh andq)can be understood as an effective radius of the non-localinteraction.

2.3. Constitutive relations

The stressσ consists of three contributions,

σ = σ + ωσ + σωT + σ′, (5)

where the second contribution (the second and third termon the right-hand side) reflects the fact that the pre-stressfield (the applied stress)σ has been rotated by the materialrotationω. The contributionσ′ is caused by the incrementaldeformation that is coupled toτi by relations

τ1 + τ2 = sin 2φ(σ + σ′yy − σ′

xx),

τ2 − τ1 = 2 cos 2φ(ωpσ + σ′xy).

(6)

Combination ofEqs. (4) and (6)results in the constitutiveequations

σ′xx − σ′

yy = µ1(εxx − εyy)+ µεxy,

σ′xy = −µ(εxx − εyy)

4+ µ2εxy ,

(7)

where the instantaneous tangent-hardening coefficients,µ1,µ, andµ2 are the operators

µ1 = h+ q

sin2 2φ+ (h− q)sin2 φ

sin2 2φ

∂2

∂x2+ (h+ q)cos2 φ

sin2 2φ

∂2

∂y2,

µ = −4h sinφ cosφ

sin 2φ cos 2φ

∂2

∂x∂y, (8)

µ2 = h− q+σcos 2φ

cos2 2φ+ (h+ q)sin2 φ

cos2 2φ

∂2

∂x2

+ (h− q)cos2 φ

cos2 2φ

∂2

∂y2.

The constitutiveEqs. (7)without the non-local effect, i.e.with h = 0, q = 0, represent the standard orthotropic ma-terial already studied by Biot[14]. The termσ cos 2φ ac-counts for the geometrical hardening/softening, i.e. for therotation of the applied stress field with respect to the crystallattice. As it will be seen, the anisotropy of the hardeningbehavior controlled by the coefficientsµ1, µ andµ2 is anecessary prerequisite for the appearance of an instabilityof the homogeneous deformation.

2.4. Variational formulation

The condition of the quasi-static stress equilibrium for apre-stressed sample[14] reduces in the present plane-straincase to

∂xσ′xx + ∂yσ

′xy +σ∂yω = 0,

∂xσ′xy + ∂yσ

′yy +σ∂xω = 0.

(9)

To satisfy the incompressibility of deformation,εpxx + εpyy =

∂xux + ∂yuy = 0, we make use of the stream functionϕ,so thatux = ∂yϕ, uy = −∂xϕ. To get a variational formula-tion of the problem at hand, cf. the variational formulationof crystal plasticity[15], we use the principle of virtual dis-placements (weak formulation of the problem). Supposing,the (virtual) stream function vanishes at the boundary of thedomainΩ considered, the sum of the two left-hand sidesof the equilibriumEq. (9) multiplied with the correspond-ing two components of the virtual displacementδu and in-tegrated overΩ (actually a scalar product) equals zero. Theweak formulation is equivalent to the requirement that anenergy functionalI attains a minimum. A condition for thisis that the first variation of this functional vanishes

δI = 0. (10)

From this condition and the weak formulation, one can eval-uate the functionalI that can be expressed e.g. in the form

I =∫Ω

[h(γ1 + γ2)+ 2qγ1γ2

+ σ cos 2φ(γ1 − γ2)(γ1 − γ2 + ω)

2+ h[(s1 · ∇γ1)

2

+ (s2 · ∇γ2)2] + 2q(s1 · ∇γ1)(s2 · ∇γ2)]dV, (11)

where the kinematically admissible slipsγ1, γ2 and materialrotationω are related to the stream functionϕ by

γ1 = − 2

sin2φ

∂2ϕ

∂x∂y− 1

2cos2φ

(∂2ϕ

∂y2− ∂2ϕ

∂x2

),

γ2 = − 2

sin 2φ

∂2ϕ

∂x∂y+ 1

2 cos 2φ

(∂2ϕ

∂y2− ∂2ϕ

∂x2

), (12)

ω = 1

2

(∂2ϕ

∂y2+ ∂2ϕ

∂x2

).

It should be noted that the appearance of a microstructureis related to non-convexity of the functionalI and, accord-ingly, to a non-existence of a minimizer[15]. Without thenon-local effect, i.e. withh = 0, q = 0, an optimization se-quence applied toI would develop a finer and finer subgrainstructure. However, as will be shown in the next section, thenon-local effect provides both an optimal orientation of thestructure and an optimal size of the subgrains.

70 J. Kratochv´ıl, R. Sedl´acek / Materials Science and Engineering A 387–389 (2004) 67–71

3. Results and discussion

The variational formulation results in a general and dif-ficult optimization problem: which one out of all kinemat-ically admissible deformation modes is the preferred one?The admissible modes are formed by all possible linear com-binations of the stream functionsϕ(x+ ξy). A stream func-tionϕ(x+ξy) represents an arbitrary parallel arrangement ofdeformation bands perpendicular to the directionξ = tanϑ,ϑ being the angle measured with respect to they-axis; anyϑ is kinematically admissible.

To illustrate the proposed variational method, a reducedproblem is analyzed. The set of deformation modes to beoptimized is restricted to subgrain structures arising fromsuperposition of two periodically arranged sets of paralleldeformation bands perpendicular to directionsξ and −ξ,Fig. 1. In each band, and hence, in each subgrain the in-crementsγ1, γ2 andω are supposed to be uniform. In thedirection perpendicular to the bands the incremental slipsaverage to zero. The widthb of the bands, which specifiesthe size of the subgrains, is taken to be the same for all thebands. In this case, the gradient termssi · ∇γi are replacedby the difference in slipsγi in the neighboring subgrains.The non-local terms led byh and q in Eq. (11)are thus re-duced to a contribution to the surface energy of the subgrainboundaries. In the terms of the proposed interpretation of thenon-local effect (Section 2) the contribution is related to thework needed for a dislocation to join an existing subgrainboundary. The optimization is then reduced to the problemto find out from the considered set the subgrain structurewith the preferred widthb and orientations±ξ which mini-mizes the functionalI.

Upon using relations (12) evaluated for the microstructureof Fig. 1 and withsi · ∇γi replaced by the differences inγimultiplied by the subgrain boundary surface, the functional(11) reads

I = Va2[µ2 +σ

b2(ξ4 +m1ξ

2 +m2)

+ 4µ2

b3(ξ4 +mξ2 + 1)

], (13)

Fig. 1. Sketch of the two sets of deformation bands that form the subgrainstructure. The direction of the bands is given byξ (or by the angleϑ),their width is b. The slip directionssi are inclined by an angleφ withrespect to they-axis.

whereV is the volume of the domainΩ, a is the amplitudeof the incremental displacement in the band boundaries, and

m1 = 2(2µ1 − µ2)

µ2 +σ, m2 = µ2 −σ

µ2 +σ,

m = 2(2µ1 − µ2)

µ2,

µ1 = h+ q

sin2 2φ, µ2 = h− q+σ cos 2φ

4cos2 2φ,

µ1 = h+ q

sin2 2φ, µ2 = h− q

4cos2 2φ. (14)

The subgrain structure will form if the difference betweenthe functionalI given by Eq. (13) and the functional forthe macroscopically equivalent homogeneous deformationis negative. In such a case, the inhomogeneous deformationis energetically more favorable than the homogeneous one.As the incremental deformation averages out, the functionalfor the equivalent homogeneous deformation vanishes andthe condition for appearance of the subgrain structure con-sidered isI < 0. FromEq. (13)we get

µ2 +σ

b2(ξ4 +m1ξ

2 +m2)+ 4µ2

b3(ξ4 +mξ2 + 1) < 0.

(15)

This relation delimits the instability region of material pa-rametersh, q, h, q, and loading conditionsσ, φ, underwhich the considered microstructure may occur. This find-ing is an extension of the result for symmetric double slipin plane strain with the non-local effect excluded that wasdiscussed in detail by means of bifurcation maps in[5,6].

The minimum ofI with respect to the widthb and theorientationξ indicates the most favorable structure formed.The conditions∂I/∂b = 0 and∂I/∂(ξ2) = 0 yield a cubicequation forξ2 and an equation forb, where the solutionξ2

of the cubic equation is employed,

0= 2ξ6 + (4m−m1)ξ4 + (m1m− 4m2 + 6)ξ2

+ 3m1 − 2mm2, (16)

b = − 4µ2(2ξ2 +m)

(µ2 +σ)(2ξ2 +m1).

For the orientationξ specified byEq. (16), the misorientationω∗ can be evaluated from Eq. (2). The density of geometri-cally necessary dislocations of the individual slip systems inthe subgrain boundaries follow fromEq. (3). For symmetricdouble slip without the non-local effect, the correspondingfigures of the subgrain boundary dislocation contents werepresented in[5,6]. As seen fromEqs. (16) and (14), thewidth b of the bands degenerates to zero, if the non-localeffect is excluded. This corresponds to the above mentionedtendency to form a finer and finer microstructure that appearin the standard models of hardening[15]. On the other hand,in the present formulation, the subgrain refinement leads to

J. Kratochv´ıl, R. Sedl´acek / Materials Science and Engineering A 387–389 (2004) 67–71 71

the increase of the total energy of subgrain boundaries by in-creasing their total surface. The compromise between thesetwo tendencies leads to the finite size of the subgrains. Wenote that similar argument was used in[12]. The experimen-tally observed decrease of the subgrain size with progress-ing deformation is not reflected inEq. (16). This problemprobably requires a more detailed analysis of the hardeningbehavior,Eq. (4).

4. Conclusions

Variational formulation of the model of subgrain for-mation in a crystal deformed by symmetric double slipin plane strain has been set up. The analysis delimits theregion of material and geometrical parameters in that theconsidered inhomogeneous deformation is energeticallymore favorable than the homogeneous one, that is the re-gion of parameters where an instability of the homogeneousdeformation leading to subgrain formation may occur,Eq. (15). Thanks to the nonlocal (gradient) formulation ofthe hardening law (4), the minimization procedure yieldsa finite subgrain size, i.e. the width of the deformationbandsb, as well as a preferred orientation of the bandsξ,Eq. (16).

Acknowledgements

This research has been supported by grants GACR106/03/0826 and VZ J00021.

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