Strain gradient effects in surface roughening

Transcription

1 INTITUTE OF PHYIC PUBLIHING MODELLING AND IMULATION IN MATERIAL CIENCE AND ENGINEERING Modellng mul. Mater. c. Eng. 15 ( do: / /15/1/01 tran gradent effects n surface roughenng Ulrk Borg 1 and Norman A Fleck 2 1 Department of Mechancal Engneerng, Techncal Unversty of Denmark, 2800 Kgs. Lyngby, Denmark 2 Engneerng Department, Cambrdge Unversty, Trumpngton treet, Cambrdge CB2 1PZ, UK E-mal: ub@mek.dtu.dk Receved 4 May 2006, n fnal form 28 August 2006 Publshed 6 December 2006 Onlne at stacks.op.org/mme/15/1 Abstract A thn alumnum sheet comprsng of large polycrystals s pulled n unaxal tenson and the resultng surface profle s measured n a scannng electron mcroscope. The surface profle near the gran boundares reveals a local deformaton pattern of wdth of a few mcrometres and s strong evdence for stran gradent effects. Numercal analyses of a bcrystal undergong n-plane tensle deformaton are also studed usng a stran gradent crystal plastcty theory and also by usng a stran gradent plastcty theory for an sotropc sold. Both theores nclude an nternal materal length scale. An nterfacal potental that penalzes the dslocatons n crossng the gran boundary s ncluded n the analyss. The results ndcate that the surface profle s strongly dependent upon the choce of ths potental and on the materal length scale. 1. Introducton urface roughenng of polycrystallne metals durng plastc deformaton s a common phenomenon n metal formng processes and s commonly referred to as the orange peel effect. In addton to creatng cosmetcally undesrable surfaces, the nhomogenetes can provde ntaton stes for fatgue crack ntaton and stran localzaton. The effect derves from the crystallographc lattce msmatch from gran to gran and the resultng ansotropy n dfferent grans. Incompatbltes of deformaton from the nteracton between adjacent grans result n the roughenng of the free surface. The surface roughenng of polycrystallne Al Mg alloys durng tensle deformaton has been expermentally studed by, for example, toudt and Rcker (2002. It was observed that the degree of surface roughenng ncreases lnearly wth both gran sze and level of mposed stran. However, an extrapolaton of the data to vanshng gran sze gves a fnte surface roughenng and ths suggests that an ndependent materal length scale nfluences the surface roughenng n addton to the gran sze. Wlson and Lee (2001 made observatons of alumnum alloy surfaces deformed n tenson, sheet formng and rollng. They observed the /07/ $ IOP Publshng Ltd Prnted n the UK 1

2 2 U Borg and N A Fleck formaton of valleys near gran boundares and concluded that ths was a result of the msmatch n the orentaton of slp systems of adjacent grans. The wdth of the valleys were on the order of 3 µm wth gran szes of 15 µm for a surface rolled to 30% reducton. Numercal smulatons based on crystal plastcty fnte element models have been wdely used to study surface roughenng n polycrystals (see e.g. Zhao et al (2004 usng a threedmensonal conventonal crystal plastcty model. The results ndcate that the dsperson of crystallographc texture from gran to gran plays an mportant role n surface roughenng. Most of such analyses have been based on several crystals, and have not focused on the local deformaton feld near surface gran boundares. The gran boundary obstructs the moton of dslocatons and results n local gradents n plastc stran. In stran gradent plastcty formulatons t s possble to ntroduce an energy term actng at an nterface n order to constran the plastc flow across the nterface. Afants and Wlls (2005 employ the framework of Fleck and Wlls (2004 but ntroduce an nterfacal energy potental to penalze the buld-up of plastc stran at nterfaces. Ths nduces a jump n the hgher order tractons and plastc stran gradents but a contnuty of plastc stran across the nterface. A smlar approach has been used by Gudmundson (2004 that allows for jumps n hgher order tractons and plastc strans and by Evers et al (2004 who uses an enhanced crystal plastcty framework based on varous types of dslocaton denstes allowng for jumps n dslocaton denstes at gran boundares. Furthermore, Gurtn and Needleman (2005 have dscussed the condtons for a contnuous dstrbuton of Burgers vector across a boundary. The numercal analyses n the present study are based on a rate-dependent stran gradent crystal plastcty formulaton as presented by Borg (2006. It sts wthn the framework of the phenomenologcal stran gradent theory by Fleck and Hutchnson (2001. Here, the formulaton s enhanced by ntroducng a potental that penalzes crystallographc slp at nternal gran boundares. The potental s a functon of the slps on ether sde of the gran boundary. Ths formulaton admts jumps n the hgher order tractons and slps but has contnuty of dsplacement across gran boundares. For comparson purposes some of the smulatons are obtaned usng the vscoplastc stran gradent plastcty theory for an sotropc sold based upon Fleck and Hutchnson (2001, as presented n Borg et al (2006. No nterfacal potental s used n that case, although t would be possble to formulate one. 2. Experments Unaxal tenson tests have been conducted on commercally pure alumnum n order to observe the surface profle near gran boundares durng plastc deformaton. The specmen was prepared by prescrbng 4% tensle pre-stran to an annealed, cold rolled sheet of length 120 mm, wdth 20 mm and thckness 1.2 mm. The sheet was then gven a recrystallzaton anneal at 540 C for 15 mn, and the resultng gran structure comprsed through-thckness pancakeshaped grans of average dameter 5 mm. The surface of the specmens was mechancally polshed pror to deformaton. A scannng electron mcroscope (EM was used to nvestgate the surface profle. Fgure 1 dsplays two EM mages of the surface after 10 per cent engneerng stran. On the left mage a gran boundary has been marked to emphasze the jump n surface profle across the gran boundary. The mage on the rght shows a gran boundary and slp lnes on ether sde of t. The grans on each sde clearly have dfferent crystallographc orentatons, and the densty of slp lnes s ncreased on one sde of the gran boundary.

3 tran gradent effects n surface roughenng 3 Fgure 1. EM mages of the surface after 10% stran. 3. Materal model Results wll be based manly on the stran gradent crystal plastcty theory for fnte deformatons presented by Borg (2006. The formulaton fts wthn the framework of the stran gradent theory by Fleck and Hutchnson (2001 wth the basc formulaton equvalent to Gurtn (2002 and reduces to the conventonal crystal plastcty theory by, for example, Perce et al (1983 n the absence of stran gradents. A new nterfacal potental s ntroduced to account for the gran boundary resstance to dslocatons. For comparson purposes we also present results usng a stran gradent plastcty theory for an sotropc sold. The formulaton used s a vscoplastc verson of the theory by Fleck and Hutchnson (2001 as presented by Borg et al (2006. For the sotropc phenomenologcal sold no nterfacal potental s ntroduced. However, varous hgher order boundary condtons are assumed to explore ther effect upon the surface profle. In the followng summares of the two formulatons, the small stran theory s assumed for the sake of smplcty and clarty, although a full fnte deformaton formulaton has been used n all calculatons tran gradent crystal plastcty theory Plastc deformaton by crystallographc slp represents dslocaton moton along specfc slp systems. A slp system (α s descrbed by ts lattce vectors s (α and m (α, where s (α s the slp drecton and m (α s the drecton normal to the slp plane. Upon ntroducng µ (α j = 1 2 (s(α m (α j +s (α j m (α as the classcal chmd orentaton tensor, the overall macroscopc plastc stran rate components can be expressed by the slp γ (α along slp system (α as ɛ p j = γ (α µ (α j. (1 α An effectve slp measure γ e (α dependng on the slp and the slp gradents s ntroduced as γ e (α2 = γ (α2 + (l γ (α, s (α 2 + (l M γ (α, m (α 2 + (l T γ (α, t (α 2, (2 where t (α s the transverse drecton and l, l M and l T are materal length scales ntroduced for dmensonal consstency. Here, (, = / x s the spatal dervatve. In ths work, only plane problems wth n-plane slp systems are consdered, and therefore the contrbuton from the slp gradent n the transverse drecton s neglected n the followng. To account for the gran boundary resstance to dslocatons crossng them, two surface energy potentals φ (α I (γ (α I and φ (α II (γ (α II actng at sde 1 and sde 2 of the gran boundary Ɣ,

4 4 U Borg and N A Fleck respectvely, are ntroduced. Then, the prncple of vrtual power takes the form (where φ (α I denotes φ (α / γ (α, etc V ( σ j δ ɛ j + α I I + Ɣ + ξ (α M m(α δ γ (α, dv ( Q (α τ (α δ γ (α + ( ξ (α s(α α ( ( φ (α I δ γ (α I + φ (α II δ γ (α II dɣ = T δ u + α α r (α δ γ (α d. (3 Here, Q (α s a stress feld work conjugate to the slp rate, and the hgher order stresses ξ (α and ξ (α M are work conjugate to the slp rate gradents along the slp drecton and the drecton normal to the slp plane, respectvely. T s the stress tracton and r (α denotes the hgher order tracton workng on the part of the boundary where the dsplacement rates u and slp rates γ (α are not prescrbed. The Cauchy stress s denoted σ j and τ (α = σ j µ (α j s the chmd stress. It s assumed that the total dsplacement felds are contnuous across the gran boundary but that the presence of a gran boundary energy term causes jumps n the slps. Full constrant along Ɣ wth γ (α = 0 s obtaned by lettng φ (α I and φ (α II tend to nfnty, whereas vanshng surface energy along Ɣ mmcs that dslocatons are free to cross the gran boundary. The strong form of the feld equatons s found by requrng the above relaton to hold for all admssble varatons n u and γ (α. The classcal force balance law and boundary condtons read as σ j,j = 0 T = σ j n j, (4 where n s the surface unt normal n the deformed confguraton. In addton, we have the consstency condton and hgher order boundary condtons Q (α τ (α ξ (α, s(α ξ (α M, m(α = 0 (5 ( r (α = ξ (α s(α + ξ (α M m(α n. (6 The nterface condtons are gven by [ σj N j ] = 0, (7 ( ξ (αi s (αi ( ξ (αii s (αii + ξ (αi M m(αi + ξ (αii M m(αii N = φ (α I (γ (α I, (8 ( N = φ (α II (γ (α II, (9 where the unt normal vector N on Ɣ s drected from gran 1 to gran 2 and [f ] denotes the jump f 1 f 2 across Ɣ. mple expressons are adopted for the gran boundary energy potentals of the form φ (α I φ (α II = 1 2 κ(γ(α I 2, φ (α I = κγ (α I, (10 = 1 2 κ(γ(α II 2, φ (α II = κγ (α II, (11 wth κ beng a materal parameter descrbng the strength of the gran boundary. By use of an effectve stress, denoted τ e (α, that s work-conjugate to the effectve slp rate measure, γ e (α, a power-law creep model s formulated as ( γ e (α τ (α 1/m e = γ 0, (12 g (α

5 tran gradent effects n surface roughenng 5 where γ 0 s a reference slp rate and m s a stran rate hardenng ndex. The slp resstance g (α s assumed to harden from an ntal value τ 0 accordng to ġ (α = β h αβ γ (β e, h αβ = hδ αβ + ph ( 1 δ αβ, (13 where p s the latent hardenng ndex, h s the self-hardenng modulus and δ αβ denotes the Kronecker delta symbol. Here, we use Taylor hardenng (p = 1 and lnear hardenng (h = h 0 = constant tran gradent vscoplastc theory for an sotropc sold A much smpler, phenomenologcal stran gradent formulaton exsts for an sotropc elastc plastc sold. Introduce a nonlocal measure of the effectve plastc stran rate Ė P on the bass of the conventonal effectve plastc stran rate ɛ P and ts gradent through the ncremental relaton Ė P 2 = ɛ P 2 + l 2 ɛ P, ɛp,, (14 where l s a materal length parameter. The drecton of the plastc stran rate s gven by m j = 3 2 j /σ e, where j = σ j 1 3 δ 3 j σ kk denotes the stress devator and σ e = 2 j j s the von Mses effectve stress and σ j s the Cauchy stress tensor. The plastc stran rate components can then be wrtten as a product of ts magntude, ɛ P 2 = 3 ɛp j ɛp j and ts drecton m j as ɛ j P = m j ɛ P. Now follow Fleck and Hutchnson (2001 and assume that the plastc stran gradents contrbute to the nternal work. Then, the prncple of vrtual power takes the form ( σj δ ɛ j EL + Qδ ɛ P + τ δ ɛ, P ( dv = T δ u + rδ ɛ P d (15 V n the deformed confguraton. Here, ɛj EL s the elastc stran, Q s a generalzed effectve stress whch s work-conjugate to the conventonal effectve plastc stran rate, ɛ P, and τ s a hgher order stress whch s work-conjugate to the gradent of the conventonal effectve plastc stran rate, ɛ, P. Integraton by parts of ths work equaton leads to the dentty Q = σ e + τ,. The surface stress tracton s gven by T = σ j n j, and r denotes the hgher order surface tracton. The vscous materal behavour s modelled by a power law for the effectve plastc stran rate ( 1/m Ė P σc = ɛ 0. (16 g(e P Here, m s the stran rate hardenng exponent, ɛ 0 s a reference stran rate and σ c s an effectve stress whch s work-conjugate to the effectve plastc stran rate Ė P such that σ c δė P = Qδ ɛ P + τ δ ɛ P,. (17 The hardenng functon g s taken as g(e P = σ 0 + h 0 E P, (18 where σ 0 s the ntal yeld strength and h 0 s the hardenng modulus.

6 6 U Borg and N A Fleck Fgure 2. The bcrystal consdered n the analyses wth three slp systems. 4. Numercal method and modellng detals A bcrystal undergong n-plane tensle deformaton s studed. As shown n fgure 2 the bcrystal conssts of two grans nfntely long n the x 3 -drecton separated by a gran boundary at x 1 = 0. In the crystal plastcty calculatons we use a planar model wth three slp systems at a relatve orentaton of 60 from one to the next. Ther absolute orentaton s gven by the angles θ I and θ II between slp system three and the x 2 -drecton. The slp system orentaton θ = 0 s close to an FCC materal under plane stran tenson n the 110 drecton whch can be smulated wth three slp systems orented at ±35.3 and 90 from the tensle axs (see e.g. Rce (1987. To mmc dfferent crystal orentatons, results are presented wth varyng values of θ. The appled boundary condtons are gven by u 1 = 0, Ṫ 2 = 0 at x 1 = L 0, u 1 = U, Ṫ 2 = 0 at x 1 = L 0, (19 Ṫ 1 = 0, Ṫ 2 = 0 at x 2 = w 0, Ṫ 1 = 0, Ṫ 2 = 0 at x 2 = w 0, where U s the appled end dsplacement rate. Furthermore, the hgher order tractons r (α vansh on the external boundary. The numercal solutons are obtaned usng the fnte element method. The slp rate ncrements and plastc stran rate ncrements for the crystal formulaton and sotropc formulaton, respectvely, are prmary unknowns on an equal footng wth the dsplacement ncrements. These prmary unknowns are nterpolated wthn each element between nodal ncrements as 2k l u = N N D N, γ (α = M N γ (α N (20 N=1 for the crystal formulaton, and for the sotropc formulaton as 2k l u = N N D N, ɛ P = M N ɛ N P, (21 N=1 where N N and M N are shape functons and k and l are the number of nodes used for the nterpolatons. The elements used for the dsplacements are 8-node quadrlaterals wth quadratc shape functons, and the elements used for nterpolaton of the slp rate ncrements or plastc stran rate ncrements are 4-node quadrlaterals wth lnear shape functons,.e. k = 8 and l = 4. N=1 N=1

7 tran gradent effects n surface roughenng 7 Fgure 3. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and the same materal parameters for l/w 0 = 0.01 at ɛ 1 = 5%. To allow for dscontnuty of slps at the gran boundary, double nodes for the slps must be present at the gran boundary. In ths way the slp on each slp system s uncoupled on the two sdes of the boundary. In contrast, the dsplacements are coupled to ensure contnuty of the total stran feld. 5. Results The materal parameters used for the crystal plastcty calculatons are τ 0 /E = 0.001, h 0 /τ 0 = 10 and γ 0 = s 1. For the analyses usng the sotropc plastcty theory the materal parameters are taken as σ 0 /E = 0.002, h 0 /σ 0 = 20 and ɛ 0 = s 1. Usng these materal data, the stress stran responses n plane stran tenson for ( an sotropc materal and for ( a sngle crystal wth slp systems orented at ±30 and 90 from the tensle axs are dentcal. In both cases the Posson s rato ν = 0.3, and the stran rate hardenng ndex s taken as m = 0.02 n order to mmc rate-ndependent plastcty. The dmensons of the bcrystal are chosen such that the rato w 0 /L 0 s small enough to let us neglect specmen end effects. The appled end dsplacement rate s gven by U/2L 0 = γ 0 or U/2L 0 = ɛ 0 for crystal plastcty and sotropc plastcty, respectvely. It can be argued that a slp gradent n the slp drecton can be related to edge dslocatons, that a slp gradent n the transverse drecton (whch s absent n ths plane problem can be related to screw dslocatons and that a slp gradent n the drecton normal to the slp plane, γ (α, m (α, does not nduce any geometrcal necessary dslocatons and thus t should not contrbute to the hardenng. Therefore we have used l M = 0 n all the calculatons, and the other materal length scale l s denoted as l. Unless otherwse stated, the followng results have been obtaned usng the crystal plastcty formulaton. Frst, a bcrystal s studed such that the two grans have the same orentaton of slp systems (θ I = θ II = 0 and the same materal parameters. The surface profle at an overall stran ɛ 1 = U/2L 0 = 5% s shown n fgure 3 for l/w 0 = 0.01 and n fgure 4 for l/w 0 = 0.1. Due to symmetry the profles are only shown for x 1 0. In each fgure results are dsplayed for fve values of the parameter κ characterzng the gran boundary. For κ = 0 there s a homogeneous deformaton state throughout the specmen, and the surface profle s level.

8 8 U Borg and N A Fleck Fgure 4. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and the same materal parameters for l/w 0 = 0.1 at ɛ 1 = 5. Fgure 5. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and ntal slp resstances τ II 0 /τ I 0 = 2 for l/w 0 = 0.01 at ɛ 1 = 5%. The curve marked as κ s obtaned usng γ (α = 0, smulatng a gran boundary that s mpenetrable to dslocatons. Note that an ncreased value of κ constrans the slps, and thereby the plastc strans, leadng to a decrease n the lateral dsplacement of the specmen at the gran boundary. Fgures 3 and 4 llustrate ths: there s an ncrease n the profle at the gran boundary as the value of κ ncreases. Fgures 5 and 6 show surface profles for a bcrystal where the two grans have the same crystallographc orentaton but the ntal slp resstance n gran 1 s half that of gran 2 (τ0 II/τ 0 I = 2, for l/w 0 = 0.01 and l/w 0 = 0.1. Results are presented for fve values of κ at an axal stran of ɛ 1 = 5%. The heght of the surface for κ = 0 (the sold lne decreases monotoncally from the strong gran to the weak gran for both values of l/w 0.Forκ the profles contact a re-entrant corner at the gran boundary, due to the restrcton on slps.

9 tran gradent effects n surface roughenng 9 Fgure 6. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and ntal slp resstances τ II 0 /τ I 0 = 2 for l/w 0 = 0.1 at ɛ 1 = 5%. Fgure 7. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and ntal slp resstances τ II 0 /τ I 0 = 2 for κ = 0at ɛ 1 = 5%. The senstvty of the surface profle to the materal length scale l/w 0 s shown n fgure 7 for κ = 0 and n fgure 8 for κ. These results pertan to the same bcrystal as reported n fgures 5 and 6. For the choce κ = 0 the profle at l/w 0 = and l/w 0 = 0.01 concde, and only the profle curve for l/w 0 = 1 s sgnfcantly dfferent from the others. All the profles wth κ = 0 show a monotonc decrease from the strong gran to the weak gran. In contrast, the value of l/w 0 has a sgnfcant effect upon the profles n fgure 8 at κ.for l/w 0 = the profle s almost constant wthn the strong gran and decreases wthn the weak gran, whereas for l/w 0 = 0.1 the wdth of the deformed specmen frst ncreases wthn the strong gran before t decreases n the weak gran due to the full constrant on slp. A comparson of surface profles usng stran gradent crystal plastcty and sotropc, phenomenologcal stran gradent plastcty theory s gven n fgure 9. The crystal calculatons

10 10 U Borg and N A Fleck Fgure 8. The surface profle near the gran boundary for a bcrystal wth θ I = θ II = 0 and ntal slp resstances τ II 0 /τ I 0 = 2 for κ at ɛ 1 = 5%. Fgure 9. urface profle near the gran boundary usng stran gradent crystal plastcty wth θ I = θ II = 0, ntal slp resstances τ0 II/τ 0 I = 2 and contnuty of slps and stran gradent plastcty for an sotropc materal havng contnuty of plastc strans at the nterface wth σ0 II/σ 0 I = 2. Both at ɛ 1 = 5% for two values of l/w 0. have been carred out for contnuty of slps at the gran boundary (γ (α I = γ (α II and κ = 0,.e. not usng double nodes at the gran boundary. Ths opton s only meanngful when the slp systems n the two grans have the same orentaton, whch s the case for the results presented n fgure 9 (θ I = θ II = 0. The ntal slp resstances are taken as τ0 II/τ 0 I = 2. In the calculatons based on the sotropc formulaton there s contnuty of plastc strans at the gran boundary and the yeld strength of the two grans are taken as σ0 II/σ 0 I = 2. The results show that there s no sgnfcant dfference n the surface profles usng the two dfferent formulatons, although there s a tendency for the crystal profle to have a locally steeper slope than for the sotropc phenomenologcal sold.

11 tran gradent effects n surface roughenng 11 Fgure 10. The surface profle near the gran boundary for a bcrystal wth θ I = 0 and θ II = 15 and the same materal parameters for κ and κ = 0at ɛ 1 = 5% for two values of l/w 0. Fgure 11. The surface profle near the gran boundary for a bcrystal wth θ I = 15 and θ II = 15 and the same materal parameters for κ and κ = 0at ɛ 1 = 5% for two values of l/w 0. Next, a bcrystal wth a lattce msmatch as specfed by θ I = 0 and θ II = 15 s consdered. The surface profle s shown n fgure 10 for the case wth κ = 0 (allowng for jumps n the slp and κ usng two dfferent values of l/w 0.Forκ = 0 there s only a slght dfference n the profle for the two consdered length scales, whereas the value of l/w 0 has a more major nfluence on the profles at κ. The profles are qualtatvely smlar to results already dsplayed n fgures 5 and 6 where the grans had the same orentaton but dfferent ntal slp resstances. Fnally, a bcrystal wth the gran orentatons θ I = 15 and θ II = 15 and the same materal parameters n each gran s studed. Fgure 11 shows the surface profle for the two extreme cases κ = 0 and κ usng two values of l/w 0. As dscussed already for the

12 12 U Borg and N A Fleck profle wth θ I = 0 and θ II = 15, the value of l/w 0 has only a mnor effect upon the surface shape for the choce κ = 0. An enhanced slp actvty s evdent near the nterface for κ = 0 n fgure 11, where the profle drops at the gran boundary creatng a valley. It appears that the nterface behaves as a free surface and dslocaton actvty s enhanced adjacent to t. The profle usng κ creates a tp at the gran boundary. Ths tp becomes thnner and sharper wth decreasng l/w Concludng remarks The surface profle near a gran boundary of an alumnum sheet deformed n tenson has been examned qualtatvely n a scannng electron mcroscope. A local gradent n surface profle s observed wthn a few mcrometres of the gran boundary. Fnte element calculatons are reported here on the surface profle adjacent to a gran boundary n a bcrystal. The smulatons are reported both for a stran gradent crystal plastcty formulaton and for an sotropc, phenomenologcal stran gradent plastcty sold. A penalty to dslocatons on each sde of the gran boundares has been ntroduced wthn the crystal plastcty formulaton. The predcted profles are very senstve to the choce of gran boundary barrer to slp. In order to obtan physcally realstc results for the surface profle, slp moton to the gran boundary must occur. Acknowledgments The work of UB s fnancally supported by the Dansh Techncal Research Councl n a project enttled Modelng Plastcty at the Mcron cale. References Afants K E and Wlls J R 2005 J. Mech. Phys. olds 53, Borg U, Nordson C F, Fleck N A and Tvergaard V 2006 Int. J. olds truct Borg U 2006 Int. J. Plastcty submtted Evers L P, Brekelmans W A M and Geers M G D 2004 Int. J. olds truct Fleck N A and Hutchnson J W 2001 J. Mech. Phys. olds Fleck N A and Wlls J R 2004 J. Mech. Phys. olds Gudmundson P 2004 J. Mech. Phys. olds Gurtn M E 2002 J. Mech. Phys. olds Gurtn M E and Needleman A 2005 J. Mech. Phys. olds Perce D, Asaro R J and Needleman A 1983 Acta Metall Rce J R 1987 Mech. Mater toudt M R and Rcker R E 2002 Metall. Mater. Trans. A Wlson W R D and Lee W 2001 J. Manuf. c. Eng Zhao Z, Radovtzky R and Cutño A 2004 Acta Mater