gradient deformation models-aifantis

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E. C. AifantisLaboratoryof Mechanics and Materials,

Aristotle Universityof Tliessalonii


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strain gradients were thoughtof playing not an important rolein previous treatments. The first problem is concerned withsolid interfaces and the second with adiabatic shear bands.Thesignificance of gradient terms in determining the structure ofthe interface and the shear band profile is illustrated.

2 Nanosca le : Grad ien t E las t i c i ty in D i s l o c a t i o nC o r e s an d CracJs Tips

In this section we consider gradient effects at the nanoscaleby applying a special theoryof gradient elasticity theoryto study

the structure of dislocation cores and crack tips. In particular, ascrew dislocation is considered and a non-singular strain fieldis derived. The elimination of strain singularity is due to thegradient term and the corresponding gradient coefficientis ofthe order of the lattice parameter.The derived nonsingular solutions are used to estimate the extent of dislocation core and thenature of short range dislocation interactions, thus providinginformation which cannotbe obtained by using classical elasticity theory. The results are also used to calculate the elasticenergy which is found to be finite without introducing an arbitrary cut-off radius. Finally, the derived nonsingular solutionsare used to obtain the crack opening displacementfor a modeIII crack by representing it with a continuous distribution ofvirtual dislocations. It is shown that, in contrast to the classicalelasticity prediction, the crack faces close smoothly withthestrain being zero, insteadof infinite, at the tip.

The special form of the gradient elasticity theory used, reads

a,j = keu^ij + 2ney - cV^CXejAy + 2/ie,j) . (2.1)

where (ay, ey) denote the stress and strain tensors respectively,(X., fi) are the Lame constants,V^ denotes the Laplacian and c isthe gradient coefficient w hich,for the present case of crystallinelattice, is taken of the order of the interatomic distance a ; specifically vc = a/4. A derivation of (2.1) on the basis of amixture-type formalism for two superimposed elastic phasesca n be found in Aifantis (1994) (see also Allan and Aifantis,1997) where the aforementioned estimatefor the gradient coefficient c is obtained (see also Altan and Aifantis, 1992).

The solution for the displacement field associated withascrew dislocation was obtained by Gutkin and Aifantis (1996)

on the basis of (2.1) as

w(x, y) = u(x,y) +b sign (y) C" s sin sx

2TT / :- | j ' l V ( l / c ) +

+ s'ds .

(2 .2 )

where

u{x, y)2T T

y narc tan I s ign iy)[l

x 2sign ( x ) ] | .

(2.2a)

is the classical elasticity solution for the same problem. The

constant b denotes the Burgers vector and sign y = -I-1 for y> 0 while sign y = -\ fox y < Q . The situation is depicted inFig. 1 (left sketch) where the smooth transition exhibitedbythe gradient solution is contrasted to the discontinuous profileof the classical solution.

By utilizing the above formula to consider the structure ofdislocation core in the Peierls model (e.g., Cottrell,1953; seealso Aifantis, 1985) we obtain the result

w{x, y ^ 0) = sign {y)\ - [1 - sign {x)]{ 1 - e | / V ^ ) |

which contrasts the corresponding profile obtained for thePeierls-Nabarro dislocation on the basis of classical elasticitytheory which reads

w{x,y^Q) = s i g n (>')( | arctan . (2 .3a )a/2'

The two displacem ent profiles are plottedin Fig. 1 (right sketch)which indicates thatthe gradient model givesa narrower dislocation core than the Peierls model.

The strain components associated withthe gradient elasticity

solufion given by (2.2) read (see Gutkin and Aifantis, 1996)

4n

_ b_

^.-^^K,rye \v c

- K( r^ nfc vVc

(2 .4 )

where r denotes the radial coordinate from the dislocation line,with the first term in the bracket representingthe singular classical elasticity solutionand the second term with the Bessel function Ki representing the gradient elasticity contribution. It isnoted that Ki(r/yc) -> (V c)/ r as r- 0 and, thus, the gradientterm cancels the elastic singularity as the dislocation line isapproached. Plots of the strain components givenby (2. 4) areprovided in the aforementioned publicationby Gutkin and Aifantis (1996). These plots show explicitly thatthe strain singularity disappears at the dislocation line, that a dislocation coremay be defined ai r = r^ = 1.25a, and that the strain achievesextreme values at a distance ~ 12% within this core.

Fo r the gradient elasticity model describedby (2.1) it turnsout that the stress components are identical to those singularones predicted by classical elasticity theory,i.e..

lib sin2i T r

fib cos 9

2iT r(2 .5)

where (r , B) denote polar coordinates,in terms of which (2.4)ca n be rewritten as

b sin BAirr

b cos 6

A-nr

Vc \V c

Vc \V c

(2 .6)

The corresponding gradient elasticity total strain energy definedby the expression

Jv

ca n be calculated explicitlyas

47r '"' 9 " ^ { | ; - '-\Tc\ ^ (2 .8 )

with (ro,R) denoting the limits of integration for the cylindricaldomain at hand, and Ko denoting modified Bessel functionofzero order. It is noted that

HI -In 2Vc


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u u a fr 4

(b) n/b

1 - Gradient model

2 Peierls model

xia

Fig. 1(a) Fig. K b )

Fig. 1 (a) Sctiematic of screw dislocation and displacement fields u( x, y = 0* ) and iv(x, y, 0*) for tlie classical (curvesU) and the gradient (curves


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(a) "yz0 . 5

x/Jc

i - Ji0.5.

-to

\ - 0 . 5 -

1

\10

x/Jc

Fig. 3 Schematic of two screw dislocations of the same sign at a distance d apart (top) and strain distribution (bottom) for (a) d = A'lc an d{b)d = ^. Solid lines (1) correspond to the g radient solution and dottedlines (2) correspond to the classical elasticity solution.

than that resulting from classical elasticity theory. Thus, theelementary processes of nucleation of dislocation dipoles orformation of dislocation pile-ups, which involve interdislocationspacings at the nanoscopic range, are not difficult to occur; thisbeing again in contrast to the predictions of the classical theoryof elasticity.

Finally, we conclude this section by showing how the gradient solutions (2.2) or (2.3) can be used to obtain the crackopening displac emen t for a mode III crack extended from /to / as illustrated in Fig. 4 (top sketch to the left). In the samefigure (top sketch to the right) it is also shown the re presentationof the crack in terms of a distribution of virtual dislocationsn(x) given by the expression

n(x) =

/P(2 .13)

This is, in fact, the same distribution assumed for treating theproblem within a classical elasticity theory setting. It is alsoemployed here as the stress fields in both the classical andgradient theories are the same for the model (2.1) and theassumed boundary conditions (see Aifantis, 1992; Altan andAifantis, 1997). It follows that the crack opening displacementis given by

u{x; y = 0)

4u,vc J-I

w(x s)n(s)ds

'^'ds. (2 .14)

The integral in (2.14) can be calculated explicitly forgiving

u{x; y = Q) =

(2 .15)

X < - / ,

and the corresponding plot is also given in Fig. 4 (bottomsketch). In this figure the crack opening displacement is normalized by u and the spatial coordinate by The index 1is assigned to the gradient solution, while the index 2 is assignedto the classical solution. It is shown that the gradient solution

provides a smooth closure for the crack faces and a vanishingstrain at the tip, in contrast to the classical solution which produces abruptly closing crack faces with an infinite value of thestrain at the tip.

3 M i c r o s c a l e : G r a d i e n t D i s l o c a t i o n D y n a m i c s i n C y -c l ic a n d M o n o t o n i c D e f o r m a t i o n

Diffusion-like terms in the evolution equations for dislocationdensities were first introduced by Aifantis(1981 , 1982, 1983,1984a, 1985) on the basis of an effective momentum balance

for dislocation species, for interpreting the occurrence of instabilities in dislocation ensembles and the associated patterningphenomena (s ee also Bammann and Aifantis, 1 982). A specificmodel for cyclic deformation and the associated persistent slipbands (PSBs) was proposed and analyzed in analogy to theBrusselator model of chemical kinetics by Walgraef and Aifantis (1985), and subsequently adopted and further elaboratedupon (among others) by Glazov et al. (1995). Another modelwas proposed later for monotonic deformation and the associated dislocation clusters by Romanov and Aifantis (1993). Finally, a reaction-diffusion model for dislocation popu lationswas proposed recently by Liosatos et al. (1997) to interpretpatterning phenomena of misfit dislocations in thin films. Thesereaction-diffusion type models for dislocation populations incyclic and monotonic deformation are reviewed in a unifiedmanner in this section and new results are provided especiallyin relation to the microscopic interpretation of the gradient andkinetic coefficients entering into the diffusion and reaction liketerms.

3.1 Cyclic Deform ation The WA Model. The Wal-graef-Aifantis (WA) model for the persistent slip bands (PSBs)occurring during cyclic deformation of copper single crystalsdistinguishes between immobile and mobile dislocations of densities p, an d p , which are assumed to evolve according to thereaction-diffusion equations of the form (e. g., Walgraef andAifantis, 1985, 1988)

dp i

~d td^Pi

- T

'dpm _ 5V-

= A ^ + g ( p , ) - / ( p , , p , ) ,

(3 .1 )

dtD.

dx '+ f(Pi,p..,),

with (Di, D,)denoting the diffusion-like coefficients for theimmobile and mobile dis loc at ions , / (p , ,p,) denoting the interaction between mobile and immobile dislocations, and gipt)

(a)/ / /

(b) 10

/ / /

y^Z-fo

Fig. 4 Schematic of a mode-Ill crack extended from - / to / [sketch (a)top] and its representation by means of an array of screw dislocationswith distribution n{x) [sketch (b) top]. Bottom sketch shows the resulting crack opening displacement and the smooth closure of crackfaces for the gradient elasticity solution (curve 1), in contrast to theparabolic profile (curve 2) of the classical elasticity solution and theassociated strain singularity at the crack tip.

192 / Vol. 121, APRIL 1999 Tran saction s of the ASME

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denot ing the product ion of immobi le d i s loca t ions . The in te ract io n t e r m / ( p , , p ) h as th e f o rm

fiPi, Pm) = bpi - yp,pj, ( 3 . 1 a )

whi le the product ion te rm g(pi) in i t s s imples t fo rm reads

g(pi) = ap" - api. ? ' (p ) = f l>0 . (3.1b)

The coefficient b, w hich p lays the ro le of the b i furca t ion parameter for the problem and depends (e .g . exponent ia l ly ) on thereso lved shear s t ress , represen ts the f ree ing ra te of immobi led is loca t ions for increas ing s t ress , whi le the coeff ic ien ty r e p r esen ts the p inn ing ra te of mobi le d i s loca t ions by immobi le d i -p o l e s . The coefficient a , which for s tab i l i ty purposes needs to bepos i t ive , i s a cons tan t model ing the l inear c rea t ion of immobi led is loca t ion s , whi le pf i s a cons tan t denot in g a re fe rence hom ogeneous so lu t ion for the dens i ty of immobi le d i s loca t ions .

The l inear s tab i l i ty ana lys i s o f (3 .1) ind ica tes tha t a Tur ingins tab i l i ty occurs a t

b = b,= (^

with a c r i t i ca l wavel ength \ , . g iven by

q.

(3.2)

(3.3)

w h e r e q^ i s the c r i t i ca l wave number, c = yp"^, and p',' =blyp'!.

Next , we adopt the fo l lowing es t imates for the parametersappear ing in (3 . 3) (Ai fan t i s , 1987; Walg raef and Aifan t i s ,1 9 8 8 )

D,2ypf

bp,u, iDila ~ li, ( 3 . 3 a )

w h e r e v denotes average d i s loca t ion ve loc i ty, ^ 2 .3 x lu "mdeno tes Bu rger s vector ma gni tud e, and /, sa 1.6 X 10~^m denotes a mean f ree pa th of t rapped d is loca t ions of the order ofd ipole annih i la t ion length . These es t imates g ive a dependencefor the wavelength V of the form K = Al{pi (A 10 - 16)

which , for typ ica l l eve ls o f d i s loca t ion dens i t i es , g ives va luesfor the c r i t i ca l wavelengths very c lose to those observed exper im e n t a l l y.

Another in te res t ing fea ture of the model (3 .1) i s i t s nonl inearbehav ior beyond the b i furca t ion poin t descr ib ing t rans i t ions between d i ffe ren t types of d i s loca t ion pa t te rns as the b i furca t ionparameter (ex te rna l ly imposed s t ress or s t ra in) increases . Thisbehavior can be s tud ied wi th in a two- or th ree-d imens iona lse t t ing (e .g . , Walg raef and Aifan t i s , 19 85 , 198 8) by cons ider ingthe two- t ime sca les near b i furca t ion assoc ia ted wi th the s lowmod es j , (cor r espo ndin g to the e igen value Wj == 0) and thefas t mo des r, (cor r espon ding to the e igenvalue OJ, < 0 ) . B yadiaba t ica l ly e l imina t ing the fas t modes{d,r,i f 0) , one ob ta inst h e f o l l o w i n g " s l o w m o d e d y n a m i c s " e q u a t i o n f or t he s l o wm o d e s a in rea l (as opposed to Four ie r ) space

d,cj = [e -d,(ql + W^y + d,Vl](^ va ua \ ( 3 . 4 )

w h e r e e ~ { b - bc)/b,., V^ = Vj^ -I- V^,, deno tes the Lap laci an,and (d^, dy, v, u) are cons tan ts re la ted expl ic i t ly to the modelparameters prev ious ly def ined . By cons ider ing per turba t ions{R, 4>) of the form R = R + R, 4> = 4>c. '^ ^ around anequi l ibri um ref erenc e state ( /?, < ) defined by

a = 2R o co s {q^x + 4>,>),

Ra = ve/B w , o = co nst ..

(3.5)

w i t h Ro denot ing the pa t te rn ampl i tude and 0 the pa t te rn phase ,i t turns out that

_L. _L

. yd

4(a)

Yd1

(b)

Fig. 5 Dipole exchange mech anism, dipole narrowing, and resulting dif-fusivity due to the shift of dipole's center of gravity.

R^R, d,

+ DsS% < ( 3 . 6 )

w h e r e D\\ and D^ are new cons tan ts . The phase dynamics modeled by (3 .6)2 a l lows for "spontaneous" t rans la t ions of thelayered pa t te rn to occur, as wel l as for symmetry breakings inthe form of "supe rd i s lo ca t ion s" to take p lace and pers i s t ( ford,4> = 0) as in the case of dislocations in an elast ic lat t ice. [Inthis connection i t is noted that , for steady states, (3.6) is of aform similar to that for the displacement f ield of a screw dislocation in classical elast ici ty.]

On e of the mos t c r i t i ca l i s sues in reac tion-d i ffus ion mode lsfor d i s loca t ions and o ther defec ts i s the ava i lab i l i ty of appropriate expressions for the diffusion-l ike coefficients and the ratel ike constants. In this connection, i t is pointed out that the

or ig ina l W A mode l was propo sed wi thout re fe rence to microscopic formulas or der iva t ions per ta in ing to the var ious phe-nomenologica l coeff ic ien ts involved . Spec ia l concern was expres sed, in part ic ular, for the diffusio n-l ike coefficients D , andD as well as for the reaction-l ike termp,p}. In fact, thesethree te rms a re necessary for d i scuss ing d is loca t ion pa t te rn ingand cons t i tu te the new e lem ents of the propos ed W A mo del ascont ras ted , for example , to the ex is t ing Gi lman- type d is loca t ionkine t ics mo dels . [For a rev iew of such type of models see ,for example , Bammahn and Aifan t i s , 1982] . A der iva t ion of amicroscopic express ion for the d i ffus ion coeff ic ien tD, has already been provided (Ai fan t i s , 1986; see a l so Walgraef andAifan t i s , 1988) a long wi th a mec hanica l bas i s for the d i ffus ionl ike te rms . This express ion i s l i s ted in (3 .3a) i above . For theD , t e rm, the es t imate l i s ted in (3 .30) , was ob ta ined by neglec ti n g t h e i n te r a ct i on t e r m / ( p / , p,) and, effectively, by disregard ing the p , -equa t ion . However, no def in i te microscopicmec hani sm wa s sugges ted for jus t i fy ing e i ther the d i ffusionte rm Did^pjdx^ or the reac t ion te rm yp,pj, an issue taken upin the remain ing of th i s subsec t ion .

First , we provi de a just if ic ation for the diffusion ter m D,d^pi Idx^ by ident i fy ing p, with the dens i ty of immobi le d ipo les anda s s u m i n g a " d i p o l e e x c h a n g e " r e a c t i o n m e c h a n i s m a s s h o w nin F ig . 5 ( see a l so Di ffe r t and Essmann, 1993) . I t i s seen tha tth i s reac t ion , i . e . , the in te rac t ion be tween a mobi le d i s loca t ion(p,) a n d a n i m m o b i l e d i p o l e ( p , ) w i t h h e i g h ty,, ( left sketch ofFig . 5 ) may lead to a new mobi le d i s loca t ion and a nar rowerimmobi le d ipo le wi th he ight y f ( r igh t ske tch of F ig . 5 ) . Inaddi t ion to th i s p rocess of d ipo le nar rowing under cons tan tto ta l dens i ty of mobi le and immobi le d i s loca t ions , each d ipoleexchange event l eads to a random sh i f t o f the cen te r o f g rav i tyof the d ipo le (y,i/2; w i t h y,, d e n o t i n g m e a n d i p o l e h e i g h t ) . Acorresponding diffusion coefficient may then be defined for theprocess as D, ~ yl/Stj with t,/ denot ing the average t ime between two success ive events . An es t imate oft,i can be ob ta inedby me an s of the rel atio n 1/f,, i2pivy,i (r,7 ' defines the rate ofthe process ) wi th p, denot ing average mobi le d i s loca t ion density and V average d is loca t ion ve loc i ty. The f ina l express ion forthe diffusion coefficient D, is

D, PmVy,! (3.7)

The ab ove a rgu men t leads to bo th a jus t i f ica t ion of the d i ffusion

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term for the immobile dislocations and an expression for theassoc iated diffusion coefficient.

Next, we provide a microscopic argument for the justificationof the cubic term ypmpl by again identifying pi with the densityof the immobile dipoles and considering the coupling of thedynamics with point defect agglomerates of density p. A kinetic equation for the evolution of the point defect agglomera tesmay initially be assumed of the form

dtOtoPiPm PoPoP. (3.8)

with the first term denoting point defect formation by edgedipole disintegration and the second term denoting agglomerate"clean out" by moving dislocations. It may thus be assumedthat a dipole stabilization occurs, as a result of the competitionbetween the process of agglomerate formation by mutual annihilation of dipole and mobile dislocations and the process ofagglomerate removal by their "sweeping" by moving dislocations. By further assuming that the point defect agglomeratedensity p may be adiabatically eUminating{d,po '^ 0) due tothe short-life time of these defects as compared to time-scalesover which the density of dislocations (p,, p,) evolves, wehave

p = (aj/3^)pi; (3 .9)

a relation which may be used directly in conjunction with theterm PoPiP,,, that now enters into the system of equations descr ibing the (pi, p,)-dynamics, instead of the originally used termpmPf- This combination leads to the substitution

PoPiPm PmPt, (3 .10)

i.e., to the appearance again of the necessary for patterningcubic term in an indirect manner.

The above discussion providing further justification for thediffusion and reaction terms assumed in the original WA modelwill be expanded upon in a forthcoming report by Kalaitzidouet al. (1999) where additional microscopic mechanisms areinvoked to derive explicit relations for the phenomenologicalcoefficients defining the respective reaction-diffusion terms (seealso Zaiser and Aifantis, 1999).

3 . 2 M o n o t o n ic D e f o r m a t i o n T h e R A M o d e l . The Romanov-Aifantis (RA) three-element kinetics model for mono-tonic deform ation is also of a reaction-diffusion type for thedensities of mobile dislocationsp , immobile dislocation dipoles pi , and disclinations p^ (Romanov and Aifantis, 1993).For the present discussion we will neglect the coupling withthe higher order disclination termp ,, and consider the {pm, Pt)-dynamics only, with emphasis on the physical interpretationand schematic representation of the various defect reactionsinvolved. The starting point in such a consideration is the following system of reaction-diffusion equations

^ = Ap - Bpl + yCp^Pi + -D. ^ .

^ = aAp - Kpi + PBpl - Cp^pi + D i ^ ,

(3 .11)

where p is the density of mobile dislocations, p, is the densityof immobile dislocations, D, denotes the diffusion coefficientof mobile dislocations, and D, de notes the diffusion coefficientof immobile dislocations. The coefficientA stands for the rateof mobile d islocation mu ltiplication, the coefficientK measuresthe rate of immobile dislocations disappearance, while thecoef-ficients ( 5 , C) represent the cross-sections of the respectivedefect reactions (i.e., the interactions between mobile-mobiledislocations and mobile dislocations-immobile dipoles, respectively) and they may be further determined from microscopicmodels. Finally, the constants (a ,p, y) denote fractions of the

corresponding parameters{A,B, C) and are generally assumedto obey the relations a < g l , O = < / 0 < l , O s 7 s 2 .

The various reaction mechanisms assumed in (3.11) canschematically be represented in Figs.6{a)-6{d), while Fig. 7is a schematic representation of dislocation reactions which,however, do not change the overall defect densities but thedefect mobilities described by D, and An In fact, one mayconsider a "hopping" dislocation mechanism, as depicted inFigure 8a for dislocation configurations (positive and negative)between n-th and ( + l)-th jumps. The symbol A denotes themagnitude of dislocation jump (dislocation free path length),

while P = PfSx) denotes an associated probability density distribution depicted in Fig. 8(fo). Then, a diffusion equation maybe assumed for the total dislocation densityp = p^ + p~, witha diffusion coefficient given by the expression

D = Ao{v), (3 .12)

where AQ denotes a mean free path and {i,') an average dislocation velocity. If a "cross-slip" dislocation mechanism is assumed, as depicted in Fig. 9(a) for an initial dislocation ofSurges vector b gliding in the jc-direction with velocityv beforeits cross-slip and continuing glide on a new plane lying at adistance h apart, the following expression may be assumed forthe diffusivity Dy in the y-direction

r -Dy = Ayiv), A , = ?7 ;

1 + ^ + i f ^h 2\h

(3 .13)

where ho = p,bl2iT{l - V)(T - Tf)an d h = J^ hP(h)dh. Theconstant p denotes as usual shear modulus andv Poisson'sratio, b denotes the Burgers vector mag nitude. A., denotes anaverage distance between cross-slip events in the glide plane,ho is the distance of dislocation immobilization for dipole formation, and P(h) is the probability for the cross-slip height to beh, while ( T , T /) stand for the resolved shear and friction stressrespectively. The formulae in (3.13) hold for the mobile dislocations p,. For the immobile dislocations p, it turns out that

the same formulae hold withrj replaced by (1 77). A differentand simpler expression for the cross-slip controlled diffusioncoefficient m ay be derived on the basis of Fig.9(b). By considering the exchange of mobile dislocations between slip planeswe can write

dp,

dt

dt

= Uv r [p,(y) - P.,iO)]dy

_ Ylvyl aVml

j,=o 3 ay |j,o(3 .14)

leading to a diffusion coefficient for the dislocation density inthe J-direction of the form

' 3(3 .15)

where 11 denotes the cross-slip probability p er unit glide area,1; denotes average dislocation velocity andyc denotes the maximum cross-slip distance. The calculation implied in (3.14) involves a Taylor expansion up to the second order (the integralwith the first spatial derivative vanishes due to symmetry).

More details on such derivations with appropriate referencesand related microscopic expressions for the reaction-diffusionconstants will be included in the aforementioned report by Kalaitzidou et al. (1999). By considering, for example, an effec-

194 / Vo l. 121, APRIL 1999 Transactions of the ASME



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tive mass balance equation (Aifantis, 1981 ,1987 and Bam mannand Aifantis, 1982) for both positive and negative mobile dislocations

(a)

initial mobiledislocation

V

dislocation multiplication[ dp=A pdt ; dpi=0 ]

(" ) dipole generation

[dp=0 ; dp ,=aApdt]

yi

->(i l )

dipole-dislocation interactioninitial configuration

X - v

bypassing

[dp,-=0 ; dp,=0]

recombination

[dp =0 ; dpi = 0]

( i) + ( ii ) = > d p = A p d t ; d p i = a A p d t X - VL'~*v

(b)

-V

^ . . . - 1 .

V

interaction betweentwo mobile dislocations

,0)

MH)

dipole formation

[ dp = -pBpJ dt; dpi =PBp'dt ]

direct annihilation

[ d p = - ( l - P ) B p J d t ; d p i = 0 ]

(i) + (ii) => dp =-BpMt ; dp, = pBpM t

Fig. 6 8 , b Schematic representation of dislocation reaction mechanisms assumed in the R-D model of (3.11).

(c ) i.

dipole disappearance dueto point defects

(d)

dipole-dislocation interactioninitial configuration

indirect annihilation

[ d p i = - K p i d t ]

dislocation immobilization

[ dp 5*0 ; dpi =0 ]

dipole dtsassociation[ dp ^0 ; dpi *0 ]

change of defect mobilities D ,D |

[dp = 0 ; dpi = 0]

(1) + (ii) + (iii)=> dp = 0 ; dpi = 0

Fig. 7 Schematic representation of dislocation reaction mechanismsfor the R-D model of (3.11) resulting into the same densities but differentdefect mobilities.

dtdp i 1 ,

D = - gdx 2 cp

(3 .16)

with g accounting for dislocation multiplication, c ' denotingthe mean lifetime of mobile dislocations and * =-v"' = vbeing an average constant dislocation velocity, we can derivecorresponding evolution equations for the sump, = p,t + p^ ,and the difference 6, = p^ - p,;,. Adiabatic elimination ofS,(d,6, f 0) then, results into the expressionS, = (v/c)p, which,in conjunction with (3.15), gives

dt g - cp,D, (3 .17)

where D, D^/C = vl, (/, == v/c is the mean free path of mobiledislocations); thus arriving again at the general expressions forthe effective dislocation diffusivity given in (3.12) and (3.13)iabove. Dislocation interactions of a more general form, leadingto effective dislocation diftusivities, may be assumed as a resultof long-range defect forces. Dislocation mobility is then determined not only by the applied stress but also by the long-range stress field of the ensemble of all other dislocations. Thisinfluence may be accounted for by the random effective stress

. , = &

. ^ ' " . .

dipole d isappearance

[ dp=0 ; dpi 5*0 )

( i ) + ( i i) => dp =Y Cp P idt

(ii) + (iii) => dpi = - C p p, dt

* A ; - H *

* A - , * l *A ; -

(a)

PA ( X )

(b)

Fig. 6c, d Schematic representation of dislocation reaction mecha- Fig. 8 (a) Dislocation hopping mechanism and (b) probability for thenisms assumed In the R-D model of (3.11) . associated dislocation jumps.

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0 --

y < - -

(a) (b)

Fig. 9 Dislocation cross-slip mechanisms.

fluctuations ST^n which, in turn, lead to a random fluctuation ofthe dislocation velocity in the glide x-direction. Then, in a framemoving with the average dislocation velocity u, the dislocationmay be considered as performing a random walk with a diffusion coefficient

D = - {OV )rcor, = T Jeff) vL (3 .18)

where {6v^) stands for the amplitude of dislocation velocityfluctuations, (^con,ho n = vrcorr) denotc the corresponding correlation time (dislocation velocity fluctuations) and correlationlength (effective stress fluctuation) andS = (din v/dr^fs)'^ isthe strain rate sensitivity. If dislocation ensembles extendingover several grains are considered with positive and negativedislocations gliding with an average velocityv within each grain

of an average size d and a random orientation, an effectivediffusivity D = (1/4) af(tan^ ip)v may be deduced withv beingan ave rage g lide veloc ity and (tan^ ) deno ting a num ericalfactor resulting from the averaging over all glide planes andgrain orientations.

W e conclude this section with a brief discussion of the steady-state inhomogeneous solutions of (3.11) by assuming also thatthe coefficients (a, D,) are vanishingly small. The resultinggoverning nonlinear differential equation is

d'{bp)+ bp - ibp,f +

T = Toy

(4.2)

Fig. 10 Stationary mobile (1) and immobile (2) periodic dislocation distributions for the R-D model of (3. 11).

where T is the shear stress,y is the space coordinate normal tothe interface and r " stands for the applied shear stress at infinity.The assumed gradient-dependent constitutive equation is of theform

196 / Vo l. 121, APRIL 1999 Transactions of the ASME



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Film (Material 1)

Substrate (Material 2)

Fig. 11(a)

l E - 5

-IB-S

-2E-S

Case 1 fci=0 27N, c-)=0.49Nl

Case 2 (/|=3nm, /2=35^m)

Hl=30GPa, H2=40GPa

^ _ ^ ^ ^ ^ ^ > ^

^

0.02S 0.030heu - a t raJn

Fig . 11(b)

Fig. 11 (a) Schematic of bimaterlal Interface and (b) strain distributionacross the Interface.

T = Kiiy) - CjV^y, (4.3)

where K is, in general, a nonlinear function ofy and c denotesthe gradient coefficient, while here and subsequently the indexi = 1,2 designates material 1 and 2, respectively.

For a linearly elastic bimaterial we have

Kiiy) = fJ-iy, (4.3a)

whe re /Ui and /Li2 deno te th e res pec tive shear mod uli. The n,introduction of (4.3a) and (4.2)2 into (4.3) and solution of theresulting one-dimensional differential equation yields

7 = + C,e^^' + A e " - " ^ , (4 .4)

where the constants C, and ZJ, are to be determined from theconditions assumed at infinity and at the interface. The conditions at infinity are assumed to be of the form

\y\ ydy

0 (4 .5)

while the conditions assumed to hold at the interface are

dy , (9720 7 i = 72 , ^ 1 ^ = :ixi ^ (4.6)

dy dy

where (4.6)i is a standard interfacial condition, while (4.6)2 isan extra interfacial condition associated with the gradient term.With (4.5) and (4.6), (4.4) becomes

1 -

1 -

M 2 - M l

M2

A ' l - ^ 2

VfJ-lC,

V/U1C2 + VM 2fl

'JfJ'tCl

g-W/iA' , . y Sz 0,

Ml VM1C2 + VM2C

pyiiiQici' y < 0,

(4.7)

which gives the distribution ofy across the bimaterial interface.This solution may be contrasted to the one provided by Fleck

and Hutchinson (1993) for the same problem by using their

asymmetric stress gradient theory which for the situation athand is summ arized by the relations r = T ' +2(9m/dy), T"= ny, m = l^fj,(dy/dy) where r denotes the total asymmetricstress, T ' is its symme tric counterpart and m denotes the couplestress with / being an internal material length. The equilibriumequation (4.2) is still valid and, thus, the same form of governing differential equation (with c, in the symmetric stress theorybeing formally set equal to ^,/?/2 in the couple stress theory)holds for both theories. This is also true for the standard boundary and interfacial conditions (4.5) and (4.6)|. However, theextra interfacial condition (4.6)2 is replaced by the requirementof equ al co uples at the interface, i.e., mi = OT2 =>l]lJ,,(dy,/dy) = l\fj,2i.dy2ldy) which, in turn, leads to the following strain distribution for the Fleck-Hutchinson theory

y =1 - ^ ^ " ^ ' /^ ^^ . g -^ ^/ ' . ; y ^ O ,M2 Ml^l + ^2^2

Ml - M2 M iA

/U | / L t | / i - I - ,1^2/2

(4.8)g - . Vi / / j . y ^Q

The strain distribution given by (4.7) and (4.8) are depicted inFig. 11 (bottom sketch) for typical values of the parameters involved as shown in the diagram. It is noted that the values ofc,= 0.27N and c'2 = 0.49^ for the author's symmetric stress theoryare ch osen such th at the inte rnal le ngth s Vci/jUi and VC2/M2 definedby them to coincide with the internal lengths /, = 3fim and I2 =3.5 ^m of the Fleck-Hutchinson asymmetric stress theory for theused values of /ni = 30 GPa andfj,2 = 40 GPa.

Next, it is noted that the distributions (4 .7) and (4.8 ) can bothbe obtained from the same general expression if the problem is

reformulated by introducing an interfacial stressT, , an interfacial strain -y; and an interfacial shea r mod ulus / i; defined b y therelations

r , = r , = T2 = T , 7 , = y , = -yj,

Then, (4.7) and (4.8) can be written as

T, = M/T/- (4-9)

7T

M- M'( 4 . 1 0 )

w i t h M/ = M i M 2 ( V / V ^ + VM2/C2)/ (MIVM2/C2 + M2VM1/C1) f"''the f irst case, and fi, = fi,ij,2(vjJ-iCi + VM2C2)/(MIVM2C2 +lj,2^lfJ,,Ci) w i t h Ci = fj,ilf/2 for the second case .

In fact, (4.10) can be obtained as a special case of the generalsolution of the bimaterial interface problem defined by the nonlinear inelastic constitutive equation (4.3), the conditions (4.5)at infinity, and the general interfacial condition (4.9)2. Thesolution of this problem can be exp ressed in terms of the analytical solution obtained for fluid interfaces by Aifantis and Serrin(1983) as

y J ' dyVf(-y) F(y)2 r

(y) - Ki{yT)]dy ,

(4 .11)

where , (y T) = T " and the values of the strain at the interfacey, and at the two outer boundaries of the bimaterial yT and

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c,=0.01N, C2=0.001NHl=30GPa, nj=40GPatm=10MPa, 0=0.002

4 & -

- 4 & < l .

0.8 T

0.9X

Fig. 12(a)

) i

^ . ^

8E-6 -

4B-tf -

4B-6 -

-IB-6

0.8 x

0.9 T

^ ^^"^/'

1111

Fig. 1 2 ( b )

i

"==-

Fig. 12(c)

Fig. 12 (a) Softening type of interfacial stress-strain graph and corresponding strain distribution for (d ) y , < y and (c) y, > y for differentT"/Tm ratios.

72 should satisfy the "equal area" or "Maxwell 's rule" condition (Aifantis and Serrin, 1983)

[Kiiy) - T]dy = 0 , (4 .12)for a nonconve x or softening type graph r =K, (7 ) . For a l inearelastic bimaterial of the type (4.3o), the solution (4.11) reads

7 = Ji 1 (4.13)

which for an elastic interface, i.e.T, = ij,,yi, is reduced to (4.10 )discussed above.

For an inelastic interface of the form, for example,

r , = T, - a{y, - yj^ (4 .14)

where T , denotes the maximum (taken as 10 MP a) of the r -7 graph depicted in Fig. 12 (top sketch) and a is a numericalcoefficient (taken as 0.002), the smooth transition profiles depicted in Fig. 12(b) or the strain localization profiles depictedin Fig. 12 (c ) can be obtained by utilizing the solution (4. 11)and the condition y, < y, for the case (fe) or the condition 7,

> 7m for the case ( c ). Th ese qualitative plots are given fordifferent ratios of T^IT,,, and the values C\ = 0.01 N and c, =0.001 N respectively, while the values ofjii and ^2 are takento be /^i = 30 GPa and //2 = 40 GPa as before.

Finally, it should be pointed out that the solution (4.11) andcondition (4.12) can be used, in particular, for discussing thestructure and strain distribution of thin films on rigid substrates.Some preUminary results are listed below for this case by utilizing atomistic calculations of Rose et al.(1981, 1984) to motivate the Thorn = K{y) softening curve modeling the "homogeneo us' ' portion of the grad ient-depende nt flow stress T = Thom

cV^ 7 defining the co nstitutive respons e of the thin film.Typical graphs for the adhesive energy and the correspondingstress-strain relation are given in Fig. 13, whe re the corre sponding strain profiles for an Al- and Mg-film on a rigid substrateare also provided. These profiles are based on the followingsoftening type graph

T|,cX

(4 .15)

where the atomistic material parametera (not to be confused

y(nm)

14

12

10

8

fa

4

2

Alfilm

10 20

Mgfilm

/

30 40 50stralndO")

Fig, 13(c)

Fig. 13 (a) Schem atic of binding (or adhesion) energie s, (b) corresponding stress-strain diagram , and (c) strain distribution in a thin f i imrigid substrate system.

198 / Vol. 121, APRIL 1999 Tran saction s of the ASM E



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with the symbol used in (4.14) above) and strain parameter yare defined by a = e\T,/Eo and y = (5 - 5o)/\.

In these definitions, e denotes the usual natural logarithmicbase and T ^ the maxim um value of Thom (wh ich is simply denoted by T in Fig.13(b)), 6 denotes the actual interface separation distance and 5o its equilibrium va lue,EQ is the equilibriumvalue of the adhesive energyE = Eo(l -H a y ) exp( ay) [ i tis noted that typical values forEo are 700 erg/cm^ for Al and500 erg/cm^ for Mg ], and \ denotes the range over whichstrong interatomic forces act (0.336 A for Al and 1.77 A forM g ) . These, rather qualitative results, will be expanded upon

for a variety of real bimaterial interfaces and thin film-substratesystems in a forthcoming report by Karagiannis et al. (1999).

4.2 Strain Gradien ts in Adiab atic Sliear Band ing. Thefinal topic to be considered here within a strain gradient framework of macroscopic plasticity is the evolution of adiabaticshear bands. The general view prevailing so far for this problemwas that introduction of higher-order gradients is not necessary,since the heat conductivity offers a natural way for capturingspatial features of shear bands such as widths and spacings. Analternative view was expressed by the author (Aifantis, 1992)who argued that it is the heterogeneity of deformation thatinduces inhomogeneous heating which, in turn, softens the material inducing additional plastic deformation and giving rise toan autocatalytic process leading to instability. Within such aframework, heat conduction can be neglected due to rapid deformations (adiabatic heating) with the internal length scale againprovided by the higher-order strain gradients. In this connection,it is pointed out that the energy equ ation should now be modifiedto include the extra work done by the higher-order gradients.Thus, the modified equations of continuum thermomechanicswhich are appropriate for describing adiabatic shear bandingare

T = K(y, y , 9) cV^y,

e = aSI^B + bTy + e,(4 .16)

where the thermal diffusivity a, the gradient coefficient c, andthe plastic work coefficient b may generally depend on thetemperature d, while the extra work term e depends, amongother things, on the gradients of y in a way compatible withthe second law of thermodynamics (Aifantis, 19 84b). The " homogeneous" part of the flow stress Ti,om = '


12/14

where (q, oj) denote the wave number of the fluctuation andits growth rate, a superimposed " o " denotes uniform (time-dependent) solution and a superimposed " " denotes thedeparture of the respective quantity from the uniform state.Moreover, this departure is assumed to be infinitesimal whenis normalized with respectto its uniform counterpart.The corresponding dispersion equation reads

LO ^ + A L O + Boj + C = 0, (4 .19)

where A = (l/pcMq\k + c,,s) - P^y], B = ( 1 /P^c^)q\pc^iH + cq^) + q'^ks]and C = (k/p^c^)q'*ih + q^c)with y and T evaluated at the uniform time-dependent state.The quantities in (4 .19) not defined earlier arethe strain hardening modulus h = {dKldy) > 0, the strain rate sensitivity s =(dK/dy) > 0, and the thermal softening parameter$ = (OK/dO ) 0] when any of the following four conditionsaresatisfied

A < 0 or B < 0 or C < 0 or A5 - C s 0. (4 .20)

However, in our case, A and C are always positive. It alsoturns out that periodic spatio-temporal patterns correspondingto condition (4.20)3 are possiblefor the following critical valuesof the total strain hardening coefficientH and wave numberq

sy)ql]H =

$

, k[s(k + cs)qt +

pc^sq 4- pc^y

. 2 , ycArky +

sky - pc^y)k + kc^s + pclc

(4.21)

In fact, the eigenvalue of (4.19) associated with conditions(4.21) is complex with a non-negative real partbut the numerical analysis shows that it becomes quickly real as the shearstrain y increases. It is also noted that the above estimate ofpositive H for the initiation of instability is in contrast toprevious instability analyses which,in the absence of straingradient effects, always predict vanishing valuesfor H and


13/14

3

J

" ^ 1 .5-

as

0.5

/

c=oc=1c= 2

~ - - _ ^

2 3

Fig. 15(a)

. /k=0

- - - k=54k=100

2.5

V 1.5'0:5

0.5

0 1 2 3 4 5

Fig. 15(b)

Fig. 15 The effect of gradient coefficient c and heat conductivity k onthe preferred wavelength: (a) effect of c (k = 0) and (b) effect of k (c =0 ). The coordinates are normalized again as q = qL and 6) = laly.

0k + c-s

(4.29)

and the following estimate for the characteristic time of theprocess /,.

tc1 k + c^s s

HmrHy

(4.30)

which can be used to estimate, in turn, shear band spacingswhen the rate of the overall deformation process is known. Forstrictly adiabatic conditions (fc = 0), it can easily be shownfrom (4.28) that

H + Icql,

c(4pc - s^ )2pcH + sy-^V

cX'^pc - s"-) )

H(pcH + .yy$)cCiX^pc s^)

(4.31)

It follows that for c = 0,tOp^ = -H/s an d qp,. -> TO. Theseconclusions can also be deduced from the numerical resultsdepicted in the diagrams of Fig. 15. The top diagram showsthat (for fe = 0) a preferred wave n umb er is clearly defined forc * 0, in contrast to the case c = 0. The bottom diagram showsthat (for c = 0) the effect of thermal conductivity is negligiblein selecting a preferred wave number which is, thus, left undetermined.

AcknowledgmentsThis work was completed as a part of the US Air Force grant

No. F49620-95-1-0208 under the Mechanics and Materials Program directed by Walter Jones, with partial support of the USAFOSR grant No. F4962G-96-1-0478. The support of the EUCommission of European Communities under contracts No.ERB FMRX-CT96-0062 (TMR Research Network) and No.FI4S-CT96-0024 (Revisa) is also acknowledged. Discussionswith my co-workers M. Zaiser and M. Gutkin, my graduatestudents I. Mastorakos and J. Huang and my undergraduatestudents K. Kalaitzidou and M. Karagiannis contributed to thepresentation.

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