DISLOCATION SEGMENT DISTRIBUTION EFFECTS ON DYNAMIC MODULUS AND INTERNAL FRICTION

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DISLOCATION SEGMENT DISTRIBUTION EFFECTS ON DYNAMIC MODULUS AND

INTERNAL FRICTION

E. Bonetti, P. Gondi, A. Sili

To cite this version:

E. Bonetti, P. Gondi, A. Sili. DISLOCATION SEGMENT DISTRIBUTION EFFECTS ON DY-

NAMIC MODULUS AND INTERNAL FRICTION. Journal de Physique Colloques, 1983, 44 (C9),

pp.C9-785-C9-790. �10.1051/jphyscol:19839119�. �jpa-00223354�

DISLOCATION SEGMENT DISTRIBUTION EFFECTS ON DYNAMIC MODULUS AND INTERNAL FRICTION

E . B o n e t t i , P . ~ o n d i * and A. s i l i r *

I s t i t u t o d i Fisica deZZ ' U n i v e r s i t d , Unitd G.N.S.M. deZ C . N.R., Via I r n e r i o , 46, 40126 Bologna, I t a l y

* ~ s t i t u t o d i Fisica deZZrUniversitd d i BoZogna, Unitd G.N.S.M. deZ C.N.R.

and I s t i t u t o d i MetaZZurgia deZZruniversitd, Via mdossiana, 18, 00184 Roma, ItaZy

* ' ~ s t i t u t o d i MetaZZurgia deZZrUniuersitd, Via EUdossiana, 18, 00184 Roma, ItaZy

Resume

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Module dynamique e t f r o t t e m e n t i n t e r i e u r mesures directement pendant l a d € f o r m a t i o n p e r a e t t e n t d ' a n a l y s e r l ' e v o l u t i o n de l a d i s t r i b u t i o n des se- gments de d i s l o c a t i o n s . On a obtenu a i n s i des i n d i c a t i o n s s u r l e s transforma- t i o n s des s t r u c t u r e q u i accompagnent s o i t l e passage e n t r e l e proces de f l u - age determines p a r l ' e n e r g i e de a u t o d i f f u s i o n massive ou p a r l ' e n e r g i e i n f e - r i e u r e , s o i t l e passage e n t r e c o n d i t i o n s de dechargement e t rechargement.

A b s t r a c t - By means o f analyses of t h e d i s l o c a t i o n segment d i s t r i b u t i o n s obtained from dynamic modulus and i n t e r n a l f r i c t i o n measurements made d i r e c t l y d u r i n g deformation, i n d i c a t i o n s have been obtained on t h e d i s l o c a t i o n s t r u c - t u r e changes accompanying b o t h t h e passage from lower t o b u l k s e l f - d i f f u s i o n a c t i v a t i o n energy creep processes as w e l l as t h e t r a n s i t i o n o c c u r r i n g by un- l o a d i n g and reloading.

I n t r o d u c t i o n

-

I n previous papers /1,2,3/ i t has been shown t h a t from dynamic modulus (Md) measured d i r e c t l y d u r i n g deformation t o g e t h e r w i t h i n t e r n a l f r i c t i o n c o e f f i c i e n t

( Q - l ) , deformation r a t e

(2)

and e v e n t u a l l y d i s l o c a t i o n d e n s i t y ( p ) i t i s p o s s i b l e t o o b t a i n i n d i c a t i o n s on t h e e v o l u t i o n o f t h e d i s t r i b u t i o n f u n c t i o n @(R) /4,5,6/ g i v i n g t h e d e n s i t y per dR o f d i s l o c a t i o n segments i n t h e range RtR+dR. I n p a r t i c u l a r r e s u l t s o f previous researches concern t h e v i b r a t i o n frequency which i s e f f e c t i v e i n t h e t h e r m a l l y a c t i v a t e d processes o f l i n k overcoming by t h e d i s l o c a t i o n segments, t h e average area swept o u t per elementary s l i p a c t by each d i s l o c a t i o n segment and t h e c o n t r i b u t i o n t o dynamic modulus and t o deformation o f t h e l o n g l e n g t h t a i l s of t h e d i s t r i b u t i o n f u n c t i o n s .

Taking advantage o f those r e s u l t s f u r t h e r problems a r e considered here, i.e. t h e questions o f t h e t r a n s i t i o n mechanism r e s p o n s i b l e f o r t h e passage from 87 KJ/mol t o 125-142 KJ/mol as dominating a c t i v a t i o n energies f o r s t a t i o n a r y creep above o r below

-

425 K i n ~ 1 / 7 / , and o f t h e s t r u c t u r e changes i n unloading and reloading.

Experimental

-

Rectangular sheets o f A1 99.49%, I x I O - ~ ~ t h i c k , 3 ~ 1 0 - ~ m wide, w i t h l e n g t h s r a n g i n g from 1 t o 1 . 5 ~ 1 0 - ~ r n , were used f o r t h e experiments, w i t h g r a i n sizes

-

^{10-3 m.}

I n t e r n a l f r i c t i o n and dynamic modulus were measured d i r e c t l y d u r i n g constant l o a d de- formation by means o f an i n v e r t e d pendulum, as a l r e a d y described by us and by o t h e r authors / l , 8 / . Loads were chosen so as t o have a t v a r i o u s temperatures comparable de- f o r m a t i o n r a t e s , i n t h e range I 0-6sec-1. T o r s i o n a l frequencies ranged from 5 t o 10 Hz. The amp1 i t u d e s were s m a l l e r than E r r o r s r e l a t i n g t o i n t e r n a l f r i c t i o n c o e f f i c i e n t s were

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^I%, t o dynamic modulus 1-2%.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19839119

C9-786 JOURNAL

DE

PHYSIQUE

The resonance frequencies served t o determine t h e t o r s i o n modulus : t h e modulus changes were determined by d i f f e r e n t i a l measurements, w i t h r e f e r e n c e samples non de- formed and w i t h equal lengths.

TEM observations were made a f t e r some deformations a t some temperatures. For TEE t h e specimens were prepared by means o f e l e c t r o l y t i c p o l i s h i n g .

Results

-

As a l r e a d y discussed i n previous papers /1,2/ i n t e r n a l f r i c t i o n a n d d y n a m i c modulus show behaviour w i t h correspondences which may be expressed by t h e r e l a t i o n

Q ~ ; ~ + Q v ~ + ~ i l = Q;;~ + CE+BAM ( 1

where

s1

i s t h e ba$kground c o n t r i b u t i o n taken as a c o n s t a n t and equal t o Q;' b e f o r e deformation, = CE i s t h e i n t e r n a l f r i c t i o n component due t o viscous deformation, which r e s u l t e d n e g l i g i b l e ,

Qdl

t h e component depending on d i s l o c a t i o n damping. L a s t Qalis considered p r o p o r t i o n a l t o aM=(Md-M,)/M, (corresponding t o a modulus d e f e c t ) through t h e c o e f f i c i e n t f3 which expresses t h e amount o f r e l a x a t i o n ( f o r S.L.S., f3

<

1 i f r e l a x a t i o n p r e v a i l s ) .

The diagrams i n F i g . l a ) - b ) - c ) i l l u s t r a t e s t h e s a i d behaviour f o r some temperatures i n evidence.

A t t h e v a r i o u s temperatures e f f e c t o f u n l o a d i n g , d u r i n g creep have a l s o been examined.

Such unloading e f f e c t s d u r i n g deformation can be b e t t e r f o l l o w e d through t h e v a r i - a t i o n s o f

Qil,

i n s t e a d o f those o f t h e modulus d e f e c t which becomes small by i n c r e a s - i n g deformation, w i t h l a r g e r e l a t i v e e r r o r s . O f course t h e v a r i a t i o n s o f Qdlcan be t r a n s l a t e d t o those o f AM through t h e r a t i o '6, which i s assumed t o remain constant as d u r i n g deformation.

Fig. l a ) - b ) - c )

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I n t e r n a l f r i c t i o n c o e f f i c i e n t (Qel) and modulus d e f e c t (AN) measured d i r e c t l y d u r i n g creep deformation a t 297K(a), 423K(b), 623K(c).

Fig. 2a1)-b3-cl)

-

Q - ' v a r i a t i o n s a f t e r unloading and r e l o a d i n g a t 297K(a), 423K(b), 623K(c). Values o f B = Q$/AM a r e v a l i d both f o r a,al; b,bl; 'c,cl r e s p e c t i v e l y . On unloading Q;il(B AM) shows a d i f f e r e n t behaviour a t d i f f e r e n t temperatures, as v i s - i b l e i n Fig. 2a')-bl)-cl). A t lower temperatures, e.g. room temperature and 423K, unload- i n g i s accompanied by r a p i d l a r g e ^Q-l decreases, f o l l o w e d by smaller, progressive ad- d i t i o n a l decreases: on r e l o a d i n g previous

Q-'

v a l ues a r e r e s t o r e d p r o g r e s s i v e l y .

small p o l y g o n i z a t i o n peak, r e l o a d i n g causes a s i m i l a r behaviour.

In

various papers /9,10,11,12/ i t has been assumed t h a t t h e v a r i a t i o n s o f Q-'and ivld i n t h e medium temperature range, from - 4 5 0 K t o - 7 5 0 K , are due t o d i s l o c a t i o n r e l a x - a t i o n . The decreases o f Md occur i n a broad temperature range, owing t o a broad spec- trum of r e l a x a t i o n times. The phenomena can be b e t t e r l o c a l i z e d by c o n s i d e r i n g t h e Q-I

and Md behaviour together, as shown i n F i g . 3. F u l l r e l a x a t i o n can be assumedtooccur a f t e r t h e K2 peak, i.e. a t Tc.

A t l o w e r temperatures t h e m d u l u s d e f e c t s c a n be i n t e r p r e t e d w i t h r e f e r e n c e t o two ex- treme cases, i.e. p a r t i a l r e l a x a t i o n o f a l l t h e d i s l o c a t i o n s o r complete r e l a x a t i o n o f p a r t o f a l l t h e d i s l o c a t i o n s . Since B < 1 ( F i g . 1) t h e second case i s admitted; so t h e f r a c t i o n

5

o f t h e r e l a x e d d i s l o c a t i o n s h a s t o be taken i n t o c o n s i d e r a t i o n and t h i s i s obtained by c o n s i d e r i n g

5

equal t o t h e r a t i o AMTd/AMTc where AMT~ i s t h e modulus decrease from room temperature up t o t h e deformation temperature Td, AMT, t h e modulus decrease up t o Tc7both r e f e r r i n g t o c o n d i t i o n s b e f o r e deformation.

F i n a l l y a TEM image i s given i n Fig. 4 i l l u s t r a t i n g d i s l o c a t i o n s t r u c t u r e s o c c u r r i n g b e f o r e t h e ^Q-l, AM peak experienced d u r i n g deformation a t 423K: such s t r u c t u r e s are o f i n t e r e s t f o r t h e d i s c u s s i o n

Fig. 3

-

I n t e r n a l f r i c t i o n c o e f f i c i e n t (Q-l) and dynamic modulus (\Id) vs temperature b e f o r e deformation; Tc temperature o f f u l l d i s l o c a t i o n r e l a x a t i o n . Fig. 4

-

TEM image o f an A1 specimen deformed j u s t b e f o r e t h e Q-'peak d u r i n g creep a t 4 2 3 ~ ( r e f . F i g . l b ) . Fig. 5 a ) - A c t i v a t i o n energies c h a r a c t e r i s t i c o f s t a t i o n a r y creep o f A1 a t v a r i o u s tan- peratures ( r e f . /7,13/. Fig. 5b)

-

Reference model f o r t h e passage from stage H: t o stage Hz i n Fig. 5a).

Discussion

-

F i r s t t h e t r a n s i t i o n between the two medium temperature processes gov- erned by d i f f e r e n t a c t i v a t i o n energies i s discussed.

As an i n t r o d u c t i o n t o t h e problem w e l l known experimental r e s u l t s from /13/, subse- q u e n t l y s p e c i f i e d i n /7/, a r e r e c a l l e d . These r e s u l t s are i l l u s t r a t e d by the diagram i n F i g . 5a). The transition between the two processes c h a r a c t e r i z e d by HI 87 KJ/mol and H2 = 125-542 KJ/mol m i g h t be understood i n terms o f two temperaturedependent r a t e processes as sketched i n Fig. 5b). Assuming t h a t t h e two processes a c t i n s e r i e s l a r g e r r a t e s w i l l be dominant. According t o t h i s scheme, t r a n s i t i o n between each o t h e r dominating proces w i l l occur a t temperature Tt, being

I n t r o d u c t i o n o f t h e e x p e r i m e n t a l l y known q u a n t i t i e s , i .e. H j = 87KJ/mol, H2 = 142 K J h l , Tt = 423K leads t o t h e pre-exponential r a t i o

. .

E ~ ~ / E , , ,

l o - '

( 3 )

C9-788 JOURNAL DE PHYSIQUE

The deformation r a t e can be expressed /5/ by

CO f ~ ~ b ' l a b / k T

; =

j

b s v d j a - - I exp

[- ^L]

^-i(a)da

Ja Gb/o R I kT,

where vd and H a r e v i b r a t i o n frequency and a c t i v a t i o n energy f o r t h e t h e r m a l l y a c t i - vated d i s l o c a t i o n motion, S=c

R2

i s t h e average area swept o u t i n each elementary s l i p s t e p p e r d i s l o c a t i o n segment, thus expressed as a f u n c t i o n o f

R2

through t h e p r o p o r t i o n a l i t y c o e f f i c i e n t c, oe t h e e f f e c t i v e s t r e s s which i s taken equalto a-aGb/!L ( a a p p l i e d s t r e s s , a s t r e n g t h constant taken equal t o I ) , a=A/oe expresses t h e propor- t i o n a l i t y o f a c t i v a t i o n area A t o e f f e c t i v e s t r e s s .

According t o 4b) t h e p o s s i b i l i t y t h a t t h e pre-exponential r a t i o i n ( 3 ) depends on d i f f e r e n t v d t s i s considered f i r s t . The lower energy H1=87KJ/mol has been r e l a t e d t o p i p e d i f f u s i o n along t h e d i s l o c a t i o n s /7,14/, whereas t h e h i g h e r energy H 2 i s r e l a t e d t o b u l k s e l f - d i f f u s i o n . So i t may be reasonable t o assume t h a t the v i b r a t i o n f r e - quency e f f e c t i v e i n p i p e d i f f u s i o n i s t h e one o f a t t a c k per d i s l o c a t i o n segment o f l e n g t h R , i . e . vdl=vD b/R, whereas vd =vD f o r l a t t i c e s e l f - d i f f u s i o n , vD being t h e Debye frequency. Since r e 1 a t i o n s (4a-6) r e f e r t o e l e m n t a r y processes the a c t i v a t i o n energy f o r l a t t i c e s e l f - d i f f u s i o n w i l l be i n t r o d u c e d i n t o (4a) as h i g h e r energy, i .e.

Hz = 125 KJ/mol /15/.

To check t h e e f f e c t s o f t h e s a i d d i f f e r e n c e s i n vd i t i s worth c o n s i d e r i n g a l s o t h e f o l l o w i n g r e l a t i o n s , r e g a r d i n g t h e dynamic modulus and d i s l o c a t i o n d e n s i t y both given as a f u n c t i o n o f t h e d i s l o c a t i o n segment d i s t r i b u t i o n f u n c t i o n :

r"

Im

AG/G =

< 1

n R 3 -i(R)dR

"

AM ( 5 ) ; p =

I

R @ ( R ) ~ R (6)

J 0 J o

F o l l o w i n g previous con- s i d e r a t i o n s /3/ @(!L)=AR2 exp(-BR2) wi 11 be used as d i s t r i b u t i o n function.In- t r o d u c t i o n o f t h i s func- t i o n i n t o ( 5 ) a n d ( 6 ) a l - lows us t o o b t a i n AandB as a f u n c t i o n o f a a n d AM assuming AM=AG/G.Results o f r e 1 a t i n g c a l c u l a t i o n s are. ex-pressed by diagrams 10" 1012 10" lot6 10" 10''

ldL

1d6 10'' 1012 10"

1d6

of &*=& ( n F j / n ~ ) ^{- 1} as a fun-

f'(rn ') ye(rn-') y"(rn-2) c t i o n o f p * = p ( ~ ~ / < n ) - ' .

F i g . 6a)-b)-c) Diagrams o f

;* AM/<^)-^

vs p* p ( ~ ~ / t n ) - l c a l c u l a t e d by s o l v i n g eqs.

(4a) ,(5) , ( 6 ) . I n F i g . 6a) T=423K; diagram 1, vd=vDb/R, H=87KJ/mol

,

c=47i; diagram 2, V~=VD, H=125KJ/mol, c=4n; diagram 3, Vd'v~, H=87KJ/mol, c=47i. I n Fig. 6b) T=297K; dia- gram l , vd=vDb/!L, H=87KJ/mol, c=47; diagram 2,vdZvD, H=87KJ/mol, c=4n. I n Fig. 6c) T=423KJ/mol; diagram 1, vd=vD, c = l ; diagra? 2, v d = v ~ , c = l r 3 ; diagram 3 vdzvD/5, c = I O - ~ . The d o t t e d l i n e s show experimental ^E* f o r t h e deformations E i n d i c a t e d . According t o t h e assumption i n q u e s t i o n t h e experimental creep r a t e s a t T=423K should be c o n s i s t e n t w i t h values c a l c u l a t e d by i n t r o d u c i n g i n t o (4a) e i t h e r vd, and HI o r vd2 and Hz. Diagrams so c a l c u l a t e d a r e drawn i n F i g . 6a); t h e y are c o n s i s t e n t w i t h t h e experimental data represented by f a i n t l i n e s , e v e n t u a l l y w i t h small changes i n c (causing o r d i n a t e s h i f t ) o r o f

5

(causing s h i f t o f both o r d i n a t e s and abscissae) f o r

However, i s used a t t h e lower temperatures, ~ . g . a t room tempe- r a t u r e , diagram I o f Fig. 6b) i s obtained, displaced towards low E* values, whereas w i t h v d = v ~ good correspondence w i t h t h e experimental deformation r a t e i s obtained (diagram 2 i n Fig. 6b). Large increases o f the o t h e r terms, besides vd, t h a t c o n t r i - bute t o

B o

i n 4b), a r e u n l i k e l y : so t h e e x p l a n a t i o n based on t h e reduced a t t a c k f r e - quency i s abandoned. I n t h i s r e s p e c t i t i s a l s o n o t i c e d t h a t , i n a t h e o r e t i c a l t r e a t - ment, Granato e t a l . /16/ conclude t h a t t h e e f f e c t i v e frequencies can d i f f e r from those o f a t t a c k , tending t o vD f o r s t r o n g p i n n i n g p o i n t s .

O f course i f V ~ = V D i s maintained w i t h HI, diagrams c a l c u l a t e d a t 423K w i l l r e s u l t d i s p l a c e d towards deformation r a t e s h i g h e r than those experienced, as shown i n Fig.6a);

i.e. t h e r e occurs the i n v e r s e o f t h e s i t u a t i o n considered p r e v i o u s l y w i t h reference t o room temperature, w i t h t h e d i f f e r e n c e t h a t now decreases o f o t h e r terms c o n t r i b u t - i n g t o E o i n 4b),seem reasonable, i n p a r t i c u l a r o f t h e c c o e f f i c i e n t s i n c e t h e r a t i o o f S over 9,' should change w i t h d i f f e r e n t d i s l o c a t i o n s t r u c t u r e s . For example i f the moving d i s l o c a t i o n s f i n d t h e i r p i n n i n g p o i n t s i n grooves o f d i s l o c a t i o n s , as i n the case o f Fig. 4, t h e c r a t i o w i l l be a/R where

a

i s t h e distance between d i s l o c a t i o n s i n t h e groove.

This a / R r a t i o may be o f t h e order o f l a r g e r t h a n t h e r a t i o between t h e pre-ex- p o n e n t i a l s i n ( 3 ) . F u r t h e r adjustments can be obtained by c o n s i d e r i n g d i f f e r e n t d i s - tri b u t i o n f u n c t i o n s (see a1 so /3,5,6/) s m a l l e r i n t e r n a l back-stresses ( w i t h minor c o n t r i b u t i o n s ) as w e l l as e f f e c t i v e frequency decreases f o r weak p i n n i n g p o i n t s . Curves f o r d i f f e r e n t c1s,down t o t h e range o f t h e experimental deformation r a t e s a r e drawn i n Fig. 6c).

According t o t h i s view t h e t r a n s i t i o n between t h e stages c h a r a c t e r i z e d by t h e two a c t i v a t i o n energies HI-2 should depend on v a r i a t i o n s o f t o , m a i n l y dependent on t h e d i s l o c a t i o n s t r u c t u r e , so t h a t above Tt t h e H2 process p r e v a i l s . A d e t a i l e d a n a l y s i s o f t h e processes governed by two energies i s n o t considered here: mention i s o n l y made t o paper 1171 where c o n t r i b u t i o n s o f g.b. and l a t t i c e d i f f u s i o n a r e t r e a t e d w i t h p a r t i c u l a r reference t o t h e t r a n s f e r o f g l i d e from one g r a i n t o t h e o t h e r across t h e g r a i n boundary: r e s u l t s o f a previous work /2/ are a l s o r e c a l l e d according t o which vdl, HI and vd2, H2 r e f e r r e s p e c t i v e l y t o the c o n d i t i o n s b e f o r e and a f t e r t h e Q-',AM peak encountered d u r i n g creep a t t h e temperature i n evidence.

Discussing now t h e unloading and r e l o a d i n g e f f e c t s r e f e r e n c e i s made t o t h e e x p e r i - mental r e s u l t s given i n Fig. 2a)-b)-c).

The phenomena a r e analyzed by assuming t h a t t h e d i s l o c a t i o n d e n s i t y p doesnotchange on unloading and r e l o a d i n g , b u t on1 y t h e d i s l o c a t i o n segment d i s t r i b u t i o n . T h i s assumption i s based on t h e f a c t t h a t r e l o a d i n g causes t h e recovery o f t h e same relax- a t i o n c o n d i t i o n s present before unloading.

The v a r i a t i o n s o f the d i s t r i b u t i o n f u n c t i o n may thus be f o l l o w e d by s o l v i n g t h e sys- tem of equations ( 5 ) and ( 6 ) f o r constant p ( t h e p has been taken corresponding f o r each temperature and AM ( b e f o r e unloading) t o t h e maximum o f t h e diagrams o f Fig.6a).

The s t r u c t u r e e v o l u t i o n c a n be f o l l o w e d through t h e average segment l e n g t h

ji =

.r:

a @ ( a ) ~RII~~ @ ( a ) d~ ( 7 )

The diagrams r e p o r t e d i n Fig. 7 have been obtained i n t h i s way

A t t h e lower temperatures t h e decreases i n ji a r e understood c o n s i d e r i n g t h a t p i n n i n g p o i n t s which are overcome under l o a d become e f f e c t i v e again a f t e r unloading.

C9-790 JOURNAL DE PHYSIQUE

Fig. 7a)-b)-c)

Behaviour o f

R

vs time a f t e r unloading and r e - l o a d i n g d u r i n g creep a t 297K (a), 423K ( b ) , 623K ( c ) , as represented i n Fig. 2a)-b)-c).

Reloading causes new unpinning w i t h o u t r e l e v a n t v a r i a t i o n s i n t h e d i s l o c a t i o n den- s i t y . So t h e e v o l u t i o n on r e l o a d i n g appears d i f f e r e n t from t h e one a f t e r t h e f i r s t l o a d i n g (see f o r ex. / I / ) , when increases o f t h e d i s l o c a t i o n d e n s i t y by mu1 t i p l i c a - t i o n p l a y a fundamental r o l e .

A t h i g h e r temperatures, instead, d i s l o c a t i o n s , which a r e pushed a g a i n s t l i n k s o r o t h e r p i n n i n g s i t e s by t h e load, become f r e e when t h e l o a d drops, thus a c q u i r i n g l o n g e r f r e e l e n g t h s and g i v i n g g r e a t e r c o n t r i b u t i o n s t o t h e p e r t a i n i n g r e v e r s i b l e deformation; they then r e t u r n t o t h e lengths corresponding t o the average n e t o f p i n n i n g distances by rearrangement processes d r i v e n by t h e i n t e r n a l stresses. Reload- i n g has a s i m i l a r e f f e c t , f i r s t o f unpinning and subsequent pushing t h e d i s l o c a t i o n s a g a i n s t s t r o n g e r l i n k s , i n p a r t due t o t h e development o f new subboundaries.

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