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DISLOCATION MECHANICS BASED ANALYSIS OFMATERIAL DYNAMICS BEHAVIOR

R. Armstrong, F. Zerilli

To cite this version:R. Armstrong, F. Zerilli. DISLOCATION MECHANICS BASED ANALYSIS OF MATERIALDYNAMICS BEHAVIOR. Journal de Physique Colloques, 1988, 49 (C3), pp.C3-529-C3-534.<10.1051/jphyscol:1988374>. <jpa-00227797>

JOURNAL DE PHYSIQUE Colloque C3, Supplement au n 0 9 , Tome 4 9 , septembre 1 9 8 8

DISLOCATION MECHANICS BASED ANALYSIS O F MATERIAL DYNAMICS BEHAVIOR

R.W. ARMSTRONG*," and F.J. ZERILLI'

' ~ a v a l Surface Warfare Center, White Oak, Silver Spring, MD 20903-5000, U.S.A. "*university of Maryland, College Park, MD 20742-3035, U.S.A.

R ~ S U M ~ : Les mesures d e l ' i n f l u e n c e d e l a t a i l l e d e g r a i n s u r l a c o n t r a i n t e d e maclage du f e r Armco e t d ' u n a c i e r au carbone montrent qu 'une t r a n s i t i o n e s t v i s i b l e aux g randes v i t e s s e s d e deformat ion. On p a s s e d e l a deformat ion p a r g l i s s e m e n t b a s e e s u r un modele f a i s a n t i n t e r v e n i r des d i s l o c a t i o n s , a une de fo rma t ion p a r maclage. Des c a l c u l s d e l a deformat ion p a r maclage l o r s de l ' i m p a c t d ' un c y l i n d r e ( e s s a i s d e Tay lo r ) s o n t compares avec les r e s u l t a t s des e s s a i s . Pour du c u i v r e p o l y c r i s t a l l i n , l ' i n c o r p o r a t i o n des e f f e t s de f r o t t e m e n t v i squeux e t d e l a t a i l l e d e g r a i n dans un modele c o n s t i t u t i f base s u r les mecanismes d e d i s l o c a t i o n donne des r e s u l t a t s en bon accord avec ceux recemment r a p p o r t e s conce rnan t des e s s a i s d ' expans ion de bague. Les equa t ions ob tenues pour du f e r ( c . c . ) e t du c u i v r e ( c . f . c . ) pe rme t t en t de r ep rodu i r e des r e s u l t a t s d ' e s s a i s d ' impac t e n t r a c t i o n obtenus s u r du f e r o: e t du c u i v r e p a r Von Karman e t Duwez. Des d i f f e r e n c e s d e fond dans l e s l o i s d e comportement pour l e s C . C . ou i e s c . f . c . , p rodu i sen t pour c e s mater iaux des d i f f e r e n c e s i m p o r t a n t e s dans ies p r o p r i e t e s d ' i n s t a b i l i t k p l a s t i q u e en t r a c t i o n precedant l e declenchement d e l a r u p t u r e d u c t i l e .

Abs t r ac t - Grain s i z e dependent twinning s t r e s s measurements repor ted f o r Armco i r o n and carbon s t e e l m a t e r i a l s g ive a high s t r a i n r a t e t r a n s i t i o n t o twinning from t h e s l i p behavior desc r ibed by a d i s l o c a t i o n mechanics based c o n s t i t u t i v e equa t ion a n a l y s i s . Consequent model c a l c u l a t i o n s of deformation twinning in a c y l i n d r i c a l impact (Tay lo r ) t e s t r e s u l t a r e compared with r e s u l t s repor ted i n t h e o r i g i n a l Taylor t e s t exper iments . For copper p o l y c r y s t a l s , d i s l o c a t i o n d rag and g r a i n s i z e e f f e c t s i nco rpo ra t ed wi th in a d i s l o c a t i o n mechanics based c o n s t i t u t i v e model a n a l y s i s show agreement with r e s u l t s repor ted r ecen t ly from expanding r i n g t e s t exper iments . The d i f f e r e n t (bcc) a- i ron and (f c c ) copper equa t ions a r e shown t o f i t a - i ron and copper t e n s i l e impact r e s u l t s repor ted in a p ionee r ing s tudy by von Karman and Duwez. Bas ic d i f f e r e n c e s i n the bcc and f c c c o n s t i t u t i v e behaviors produce impor tant d i f f e r e n c e s i n t he t e n s i l e p l a s t i c i n s t a b i l i t y p r o p e r t i e s of t hese m a t e r i a l s preceding the onset of d u c t i l e f r a c - t u r i n g .

1 - DISLOCATION STRAIN RATE EQUATIONS

Z e r i l l i and Armstrong /1 / have r epo r t ed on the fo l lowing c o n s t i t u t i v e equat ions developed f o r t h e p l a s t i c deformat ion of body-centered-cubic (bcc) a- i ron and face-centered-cubic ( f c c ) copper m a t e r i a l s :

bcc a - i ron - o = AoG + Boexp[(-Do + Blln;)T] + K ~ E " + kEL -112

(1)

-1/2 f c c copper - oc = AoC + Blr 112exp[ ( -~o + k l l n t ) T l + kEL ( 2 )

where T i s t empera tu re , i s s t r a i n r a t e , E i s s t r a i n , L is the po lyc rys t a l g r a i n diameter, and AoG, Bo, B y , R 0 , K O , n , and kE a r e exper imenta l cons t an t s based on a d i s l o c a t i o n mechanics a n a l y s i s of the p l a s t t c deformation mechanisms ope ra t ive i n the two d i f f e r e n t c r y s t a l l a t t i c e s t r u c t u r e s . The c o n s t i t u t i v e equat ions were applied t o computing t h e deformat ion shapes of cy l inde r impact (Taylor) t e s t r e s u l t s obtained by Johnson and Cook 121, u t i l i z i n g t h e i r Lagrangian ma te r i a l dynamics code, EPIC-2, and, a l s o , u t i l i - z i n g independent t e s t r e s u l t s they repor ted f o r de termining the var ious m t e r i a l c o n s t a n t s of o t h e r empir ica l ly-based input equat ions . While

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988374

C3-530 JOURNAL DE PHYSIQUE

comparing t h e s e numerically-based and d i s l o c a t i o n mechanics based c o n s t i t u t i v e e q u a t i o n s , Johnson and Holmquist 131 have looked i n t o t h e u s e f u l n e s s of Taylor impact t e s t s , themselves , f o r de t e rmin ing t h e m a t e r i a l c o n s t a n t s of c o n s t i t u t i v e e q u a t i o n s proposed t o be employed f o r computa t ional mechanics purposes.

Fig. 1 - Comparison of twinning s t r e s s measurements 16-81 and computed s l i p s t r e s s e s / I / a t v a r i o u s t empera tu re s and s t r a i n r a t e s .

I n a f u r t h e r a n a l y s i s of t he Taylor impact t e s t r e s u l t s on (Armco) a - i r o n , Z e r i l l i and Armstrong /4 / proposed t h a t deformat ion twinning had occurred a t t he heav i ly deformed c y l i n d e r impact end and t h i s would have produced s t r e n g t h e n i n g of the m a t e r i a l through e f f e c t i v e g r a i n s i z e ref inement . Evidence of deformat ion twinning was found i n low magn i f i ca t ion photomicrographs s u p p l i e d by Johnson. Incorpora t ing a s imple model f o r twinning wi th independent e s t i m a t e s of t he twinning s t r e s s and number of twins per g r a i n , provided e x c e l l e n t agreement between computed and measured c y l i n d e r deformat ion shapes. One purpose of t h e p r e s e n t r e p o r t i s t o provide f u r t h e r i n fo rma t ion concerning t h e twinning de fo rma t ion , e s p e c i a l l y , with r ega rd t o t h e compe t i t i on between twinning and the d i s l o c a t i o n mechanics based c o n s t i t u t i v e behavior f o r s l i p . I n a d d i t i o n , f u r t h e r computa t ional r e s u l t s a r e provided f o r t h e h igh s t r a i n r a t e deformat ion behavior of copper concerning d i s l o c a t i o n drag and g r a i n s i z e e f f e c t s . The c o n s t i t u t i v e equa t ions f o r both copper and a- i ron have been app l i ed t o p ionee r ing t e n s i l e impact r e s u l t s r epo r t ed by von Karman and Duwez 151. The b a s i c d i f f e r e n c e s i n t h e f c c and bcc c o n s t i t u t i v e r e l a t i o n s produce impor tant diEEerences as wel l i n t he t e n s i l e p l a s t i c i n s t a b i l i t y p r o p e r t i e s of t h e s e m a t e r i a l s preceding the onse t of d u c t i l e f r a c t u r i n g .

2 - DEFORMATION T W I N N I N G I N a-IRON

Moiseev and T r e f i l o v /6 / measured the dynamic t e n s i l e twinning s t r e s s e s exh ib i t ed by Armco i r o n po lyc rys t a J specimens wi th d i f f e r e n t average g r a i n d i ame te r s t e s t e d a t a s t r a i n r a t e oE 1 . 1 ~ 1 0 s over t h e tempera ture range from -185'C t o 20°C. Thei r c o l l e c t e d exper imenta l measurements of t he p ropor t iona l l i m i t s t r e s s e s a r e shown in Fig . 1 a long wi th i n d i v i d u a l exper imenta l t e s t r e s u l t s ob ta ined on shock loaded Fe r rovac E i r o n a t 300K, by Rohde 171, and on 1010 s t e e l compressed a t 4.2 K, by Madhava, Worthington, and Armstrong 181. The twinning s t r e s s ( aT) vs g r a i n diameter r e l a t i o n s h i p employed by Z e r i l l i and Armstrong /4 / i s shown i n Fig. 1, a l s o , i n accordance wi th t he equa t ion :

where oOT = 330 MPa and k~ = 90 ~ ~ a - m m l / ~ . The e s s e n t i a l l y a thermal n a t u r e of the twinning s t r e s s has been d i scussed p rev ious ly by Armstrong and Worthington /9/ i n te rms of t h e e n e r g e t i c s involved i n n u c l e a t i n g deformat ion twins .

EXP'T . MODEL

M ~ l d steel cyl~nder Str~klng veloc~ty 338 m/s

Fig. 2 - Comparison of computed Taylor t e s t twinning zone i n Armco i r o n /4/ and t h a t observed i n mild s t e e l / l o / .

On the o t h e r hand, t h e tempera ture and s t r a i n r a t e dependent s l i p s t r e s s , US = a, a t E'O i n Eqn. ( I ) , i s shown i n Fig. 1 a t va r ious E va lues i n accordance with t he constants g iven by Z e r i l l i and Armstrong /1 / a s h a ~ = O , B0=1033 MPa, R0=0.00698 K-', 81=0.000415 K::, %=266 MPa, n=0.289, and kE=22 ~ ~ a - m r n l f ~ . Beginning from a t y p i c a l low E=0.014 s a t 300 K, s l i p i s p r e f e r r e d over twinning and remains so a t d i f f e r i t g g f a i n s i z e s u n t i l r e l a t i v e l y l a r g e values of 5 a r e reached. For example, a t ;= lo s , twinning has occurred r a t h e r than s l i p f o r g r a i n d iameters l a r g e r than 100 u m . Th i s v e n d is shown c l e a r l y i n t he o r i g i n a l r e s u l t s of Moiseev and T r e f i l o v where twinning was observed a t r e l a t i v e l y high tempera tures only f o r specimens with r e l a t i v e l y l a r g e average g r a i n d iameters .

The foregoing c o n s i d e r a t i o n of twinning occu r r ing before s l i p i f an app ropr i a t e ly high s t r a i n r a t e i s reached was used by Z e r i l l i and Armstrong /4 / t o ob ta in t he agreement i n d i c a t e d i n Fig. 2 between model and exper imenta l r e s u l t s f o r an impact velocity of 221 m/s app l i ed t o a p o l y c r y s t a l Armco i r o n specimen with an average grain d iameter of 110 u m . Four twins per g r a i n were roughly es t imated t o be contained i n t h e s e c t i o n e d specimen photomicrograph s e n t t o us by Johnson. The consequent g r a i n s i z e refinement r a i s e d the twinning s t r e s s s u f f i c i e n t l y h igher than the s l i p s t r e s s t o cause a l l subsequent deformat ion t o occur by s l i p , c o n s i s t e n t with the app rec i ab ly deformed shapes of twins i n t he impacted specimen. The computed zone of twinning shows an encouragingly s i m i l a r appearance t o t h e twinned cross-sec t ion a l s o shown i n Fig. 2, as produced i n mild s t e e l by Carr ington and Gayler / l p / i n t h e i r o r i g i n a l work with Taylor . Vickers hardness numbers ( i n u n i t s of kgf /mm ) a r e shown t o demarcate t he s t r e n g t h e n i n g e f f e c t a s soc i a t ed with t he occurrence of tw ins , l abe l ed a s Neumann l ame l l ae .

3 - DISLOCATION DRAG I N COPPcR_

Follansbee, Regazzoni, and Kocks /11/ have produced evidence t h a t d i s l o c a t i o n drag e f f e c t s e n t e r i n o t h e s t r a i n r a t e dependence of t he flow s g r e q of pure copper when deformed a t ~ > l $ s' a t 300 K. S t r a i n r a t e s as high as 10 s were es t imated by Z e r i l l i and Armstrong /1/ t o have occurred a t t h e copper c y l i n d e r impact f a c e s i n the Taylor t e s t r e s u l t s of Johnson and Cook 121. Therefore , i t seemed worthwhile t o rework t h e d i s l o c a t i o n mechanics based c o n s t i t u t i v e equat ion a n a l y s i s l ead ing t o Eqn. ( 2 ) so a s t o i nc lude t h e neg lec t ed d rag e f f e c t . The drag cons ide ra t ion e n t e r s into the time-of -f l i g h t of d i s l o c a t i o n motion between thermal a c t i v a t i o n b a r r i e r s . An approximate d e s c r i p t i o n of t he complete process g ives a new drag-affec ted thermal s t r e s s , a* a s

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(L

A FOLLANSBEE, REGAZZONI, 3 300 - 8 KOCKS (19841 OFE COPPE z Z

Co= 0

r 200 C

STRAIN RATE E (s-'] - a" 10400 8600 6500 4450 1930

-- INCLUDING PHONON DRAG 3

z - h

4 0 0 -

W LZ

W

rn g 2 0 0 - ("7,

EOUIVALENT TENSILE STRAIN E

I I I I

OFHC COPPER

+ +

Fig. 3 - Computed e f f e c t of phonon d rag i n copper on v a r i a t i o n of f low s t r e s s with s t r a i n r a t e , i n comparison with exper imenta l measurements 1111.

I- J = ~ ~ ~ ~ ~ m E X P A N D I N G RING TEST DATA W H GDUROIN 11987)

Fig. 4 - Computed copper s t r e s s - s t r a i n behav io r , i n c l u d i n g d i s l o c a t i o n d rag and g r a i n s i z e dependences, i n comparison wi th e l e c t r o m a g n e t i c expanding r i n g t e s t measurements 1131.

5 1 00,-T=423K - - THERMAL ACT!VATI&Y MODEL -

where o ~ h i s t h e T and E dependent component of s t r e s s i n Eqn. ( 2 ) and 2 2

co=(BIBm /Nb ) i n which B is t h e phonon d rag c o e f f i c i e n t , m i s t he Taylor o r i e n t a t i o n f a c t o r , N is t h e n o b i l e d i s l o c a t i o n d e n s i t y , and b is t h e d i s l o c a t i o n Burgers vec to r .

F i g u r e 3 shows an a p p l i c a t i o n of Eqns. ( 2 ) and ( 4 ) t o t he t o t a l copper s t r a i n r a t e dependence of f low s t r e s s measurements r epo r t ed by Fo l l ansbee , Regazzoni, and Kocks a t ~ = 0 . 1 5 . The exper imenta l c o n s t a n t s employed i n Eqn. ( 2 ) o r Eqn. ( 4 ) wi th co=O, a r e those-:eported by Z e r i l l i and Armstrong / I / : Aac=46. 5 MPa, B1=890 MPa, B0=0.0028 K , B1=0.000115 K-' and k=5 P l ~ a * m m l / ~ . Fol ansbee , Regazzoni, and Kocks -5 r e p o r t e d a g r a I n d iameter of 37 um. A value of co=10 M P ~ * ~ . K - ' g i yes-fgreernent w i th t h e d i s l o c a t i o n d r a g a f f e c t e d flow s r e s s measurments a t 2 > 10 s . With b-2.56 1, m=3. I , and a va lue of B=L.7xlO-' P a - s a t 300 K as r e p a r q d by Jassby and Vreeland 1121, a r ea sonab le mobile d i s l o c a t i o n d e n s i t y , N=3x10 m , i s obta ined.

4 - COMPUTED COPPER R I N G TEST R E S U E

Gourdin / I ) / has r epo r t ed i n i t i a l r e s u l t s on the sfr:z:-strain behavior of d i f ferent g r a i n s i z e OFHC copper p o l y c r y s t a l s t e s t e d a t ; > l o i n an e l ec t romagne t i ca l ly - d r i v e n expanding r i n g t e s t /14/. The d a t a which a r e shown i n Fig. 4 provide an a d d i t i o n a l i n t e r e s t i n g case f o r a s s e s s i n g Eqns. ( 2 ) and (4 ) . As i n d i c a t e d , two e f f e c t i v e p o l y c r y s t a l g r a i n d iameter m a t e r i a l s were t e s t e d : one reasonably uniform a t 10 u m and one wi th an average value of 172 u m f o r a measured range i n diameters between 150 and 200 pm. The s c a l e a t t he t op of t he f i g u r e shows t h e varying s t r a i n r a t e which a c t u a l l y app l i ed du r ing t h e r i n g t e s t deformat ion. The cross-hatched bands a r e t h e envelope of i n d i v i d u a l s t r e s s - s t r a i n curves which were r epo r t ed . The lower computed dot-dashed curve i n each case i s t h a t ob t a ined by our p rev ious ly r e p o r t e d m a t e r i a l c o n s t a n t s put i n t o Eqn. (21 , w i th t h e varying s t r a i n r a t e taken i n t o account. The_5upper cornputed dashed curves a r e t hose ob ta ined from Eqn. ( 4 ) , a g a i n , wi th co=10 MPa.s*K . Reasonable agreement between computed and expe r imen ta l r e s u l t s is i n d i c a t e d . Support is provided f o r t h e importance of accoun t ing f o r t h e d i s l o c a t i o n drag e f f e c t a t t h e s e s t r a i n r a t e s , a s proposed by Fo l l ansbee , Regazzoni , and Kocks /11/ .

5 - COPPER AND a-IRON TENSICE IMPACT RESULTS

A s is well-known, when Taylor and co l l eagues were deve lop ing t h e i r c y l i n d e r impact t e s t i n g procedure and, a l s o , developing a p l a s t i c wave d e s c r i p t i o n of t he r e su l t an t

P - ADIABATIC CURVES E - ,o - To=300 K - >

0 0 .05 0.10 STRAIN €, STRAIN E

Fig. 5 - Computed t e n s i l e impact v e l o c i t y - s t r a i n dependencies f o r mild s t e e l and copper i n comparison wi th measurements r epo r t ed by von Karman and Duwez 151. Fig. 6 - Computed p l a s t i c wave v e l o c i t y - s t r a i n dependence f o r copper i n comparison with " s t a t i c " cu rve e s t i m a t e s r epo r t ed by von Karman and Duwez / 5 / .

dynamic deformat ion behav io r , von Karman and Duwez /5/ produced a compf ementary dynamic p l a s t i c i t y a n a l y s i s of t e n s i l e impact t e s t r e s u l t s they obta ined on copper and mild s t e e l w i r e m a t e r i a l s . I n t h e i r a n a l y s i s , an impact v e l o c i t y , v l , of s u f f i c i e n t magnitude t o produce a p l a s t i c wave t r a v e l i n g a long t h e wire rod is connected wi th t h e work hardening of t h e m a t e r i a l through the r e l a t i o n s h i p

where p is t h e m a t e r i a l d e n s i t y and €1 is t h e cons t an t p l a s t i c s t r a i n behind t h e p l a s t i c wave f r o n t . The p l a s t i c wave v e l o c i t y i t s e l f , which is more d i f f i c u l t t o measure, is g iven by

Kuscher, Hohler, and S t i l p /15/ have r epo r t ed r ecen t measurements of p l a s t i c wave v e l o c i t i e s de termined i n compressive impact experiments on a-iron.

Equations (5 ) and ( 6 ) provide ano the r method of a s s e s s i n g the use fu lnes s of t he d i s loca t ion mechanics based c o n s t i t u t i v e r e l a t i o n s expressed i n Eqns. (1) and ( 2 ) . For example, F ig . 5 shows t h e r e s u l t of s u b s t i t u t i n g r e l evan t work hardening r a t e s derived from Eqns. ( 1 ) and ( 2 ) , w i th t he s t r e s s e s themselves, i n t o Eqn. (5) and computing t h e v l dependence on s t r a i n . For mild s t e e l , va lues of K0=492 MPa and n=0.314 were de r ived from an a n a l y s i s r epo r t ed by Liu, Armstrong, and Gurland 1161 of var ious s t r e s s - s t r a i n r e s u l t s on s t e e l m a t e r i a l s . Improved agreement was obtained wi th t h e s e s t e e l c o n s t a n t s a s compared wi th computations made with Armco iron parameters. The r e s u l t s i n t h i s case a r e independent of s t r a i n r a t e and temperature. For annealed copper , t h e r e s u l t s r epo r t ed by von Karman and h w e z a r e Compared wi th t h e r e s u l t s p red ic t ed by us parameters f o r OFHC copper. There f s good agreement f o r a s t r a i n r a t e of 10 i s a reasonable average s t r a i n

ra te f o r t he von Karman-Duwez experiment.

Figure 6 compares von Karman and Duwez's c a l c u l a t i o n s of the p l a s t i c wave v e l o c i t y in annealed copper wi th p r e d i c t i o n s ob ta ined by s u b s t i t u t i n g Eqn. (2) i n Eqn. ( 6 ) -

von Karman and Duwez r e s u l t s were obta ined from a " s t a t i c " S t r eSS-s t r a in curve and f i t f o r t h e most p a r t w i t h i n a range d e l i n e a t e d by the d i s l o c a t i o n mechanics

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0 -1 based a d i a b a t i c c u r v e s f o r s t r a i n rtftes_,of 10 and LO-' s . The main e f f e c t computed f o r a n i n c r e a s e of i t o 10 s i s s imp ly t o r a i s e t h e l e v e l of t h e v p vs dependence , t h u s i n d i c a t i v e of an expec t ed i n c r e a s e i n s t r a i n ha rden ing .

6 - COPPER AND a-IRON TENSILE INSTABILITIES

An advan t age of t e n s i l e impact t e s t i n g , a s u t i l i z e d by von Karman and Duwez, i s that u s e f u l i n f o r m a t i o n is p o t e n t i a l l y p rov ided on t h e d u c t i l e ' f r a c t u r i n g p r o c e s s under dynamic c o n d i t i o n s . For example, t h e p l a s t i c i n s t a b i l i t y c r i t e r i o n a t t h e onse t of neck ing

i s a we l l - de f i ned c o n d i t i o n a b l e t o be e v a l u a t e d f o r any proposed c o n s t i t u t i v e e q u a t i o n .

Fo r Armco i r o n , and r e l a t e d s t e e l s , t h e s e p a r a t i o n of s t r a i n h a r d e n i n g from t h e t h e r m a l l y - a c t i v a t e d component of s t r e s s l e a d s t o a reduced uni form s t r a i n t o t he p l a s t i c i n s t a b i l i t y p o i n t w i t h i n c r e a s i n g s t r a i n r a t e . The e f f e c t of a d i a b a t i c h e a t i n g , which is an impor t an t concern f o r a - i r o n , is t o a l l e v i a t e t h e r a p i d r e d u c t i o n i n uni form s t r a i n o t h e r w i s e produced. For coppe r t h e s i t u a t i o n is t o t a l l y r e v e r s e d . The coupled s t r a i n , t e m p e r a t u r e , and s t r a i n r a t e dependences l e a d t o an e q u a l l y s t r o n g , bu t o p p o s i t e , i n c r e a s e i n uni form s t r a i n t o t h e neck ing i n s t a b i l i t y under i n c r e a s i n g s t r a i n r a t e . I n t h i s c a s e , however, t h e a d i a b a t i c h e a t i n g e f f e c t , which is of r e l a t i v e l y l e s s impor t ance , a c t s t o reduce t h e i n c r e a s e i n uni form s t r a i n .

7 - ACKNOWLEDGMENTS

The a u t h o r s a r e g r a t e f u l t o c o l l e a g u e s K. Wayne Reed, K. Kim, and P. A. Wa l t e r for a d m i n i s t r a t i v e s u p p o r t and guidance . G. R. Johnson, Honeywell Inc . , and P. S. F o l l a n s b e e , tos Alamos Na t i ona l Labo ra to ry , have p rov ided h e l p f u l d i s c u s s i o n s .

8 - REFERENCES

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