Transient stress states in composite creep
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Transient Stress States in Composite Creep G. GARMONG In 1968 de Silva 1 d e m o n s t r a t e d that t r a n s i e n t s t r e s s e s m a y o c c u r in the p h a s e s of a r e i n f o r c e d c o m p o s i t e m a t e r i a l which is e l a s t i c a l l y l o a d e d and then a l l o w e d to c r e e p . The s t r e s s r e d i s t r i b u t i o n is a c o n s e q u e n c e of the condition that the e l a s t i c s t r e s s e s m a y d i f f e r f r o m the c r e e p s t r e s s e s in e a c h c o n s t i t u e n t . Since then o t h e r a u t h o r s 2'3 have a l l u d e d to the s a m e effect. de S i l v a ' s a n a l y s i s , which d e a l s with an e l a s t i c r e i n f o r c e m e n t and a m a t r i x that is f i r s t e l a s t i c and then c r e e p p l a s t i c , i s s o m e w h a t difficult to apply a n a l y t i c a l l y to m a n y s p e c i f i c c a s e s . H e r e we p r e s e n t a n o t h e r m e t h o d of f o r m u l a t i n g the t r a n s i e n t s t r e s s s t a t e of m e t a l - m a t r i x c o m p o s i t e s c o n t a i n i n g continuous r e i n f o r c e m e n t . T h i s m e t h o d is e a s i e r to u s e and, in a d d i tion, shows m o r e c l e a r l y the d i f f e r e n c e b e t w e e n f u n c t i o n a l d e p e n d e n c e s of c r e e p and " c r e e p r e s i s t a n c e " in d e t e r m i n i n g s t r e s s r e d i s t r i b u t i o n . F o r a c o m p o s i t e d e f o r m i n g p a r a l l e l to the r e i n f o r c e m e n t the s t r e s s e q u i l i b r i u m i s f o r m u l a t e d a s
ac = a r V r + a m Vm
[la]
w h e r e a is s t r e s s , V is v o l u m e f r a c t i o n , and c, r , and m r e f e r to c o m p o s i t e , r e i n f o r c e m e n t , and m a t r i x . R e s i d u a l s t r e s s e s a r e t a k e n to be z e r o . F o r a c o n s t a n t s t r e s s c o n d i t i o n Eq. [ l a ] b e c o m e s
0 = 6 c = ~rVr + ~mVm
[lb]
w h e r e a dot a b o v e a s y m b o l d e n o t e s t i m e d i f f e r e n t i a tion d/dt. H e r e it is a s s u m e d that Vr = fZm = 0. S t r a i n G. GARMONG is Member of Technical Staff, Science Center, Rockwell International, Thousand Oaks, Calif. 91360. Manuscript submitted October 18, 1973. METALLURGICAL TRANSACTIONS
continuity during deformation may be expressed by the e q u i v a l e n t s t a t e m e n t s % : 9
-- ~ m
[2a]
~c = g r = gm
[2b]
w h e r e c is s t r a i n . The c o n s t i t u t i v e e q u a t i o n s can u s u ally be written as e = 9 (a, ~, t) = e ( a ) . 9 ( ~ ) . 9 (t)
[3a]
or : ~ (a, 9 t) : ~ ( a ) . ~ ( 9
(t)
[3b]
w h e r e t i s t i m e and c o n s t a n t t e m p e r a t u r e is a s s u m e d . To t e s t f o r p o s s i b l e s t r e s s r e d i s t r i b u t i o n d u r i n g c r e e p , it is n e c e s s a r y o n l y to s u b s t i t u t e Eq. [3] f o r e a c h p h a s e into w h i c h e v e r f o r m of Eq. [2] is convenient. The v a l u e s and s t r a i n o r t i m e d e p e n d e n c e m a y then b e found b y f u r t h e r s u b s t i t u t i o n into e i t h
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