Interface morphology development during stress corrosion cracking: Part I. Via surface diffusion
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It NECESSARY AND S U F F I C I E N T CONDITIONS FOR CRACK GROWTH The second law of t h e r m o d y n a m i c s n a t u r a l l y p r o v i d e s us with the n e c e s s a r y condition for c r a c k growth. T h i s one condition is that the e n t r o p y of an i s o l a t e d s y s t e m (in this c a s e the e l a s t i c body p l u s its loading m e c h a n i s m s ) be m a x i m i z e d . This i s , of c o u r s e , e q u i v a l e n t to specifying e x t r e m u m conditions on the f r e e e n e r g y of the e l a s t i c body while i m p o s i n g c o n s t r a i n t s on the loading m e c h a n i s m s . A t h e o r e m of this s o r t is well known in the t h e o r y of e l a s t i c i t y - - t h e t h e o r e m of m i n i m u m p o t e n t i a l e n e r g y . 5 In Appendix I, a f o r m a l d e v e l o p m e n t p r o d u c e s a r e s u l t in c o m p l e t e h a r m o n y with c l a s s i c a l t h e r m o d y n a m i c s , ~'8 n a m e l y that, for a s p e c i e s r e s i d i n g in the l a t t i c e , the c h e m i c a l potential defined only at the s u r f a c e is given by = ~o + V ~
V~
[1]
w h e r e A F i s the s t r a i n e n e r g y d e n s i t y (Helmholtz f r e e e n e r g y change), V ~ the a t o m i c v o l u m e , and P the n o r real p r e s s u r e at that point on the s u r f a c e . A somewhat s i m p l e h e u r i s t i c a r g u m e n t can a l s o be u s e d to c a l c u l a t e the c h e m i c a l p o t e n t i a l at a f r e e s u r f a c e a s well a s at a s u r f a c e w h e r e a n o r m a l t r a c t i o n T n a c t s . It i s that the two t y p e s of work we m u s t c o n s i d e r in adding an e l e m e n t of m a s s , f r o m a n u n s t r a i n e d r e f e r e n c e s t a t e , to the s t r e s s e d body a r e i) s t r a i n e n e r g y or w o r k n e c e s sar__y to p r o d u c e a c o h e r e n t m a t c h and ii) a P V o r T n • V t e r m which is the work r e q u i r e d to " p u s h b a c k " the loading m e c h a n i s m . T h i s l e a d s a g a i n to Eq. [1] or in the case of a f r e e s u r f a c e to Eq. [12] in Appendix I. It should b e r e a l i z e d at this point just how useful is the concept of c h e m i c a l p o t e n t i a l in t r e a t i n g the e n e r g e t i c s of c r a c k growth s i n c e , in u s i n g the c o m m o n l y a c c e p t e d a p p r o a c h of m i n i m u m p o t e n t i a l e n e r g y as is c o m m o n l y done, the a n a l y s i s i s l i m i t e d to specific geo m e t r i e s . Indeed, a u t h o r s ~ have found it difficult to set " p r e c i s e c o n d i t i o n s for m o r e g e n e r a l c o n f i g u r a t i o n s " . T h e s e p r o b l e m s a r e p u r e l y c o m p u t a t i o n a l and a r e e l i m inated by the u s e of the d i f f e r e n t i a l q u a n t i t y p. It i s i n t e r e s t i n g to note that the s p e c u l a t i o n s c o n c e r n i n g g e n e r a l c o n f i g u r a t i o n s of Rice and D r u c k e r 6 s e e m to b e in h a r m o n y with the t h e r m o d y n a m i c f o r m u l a t i o n
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