Strain-Hardening equations and uniform strain
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Strain-Hardening Equations and Uniform Strain
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KANJI ONO c a p a c i t y of a s t r u c t u r a l m e m b e r in t e n s i o n i s d i c t a t e d by i t s w o r k - h a r d e n i n g c h a r a c t e r i s t i c s . It can b e e a s i l y shown x that, when the m a x i m u m load is reached [1]
h o l d s w h e r e Y and ~ a r e t r u e - s t r e s s and t r u e - s t r a i n , respectively. Further plastic deformation proceeds l o c a l l y ; i. e., the n e c k i n g s t a r t s . The a m o u n t of t r u e s t r a i n b e f o r e the n e c k i n g o c c u r s i s c a l l e d " u n i f o r m s t r a i n " , Eu. In t e r m s of e n g i n e e r i n g s t r a i n , t h e c o r r e sponding quantity i s " u n i f o r m e l o n g a t i o n " . It i s w e l l known that when t h e s t r e s s - s t r a i n c u r v e of a m a t e r i a l can be d e s c r i b e d by t h e p o w e r law
-~
=
g-~ m
[2]
w h e r e K and m a r e c o n s t a n t s , the u n i f o r m s t r a i n i s equal to the exponent; t h a t i s , ~u = m . T h i s p o w e r - l a w r e l a t i o n s h i p h a s b e e n ~ m o s t c o m m o n l y e m p l o y e d in d e s c r i b i n g e x p e r i m e n t a l d a t a and in t h e o r e t i c a l c a l c u lations. A n u m b e r of s t r e s s - s t r a i n r e l a t i o n s h i p s cannot b e a p p r o x i m a t e d b y Eq. [2]. O n e group of m a t e r i a l s i n c l u d e s a f a m i l y of T R I P s t e e l s . 2 A s shown in F i g . 1, the s t r e s s - s t r a i n c u r v e s have the f o r m of = (r0 + gz . ~
[3]
w h e r e ~o and K, a r e c o n s t a n t s . F r o m Eq. [1], t h e u n i form strain for this case is
-Eu = 1 - ao/gz
[4 ]
Note that Eq. [4] p r e d i c t s a s m a l l e r Eu f o r a l o w e r w o r k - h a r d e n i n g r a t e , Kz, a n d / o r for h i g h e r y i e l d s t r e n g t h , no. When (to = K1, z e r o u n i f o r m s t r a i n i s p r e d i c t e d . T h e v a l u e s of -~u p r e d i c t e d b y Eq. [4] a r e shown by a r r o w s in F i g . 1. In one of the t h r e e e x a m p l e s shown, a g r e e m e n t i s e x c e l l e n t . In two o t h e r c a s e s , the s p e c i m e n s n e c k e d b e f o r e r e a c h i n g the p r e d i c t e d v a l u e s of ~u. It i s e x p e c t e d , h o w e v e r , that s t r e s s - c o n c e n t r a tion at m a c r o s c o p i c and m i c r o s c o p i c f l a w s p r o d u c e s p r e m a t u r e necking. T h e r e f o r e , the d u c t i l i t y of t h e T R I P s t e e l s m a y f u r t h e r b e i m p r o v e d to the l e v e l s p r e d i c t e d by Eq. [4]. M o d i f i e d AISI 201 s t a i n l e s s s t e e l s a l s o belong to t h i s t y p e of m a t e r i a l that p o s s e s s the s t r a i n - h a r d e n i n g c h a r a c t e r i s t i c s of Eq. [2]. T a b l e I l i s t s t h e y~eld s t r e n g t h , l i n e a r s t r a i n - h a r d e n i n g r a t e , KANJI ONO is Associate Professor of Engineering, Materials Department, School of Engineering and Appli
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