On the calculation of the tensile strain associated with mechanical twinning
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Under the above condition it follows directly that the instantaneous length, L, of a tensile specimen should be a function of f the volume fraction transformed. If Lo is the initial length of the gage of the specimen it is possible to write; L = Lo + ~b*(f)
[1]
w h e r e ~ * e x p r e s s e s t h e f u n c t i o n a l d e p e n d e n c e of L u p o n f . It is s i m p l e to s h o w t h a t ; dL _ d~*ff) df df
-
dL
L
_
d ~ * ( f )
[3]
Discussion of "On the Calculation of the Tensile Strain Associated with Mechanical Twinning"* R. E. R E E D - H I L L
E : I n (1
Lo + qS*~f) "
• = In (1 + qS*(f) ~ Lo /
=
~bf
[4]
[5]
w h e r e dp w o u l d be a n a v e r a g e v a l u e of t h e o r i e n t a t i o n f a c t o r m u l t i p l i e d by the t r a n s f o r m a t i o n s t r a i n . J. R. C. GU1MARAES is Professor, Instituto Militar de Engenharia, Pqa Gen Tibflrcio, s/n °, Rio de Janeiro, Brazil, and R. J . DE ANGELIS is O . A . S . Visiting Scientist, on leave from the University of Kentucky, Lexington, KY. Manuscript submitted July 2 0 , 1976. METALLURGICAL
TRANSACTIONS A
~*(f) +---L-70 ]
[1]
a n d that the f u n c t i o n ~ * 0e) m u s t be a c o m p l e x f u n c t i o n . H o w e v e r , t h e p o i n t o f d i s c u s s i o n i n t h e o r i g i n a l p a p e r1 w a s t h e u s e of t h e a p p r o x i m a t i o n
w h i c h c l e a r l y d e m o n s t r a t e s t h e v a l i d i t y of u s i n g a logarithmic law t o e x p r e s s t h e tension s t r a i n equival e n t o f t w i n n i n g a v o l u m e f r a c t i o n of m a t e r i a l . T w o c r i t i c a l p o i n t s a p p a r e n t l y o v e r l o o k e d by R e e d H i l l e t a l2 a r e w o r t h n o t i n g : i) t w i n n i n g s t r a i n a f f e c t s t h e s p e c i m e n l e n g t h a n d c o n s e q u e n t l y m u s t be t a k e n into a c c o u n t i n c a l c u l a t i n g t h e c o r r e s p o n d i n g t r u e s t r a i n ; a n d ii) t h e r e i s a n a p p r o x i m a t i o n b u i l t into t h e e q u a t i o n a d v a n c e d in R e f . 3 : specifically d p * ~ ) / L o w a s t a k e n t o be l i n e a r l y r e l a t e d t o f ; i . e . , Lo"
1. M.K. Keshavan,G. Sargent, andH. Conrad: Met. Trans. A, 1975, vol. 6A, pp. 1291-92. 2. R. Reed-Hill, J. R. Donoso, and A. M. Gard: ibid, pp. 1292-94. 3 . J. R. C. Guimar~es andR. J. De Angelis: Mater. Sci. Eng., 1974, vol. 13, pp. 109-1 i.
T h e s e a u t h o r s a g r e e with G u i m a r ~ e s a n d De Angelis w h e n t h e y s a y that
E q . [3] c a n be e a s i l y i n t e g r a t e d f o r t h e b o u n d a r y c o n d i t i o n s ; L = Lo a t f = e = 0 t o o b t a i n ;
~*(f)
in item it). Finally it is worth mentioning that qS*(f) must be a complexfunction of f, probably containing difficult to specify microstructural parameters, except in the most simple cases of single crystal tensile deformation. This work was supported by the Brazilian Army, FINEP and BNDE. One of us (RJD) is grateful to the Organization of American States for a travel grant.
F . J . M . B O R A T
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