On the stress state dependence of the strain-hardening of anisotropic sheet steel
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Nv[1rR27]
S V ~ NV[27rR 2] M V ~ N V [ 7r2R]
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TrNvrG2t 2
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27rNvG212
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[12]
~r2Nv G t
t r u e r a d i a l growth r a t e of the p l a t e s . B e c a u s e the i n t e g r a l m e a n c u r v a t u r e is p r o p o r t i o n a l to the p e r i m e t e r length for p l a t e s , it p r o v i d e s a useful m e a s u r e of the s c a l e of the s t r u c t u r e in the plane of the p l a t e s . The c u r v a t u r e a v e r a g e d i n t e r f a c e v e l o c i t y is a l s o shown to be s i m p l y r e l a t e d to the a v e r a g e edge v e l o c i t y for p l a t e s , and is thus the a p p r o p r i a t e m e a s u r e for growth in the plane of the p l a t e . The N a t i o n a l Science F o u n d a t i o n s u p p o r t e d this work: this s u p p o r t is g r a t e f u l l y acknowledged.
The C a h n - H a g e l growth r a t e is: 1 d V V _ 27rNvrGZt T vS = S V dt 27rNvG2t~ = t-
while the c u r v a t u r e a v e r a g e d growth r a t e i s : 1 VH = 2 M v
~. V _ 4 n N v G~t 2 dt - 20r2NVGt) = ~ G
[14]
It is c l e a r that a t t e m p t s to i n t e r p r e t p l a t e l e t growth f r o m the C a h n - H a g e l growth r a t e would be m i s l e a d i n g i n this case; one might e r r o n e o u s l y conclude f r o m Eq. [13] that the growth r a t e slows down with t i m e , and is i n d e p e n d e n t of t e m p e r a t u r e ] The c u r v a t u r e a v e r a g e d growth r a t e , on the other hand, is p r o p o r t i o n a l to the
On the Stress State Dependence of the Strain-Hardening of Anisotropic Sheet Steel
1. R. T. DeHoffand F. N. Rhines:QuantitativeMicroscopy, McGraw-HillBook Company,NY, 1968. 2. R. T. DeHoff:Microstructural Science, J. D. Braun,H. W.Arrowsmith,and J. L. McCall,eds., vol. 5, p. 331, ElsevierScientificPublishingCompany, NY, 1977. 3. E. E. Underwood:Quantitative Stereology, Addison-Wesley,Reading,MA, 1970. 4. J. W.CahnandW.C. Hagel:Decomposition of Austenite by Diffusional Processes, p. 131,IntersciencePub.,NY, 1962. 5. A. M.Gokhale:DoctoralDissertation,Universityof Florida,Gainesvill,FL, 1977. 6. R. T. DeHoff:Treatiseon Materials Science, H. Herman,ed., vol. 1, p. 247, AcademicPress, Inc.,NY, 1972. 7. G. R. Speichand R. M. Fisher:Recrystallization Grain Growth and Textures, p. 563,ASM,MetalsPark,OH, 1966.
be c a l c u l a t e d in u n i a x i a l t e n s i o n f r o m : ae =
3~/2 4 (r + I)/(~ + 2) a~,
ee = 2 4 ~
~/(r + 2 ) / ( r + 1) s
[2]
A . J . RANTA-ESKOLA and in b a l a n c e d b i a x i a l t e n s i o n f r o m : The s t r e s s state is b i a x i a l in most f o r m i n g o p e r a t i o n s . C o n s e q u e n t l y , it is of g r e a t i m p o r t a n c e to d e t e r m i n e the s t r a i n - h a r d e n i n g p r o p e r t i e s of the sheet under b i a x i a l s t r e s s . The most s u i t a b l e method for this is the h y d r a u l i c bulge t e s t . ~ The a v e r a g e s t r e s s and s t r a i n at the top of the d e f o r m i n g bulge is given by: 2 I~ t I = p2-~t ' [etl = 2 e %
[1]
where p is the bulging p r e s s u r e , t the t h
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