9-27-11_On-the-strain-hardening-of-metals

9-27-11_On-the-strain-hardening-of-metals - On the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: On the Strain-Hardening Parameters of Metals H . J. KLEEMOLA AND M. A. NIEMINEN T he a p p l i c a b i l i t y of the Ludwik, Hollomon, Swift and Voce equations in d e s c r i b i n g the s t r e s s s t r a i n c u r v e s of m e t a l s was i n v e s t i g a t e d . C a l c u l a t e d u n i f o r m s t r a i n v a l u e s w e r e found to d e p end on the equation used. Even when the Hollomon equation gave a high l i n e a r c o r r e l a t i o n c oefficient in l o g - l o g c o o r d i n a t e s , the s t r a i n - h a r d e n i n g exponent n could give an e r r o n e o u s u n i f o r m s t r a i n . The equation with the lowest s t a n d a r d e r r o r of e s t i m a t e gave the u n i f o r m s t r a i n n e a r e s t to the v a l u e obtained b y d i r e c t m e a s u r e m e n t f r o m the l o a d - e l o n g a t i o n c u r v e . T H E d e s c r i p t i o n of the s t r e s s - s t r a i n c u r v e s and s t r a i n - h a r d e n i n g of m e t a l s by m a t h e m a t i c a l e x p r e s s i o n s is a f r e q u e n t l y used approach. This is b e c a u s e i t allows the p l a s t i c p a r t of the c u r v e to be t r e a t e d by c e r t a i n p a r a m e t e r s which can be applied to the study o f f o r m a b i l i t y and d e f o r m a t i o n m e c h a n i s m s . T he m o s t i m p o r t a n t and widely used a p p l i c a t i o n is t he e v a l u a t i o n of s t r e t c h - f o r m a b i l i t y b y the n value, w hich is the exponent of the Hollomon 1 equation, = Ke n. C a i r n s e t a l . z h ave also used the n v a l u e in the a s s e s s m e n t o f ' r e s i d u a l d u c t i l i t y of h e a v i l y d e f o r m e d c opper. The s t r a i n - h a r d e n i n g exponent, n, can be d e t e r m i n e d b y a s i m p l e t e n s i o n t e s t o r f r o m the m e a s u r e m e n t of s t r a i n in a s p e c i a l s p e c i m e n , s The u n i f o r m s t r a i n , eu, which is equal to n when the Hollomon e q u a t ion is valid, has also f r e q u e n t l y b e e n used as a m e a s u r e of f o r m a b i l i t y . H owever, n is the exponent of a n e m p i r i c a l equation a nd it is not s u r p r i s i n g that this equation can not a c c u r a t e l y d e s c r i b e the s t r e s s - s t r a i n c u r v e s for all m e t a l s . T hus s e v e r a l e m p i r i c a l r e l a t i o n s without any p h y s i c a l m e a n i n g , but giving a b e t t e r fit to e x p e r i m e n t a l data in s ome c a s e s , have b e e n developed u s i n g c o m p u t e r s . 4,s I n d e f o r m a t i o n m e c h a n i s m s t u d i e s d i f f e r e n t equations h ave b e e n used to d e s c r i b e d i f f e r e n t s t r a i n r e g i o n s of t he u n i f o r m s t r a i n . 6'7 A v e r y c o n v e n i e n t method for f i n d i n g s t a g e s in the s t r e s s - s t r a i n c u r v e s is the C r u s s a r d a - J a o u l 9 (C J) a n a l y s i s w h e r e d a / d e - e - v a l u e s are p lotted in l o g - l o g c o o r d i n a t e s . R e c e n t l y H e e d - H i l l e t a l . I~ h ave p r e s e n t e d a modified C J - m e t h o d for a n a l y z i ng the s t a g e s . T he a i m of the p r e s e n t work was to study the a p p l i c a b i l i t y of four widely used s t r e s s - s t r a i n r e l a t i o n s for t he e s t i m a t i o n of the u n i f o r m s t r a i n ( i . e . , f o r m a b i l i t y ) of m e t a l s . C OMPARISON OF THE STRESS-STRAIN R E LATIONS S e v e r a l e m p i r i c a l equations have b e e n used to d e s c r i b e e x p e r i m e n t a l s t r e s s - s t r a i n c u r v e s , but h e r e H. J. KLEEMOLA is Senior Fellow of the Finnish Academy, Departm ent of Mining and Metallurgy, Helsinki University of Technology, O taniemi, Finland. M. A. NIEMINEN is Research Metallurgist, Metal L aboratory, Pori Works, Outokumpu Oy, Pori, Finland. M anuscript submitted June 8, 1973. M ETALLURGICAL TRANSACTIONS o nly the following four which a r e the m o s t c o m m o n are treated: a = a o + he m ( Ludwik) 11 [1] a = K 1e n t (Hollomon) 1 [2] a = K z (e + Co)n2 (Swift) 1~ [3] = B - (B - A ) exp ( - n s e ) (Voce) Is [4] w here a and e a r e the t r u e s t r e s s and t r u e p l a s t i c s t r a i n r e s p e c t i v e l y , and the other p a r a m e t e r s a r e c o n s t a n t s . The d o u b l e - n method for s t e e l s i n t r o d u c e d b y M o r r i s o n 6 c o n s i s t s of the use of the two Hollomon e q u a t i o n s . Man e t a / . T d e s c r i b e d the t e n s i l e c u r v e s of copper u s i n g a s i m i l a r method, but u s i n g two equations of type. ~ T his kind of a n a l y s i s is b a s e d on the a s s u m p t i o n that a c hange in d e f o r m a t i o n m e c h a n i s m o c c u r s d u r i n g d e f o r m a t i o n . Zankl, 14 and Schwink and V o r b r u g g Is found t h e s e k ind of stages a l s o in the t e n s i l e c u r v e s of a n n e a l e d n i c k e l and copper at low s t r a i n s . T he u n i f o r m s t r a i n of the equations m e n t i o n e d above c an be c a l c u l a t e d u s i n g the r e l a t i o n d a / d E = a. C OMPUTATIONAL METHODS WITH E XAMPLES a ) C o m p u t a t i o n a l Methods T he solution of the p a r a m e t e r s of Eq. [2] is the b a s i s o f the c u r r e n t c a l c u l a t i o n methods. This equation can b e l i n e a r i z e d by taking l o g a r i t h m s on both s i d e s and t hus a l i n e a r r e g r e s s i o n line can be d e t e r m i n e d and t he p a r a m e t e r s KI and nl, and the coefficient of l i n e a r c o r r e l a t i o n , r , a r e e a s i l y calculated. The d e t e r m i n a t ion of the p a r a m e t e r s of Eq. [1] is p o s s i b l e by a n i t e r a t i v e method. .6 F i r s t a o is moved to the left side and l o g a r i t h m s a r e t a k e n on both s i d e s , then the a e v a l u e s a r e e s t i m a t e d and the c o r r e s p o n d i n g v a l u e s of h and m a r e c a l c u l a t e d b y the r e g r e s s i o n method m e n t i o n e d a bove. Then in the v i c i n i t y of the a o v a l u e giving the l e a s t s u m of s q u a r e s of d e v i a t i o n s a new c a l c u l a t i o n r o u n d u s i n g a s m a l l e r step is p e r f o r m e d . A s i m i l a r p r o g r a m m i n g can be used to solve the p a r a m e t e r s of Eq. [3], but in this case e o v a l u e s a r e e s timated. E q. [4] can be r e a r r a n g e d to give In(a-B) =In (A-B)-n se [5] VOLUME 5, AUGUST 1974-1863 w hich is again a linear expression when A and B are c onstants, and permits the determination of n s from t he experimental data by the iterative least squares m ethod when both A and B are estimated. T he calculation methods were applied to the analys is of the s t r e s s - s t r a i n Curves of steel and copper u sing an HP2000 B computer. Some of the results are t reated at the end of this section. b ) Examples I n Fig. 1 the calculated s t r e s s - s t r a i n curves of Eqs. [1], [3], and [4] have been used as " e x p e r i m e n t a l " data f or Eq. [2]. The solid lines are " e x p e r i m e n t a l " curves, r epresenting 40 points which were calculated by the e quations given in the figure. A constant step in the l ogarithm of strain in the range of 0.005 to 0.3 was app lied. The Hollomon equation was fitted to them using t he method mentioned above. The p a r a m e t e r s obtained, t he corresponding ~u value, and the linear correlation c oefficient, r, have been given in the figure. The der ivatives of the " e x p e r i m e n t a l " and Hollomon curves h ave also been drawn to reveal the difference in the u niform strains determined by the equations. I n the strain range investigated, the e u v alues of Eqs. [1], [3], and [4] differ by 10 to 35 pct from those of the H ollomon equation (n = eu) calculated from the same d ata points. However, high linear correlation coeffic ients (better than 0.99) were obtained, in the cases i nvestigated, when the calculated data of the Ludwik, Swift, and Voce equations were described by Hollomon e quations. It is expected that the Hollomon equation w ill fit with high correlation to the data of Eqs. [1] to [4] when typical p a r a m e t e r s and strain range are app lied. The use of the Hollomon equation gives a misl eading picture of the strain-hardening properties of t he material because the strain-hardening exponent is n ot equal to the c o r r e c t Eu value. S t r e s s - s t r a i n curves for pure copper in an annealed s tate and after 40 pet deformation are given in Figs. 2 a nd 3 respectively. The p a r a m e t e r s of the curves in t he figures were calculated by Eqs. [1] to [4]. The c o r r esponding e u v alues, sums of the squares of deviations (SSD), and standard e r r o r s of estimate (SE) have been l isted in Table I. T he " m e a s u r e d " eu value was determined by applyi ng a least squares parabola fit to the last part of the l oad-elongation curve. Then Eu is the strain at the m aximum point of the parabola. The best correlation w ith the experimental data in these cases is obtained b y Eq. [4], the others being quite similar and not c a p able of describing the data accurately. For example t he Hollomon equation, when applied to the deformed c opper, gives a uniform strain which is 40 times g reater than the measured value (Table I). T his is depicted in Figs. 4 and 5 where the data of F igs. 2 and 3 are presented using the CJ analysis. The d erivative curves were determined using the c o r r e s ponding equations of Figs. 2 and 3. It is clearly seen t hat Eqs. [1] to [3] are not capable of giving the c o r r e c t u niform strain, although they have good correlation par a m e t e r s . These equations gave too high values of Eu w hich means a too optimistic value of formability e s 6 00 5 00 -- 0 = 2 0 0 + 5 0 0 E 02 Cu = 0.12/, d = 6 70.651C 0~131)3 ,~00 ~E r = 0 .999 300 -- 0 =SOO(e+O.01) ~" o ~ . e u = 0 190 2 00 ~~" Or = 0 ./9 9= / * ~ 8 ' 1 ~ ~ E ~ ~ 3 0~ 0 = 300-(300-100)expl-10r Cu = 0.199 . . 0 = /,0"/.215s ~ 9 / /J" ~ / r = 0 .092 1 00 I [ I I'lllLI 0 .001 I I I I IIIll I 0.01 I I I I II 0.1 e: F i g . 1 - - E x a m p l e of how t he H o l l o m o n e q u a t i o n f i t s th e L u d w i k , S wift, and V oc e e q u a t i o n s . 4 00 3 00 -. o nneoled copplr --H ollomon, Swift 200 .... / ~.~,o;,,k /" . ++ ff/ 1 00 50 0.001 ," i 0.01 0.1 e: F i g . 2 - - A n a l y s i s of a s t r e s s - s t r a i n u s i n g E q s . [1] t h r o u g h [4]. c u r v e of a n n e a l e d c o p p e r Table I. The Uniform Strain and Accuracy of Fit Values of Figs. 2 and 3 ~,OC M easured Fig. 2 Annealed Copper Uniform strain SSD SE Fig. 3 Deformed Copper Uniform strain SSD SE Ludwik Hollomon Swift Voce 30C 0.340 - 0.434 3.943 0.375 0.449 4.463 0.386 0.447 0.302 4.305 0.897 0.386 10.179 9 . 2 0C ~ / I --- Holtomon, swift ,o 0.00309 - 0.0983 31.262 0.945 0.134 36.707 0.996 0.132 35.582 1.008 0.00204 4.764 0.369 SSD is the sum of the squares of deviations and SE the standard error of estimate. 1 8 6 4 - V O L U M E 5,AUGUST 1974 1 00 1 0-s , , ,,~,,,, 10-5 , , R,,,,,, 10-~ h , ,,,,,,, , 10-3 , ,,,,,~ 10-2 r F i g. 3 - - A n a l y s i s of a s t r e s s - s t r a i n u s i n g E q s . [1] t h r o u g h [4]. c u r v e of d e f o r m e d c o p p e r METALLURGICAL TRANSACTIONS 2 500 2 000 I 10 ? S wif t ond Hottomo 1500 Voce Ludwik o //2 dd o 1 000 ~o z 500 ~ 300 "D ~ r o 200 ~ 9 V oce 10 s Swift and Hottomon u" 100 9 10~" 50 i 0 .001 i , ,e,,,,, , i , , ,1,,, 0.01 i i Ludwik io i iiiL 0,1 r F ig. 4--Demonstrating the d(~/de v alues of the data of Fig. 2. p e c i a l l y in the d e f o r m e d s t a t e . The Voce equation had t he b e s t fit to the e x p e r i m e n t a l p o i n t s ( F i g s . 2 and 3) i n t h e s e c a s e s and it a l s o gave the m o s t c o r r e c t eu ( Table I). T he d e r i v a t i v e s of the e x p e r i m e n t a l p o i n t s show quite a c ontinuous change with s t r a i n without any d i s t i n c t s t a g e s . H o w e v e r , c h a n g e s in the d e f o r m a t i o n m e c h a n i s m w e r e not the s u b j e c t of the p r e s e n t w o r k . W hen the s t r e s s - s t r a i n c u r v e s of l o w - c a r b o n s t e e l s , w hich w e r e m o s t l y of the Hollomon t y p e , w e r e f i t t e d to E q. [4], it was not p o s s i b l e to find r e a s o n a b l e p a r a m e t e r s with a low s u m of s q u a r e s of d e v i a t i o n s . D ISCUSSION A s u r v e y of the l i t e r a t u r e r e v e a l s s o m e m i s c o n c e p t i o n s c o n c e r n i n g the a p p l i c a t i o n of the Hollomon and L udwik e q u a t i o n s . It was s t a t e d b y M o n t e i r o and R e e d H ill 1~ that b y t a k i n g the d e r i v a t i v e of both e x p r e s s i o n s , t he v a l u e of n can be o b t a i n e d i n d e p e n d e n t l y of the e x i s t e n c e of a o. As shown above, h o w e v e r , this is not c o r r e c t b e c a u s e the two exponents a r e not e q u i v a l e n t . This i s b e c a u s e the exponent of the Hollomon equation is e qual to the u n i f o r m s t r a i n while that of the Ludwik e quation is not. The w o r k of M o n t e i r o and R e e d - H i l l w as d i s c u s s e d b y M o r r i s o n , la who a r g u e d that a o is z e r o when the Hollomon equation d e s c r i b e s the s t r e s s strain data satisfactorily. Furthermore, Morrison ment i o n e d that m o s t t e n s i l e d a t a can b e f i t t e d e i t h e r to the H ollomon o r to the Ludwik equation, but in the o t h e r c a s e s both a r e i n a d e q u a t e . Ono ~~ s t a t e d that the H o l l o m o n equation can not a p p r o x i m a t e 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 , including the Ludwik e q u a t ion. However, a s shown in F i g . 1, a Ludwik equation c an b e d e s c r i b e d quite a c c u r a t e l y b y a Hollomon e q u a t ion o v e r a t y p i c a l r a n g e of s t r a i n . T h e r e is a v e r y s m a l l d i f f e r e n c e b e t w e e n a = 200 + 500 co.2 and a = 6 70.651~~ a c o r r e l a t i o n c o e f f i c i e n t 0.998796 w as o b t a i n e d in t h i s c a s e . It should b e e m p h a s i z e d t hat the exponents a r e d i f f e r e n t and a o d i f f e r s f r o m z e r o (a o = 200 M N / m 2) even though a Hollomon e q u a t ion f i t s the d a t a with a high l i n e a r c o r r e l a t i o n c o e f f i c ient. T he v a l u e s of the p a r a m e t e r s in the H o l l o m o n e q u a t ion depend on the s t r a i n r a n g e u s e d in the c a l c u l a t i o n s . T h i s is g e n e r a l l y t r u e when an equation is not a s t r a i g h t l lne u s i n g l o g - l o g c o o r d i n a t e s . Short s t r a i n i n c r e m e n t s c an be v e r y a c c u r a t e l y d e s c r i b e d b y the Hollomon e x METALLURGICAL TRANSACTIONS 1 03 10 ~ i 1 0-~ I i Jl[ilJ i i iiiiiii 10-S i i Iltlill 10 -~ i 10-3 i iiiJiii 10-2 i i i iii 10 -1 c F ig. 5--Demonstrating the da/de v alues of the data of Fig, 3. p r e s s i o n , but then the exponent does not give the c o r r e c t u n i f o r m s t r a i n . Only when the s t r a i n i n c r e m e n t i s s u f f i c i e n t l y s m a l l and c o n t a i n s the a c t u a l u n i f o r m s t r a i n point d o e s the Hollomon exponent a p p r o x i m a t e t he v a l u e of eu. However, it is not known b e f o r e h a n d w hat the c o r r e c t s t r a i n i n c r e m e n t is ( F i g s . 4 and 5). W hen an e x p e r i m e n t a l c u r v e is d e s c r i b e d b y d i f f e r e nt e q u a t i o n s e a c h will give a d i f f e r e n t Cu v a l u e (exa m p l e s of T a b l e I), although the fit is good f o r a l l the e q u a t i o n s . Thus a s e e m i n g l y high c o r r e l a t i o n d o e s not g u a r a n t e e a c o r r e c t u n i f o r m s t r a i n and a s i g n i f i c a n t l y b e t t e r e s t i m a t i o n of f o r m a b i l i t y m a y s t i l l b e o b t a i n e d b y o t h e r f o r m u l a e (Table I). E s p e c i a l l y in the s t u d y of t he r e s i d u a l d u c t i l i t y of d e f o r m e d m e t a l s ( e x a m p l e in F ig. 3) m i s l e a d i n g l y high u n i f o r m s t r a i n s m a y be o b t a i n e d using the H o l l o m o n equation. However, C a i r n s e t alfl w e r e s t i l l a b l e to c o r r e l a t e c h a n g e s in t h e i r u n i f o r m s t r a i n v a l u e s with c h a n g e s in the m i c r o s t r u c t u r e after heavy deformation. S UMMARY AND CONC LUSIONS I n the p r e s e n t w o r k the a p p l i c a b i l i t y of the four m o s t w i d e l y u s e d e m p i r i c a l s t r a i n - h a r d e n i n g e q u a t i o n s to the u n i f o r m s t r a i n d e t e r m i n a t i o n was i n v e s t i g a t e d . C o m p u t a t i o n a l m e t h o d s w e r e u s e d f o r the d e t e r m i n a t i o n of the p a r a m e t e r s f r o m e x p e r i m e n t a l d a t a and two e x a m p l e s o f the a n a l y s i s of s t r e s s - s t r a i n c u r v e s w e r e p r e s e n t e d . I t was shown that the s t r a i n - h a r d e n i n g equations i n v e s t i g a t e d give d i f f e r e n t v a l u e s of u n i f o r m s t r a i n even i f t h e y s e e m to fit e x p e r i m e n t a l d a t a v e r y c l o s e l y . The m o s t c o r r e c t v a l u e i s o b t a i n e d with the equation giving t he s m a l l e s t s t a n d a r d e r r o r of e s t i m a t e . F o r d e s c r i b i ng the s t r e s s - s t r a i n c u r v e s of a n n e a l e d and d e f o r m e d c o p p e r the Voce equation is b e s t . The Hollomon e q u a t ion g i v e s a v e r y i n c o r r e c t m e a s u r e of the r e s i d u a l d u c t i l i t y of a m a t e r i a l e s p e c i a l l y when a p p l i e d to the a n a l y s i s of the s t r e s s - s t r a i n c u r v e s of d e f o r m e d s p e c i m e n s . T his h a s a l w a y s to b e t a k e n into account, when the n v a l u e is u s e d f o r the a s s e s s m e n t of f o r m a b i l i t y . T he l e s s c o m p l e x Hollomon equation is b e t t e r than t he Voce equation f o r the d e s c r i p t i o n of the c u r v e s of steel. VOLUME 5, AUGUST 1974-1865 REFERENCES I . J. H. Hollomon: Trans. AIME, 1945, vol. 162, pp. 268-90. 2. J. H. Cairns, J. Clough, M. A. P. Dewey, and J. Nutting: J. Inst. Metals, 1971, vol. 99, pp. 93-97. 3. J. L. Duncan: S heet Metal lnd., 1967, vol. 44, pp. 482-89. 4 . T. Gladman, B. Holmes, and F. B. Pickering: J. Iron Steellnst., 1 970, vol. 2 08, pp. 172-83. 5. D. C. Ludwigson: Met. Trans., 1 971, vol. 2, pp. 2825-28. 6. W. B. Morrison: Trans. ASM, 1 966, vol. 59, pp. 824-46. 7. J. Man, M. Hohmann, and B. Vlach: Phys. Status Solidi, 1 967, vol. 19, p p. 543-53. 8. C. Crussard: Rev. Met., Paris, 1953, vol. 10, pp. 697-710. 1 8 6 6 - V O L U M E 5, AUGUST 1974 9. B. Jaoul: J. Mech. Phys. Sol., 1957, vol. 5, pp. 95-114. 10. R. E. Reed-Hill, W. R. Cribb, and S. N. Monteiro: Met. Trans., 1 973, vol. 4, p p. 2665-67. 11. P. Ludwik: Elemente der Technologischen Mechanik, V erlag yon Julius S pringer, Berlin, 1909. 12. H. W. Swift: J. Mech. Phys. Solids, 1952, vol. 1, pp. 1-18. 13. E. Voce: J. Inst. Metals, 1948, vol. 74, pp. 537-62, 14. G. Zankh Z. Naturforsch., 1963, vol. 18a, pp. 795-809. 15. Ch. Schwink and W. Vorbrugg: Z. Naturforsch., 1 967, vol. 22a, pp. 626-42. 16. W.Oldfield and A. J. Markworth: S criptaMet., 1971, vol. 5, pp. 235-40. 17. S. N. Monteiro and R. E. Reed-Hill: Met. Trans., 1 971, vol. 2, pp. 2947-48. 18. W. B. Morrison: Met. Trans., 1 971, vol. 2, pp. 2948-49. 19. K. Ono: Met, Trans., 1 972, vol. 3, pp. 749-51. METALLURGICAL TRANSACTIONS ...
View Full Document

Ask a homework question - tutors are online