Some aspects of anisotropic plasticity in sheet metals

CoA NOTE MAT. No. 15

Kfuvv^-?y6. HS DELFT

THE COLLEGE OF A E R O N A U T I C S

C R A N F I E L D

SOME ASPECTS OF ANISOTROPIC PLASTICITY IN SHEET METALS by

R. P e a r c e

THE_COLLEGE_OF AERONAUTICS DEPAROMEWT OF MATERIALS Some a s p e c t s of a n i s o t r o p i c p l a s t i c i t y i n s h e e t m e t a l s by -Roger P e a r c e , B . A . , B . S c , F . I . M . S_U_M_M_A R_Y

The p r e d i c t i o n of t h e y i e l d i n g and flow b e h a v i o u r of m a t e r i a l s u n d e r coïïiplex s t r e s s systems from t e n s i l e t e s t o r o t h e r e a s i l y d e t e r m i n e d d a t a h a s b e e n t h e aim of e n g i n e e r s f o r many y e a r s . The y i e l d c r i t e r i a of T r e s c a and then Mises f o r i s o t r o p i c m e t a l s a r e u s e f u l , b u t t h e r e a l i s a t i o n t h a t a n i s o t r o p y i s t h e r u l e r a t h e r t h a n t h e e x c e p t i o n , e s p e c i a l l y i n s h e e t m e t a l s l e d t o t h e exajTiination of H i l l ' s a n i s o t r o p i c t h e o r y by v a r i o u s w o r k e r s . I n the p r e s e n t p a p e r t h e s t r e s s - s t r a i n c u r v e s of v a r i o u s s h e e t m e t a l s a r e

d e t e r m i n e d i n u n i a x i a l and b a l a n c e d b i a x i a l t e n s i o n . As f a r as y i e l d i n g b e h a v i o u r i s concerned i t i s concluded t h a t the t h e o r y i s r e a s o n a b l y s a t i s -f a c t o r y -f o r m a t e r i a l s where a n i s o t r o p y i s d e s c r i b e d w i t h r > 1 , w i t h c e r t a i n anomalies f o r m a t e r i a l s w i t h r < 1 . As f a r a s flow b e h a v i o u r i s c o n c e r n e d , t h e t h e o r y o n l y a p p l i e s f o r m a t e r i a l s f o r r > 1 . C r o s s i n g of the u n i a x i a l and b i a x i a l c u r v e s i s o b s e r v e d f o r c e r t a i n m e t a l s a t low s t r a i n s and t h i s i s n o t p r e d i c t e d by t h e t h e o i y . More work i s n e c e s s a r y on l o w - r m a t e r i a l s t o r e s o l v e t h e s e m a t t e r s .

C o n t e n t s Page No. Svanmary I n t r o d u c t i o n 1 E x p e r i m e n t a l 3 R e s u l t s ^ C o n c l u s i o n s ^ Ac knowle dgeme nt s 5 T a b l e ^ R e f e r e n c e s T F i g u r e s

1

-Introduction

The mechanical p r o p e r t i e s of sheet metal can vary with the d i r e c t i o n of t e s t i n g , not l e a s t in the thro ugh-thiclcness d i r e c t i o n . Anistrojy can be due to mechanical f i b e r i n g - i n c l u s i o n s , p o r o s i t y - b u t , t h i s i s not the concern of t h i s paper. P l a s t i c anisotropy - flow s t r e s s , work-hardening behavioior - which r e s u l t s from c r y s t a l l o g r a p h i c preferred o r i e n t a t i o n , giving the metal a ' t e x t u r e ' , can be varied in a sheet metal by a l t e r i n g the sequence and nature of the thermal and mechanical treatments T*iich are used in i t s manxifacture . In ffeict when i t i s r e a l i s e d t h a t in r o l l i n g a c o i l from an i n g o t , the dispostion of the metal to the processing machinery i s generally unchanged throughout the defomiation and this consists of

reducing the metal thickness by a factor of ~ 500 i t i s not surprising that anisotropy i s the r u l e r a t h e r than the exception.

Conventional t e n s i l e t e s t i n g w i l l not r e v e a l p l a s t i c anisotropy as defined above. However, measurements of the changes in width s t r a i n and thickness s t r a i n during u n i a x i a l p l a s t i c deformation w i l l i n d i c a t e a n i s o -tropy, and t h e i r r a t i o ;

e w

i s called the s t r a i n r a t i o , or commonly the r v a l u e . I f t h i s value i s unity the m a t e r i a l i s i s o t r o p i c and deviation i s recognised as anisotropy.

Variation i n r with d i r e c t i o n of t e s t i n g i n the sheet plane i s termed planar anisotropy (Ar) and manifests i t s e l f as ' earing' in a drawn axisymmetrical cup.

I n metals with r > 1 an increased r e s i s t a n c e t o yielding in b i a x i a l tension i s found and the converse for r < 1 m a t e r i a l s . Hosford and Backofen-'-have coined the phrases ' t e x t u r e hardening' and'texture softening to describe these phenomena. C l e a r l y , the number of s l i p systems which can operate w i l l affect the value of the s t r a i n r a t i o a t t a i n a b l e . A m a t e r i a l with an i n f i n i t y of systems w i l l be i n e v i t a b l y i s o t r o p i c , while a metal with one aLip d i r e c t i o n , i n one s l i p plane, and t h i s , say, aligned p a r a l l e l to the plane of t h e sheet w i l l show r = w. Additionally, t h i s metal would give r t= o for a specimen

cut normal to the sheet surface. Face-centred cubic metals deform p l a s t i c a l l y on 12 equivalent systems, and in conmercial p o l y c r y s t a l l i n e sheet, r = 1

max

can be obtained^. G3fterally, the s t r a i n r a t i o value i s around 0.6^.

In body-centred cubic metals, four s l i p d i r e c t i o n s can 'choose' fran a packet of s l i p planes^ a l l of s i m i l a r type. Although more s l i p systems can be w r i t t e n down for b . c . c . deformation, the modes are in f a c t more r e s t r i c t i v e

than in f . c . c . m e t a l s , and consequently higher and lower s t r a i n r a t i o s can be obtained. Values from O.J to 2.2 have been measured for extra-deep-drawing s t e e l s ^ . In close-packed hexagonal metals a s i n g l e s l i p d i r e c t i o n i s normally operative in one of three p l a n e s , and so, depending on the angle between the s l i p d i r e c t i o n and the sheet p l a n e , very high of' very low s t r a i n -r a t i o s w i l l "he obse-rved. Values of 0.12 have been -repo-rted in zinc^, while in zirconium values as high as 9-0 have been quoted^.

2

-Texture hai^ening i s a useful phenomenon. A recent study conducted by an a i r c r a f t company and reported by Babel and Corn"^showed that a ik.J'fo weight saving could be achieved by a redesign involving a change from an aluminiiom alloy t o a titanium one . However, by taking into account the b i a x i a l p r o p e r t i e s of titanium i n a d d i t i o n , t h i s saving could be increased to 21.C^o.

The purpose of the present work i s to i n v e s t i g a t e the behaviour of a number of metals exhibiting a wide range of s t r a i n r a t i o s in u n i a x i a l and b i a x i a l tension to t e s t present anisotropic p l a s t i c i t y theory, and to show

the advantages and disadvantages of t h i s phenomenon in engineering a p p l i c a t i o n s

The Mises c r i t e r i o n s t a t e s :

( a i - a^)^ + (a^ - a^)^ + (a^ - a j ^ ^ g a ^ ( l )

where a i , era and 03 are the s t r e s s e s along orthogonal axes as shoim in Figure l a and a^ i s the y i e l d s t r e s s in u n i a x i a l tension. Far sheet loading (03 = O) t h i s reduces to:

erf - aiaa + of = a^^ (2)

H i l l ' s y i e l d c r i t e r i a for anisotropic m a t e r i a l s (again a^ = O) can be w r i t t e n :

a^ + ff2 - <^i°'2 [ 7 ) = ^ } where r i s the s t r a i n r a t i o previously

discussed. This equation assumes planar isotropy, i . e . Ar = 0 which is not t r u e for mar^y m e t a l s . The problem of a value for r in a b i a x i a l s i t u a t i o n then must be solved, and a t preser.t the most s a t i s f a c t o r y solution is to define i t as:

r + 2r45 + rso

r = _k

(M

or some v a r i a n t on t h i s theme depending on the number of d i r e c t i o n s in which t e s t s are m.ade. Figure lb shows p l a n e - s t r e s s yiei.d l o c i for various values of r(Ar = 0) and the t e x t u r e hardening i n the t e n s i o n - t e n s i o n quadrant and textujre softening i n the tension-compression quadrant should be noted.

The effect of r on y i e l d and flow in u n i a x i a l and balanced b i a x i a l tension i s the subject of the present i n v e s t i g a t i o n , and for a s t r e s s r a t i o

'la

(5)

of unity, it can be shown that;

% /rfl

a

y

where a, is üie biaxial yield stress. b

^3 = , ^ 3 = ,

J^

1

2

I t should be remembered t h a t yield c r i t e r i a are formulated for i d e a l e l a s t i c / p l a s t i c m a t e r i a l s ; the p r a c t i c a l determination of a^^ and Oy -e s p -e c i a l l y th-e fonn-er - ar-e v i r t u a l l y impossibl-e; hov/-ev-er, i t i s possibl-e to talce values of s t r e s s at low s t r a i n s and measure ti:e r a t i o of proof strengths a t a specified s t r a i n .

V/ith regard to the p o s t - y i e l d p a r t of tiie s t r e s s - s t r a i n curves, Mellor and Braraley^ave suggested the p r e d i c t i o n of the balanced b i a x i a l curve from the u n i a x i a l data using the following expressions:

(6)

(7)

a i s the average true s t r e s s a t s t r a i n ^^y, the average t r u e s t r a i n ,

the averages being determined fran the u n i a x i a l data and 'weighted' according to an analogous foimula to t h a t for r .

Experimental

The uniaicial t e n s i l e t e s t s were carried out on an I n s t r o n machine a t a crosshead speed of 0.2 in/uin-. Specimens were machined according to BSl8, and a load-extension curve was dra-i-m, using a 2" gauge-length strain-gauge extensometer on the specimen up to f r a c t u r e . a was c a l c u l a t e d from the r e l a t i o n s h i p :

a = | - (1+e) o

where P i s the instantaneous load, AQ the o r i g i n a l c r o s s - s e c t i o n a l area and e the engineering s t r a i n , and e

from:-e = In (l+from:-e)

The b i a x i a l curves were detennined using a Mand Precision Engineering Co. L t d . , t e s t extensometer, developed by Johnson and Duneanywhere o i s

computed from:

"-i

where p i s the radius of cu-rvatiure and t the current thiclaiess, and e from:

~ k

-Results

Tlie r e s u l t s are shown in Figures 2 , 3 , h, 5 , 6 , 7, S,^9 and 10. I t w i l l be seen (Figure 2) that the r e l a t i o n s h i p between J^ and r

«Jy

( a t a s t r a i n of O.Ol) i s similar in shape to the predicted one, but displaced to a higher s t r e s s l e v e l . The exceptions here are the tv/o samples of

commercial-purity aluminl.um t e s t e d , one soft and one h a r d - r o l l e d . In Figi-'.res 3 through 10 the experimental and predicted curves are drawn. For materials with r > 1 the predicted c\irves are quite good approximations to t h e determined ones, e s p e c i a l l y titani-um and zirconiimi. For r < 1 the discrepancies are very marked. In no cases are the b i a x i a l

curves below the u n i a x i a l , as p r e d i c t e d , but in some cases the b i a x i a l curves s t a r t lower and then c r o s s . In the case of commercial-purity z i n c , thu u n i a x i a l curves themselves are so widely spaced t h a t the use of an averaging procedure i s a doubtful operation.

Conclusions

Present anisotropic p l a s t i c i t y theory f a l l s down for m a t e r i a l s exhibiting r < 1 arid even for m a t e r i a l s of r - 1; the superposition of u n i a x i a l and

b i a x i a l curves i s not good. In general, t h e flow curve i n b i a x i a l tension f a l l s above the u n i a x i a l one regardless of the s t a t e of anisotropy. However, a t the low - s t r a i n end of the curves there i s a tendency for the curves to c r o s s , e s p e c i a l l y the curves for the low-r m a t e r i a l s . Explanations for t h i s must be m i c r o s t r u c t u r a l r a t h e r than phenomenological and the following

t e n t a t i v e q u a l i t a t i v e p i c t u r e i s offered. I t i s permissible to equate

balanced b i a x i a l tension with through-thickness compre.^sion plus a h y d r o s t a t i c tension which does not affect y i e l d i n g . I n a lowr metal the operative s l i p -systems w i l l tend to be aligned t o cause s t r a i n in the thickness d i r e c t i o n and so under t h i s s t r e s s sytem s l i p w i l l occur readily on many planes, many d i s l o c a t i o n s w i l l move and entangle and a high i n i t i a l r a t e of woik-hardening w i l l r e s u l t .

I t i s not possible to allow for t h i s sort of behaviour in p l a s t i c i t y oheory. In the case of z i n c , i t w i l l be observed that the ^, ^, cur\re f a l l s a t high s t r a i n s . This i s due to the formation of voids during s t r a i n i n g ( F i g . l l ) and the consequent v i o l a t i o n of the volume-constancy exiom. .

Again, during deformation - p a r t i c u l a r l y of face-centred cubic metals, g r a i n r o t a t i o n occurs, and r may a l t e r during defoimation. This i s another uncharted v a r i a b l e .

A liiflitation of these experiments i s the i n a b i l i t y to measure b i a x i a l s t r e s s - f s t r a i n a t very low s t r a i n s , due to t h e uncertainty of radius of curvature measurements a t low bulging p r e s s u r e s .

I t should be remembered t h a t yield c r i t e r i a are formulated for i d e a l e l a s t i c / p l a s t i c m a t e r i a l s ; the p r a c t i c a l determination of a.^ and Oy -e s p -e c i a l l y th-e form-er - ar-e v i r t u a l l y impossibl-e; hcn/-ev-er, i t i s possibl-e to talce values of s t r e s s at low s t r a i n s and measure tlie r a t i o of proof s t r e n g t h s a t a specified s t r a i n .

With regard to the p o s t - y i e l d p a r t of the s t r e s s - s t r a i n curves, Mellor and Braraley^are suggested the p r e d i c t i o n of the balanced b i a x i a l curve from the u n i a x i a l data using the following expressions:

<^3

=

J^-^

•

V

(^^

• e (7)

av ^ '

a i s the average true s t r e s s a t s t r a i n ^gy, the average t r u e s t r a i n ,

the averages being determined fran the u n i a x i a l data and 'weighted' according to an analogous formula t o that for r .

Experimental

The ujiiaicial t e n s i l e t e s t s were c a r r i e d out on an I n s t r o n machine a t a crosshead speed of 0.2 in/rain-. Specimens were machined according to BSl8, and a load-extension curve was dra>ni, using a 2" gauge-length strain-gauge extensometer on the specimen up to f r a c t u r e . a was c a l c u l a t e d from the r e l a t i o n s h i p :

0 = | - (l+e)

o

where P i s the instantaneous load, AQ the o r i g i n a l c r o s s - s e c t i o n a l area and e the engineering s t r a i n , and e

from:-e = In (l+from:-e)

The b i a x i a l curves were determined using a Mand Precision Engineering Co. L t d . , t e s t extensometer, developed by Johnson and Duncan^where o i s computed from:

0 = ££ 2t

where p i s the radius of curvature and t the cu-rrent thiclmess, and e from:

e = In •^d

„ 1;

-Results

The r e s u l t s are shown in Pigvires 2 , 3> ^} 5, 6, 7; 3;a9 ^^^ lO» I t w i l l be seen (Figure 2) that the r e l a t i o n s h i p between _k and r

( a t a s t r a i n of O.Ol) i s similar in shape to the predicted one, but displaced to a higher s t r e s s l e v e l . The exceptions here are the tv;o samples of

conmercial-purity aliauini.um t e s t e d , one soft and one h a r d - r o l l e d .

In Figures 3 through 10 the experimental and predicted curves are drawn. For materials with r > 1 the predicted curves a r e quite good approximations to t h e determined ones, e s p e c i a l l y titanium and zirconium. For r < 1 the discrepancies are very marked. In no cases are the b i a x i a l

curves below the u n i a x i a l , as p r e d i c t e d , but in some cases the b i a x i a l curves s t a r t lower and then c r o s s . In the case of commercial-p-urity z i n c , the u n i a x i a l cxxrves themselves are so widely spaced t h a t the use of an averaging procedure i s a doubtful o p e r a t i o n .

Conclusions

Present anisotropic p l a s t i c i t y theory f a l l s down fcr m a t e r i a l s exhibiting r < 1 arid even for m a t e r i a l s of r ~ 1; the superposition of u n i a x i a l and

b i a x i a l curves i s not good. In general, t h e flow curve in b i a x i a l tension f a l l s above the u n i a x i a l one regardless of the s t a t e of anisotropy. However, a t the low - s t r a i n end of the curves there i s a tendency for the curves to c r o s s , e s p e c i a l l y the curves for the low-r m a t e r i a l s . Explanations for t h i s must be m i c r o s t r u c t u r a l r a t h e r than phenomenological and the following

t e n t a t i v e q u a l i t a t i v e p i c t u r e i s offered. I t i s permissible to equate

balanced b i a x i a l tension with through-thickness compression plus a h y d r o s t a t i c tension which does not affect y i e l d i n g . In a lowr metal the operative s l i p -systeras w i l l tend to be aligned t o cause s t r a i n in the thickness d i r e c t i o n and so under t h i s s t r e s s sytem s l i p w i l l occur readily on many planes, many d i s l o c a t i o n s w i l l move and entangle and a high i n i t i a l r a t e of woric-hardening w i l l r e s u l t .

I t i s not possible to allow for t h i s sort of behaviour in p l a s t i c i t y Gheory. In the case of z i n c , i t w i l l be observed that the i^, ^ , curve f a l l s a t high s t r a i n s . This i s due to the formation of voids during s t r a i n i n g ( F i g . l l ) and the consequent v i o l a t i o n of the volume-constancy &x:iom. ,

Again, during deformation - p a r t i c u l a r l y of face-centred cubic metals, g r a i n r o t a t i o n o c c u r s , and r may a l t e r during defoimation. This i s another uncharted v a r i a b l e .

A l i m i t a t i o n of these experiments i s the i n a b i l i t y to measure b i a x i a l s t r e s s - s t r a i n a t very low s t r a i n s , due to the lancertainty of radius of curvature measurements a t low bulging pressiAres.

w i l l r a i s e the b i a x i a l flow curve, and in p r a c t i c a l engineering applications i t i s in t h i s d i r e c t i o n that the b e n e f i t l i e s ; approximate c a l c u l a t i o n s of

a from a and r would be in order. An a d d i t i o n a l benefit i s that cr i s ,

b y - °

in genei-al, most l i k e l y to be g r e a t e r than a f o r a l l m e t a l s , but here a^ must be measured experimentally. However, "^the bulge t e s t ptrocedurc used

i n t h i s work i s not tedious and tends t o be quite reproducible .

Some of those i n t e r e s t e d in sheet metal formability and p a r t i c u l a r l y the f o r n a b i l i t y of sheet s t e e l regard n, the exponent in the expression a = Ke , as a useful parameter. This can be obtained q u i t e e a s i l y from the tia3cial curves (those which conform to t h i s r e l a t i o n s h i p ) and in general appears to be of s i m i l a r value to t h a t obtained in uniaxial t e n s i o n . Maximum s t r a i n s obtained in t h i s work accord approximately with H i l l ' s p r e d i c t i o n for a circ\ilar diaphragm and are given in Table 1. The good agreement of tlie mild s t e e l s in p a r t i c u l a r should be noted.

More work i s needed p a r t i c u l a r l y with metals of r ^ 1. Bott and Pearce shewed t h a t , with commercial-purity aluminium, q u i t e l a r g e changes in s t r a i n could be affected by cold r o l l i n g , without s i g n i f i c a n t changes in r . I t i s with m a t e r i a l of t h i s sort that furtlier work w i l l be c a r r i e d out.

Acknowle dgement s

I would l i k e to thank Mr. I . C . Drinkwater for his careful work on the s t r e s s - s t r a i n curves and Dr. P.B. Mellor for a helpful d i s c u s s i o n .

- 6

TABLE 1

P r e d i c t e d and measured e f f e c t i v e s t r a i n s a t i n s t a b i l i t y i n a c i r c u l a r diaphragm.

M a t e r i a l s ( a l l i n t h e a n n e a l e d c o n d i t i o n s ) . n _u 7C/30 B r a s s O.I13 Cur-containing s t e e l 0.24 T i ' . c o n t a i n i n g s t e e l 0 . I 8 Riiüming s t e e l 0.26 C c m m e r c i a l - p u r i t y aluminium 0.55 Commercial-p-urity z i n c O.36 balanced ^ b i a x i a l ^ ^ ^ ' - t e n s i o n 0.73 0.59 0.50 0.51 0.54 0.57

' -

References

Hosford, V7.F. J n r . , and Bo.c;..'0--en, W.A.

Pearce, Roger and J o s h i , P.G.

Pearce, Roger and B o t t , C.H,

Heyer, R. and Newby, J .

Atld.nson, M.

Larson, F.R.

Babel, H.¥. and Corn, D.L.

Mellor, P.B. and Bramley, A.N.

Johnson, W. and Duncan, J . L .

Strength and p l a s t i c i t y in textured m e t a l s . Proceedings of the ninth Sagamore ordnance m a t e r i a l s research conference, Syracuse University P r e s s , I965•

Relationship between the behaviour of a range of alumini-um/magneslLin a l l o y s in u n i a x i a l and b i a x i a l tension.

A.S.M. Trans. Quart. 19èk, ^ J ,

599-Tlie effect of varying s t r a i n r a t i o on the hydraulic bulging behaviour of aluminium

s h e e t .

J . Aust. I n s t . Metals. I967, 12, I I 9 .

Effect of mechanical p r o p e r t i e s on b i a x i a l s t r e t c h a b i l i t y of low caib on s t e e l s .

Paper to be presented at the I968 National S.A.E. Congress, D e t r o i t . (January, I968),

Assessing noiinal aniEotrop.ic p l a s t i c ! ty of sheet metals.

Sheet Metal I n d u s t r i e s , 1967.1 M ; 1^7•

Anisotropy of titanium sheet in unaxial tension.

A.S.M. Trans. Quart. I962, ^ , 26U.

A comparison of methods for c o r r e l a t i v e text-uring with the b i a x i a l strengths of titani-um a l l o y s .

Metals Eng. Quart. I967, Jj ^ 5 .

Contribution to a discussion on 'Mechanical anisotropy in sheet metals' .

March, 1965^ London.

The use of the b i a x i a l t e s t extensometer. Sheet Metal I n d u s t r i e s , 1965, kg, 2 8 l .

cr2

era

FIGURE 1a.

M

Tension - Tension

oC=1, Sphcricat pressure vessel.

' • ( ï ^ ( P l a n e strain, d £ y : 0 ,

«4.

o<=p PUI» tension 1 In plane of sheet

o£ = 1,Plcine strain _{. "£1=0 ^X>y}

Terfiien -Compression

, . -01 r -6 I e ^ ^ j L 1 1 ) S 1 i 1 11 1 e z ^ ^ l 0 _

/ - ^ ^ ï

80,000 70,000 60,000 50.000 40,00'j -

e-FIG. 3

40,000 30,000 20,000 lOflOO L o . 90° BIAXIAL

FIG.

i,

ff ,£ CURVES FOR CP ZINC (1=0-18) 30,000 20,000 CP A l . ~lo UNIAXIAL ,^S«?5°^__x—«0° ]/Zjr* Al (hard) l/^ , -PREOCTED 10,OOOA-CP A l ( h a r t ) -•BIAXIAL CPAljQnn(Kiled)_ BIAXIAL — — © A l (soft) PREDICTED

Cr, eCurvds tor CP Al, ? » 0 - 4 3 ( h a r t ) Z = 0-68 (aiïwaled)

taooo

300»

O'.E Curvss tor anncaliid 7 0 / » Brass I f . O 77)

Taooo 60,000 BIAXIAL SDfflD iOfCO 30AX)

a , £ CURVES FOR A

TITANIUM-C O N T A W I N O StEEL (X = 1-3i)

20A10 _-I

80,000 70000 60.000 50000 40000 3OJ0OO aojDOo BIAXIAL

( r . £ CURVES FOR A COPPER-CONTAINING STEEL ( F = 1 - 6 2 )

•4

120000 now» lOQDOO 90000 ( J , e CURVES POR Ti 115 l ï = 3 f l ) 80000 .,. '~0° 70000 0 45° UM/WIAL 60000 .-^^ SOflOO

FIG

9

Tom

30000

20000

(J, ÊCUWES F0« Z, C 11'3-8)