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Variable

Cases

Y+t (=Y~a)

Y*s

Y-I (=Y~)

Y~

I. Isotropy II. Anisotropy III. Anisotropy and SD effect

1.0 1.0 1.0

1.0 1.05 1.6

1.0 1.0 1.2

1.0 1.05 1.05

ao/Ou

1.0 1.05 1.05

b. Computed Material Variables

Variable Cases

k

Mzz (=M22)

I. Isotropy II. Anisotropy III. Anisotropy

1.0 1.0 1.1

1.0 1.0 1.0

tx,

Maa

M13 (=M23)

MI~

1.0 0.91 0.71

0.5 -0.45 -0.35

-0.5 -0.55 -0.65

r+1

(=or2)

a3

(=r,2)

r§

0 b / 0u

0 0 0.09

0 0 0.19

1.0 1.20 0.98

1.0 1.0 1.0

1.0 1 05 1,05

and SD effect

[6j, t h r e e s i m p l e c a s e s a r e c o n s i d e r e d , a s s h o w n in T a b l e I. A n i s o t r o p i c m a t e r i a l i s i l l u s t r a t e d b y C a s e I, t h e t h r o u g h - t h i c k n e s s y i e l d s t r e s s i s i n c r e a s e d b y 5 p c t in C a s e II a n d t h e s t r e n g t h d i f f e r e n t i a l e f f e c t i s i n t r o d u c e d in C a s e III b y i n c r e a s i n g t h e v a l u e of Y-1 b y 20 p c t a n d c o m p u t i n g t h e v a l u e of Y.3 f r o m E q . [4j. N o t e t h a t i n C a s e IH, a b / a u > 1 a n d r§ < 1, c o n s i s t e n t w i t h s o m e of t h e t r e n d s t h a t h a v e b e e n o b s e r v e d . 6-9 T h e a b s o l u t e v a l u e of r**, h o w e v e r , d e p e n d s on t h e r e l a t i v e m a g n i t u d e of Y_~ w i t h r e s p e c t to Y.I. I f t h e d i f f e r e n c e b e t w e e n t h e two y i e l d s t r e n g t h v a l u e s ( C a s e III) i n c r e a s e s w i t h i n c r e a s i n g s t r a i n , r - v a l u e decreases. The corresponding yield loci for these t h r e e c a s e s a r e i l l u s t r a t e d i n F i g . 1. A n i m p o r t a n t f e a t u r e of t h e p r o p o s e d t h e o r y i s t h a t t h e y i e l d s u r f a c e i s d e f i n e d b y t h e c u r r e n t v a l u e of t h e material parameters, k, M i j , a n d eel, s o t h a t t h e e f f e c t of p r e v i o u s l o a d i n g h i s t o r y m u s t b e i n c o r p o r a t e d in t h e a n a l y s i s , 12 S u c h a f e a t u r e i s c o n s i d e r e d to b e n e c e s s a r y f o r t h e c o m p l e t e d e s c r i p t i o n of t h e s h e e t metal forming process. The theory in the present form has also been verified with experimental data obtained under complex loading conditions, n checked w i t h d i f f e r e n t t y p e s of a n i s o t r o p i c m a t e r i a l s , ~3 a n d e x t e n d e d to t h e g e n e r a l i z e d s t r a i n r a t e d e p e n d e n t framework./4 I n s u m m a r y , i t is d e m o n s t r a t e d t h a t a n i m p r o v e d a n i s o t r o p i c p l a s t i c i t y t h e o r y i s c a p a b l e of a c c o u n t i n g f o r t h e t r e n d in a n o m a l o u s y i e l d i n g o b s e r v e d u n d e r biaxial loading conditions, Specifically, the continuum theory predicts an anomalous yielding behavior when the sheet has not only the through-thickness anisotropy b u t a l s o t h e s t r e n g t h - d i f f e r e n t i a l (SD) e f f e c t i n t h e p l a n e a s w e l l a s in t h e t h i c k n e s s d i r e c t i o n . S h e e t