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NATIONAL RESEARCH LABORATORIES Ottawa, Canada TSCHNICAL TRANSLATION D i v i s i o n o f Mechanical Z n g i n e e r i n g Pages
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P r e f a c e-
3 T e x t-
2 7 F i g u r e s-
1 0 Tech, T r a n s , TT-86 Date-
26 J a n u a r y 1949 Lab, Order Noo-
54248 F i l e-
12-R4-22T i t l e : Allgerneine L8sung d e s ebenen P l a s t i z i t a t s - p r ~ b l e m s f f i r b e l i e b i g e s i s o t r o p e s o d e r a n i s o t r o p e s F l i e s s ~ e s e t z
By: H o Neuber, Brunswick
Reference: M , O o S o ( A i r ) VBlkenrode, R e p o r t s and T r a n s l a t i o n s No, 610, May 1947
S u b j e c t : THE GEN3RAL SOLUTION OF THE TWO-DIMENSIONAL PLASTICITY PROBLEM
S u b m i t t e d by: VJ,
J ,
Cox, T r a n s l a t e d by:He ad,
H,
A , G, NathanS t r u c t u r e s L a b o r a t o r y . Approved by: S o H , P a r k i n ,
Page
-
(If)
Tech, T r a n s , TT-86 TABLEOF
CONTENTS 1, I n t r o d u c t i o n 2 , D e r i v a t i o n of t h e Basic S q u a t i o n s i n C a r t e s i a n Coakdfnates 3 , I n t e g r a t i o n of t h e B a s i c Equations i n C a r t e s i a n C o o r d i n a t e s . ' 4 , D e r i v a t i o n of t h e Basic Equations a l o n g Lines of P r i n c i p a l Stressea 5 , I n t e g r a t i o n of t h e Basic Equations a l o n g Lines o f P r i n c i p a l S t r e s s e s6 , General Theory of " S l i d i n g Linesw B f b l f ography
Page 1
Page
-
( i i i ) Tech, T r a n s , TT-86 LIST OF ILLUSTRATIONS F i g u r e E f f e c t of f o r c e s i n C a r t e s i a n c o o r d i n a t e s T r a n s f o r m a t i o n of c h a r a c t e r i s t i c s The l i n e v=
c o n s t , on t h e p l a s t i c i t y s u r f a c e a l ~ d i n t h e drawing p l a n e T r a n s f o r m a t i o n of s t r e s s components E f f e c t of f o r c e s on a n e l e m e n t c u t o u t a l o n g t h e l i n e s of p r i n c i p a l s t r e s s e s I s o m e t r i c r e p r e s e n t a t i o n of a n a n i s o t r o p i c p l a s t i c i t y s u r f a c e P l a n o f t h e a n i s o t r o p i c p l a s t i c i t y s u r f a c e shown i n F i g u r e 6 w i t h t h e s l o p e s i n d i c a t e d Diagram i l l u s t r a t i n g t h e " s l i d i n g l i n e " t h e o r y M o h r f s boundary c u r v e f o r t h e i s o t r o p i c body Example of a n a n i s o t r o p i c boundary a r e aPage
-
1Tech, T r a n s , TT-86 THE G3NSRAL SOLUTION OF THE
TWO-EI~ENSIONAL
-PLASTIC
ITYBLEM
1, I N T R O D U C T I O N
Prime importance i s a t t a c h e d t o t h e s c i e n t i f i c s t u d y of t h e a f f e c t of f o r c e s on s o l i d b o d i e s , n o t o n l y from t h e p o i n t of view of p h y s i c s , b u t even more from t h e t e c h n i c a l s t a n d p o i n t , I b r e , t h e p l a s t i c s t a t 3 o c c u p i a s a s p e c i a l p o s i t i o n , s i n c e prob- lems a r e i n v o l v e d from whose s o l u t i o n fundamental i n f o r m a t i o n i s e x p e c t e d on t h e l i t t l e known t h e o r y of f a i l u r e of m a t e r i a l s , and hence on t h e c o n n e c t i n g l i n k between t h e t h e o r v of e l a s t i c i t y
and t h e s c i e n c e of s t r e n g t h of m a t 3 r i a l s , T h i s e x p l a i n s t h e e x t r a o r d i n a r i l y l a r g e number of i n v e s t i g a t i o n s which hav
c a r r i e d out i n t h i s f i e l d s i n c e t h e t u r n of t h e c e n t u r y o
!l
) baen These i n v e s t i g a t i o n s can be d i v i d e d i n t o two groups, one d e a l i n g w i t h t h e problem more from t h e p o i n t o f view ofp h y s i c s , and d e s c r i b i n g mainly t h e s t r u c t u r e of m a t e r i a l s and i t s v a r l a t i o n due t o t h e e f f e c t of e x t e r n a l f o r c e s , and t h e o t h e r , t o which t h e p r e s a n t r e p o r t b e l o n g s , approaching t h e problem
p r i m a r i l y from t h e e n g i n e e r ' s s t a n d p o i n t , i , 3 ,, d e a l i n g w i t h t e c h - n i c a l , mechanical a s p e c t s , Due t o t h e imnodiate importance of t h a s a f e t y f a c t o r f o r the problsrn i n g e n e r a l , t h e kno.ui13dge of t h e e x t s r n a l f o r c e s which detsrnline t h e occurrence of p l a s t i c flows i s of p a r t i c l ~ l a r i n t e r s s t f o r t h e a n g i n e a r , Thus t h e prob- lem of p l a s t i c s t r e s s d i s t r i b u t i o n bacomes of prime importance, I n t h e two-dimensional c a s s , from tho e n g i ~ ~ e e r ' s p o i n t of view, t h e problem i s t o deduce t h e t h r e e components of t h e two-dimen- s i o n a l s t r e s s t e n s s r s , w i t h a d a p t i o n t o t h e boundary c o r ~ d i t i o n s i n such a manner t h a t t h r e e b a s i c e q u a t i o n s a r e s a t i s f i s d , These a r e t h e two e q u a t i o n s of a q u i l i b r i u m and a n o t h e r r e l a t i o n , c a l l a d t h e "law of p l a s t i c flow" o r t h e c o n d i t i o n of p l a s t i c i t y , which i s c h a r a c t e r i s t i c of t h e m a t e r i a l i n q u e s t i o n ,
Up t o t h e p r e s e n t , s o l u t i o n s of t h i s problem were found o n l y f o r i s o t r o p i c m a t s r i a l s and t h e n o n l y f o r a l i n e a r form of t h e e q u a t n of p l a s t i c (She r i n g s t r e s h y p o t h e s i s , r e p o r t s by T r e s a h ? , S t VenantJf yvHenky5T, P r a n d t l 6 7 , ~ a r a t h e o d o r y and 3rhard7T, * a d a i d ) , and o t h e r s . )
The p r e s a n t r e p o r t g i v e s t h e complste s o l u t i o n of t h e problem f o r any c o n d i t i o n of p l a s t i c i t y , f o r i s o t r o p i c a s w e l l a s f o r a n i s o t r o p i c m a t e r i a l s , A t t h e same time a g e n e r a l mathema- t i c a l e x p r e s s i o n i s g i v e n f o r t h e s l i d i n p r o c e s s of i s o t r o p i c
4 8
materials, based on Mohr l s p r e s e n t a t i o n
,
which opens t h e way t o t h e e x t e n s i o n t o a n i s o t r o p i c p l a s t i c i t y , T h i s ~ x p r e s s i o n p e r m i t s a g e n e r a l " s l i d i n g l i n e " t h e o r y t o be f o r n i u l a t e d , p r o v i n g t h a t t h e s l i d i n g l i n e s always c o i n c i d e w i t h t h e c h a r a c t e r i s t i c l i n e s of t h e b a s i c e a u a t i o n s and t h a t t h e y c a n be r e p r e s e n t e d a s s l o p e s of an a r e a , The p l a n of t h i s a r e a , w i t h l i n e s of e l e v a t i o n drawn on i t , i s a diagram f o r r e p r e s s n t i n g t h e p r o p 3 r t i 3 s of a n i s o t r o p i cPage
-
2 Tech, T r a n s , TT-86 p l a s t i c m a t s r i a l s , I t c a n a l s o be u s e d t o c h a r a c t e r i z e t h e two-dimensional p l a s t i c i t y c o n d i t i o n and t o f a c i l i t a t e t h e g r a p h i c a l - n u m e r i c a l s o l u t i o n of t h e problems of s t r e s s d i s - t r i b u t i o n ,2, OERIVATIOH OF THE SASIC EQUATIONS
-
I N CARTESIAN COORDINATESI n t h e system of C a r t e s i a n c o o r d i n a t s e x , y t h e two- d i m e n s i o n a l s t r e s s t e n s o r h a s t h e component ax cy (normal s t r e s s e s ) andrx ( s h e a r i n g s t r e s s ) , The f o l l o w i n g c o n d i t i o n s of e q u i l i b r i u m Ean e a s i l y be r e a d from t h e e f f e c t of f o r c e s shown i n F i g u r e 1, v i z , , F o r t h e e l a s t i c body t h e r e i s a t h i r d , t h e s o - c a l l e d " c o m p a t i b i l i t y c o n d i t i o n " ( g e o m e t r i c a l p o s s i b i l i t y of daforrna-, t i o n ) , I n t h e p l a s t i c c a s e i t i s r e p l a c e d by t h e c o n d i t i o n of p l a s t i c i t y o r t h e law of p l a s t i c f l o w , i n which t h e s t a t e of s t r e s s i s a l i m i t i n g one, i , e , , i f t h e s t r e s s e s exceed t h i s l i m i t i n g s t a t e even s l i g h t l y , s l i d i n g o f i n d i v i d u a l l a y 3 r s w i l l o c c u r , T h i s c o n d i t i o n i s a f u n c t i o n a l r e l a t i o n s h i p betwsen t h e t h r e a s t r e s s components which can g e n e r a l l y be w r i t t e n i n t h e f o l l o w i n g form, and which may a l s o be a p p l i e d t o a n i s o t r o p i c m a t s r i a l s : 3 y means of t h i s r e l a t i g n s h f p t h e d e r i v a t i v e of one of t h e t h r e e s t r e s s components, f o r i n s t a n c e t h a t o f 7 X Y may be e l i m i n a t e d i n e q u a t i o n ( 1 ) " To f a c i l i t a t e t h e subsequent c a l c u l a t i o n s t h e f o l l o w i n g a b b r e v i a t i o n s a r a i n t r o d u c e d : E q u a t i o n ( 2 ) i s now w r i t t a n i n t h e f o l l o w i n g form: and by S i f f s r e n t i a t i o n , u s i n g f , A i n s t e a d of
dL
t h e f o l l o w i n g i s o b t a i n e d : d A ' T h e r e f o r e f P C dC =-
f j A dA-
f S B dB;Page
.-.
3 Tech, T r a n s , TTc.86 and, c o n s e q u e n t l y , S u b s t i t u t i n g t h e a b b r e v i a t i o n s ( 3 ) i n e q u a t i o n s (1) t h e con- d i t i o n s o f e q u i l i b r i u m t h e n become from which, by ( 7 ) t h e f o l l o w i n g e q u a t i o n s a r e o b t a i n e d : f y C A s x-
f s A A J y f 9 B B 9 y=
f , C B9y-
f S A A,,-
f q B B S x = 0 , The c a l c u l a t i o n i n C a r t e s i a n c o o r d i n a t e s b e g i n s w i t h t h e s e two e q u a t i o n s ,3,) I N T E G R A T I O N OF THE BASIC EGUATIONS I N CARTESIAN COORDINATES To make t h e i n t e g r a t i o n of t h e b a s i c e q u a t i o n s a s s i m p l e a s p o s s i b l e t h e t r a n s i t i o n t o a s y s t e m o f o b l i q u e co- o r d i n a t e s u , v i s c a r r i e d o u t i n such a way t h a t o n l y d e r i v a - t i v e s w i t h r e s p e c t t o one of t h e two v a r i a b l e s o c c u r i n e a c h of t h e two e q u a t i o n s t r a n s f o r m e d ( t r a n s f o r m a t i o n of c h a r a c - t e r i s t i c s ) . I f aU d e n o t e s t h e arlgle which t h e
x
a x i s makes w i t h t h e u axis ( t a n g e n t t o t h e l i n e v = c o n s t a n t ) , and avt h e a n g l e which t h e
x
a x i s makes w i t h t h e v a x i s ( e f , F i g , , 2), t h e n t h e f o l l o w i n g e q u a t i o n s h o l d f o r t h e components d x and dy o f t h e l i n e e l e m e n t s d s u and d s v r e s p e c t i v e l y :d x
a9u
= cos a,&
= s i n a,= c o s a v c = s i n a ,
"J
ds,
Hence t h e f o l l o w f n g t r a n s f o r n a t i o n fomnulae r e s u l t f o r t h e d o p e r a t o r s-
and d . s u? q -
Page
-
4 T e c h , T r a n s , TT-86 d S o l v i n g t h e s e two e q u a t i o n s f o r t h e o p e r a t o r s-
d x a n days
a
t h e n d-
1-
dE
-
I ( a v s i n a v - = s i n a u k ] ,1
A p p l y i n g ( 1 2 ) t o t h e b a s i c e q u a t i o n s ( 9 ) o f t h e p r o b l s m , t h e n ( f g C s i n a + f S A cos a V ) A j s + ( - f S C 91x1 a U - f , A c o s a,)A,, v u v f S B C 0 s av B 9 ,-
f S B C O s UU B , s = 0,
u v ( - f g C c o s a *>f v ' B s i n a v ) B r s u + ( f s C C 0 s au + f , ~ s i n41)
B S 8 v-
f S A s i n a, A y s s f g A s i n uu A, s = 0 , u vwhere A95 3s u s s d f o r d A e t c , I f now e q u a t i o n ( 1 3 ) i s m u l t i -
U =u
p l i e d by t h e f a c t o r D l 1 and ( 1 4 ) by Dq.2, t h e n , a f t e r a d d i t i o n , a n e q u a t i o n i s o b t a i n e d i n v h i c h d s r i v a t i v a s o c c u r w i t h r e s p e c t t o su and w i t h r e s p e c t t o s,, However, t h e u n d e t e r m i n e d f a c t o r s Dll & ~ d D 1 z 9 and t h a a n g l e s aU and a v , may be made t o s a t i s f y t h e c o n d i t i o n t h a t d e r i v a t i v e s o c c u r w i t h r a s p e c t t o
-
one v a r i a b l e o n l y , e g,, w i t h r e s p e c t t o u , S i n c e t h e c o e f f i c i e n t s of A sv and B , must v a n i s h i n o r d e r t o s a t i s f y t h e a b o v e r e q u i r e r i l e n t , s v t h e f o l l o w i n g c o n d i . t f o n a 1 e q u a t i o n s r a s u l t : ~ ( f s i n , ~%
+ f , A c o s aU)Dll + f S A s i n aUD12 = 0 , c f p B c o s a u D 11+
( f P C c o s aU + f y B s i n aU)D12= 0 ,Page c . 5
Tech, T r a n s , TT-86 These a r e two l i n e a r , homogeneous e q u a t i o n s , Hence, v a l u e s o f
and Dl2 which d i f f e r from z e r o e x i s t o n l y i f t h e d e t e r m i n a n t t h e c o e f f i c i e n t s v a n i s h e s , i , e , , i f t h e c o n d i t i o n 2 2 2 - f s C s i n aU c o s au - f g A f S C cos a, - f ! , ~ f , C s i n a, = 0 ( 1 6 ) i s s a t i s f i e d , D i v i d i n g by
-,gC
and i n t r o d u c i n g t h e double a r g u - ment r e s u l t s i nr s C
s i n 2aU *fPA + f S B s ( f , A o f , B ) c o s 26, = 0. ( 1 7 ) ~ h u s a r e l a t i o n f o r t h e a n g l e a u h ~ s been o b t s i n e d , I n o r d e r t o s i m p l i f y t h e above e q u a t i o n an a u x i l i a r y a n g l e 7 i s i n t r o - duced s t i p u l a t e d by t a n L Y = I'c
f D A-
f S BThen, by means of t h e a d d i t i o n formulae
f 9 A
*
f g Bc o s 2 ( a u - Y =
-
c o s 2 y o, A
-
f ;;BA n e q u a t i o n f o r t h e a n g l e a v can be d e r i v e d i n t h e same manner, I f now e q u a t i o n ( 1 3 ) i s m u l t i p l i e d by t h e f a c t o r D21 and ( 1 4 ) by D22 t h e n , a f t e r a d d i t i o n , an e q u a t i o n i s o b t a i n e d f o r which, by analogy w i t h t h e p r e v i o u s c a l c u l a t i o n s , i t i s s t i p u l a t e d t h a t t h e d e r i v a t i v e s w i t h r e s p e c t t o su must v a n i s h , Then t h e c o e f f i c i e n t s of A and B p s U must v a n i s h , i c e , , t h e f o l l o w i n g su l i n e a r , homogeneous e q u a t i o n s a r e o b t a i n e d f o r D21 and D22: ( f , C s i n av + f , A cos av)D21
-
f , A s i n avD22 = 0,f Y B c o s aVDZl
-
( f U C cos a v + f 9 s i n aV)D22= ~ 0 "I f now t h e d e t e r m i n a n t of t h e c o e f f i c i e n t s i s e q u a t e d t o z e r o , t h e n by means of e q u a t i o n (18)
-
o m i t t i n g , however, t h e i n t e r , = , m e d i a t e s t e p s-
t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d : f j A 9 f Y B c o s 2 ( a v - Y ) =-
7-
c o s 2 Y o 'A ,BHence t h e spme r e l a t i o n h o l d s f o r av a s f o r a,, It i s r e - q u i r e d t h a t a, q i f f e r from aU and t h a t t h e argument 2 ( a - Y )
r e s u l t i n t h e same c o s i n e v a l u e a s t h e argument 2 ( q ,
-vY,
These c o n d i t i o n s a r e s a t i s f i e d i fPage , , 6 Tech, T r a n s , TT-86 Hence Then, by ( 1 9 ) o r (211, f . 9 A + f 9 B c o s ( a
-
a U )-
-
--
-
V f -lg c o s(a,
+
a,) M u l t i p l y i n g by 1-
r e s u l t s i n c o s (a, ='a,)
c o s ( a U+
a,] f r s o l v i n g f o rB
: f g A 9~ c o s (a,-
%)
+
c o s ( a U+
a,)- -
-
'A oos ( a v - a U )-
c o s ( a u+
a,) OP a f t e r ' troansformation f g B- - -
c o t a, , c o t a,.
P A On t h e o t h e r hand, t a k i n g e q u a t i o n (18) i n t o a c c o u n t , t h e followj.ng e q u a t i o n i s deduced from ( 2 3 )v~ 'B
-
-
-
-
c o t ( c+
a ) ,L C
U v Because of e q u a t i o n ( 2 7 ) 1 1 :' B ' B 'A - 0-
- --
c o t aU c o t a, = P Af , c
, A ,CY c
s o t h a t e q u a t i o n ( 2 8 ) becomes H-
(1-
c o t a c o t a v ) = c o t (a,+
a,).,c
u F i n a l l y , a f t e r t r a n s f o r m a t i o nPage
-
7 T e c h , T r a n s , T T - 8 6 f . 9 s i n cr , s i n a A - = - u v " ,C s i n ( a u+
a v ) Because of e q u a t i o n ( 2 9 ) , t h e r e f o r e , cos a cos a2 K = -
u v ,, ,C s i n ( a u+
a v )Moreover, from e q u a t i o n ( 1 5 ) t h e f o l l o w i n g e x p r e s s i o n is ob- t a i n e d f o r t h e r a t i o of c o e f f i c i e n t s , Dl2 :
D11
1 SjA D12 s i n au+
oosq,
9 c L P ~ s f n a, -;;, Then, w i t h r e s p e c t t o e q u a t i o n ( 3 1 ) ,A t t h e same time t h i s s a t i s f i e s t h e second e q u a t i o n ( 1 5 ) s i n c e t h e d e t e r m i n a n t of t h e c o e f f i c i e n t s v a n i s h e s i n b o t h e q u a t i o n s , S i m i l a r l y , t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d from t.he f i r s t of e q u a t i o n s ( 2 0 ) : D22 s i n a v
+
cos a v f,r ' P A Then, w i t h r e s p e c t t o e q u a t i o n ( 3 1 ) , D2 2-
=-
c o t aU D2 1The same r e s u l t could a l s o have been o b t a i n e d from e q u a t i o n (201, With t h e e x c e p t i o n o f common f a c t o r s , a l l unknown
q u a n t i t i e s have now been reduced t o f u n c t i o n s of t h e a n g l e s aU and a v , so t h a t f i n a l e o u a t i o n s can be d e r i v e d , , I'his w i l l be
Page
.-
8 Tech, T r a n s , TT-86 done w i t h r e f e r e n c e t o t h e e q u a t i o n s o b t a i n e d by m u l t i p l y i n g e c u a t i o n s ( 1 3 ) by D l 1 ( o r D21) and ( 1 4 ) by Dl2 ( o r D22) and by a d d i t i o n , S i n c e , a s m e n t i o n e d a b o v e , t h e d e r i v a t i v e s w i t h r a s p a c t t o sv i n t h e f i r s t e q u a t i o n , and t h o s e w i t h r e s p e c t t o su i n t h e s e c o n d , v a n i s h , t h e f o l l o w i n g r e l a t i o n s a r e ob- t a i n e d :[
( f , j G s i n a v + f , ? ~00s aV)Dl1 - f 9 ~s i n av ~ ~A , ~ , 2 1 + [ f S B c o s avD1l - ( f , . c c o s a v + f 9 ~s i n a v ) ~ 1 2 ] B and - [ - = ~ ( f , ~ s i n au+
f , A c o s Q ) D ~ ~ * f p A 91" Q D ~ ~ ] A ~ ~ ~-
( 3 8 ) +L-f , B c o s auD21 + ( f , C c o s a, +f , B s i n a u ) ~ 2 2 1B
= 0. jSv A f t e r d i v i s i o n by f Dll and f , C D21 r e s p e c t i v e l y and e l i m i - n a t i o n o f f s A by means o f e q u a t i o n ( 3 1 ) , -f S B b y means of ( 3 2 ) , DSq
9 C2
by means o f ( 3 4 ) , and-
D22 by means o f ( 3 6 ) : t h e f o l l o w i n gDll D2 1 two e c y a t i o n s a r e f i n a l l y o b t a i n e d : r s i n a v l s i n ( z U + a v )
-
2 s i n au c o s a,] A , s U 2 C O S a v+
[ s i n ( a u + a v ) -2 s i n a v c o s aU] B S s U = 0 , s i n zV.
s i n %[
s i n ( a u + a V ) +2 s i n ~ v C O S a u ] A 9 s V 2 c o s cLu ( 4 0 )+
-
7
[ - . s i n (a, + a v ) +2 s i nautos
av] B , , ~s n Q = 0 , S i n c e t h e e x p ~ e s s i o n s i n s q u a r e b r a c k e t s c a n c e l e a c h o t h e r e x c e p t f o p t h e s i g n , s o l u t i o n r e s u l t s i n 2 . t a n a, A g U -, B , u = 0 , ( 4 1 ) I n t h e s e e q u a t i o n s , d e r i v a t i v e s o c c u r w i t h r e s p e c t t o one v a r i a b l e o n l y s o t h a t i n ( 4 1 ) t h e d e r i v a t i v e s w i t h r e s p e c t t o
sU have been r e p l a c e d by t h o s e w i t h r e s p e c t t o u and i n ( 4 2 ) t h e d e r i v a t i v e s w i t h r e s p e c t t o sv a r e r e p l a c e d by t h o s e w i t h r e s p e c t t o v , (The d e r i v a t i v e s w i t h r e s p e c t t o sv and v d i f f e r o n l y by a f a c t o r , which t a k e s t h e c u r v i l i n s a r d i s t o r t i o n i n t o a c c o u n t , , T h i s f a c t o r c a n c e l s i t s e l f , )
Pa.ge - - 9 Tech, T ~ a n s , TT--86 Gofng b a c k t o e q u a t i o n ( 6 ) , an e x p r e s s i o n c a n now be f o r m u l a t e d f o r dC a s w e l l , The f o l l o w i n g e q u a t i o n r e l a t i v e t o ( 6 ) , (31) and ( 3 2 ) i s t h e n o b t a i n e d :
P
d ~ = s i n ( a u [ s i n a, s i n a, d ~ + c o s au o o s a, d ~ ] ( 4 3 ) From ( 4 1 ) and ( 4 2 ) t h e f o l l o w f n g e x p r e s s i o n r e s - u l t s f o r dB: 2 dB = t a n av dA o r 2 dB = t a n au dA s o t h a t a f t e r t r a n s f o r n a t i o n : dC = t a n av dA o r dC = t a n au dA a l o n g v = c o n s t ,1
( 4 4 ) a l o n g u = c o n s t , a l o n g v = c o n s t , , ( 4 5 a l o n g u = c o n s t , From ( 4 5 ) a s i n g l e r e l a t i o n s h i p , which a p p l i e s t o b o t h f a m i l i e s of c u r v e s , i s o b t a i n e d :Going back now t o t h e o r i g i n a l meaning of t h e q u a n t i t i e s A , B and C as s t r e s s components, t h e n The g e o m e t r i c a l i n t e r p r e t a t i o n o f t h i s f u n d a m e n t a l r e l a t i o n becomes c l e a r i f t h e i d e n t i t y [ d ( o x
*
f l y ) ] 2- [d(rx-
u y ) ] = 4 ( d g X ) ( d o y ) ( 4 8 ) i s employed, By e l i m i n a t i o n of ( d g x ) * ( d o y ) i n e q u a t i o n ( 4 7 ) t h e P y t h a g o r e a n theorem i s o b t a i n e d , v i z , , from which a c l e a r i n t e r p r e t a t i o n o f t h e c h a r a c t e r i s t i c s c a n be d e r i v e d , Suppose t h e c o n d i t i o n of p l a s t i c i t y ( 2 ) i s c o n v e r t e d t o a r e l a t i o n between ( r x + . o y ) , (cx-
u y ) and ( 2 r X y ) by r e p l a c i n g 1 1 1 ux by -(gx+
p y )+
H ( u X-
u Y ) and U y by -(rx+
u y )-
L(rx = , r y ) - q 2 2 2 t h e n t h i s r e l a t i o n may b e i n t e r p r e t e d a s a s u r f a c e , whose p o i n t s have t h e d i s t a n c e , OF t l l e v e f (ox + u y ) f r o m t h e drawing p l a n e w i t h t h e r e c t a n g u l a r c o o r d i n a t e s ( o x-
r y ) a n d ( 2 r x y ) - o f , , F i g . 3, I f t h i s s u r f a c e f s termed m e r e l y " p l a s t i c i t y s u r f a c e " , t h e n t h c f o l l o w i n g h y p o t h e s i s c o r r e s p o n d s t o c o n d i t i o n ( 4 9 ) : h he e h a r a c t e r l s t i c s o f t h e d i f f e r e n t i a l e q u a t f o n s of t h e g e n e p a l t w o - d i m e n s i o n a l p l a s t i c i t y problem c o r r e s p o n d t o t h e 45O s l o p e s of t h e p l a s t i c i t y s u r f a c e , "Page - 1 0
Tech, T r a n s , TT
-
86The f a c t s a r e i l l u s t r a t e d i n F i g , 3 where t h s d i f f ~ r e n t i a l s d ( a x
+
a y ) d ( g x-
Q y ) and d(21xy) f o r s u c h a l i n e element a r s shown ,, Pvith ~ e g a r d t o t h e r e p r e s e n t a t i o n of t h e a n g l e s au and a v it f o l l o w s from ( 4 5 ) and ( 4 4 ) t h a t d a do N = NL ' %
= t a n zv ( o r t a n a,) ,, d7xy =x d r x yTc+i
= c o t 2 a v ( o r c o t 2 a U ) ,s o t h a t t h e a n g l s 2av ( o r 2 a u ) appears a s t h e a n g l e between
t h e t a n g e n t t o t h e image of t h e l i a e v = c o n s t , ( o r u = c o n s t , ) , p r o j e c t e d on t h e (ox
-
) , ( 2 r x y ) - p l a n e , and t h e - f l y )-
a x i s , It i s noteworthy h a t a v , the a n g l e of t h e t a n g e n t w i t h t h e l i n e u = c o n s t , i n t h ex,
y plane ( c f , F i g , 2 ) , a p p e a r s i n t h e drawing p l a n e of t h e p l a s t i c i t y s u r f a c e a s h a l f an a n g l e between t h e t a n g e n t t o t h e p r o j e c t i o n of t h e image of t h e l i n e v=
c o n s t , , which p a s s e s t h r o u g h t h e same p o i n t of r e f e r e n c e , When t h e x , y p l a n e i s p r o j s c t e d on t h e drawing p l a n e of t h e p l a s t i c i t y s u r f a c a a s u b s t i t u t i o n and d o u b l i n g of t h e a n g l e s of the t a n g e n t o c c u r s ,, I f , by means of t h e r e l a t i o n s 2rxy=
q s i n 26
I
I t r a n s f o r m a t i o n i s made t o c y l i n d r i c a l c o o r d i n a t e s p , q ( 261,
t h e n t h e a n g l e6
c o r r e s p o n d s t o t h e a n g l e between one d i r e c t i o n of p r i n c i p a l s t r a s s and t h ex
& x i s , a s i s shown i n t h e subse- quent s a c t i o n ,By d i f f a r o n t i a t i o n : d ( a , + o y ) = d p ,
d = u ) = dq cos 2
d
-# 2qdd s i n 2$,Page c , 11
Tech, T r a n s ,, TT-86 With r e s p e c t t o e q u a t i o n ( 5 1 ) a l o n g t h e c h a r a c t e r i s t i c s i t f o l l o w s from t h e sscond and t h i r d of e q u a t i o n s ( 5 3 ) t h a t
c o s 2 6
-
s i n 26. 2qd6 d ( o x-
O Y )-
-
dq = c o t 2a, ( 5 4 ) 2 d s i n 2 6 A c o s 2 6 0 d ( o r c o t 2 a U ) dq S o l v i n g f c r 2 d b 9 t h e n+-
d6 2q-
=
t a n 2 ( a v - ,6 )
o r t a n 2 ( a u6 )
.
dqOn t h e o t h e r h a n d , a d d i t i o n o f t h e two ec;uations ( 5 0 ) , which h o l d t r u e a l o n g t h e c h a r a c t e r i s t i c s , g i v e s i n c o n j u n c t f o n w i t h e q u a t i o n s ( 5 3 ) : d ( x +Q
1
= d.-
1 dq s i n 2 6?
2 q d Z B-
a i n 2av s i n 2au * ( 5 6 ) 'Taking ( 5 5 ) i n t o a c c o u n t , t h e n \ dp s i n 2aV = d q [ s i n 2 6+
t a n 2 ( c ~ ~ - 6 ) c o s 261 o r dp s i n 2au = d q [ s i n 26+
t a n 2(aU,=$) c o s 261.
whence by means o f t h e a d d i t i o n f o r m u l a e of t r i g o n o m e t r y t h e f o l l o w i n g e q u a t i o n i s irnrilediataly o b t a i n e d : dq = dpacos 2 ( a v - 6 ) [ o r d p = c o s 2 ( a u- P I ) ]
(589 T a k i n g ( 5 5 ) i n t o a c c o u n t , t h s n = d p 0 s i n 2 ( a v-6)
[ o r d p a s i n 2 ( a u .-$)I , ( 5 9 ) By s q u a r i n g and a d d i t i o n o f e q u a t i o n s ( 5 8 ) and ( 5 9 ) , t h e f o l l o w - i n g e q u a t i o n i s o b t a i n e d f o r t h e ~ h a r a c t ~ e r i s t i c s :These f a c t s a r a made c l e a r by means o f a n a n i s o t r o p i c p l a s t i c i t y s u r f a c e shown i n i s o m o t r i c p r o j s c t i o n ,
B e f o r e t h e meaning of t h e s e r e s u l t s i s e x p l a i n e d f u r t h e r , i t w i l l be shown how i n t e g r a t i o n c a n be c a r r i e d o u t i f 2 from t h e b e g i n n i n g , a l l q u a n t i t i e s ape r e l a t e d t o t h e l i n e s of p r i n c i p a l s t r e s s e s ,
Page -, 1 2
T e c h , T r a n s , TT-86
4 , -D Z R I V A T I O N O F THE -A- --- BASIC EQUATIONS ALONG -A L I N E S OF P R I N C I P A L -
-
STRZSS E S---
---
F i g u r e 1 shows t h 3 t r i a n g u l a r l i n e e l e m e n t , two s i d e s o f which a r e p a r a 1 1 3 1 t o t h e x and y a x e s r e s p e c t i v e l y and t h e t 5 i r d s i d e , on which t h e p r i n c i p a l s t r r 3 s ~ 4 2 a c t s , l i e s i n t h e d i r e c t i o n of t h e p r i n c i p a l s t r e s s 1 and n a k e s t h e a n g l e
6
w i t h t h e x a x i s By means of t h i s l i n e e l e m a n t t h e f o l l o w i n g e q u a t i o r i s of e q u i l i b r i u m a r e formed: 0 , s i n 6-
r x y c o s6
= a 2 s i n ( 6 1 1 b y cosd
= .rxy s i nd
= 0 2 c o s6
B u l t i p l y i n g t h e f i r s t e q u a t i o n ( 6 1 ) by o o sJ6
and t h e second by s i nb ,
and s u b t r a c t i n g , g i v e s t h e f o l l o w i n g e q u a t i o n f o r t h e angle8:
t a n 26
= 7xy bx-
a Y S i n c e t a n 26
rarnafns unchanged i f6
+
5
i s s u b s t i t u t e d f o r8,
n o t o n l y i s6
a d i r e c t i o n o f p r i n c i p a l s t r e s s , b u t t h e d i r s c t i o nd
+5
i s one a l s o ( d i r j c t i o n of p r i n c i p a l s t r e s s 2 ) , T h e r e f o r e , b o t h d i r e c t i o n s o f p r i n c i p a l s t r e s s a r e o b v i o u s l y p e r p e n d i c u l a rt o e a c h o t h e r and when
d
+
$
i s s u b s t i t u t e d f o r6
and o f o r@ z 9
t h e n two o t h e r e q u a t i o n s , a n a l o g o u s t o ( 6 1 ) , r e s u l t : o x c o s+
yxY s i nid
= b l cosd
,
( 6 3 ) u y s i nd
+
r x y c o sd
= u l s i n6
M u l t i p l y i n g t h e f i r s t e q u a t i o n ( 6 1 ) by s i n6 ,
and t h e f i r s t e q u a t i o n ( 6 3 ) b y c o s6,
and a d d i n g , r e s u l t s fn a r e l a t i o n f o r a,, M u l t i p l y i n g t h e second e q u a t i o n ( 6 1 ) by c o s6,
and t h e e l a t i o n s e c o n d e q u a t i o n ( 6 3 ) by s i n6
and a d d i n g r e s u l t s i n a r, f o r U F i n a l l y , m u l t i p l y i n g t h e f i r s t e q u a t i o n ( 6 1 ) by c o s6
and t g e f i r s t e q u a t i o n ( 6 3 ) by s i nd
and s u b t r a c t i n g g i v e s a r e l a t i o nOPT^^"
Thus: o x = b l C O S ' ~+
4 2 s i n 2 6 = ( D l -. 0 2 ) s i n6
c o s6
.
Page '- 13 Tech,, T r a n s , TT-86 I f t h e f o l l o w i n g q u a ~ t i t i e s a r e i n t r o d u c e d : 4 - 0 2 = p (sum of t h e p r i n c i p a l s t r e s s e s ) ,
i
a l L ' a Z = q ( d i f f e r e n c e of t h e p r i n c i p a l s t r e s s e s ) ,I
2u2 = p I - ' q 9J
t h e n , b y employing t h e t r i g o n o m e t r i c f u n c t i o n s o f t h e double argunent., e q u a t i o n s ( 6 4 ) become 20, = p+
q c o s 26
j 2 u y = p c = > q c o s 26
,
2 r x y = q s i n 2d
.
For t r a n s f o r m a t i o n of t h e e q u a t i o n s of e q u i l i b r i u m (1) t o t h e l i n e s of p r i n c i p a l s t r e s s e s t h e t r a n s f o r m a t i o n formulae f o r t h e d o p e r a t o r sd
and-
a r e r e q , u i r e d , G e n e r a l l y s p e a k i n g , t h e dx d~ f o l l o w i n g e q u a t i o n s h o l d : F o rd z
and*
s e e F i g u r e 4 where a s 1 d e q l dx-
= c o s6
anddL
= s i nd
$1 d s 1Moreover, i f
6
+
i s employed i n s t e a d of6
and t h e s u b s c r i p t2 i n s t e a d of 1, t h e n
a x
-
= - s i n = cos6
Page
-
1 4 Tech, T r a n s , TT-86 so t h a t eoua-Lions ( 6 8 ) become 7 d d-
= c o s6
A
+
s i n6 -
dy 9 d S 1 dxI
- -
-
,-- s i nd
LI-
+ c o s6
-
d d92 d x d yJ
d and t h e s o l u t i o n f o r1
and-
:tY
dx7
= s i n6
1
c c o s6
n, dy
d s l (3s I f e q u a t i o n s ( 6 7 ) a r e i n t r o d u c e d i n t o e q u a t i o n (1) t h e n t h e f o l l o w i n g e q u a t i o n s i n a b b r e v i a t e d n o t a t i o n , . u s i n g p,, i n s t e a d ofLQ
e t c , :, r e s u l t : d x I f e q u a t i o n s ( 7 2 ) w e u s ~ d w i t h t h e a b b r e v i a t i o n p,x f o r*
.
8 s l t h e n t h e f o l l o w i n g e q u a t i o n s a r e o b t a i n e d from t h e f i r s t e q u a t i o n ( 7 3 ) : + s i n 6[
p S s 2 a ( % , 2 -2q6 100s 26 + (q9s1+ 2 q P i s s 2 ) s i n 2 6 1 so
( 7 4 ) or, a f t e r employing t h e a d d i t i o n formulae of t r i g o n o m e t r y ,S i m i l a r l y , from t h e second e q u a t i o n ( 7 3 ) ,
Page
-
1 5Tech. T r a n s , TT-86 The f o l l o w i n g two r e l a t i o n s a r e o b t a i n e d d i r e c t l y from ( 7 5 ) and
( 7 6 ) :
7
These a r e t h e e q u a t i o n s of e q u i l i b r i u m i n a c u r v i l i n e a r system of c o o r d i n a t e s whose c u r v e s c o i n c i d e w i t h t h e l i n e s of p r i n c i p a l s t r a s s s s , I t would a l s o have been p o s s i b l e t o d a r i v e t h e s e
e q u a t i o n s from a c a l c u l a t i o n of e q u i l i b r i u m by v i s u a l i z i n g a l i n e element assumed t o be c u t o u t a l o n g t h e l i n e s of p r i n c i p a l s t r e s s e s , a s shown i n F i g u r e 5,
By employing t h e r e l a t i o n s ( 6 7 ) t h e c o n d i t i o n of
p l a s t i c i t y becomes a dependent f u n c t i o n of t h e q u a n t i t i e s p, q ,
6
and can be w r i t t e n i n t h e form@ ( p 3 q 9
8)
=o
( a n i s o t r o p i c p l a s t i c i t y ) . ( 7 8 ) For t h e c a s e of i s o t r o p i c p l a s t i c i t y t h i s f u n c t i o n i s no l o n g e r dependent on t h e angle6,
Hence(O = (p:, q ) = 0 ( i s o t r o p i c p l a s t j c i t y )
.
( 7 9 ) Under c e r t a i n c i r c u m s t a n c e s s o l v i n g f o r p i s advantageous mathe- m a t i c a l l y , , Provided e q u a t i o n ( 7 8 ) o r ( 7 9 ) , a s t h e c a s e may be, c a n be s o l v e d e x p l i c i t l y f o r p, t h e c o n d i t i o n of p l a s t i c i t y becomes p = p c ~ , @ ) ( a n - i s o t r o p i c p l a s t l . c i t y ) o r P = P ( q ) ( i s o t r o p i c p l a s t i c i t y ) . I f t h e c a l c u l a t i o n i s based on e q u a t i o n ( 7 8 ) t h e n t h e d i f f e r - e n t i a t i o n of t h i s e q u a t i o n i s r e q u i r e d f o r t h e e l i m i n a t i o n of p, The t o t a l d i f f e r e n t i a l of @, which v a n i s h e s e x a c t l y a s @ i t s e l f , becomeswhere t h e a b b r e v i a t i o n t#~, h a s been used f o r
@,
e t c .P d p
Hence
Page
-
16 Tech, T r a n s , TT-86 Q s P psS =-
Lq
4.9,-
Q 9 f j69,
e t c , 1 (84 1 1 T h e ~ e f o ~ e , a f t e r m u l t i p l y i n g t h e e q u a t i o n s ( 7 7 ) by9,
,
t h e quant it i e s #:, pp and @sppes r e s p e c t i v e l y c a n be elyminated.Then -I 2
The g e n e r a l s o l u t i o n of t h e two-dimensional p l a s t i c i t y p ~ o b l e m i s based on t h e s e two e q u a t i o n s ,
If t h e s o l u t i o n i s based on e q u a t i o n ( 8 0 ) , t h e n
where t h e a b b r e v i a t i o n p , q i s uaed fop
%
e t c . Equations ( 7 4 )t h e n assume t h e f o l l o w i n g form: '1
For t h e c a s e of i s o t r o p i c p l a s t i c i t y ,
)6
i n e q u a t i o n ( 8 5 ) and p o d i n e q u a t i o n ( 8 7 ) would have t o be e q u a t e d t ozero,
5, INTEGRATION OF THE BASIC EQUATIONS ALONG LINES OF PRINCIPAL
STRESSES
I n o r d e r t o c a r r y o u t t h e k r a n s f o m a t i o n of t h e b a s i c e q u a t i o n s from t h e l i n e s of p r i n c i p a l s t r e s s e s t o t h e c h a r a c - t e r i s t i c s , t h e f o l l o w i n g a n g l e s a r e i n t r o d u c e d :
BU
-
t h e a n g l e between t h e d i r e c t i o n of t h e p r i n c i p a l s t r e s s 1 and t h e ud i r e c t i o n , and
B v -.
t h e angle between t h e d i r e c t i o n of t h e p ~ i n e i p a l s t r e s s 1 and t h e v d i r e c t i o n , Then f o r t h e com- ponents of t h e l i n e elements dsu and d s v :dsl
- -
-
@ 0 Sa,
,
- - - s i n
~2F v
Page
-
1 7 Tech, T r a n s , TT-86 Hence t h e f o l l o w i n g t r a r l s f ormat i o n f o r r n u l ~ e a p p l y t o t h e d o p e r a t o r s -.- and-
d ..
s u dsv d S o l v i r l g f o r t h e o p e r a t o r s-
d s 1 d 1 d d-
=
s i n(8,
u v d 1-
d d5
= s i n (P, pu1
[ -
c o s8,
u + c o sBu
v I f t h e s e r e l a t i o n s a r e s u b s t i t u t e d i n t h e e q u a t i o n s of p l a s t i - c i t y (85) t h e l a t t e r become-@.ddvSU1-
pl:
[
(Q9p-P,q)qrs-99#69s
1
v v and+
2 4 [ - s i n ~~ ~ ~ ~ s i n~
6~
~ = 0 . ~6
~~
+~
( 9 2 )~
]
where q g s i s u s e d f o r%
.
I f e q u a t i o n ( 9 1 ) i s now m u l t i p l i e d u s u by t h e f a c t o r K 1 1 and ( 9 2 ) by K12# t h e n , a f t e r a d d i t i o n , an e c u a t i o n i s o b t a i n e d which must c o n t a i n d e r i v a t i v e s w i t h r e s p e c t t o u o n l y , So t h a t t h e f a c t o r s K 1 1 and K12 a s w e l l a s t h e a n g l e sBU
andBv
may s a t i s f y t h i s c o n d i t i o n , t h e c o e f f i c i e n t s of qYsv andd
a r e e q u a t e d t o z e r o . Hencesv
Page $8
Tech, T r a n s , TT.=86
K ~ ~ s i n [ @Bur ~ 2q@, cos BU]+~12[@sm ~ ~ 0 s
pU+
2qQ. s i nB
]
= 0 ,P P u
( 9 4 )
The v a n i s h i n g of t h e d e t e r m i n a n t of t h e s e two l i n e a r equatj-ons i s a c o n d i t i o n f o r v a l u e s of K1 and K12s which a r e n o t z e r o , f e , , t h e f o l l o w i n g e q u a t i o n ho
f
d s : By means of t h i s e q u a t i o n t h e r e l a t i o n f o r t h e a n g l eBU
h a s been o b t a i n e d , , I f an angle p i s i n t r o d u c e d w i t h t h e s t i p u l a t i o n%PI
t a n 2p =-
2qQ9t h e n , by means of t h e a d d i t i o n formulre of trigonometry, t h e f o l l o w i n g e a u a t i o n r e s u l t s :
3
c o s 2(8,
-
p ) =-
c o s 2p.
0 9 qA second p r i n c i p a l e c u a t i o n i s o b t a i n e d by m u l t i p l y i n g e q u a t i o n ( 9 1 ) by and e q u a t i o n ( 9 2 ) by K and adding..
This e q u a t i o n must s a t i s f y tha c o n d i t i o n thg? d e r i v a t i v e s occur w i t h r e s p e c t t o v o n l y , To a c h i e v e t h i s , t h e c o e f f i c i e n t s of C,sT7 and
6,
s.. a r e e c u a t 3 d t o z a r o , Hence The v a n i s h i n g of t h e d e t e r m i n a n t of t h e e o e f f i e f e n t s f s a con- d i t i o n f o p v a l u e s of t h e c o n s t a n t s K21 and K 2 which a r e n o t zero. Whence t h e r e l a t i o n f o pB v
i s o b t a i n e8
:@,d
s i n 2Bv+
2q( 9 r q
cos 2Pv + a s p ) = 0 0 (100 ) Taking e q u a t i o n ( 9 6 ) i n t o a c c o u n t , ( 1 0 0 ) r e s u l t s i n cos 2 ( P V-
p ) =-
cos 2p.
O? qPage
-
19Tech, T r a n s , TT-86
A comparison w i t h e q u a t i o n ( 9 7 ) shows t h a t t h e same c o s i n e v a l u e i s d e r i v e d from t h e argument 2 ( P U
-
p ) a s from 2 ( P v-
p ) , Thi s c o n d i t i o n i s s a t i s f i e d i f The f o l l o w i n g e q u a t i o n i s t h e n o b t a i n e d from ( 1 0 1 ) o r ( 9 7 ) : On t h e o t h e r hand, e q u a t i o n ( 9 6 ) g i v e s K12 F o r t h e r a t i o of t h e c o n s t a n t s ,-,
t h e f o l l o w i n g e q u a t i o n i s K1l o b t a i n e d from ( 9 3 ) : t a n By s u b s t i t u t i n g e q u a t i o n (104) i n (106) t h e l a t t e r becomes and a f t e r f u r t h e r t r a n s f o r m a t i o n K12 - = c o tpv
K 1 1It would have been p o s s i b l e t o o b t a i n t h e same r e s u l t by means of e q u a t i o n ( 9 4 ) , The r a t i o of t h e c o n s t a n t s K22 i s c a l c u l a t e d
R21
Page
-
20Tech, Trans, TT-86 Then
A l l q u a n t i t i e s o c c u r r i n g I n t h e t ~ a n s f o m e d b a s i c e q u a t i o n s may now be e x p r e s s e d by f u n c t i o n s of the a n g l e s
Pu
and p,,, The f i s s t of t h e s e b a s i c e c u a t i o n s , which was o b t a i n e d by m u l t i - p l y f n g e q u a t i o n ( 9 1 ) by K1l and ( 9 2 ) by K12 and adding9 nowc o n t a f n s d e r i v a t i v e s w i t h r e s p e c t t o su o n l y and h a s t h e follow- i n g form::
The second of t h e transfosmed b a s i c e q u a t i o n s was o b t a i n e d by m u l t i p l y i n g e o u a t f o n ( 9 1 ) by K 2 1 and ( 9 2 ) by K2 and a d d i n g o
It
c o n t a i n s d e r i v a t f v 3 s w i t h r e s p e c t t o s,, o n l y an
3
h a s now t h eIf eq.uat,fon ( 1 1 0 ) i s d i v i d e d by K 0 and (111) by Ql@* and moreover., 9.f e q u a t i o n s ( 1 0 4 ) : ( 1 0 & ~ , ' ? 1 0 8 ) and (109) a r e %ed t o
4
" 9 6 K12 Kz2 e l i m i n a t e2,
and-.
t h e n t h e fundamental e q u a t i o n s q sq 11 K21* ( 1 1 0 ) and (111) become + c o s ( ~ ~ - - $ ~ ) s i n B ~ } ] zqd,, = 0 (112 and 2 @ o sBu
[{cos ( ~ ~ - ~ ~ ) + c o s ( ~ ~ + ~ ~ ) ) s i n B ~ + ( - o o s ( ~ , ~ - - $ ~ ) + c o s(B,+B~))--
s f npu ] q wPage
-.
2 1 T e c h , T r a n s , T'T-86 S i n c e t h e l i n e e l e m e n t d s U v a r i e s from t h e d i f f e r e n t i a l of p a r a - m e t e r dU o n l y by a f a c t o r which d e t e r m i n e s t h e c u r v i l i n e a r d i s - t o r t i o n and t h e m e t r i c , t h e d e r i v a t i v s s ~ i t h r e s p e c t t o su i n e q u a t i o n ( 1 1 2 ) were r e p l a c e d by d e r i v a t i v e s w i t , h r e s p e c t t o u and t h o s e w i t h r e s p e c t t o s, i n ( 1 1 3 ) by d e r i v a t i v e s w i t h r e s p e c t t o v , I f i n e q u a t i o n s ( 1 1 2 ) and (113) t h e t e r m s o f common argument a r e combined, t h e n from m u l t i p l i c a t i o n by s i nP v
and s i nPu
r e s p e c t i v e l y t h e f o l l o w i n g e q u a t i o n s r e s u l t :and
I f t h e argument (Bu+ i n e q u a t i n ( 1 1 4 ) i s r e p l a c e d by
~ B ~ - ( ) V - B U ) ~ and i n ( 1 1 5 ) by 2Pu+(BV-
lu)g
t h e f ' u r t h e r r e - u c t i o n of t h t r i g o n o m a t r i c f u[
c t i o n s shows t h a t b o t h e q u a t i o n s can be s i m p l i f i e d by means of s i n ( B v -Bu):,
s o t h a t f i n a l l y t h e f o l l o w i n g s i m p l e r e l a t i o n s r e s u l t : q,, t a n 28,-
2q$,v = 0.
( 1 1 7 I t i s now p o s s i b l e t o g i v e a r e l d f i o n f o r t h e d e r i v a - t i v e of p a s w e l l , and, by m u l t i p l y i n g by 1,
t h e f o l l o w i n g e q u a t i o n s , r e l a t i v e t o ( 8 3 ) , a r e o b t a i n e d :q
A p p l i c a t f on of e q u a t i o n s (104 ) and ( 1 0 5 ) and m u l t i p l i c a t i o n by-
coa( B u + P v )
t h e n r e s u l t s i n andPage , - * 22 Tech, Trans, TT-j86 On t h e o t h e r h a n d , t h e f o l l o w i n g e q u a t i o n s ape obtained f ~ o m (116) and ( 1 1 7 ) : 2 ~ 6 , ~ = qiU tan 28, and 2q6,v = q o v t a n 28, I f t h e s e e q u a t i o n s a r e s u b s t i t u t e d i n ( 1 2 0 ) and (121) and f f t h e argument ($,
+ B v )
i n e q u a t i o n ( E r r i s peplaced by[ B B V
(BTJ
, B U ) ] and i n ( 1 2 1 ) by[
2 8 ~+ (8,
BIJ)] t h e n , a f t e r r e d u c t i o n and s i m p l i f f c a t i o n by means of cos( B v
-- BU] andr r l u l t i p l f c a t i o n by cos 2Bv and coa 28, r e s p e c t i v e l y , t h e follow-
i n g e q u a t i o n s r e s u l t :
Pru e09 2Bv q,,
and
P " V C O S 2PU = q r v ,,
I f !, however,, t a n 28, and t a n 2PU a r e t h e m u l t f p l f e ~ s ~ t h e n t h e right-hand s i d e g i v e s t h e q u a n t i t i e s 2q6,, and 2q69v -c o r r e s - ponding t o equatfona (122) and ( 1 2 3 ) , Henee
p p u s i n 28,
=
2q6,, andP ~ , s i n 28, = 2q6,, a
Squaring and adding t h e p a i r s of e q u a t i o n s (126) and (1.24), and (127) and ( 1 2 5 ) , f f n a l l y r e s u l t s f n
and
Therefore ,, along t h e c h a r a c t e r i s t i c s t h e Pythagorlean e q u a t i o n a p p l i s s :
An e q u a t i o n of t h i s type has a l r e a d y been dexnived i n S e c t i o n 3 ,,
S i m f l a ~ l y , equatfons ( 5 8 ) and ( 5 9 ) ( c f , S e c t i o n 3) a r e confirmed by ( 1 2 4 ) , ( 1 2 5 ) , (126) and ( 1 2 7 ) , s i n c e a s can be seen i n Figure
7, t h e r e l a t f o n s h i p s I
and
e x i s t between t h e a n g l e s au, a,:,
Bus
P,
and6,
The a n g l e s 2BU and 2Pv can be found a s e a s i l y a s 201, and 2av from t h e s e p r e s e n t a - t i o n of %he p l a s t i c i t y s u r f a c e a f t e r t h e s l o p e s have been drawnPage
-
2 3 Tech, T r a n s , TTc.86 I n t h e s u b s e o u e n t s e c t i o n a g e n e r a l i n v e s t i g a t i o n i s made of t h e system of l i n e s w h i c h , f o r e a c h c a s e , g i v e s t h e d i r e c t i o n of t h e p l a n e of i n t e r s e c t i o n f o r which t h e s t r e s s i s e x a c t l y a t t h e l i m i t of p l a s t i c i t y , i , e , , , t h e s o , - , c a l l e d " s l i d i n g l i n e s f 1 ,GENERAL THEORY OF "SLIDING LINES"
6,
-
Ths s l ' d i n g l i n e c o n c e p t ' , which h.as b e e n c o n f i r m e d e x p e r i m e n t a l l y
83,
i s b a s e d on Mohr 's f u n d m e n t a l i n v e s t i g a t i o n s41, Mohr found t h a t f o r a s t r e s s which i s a t t h e l i m i t o f p l a s t i c i t y t h e y e a r e two v e r y s p e c i f i c a r e a s of i n t e r s e c t i o n where, a t a s l i g h t i n c r e a s e of load:! t h e s e c t i o n a l s t r e s s w i l l c a u s e t h e d e s t $ r u c t i o n of t h e m a t e r i a l , S i n c e f o r m e r l y t h e s h e a r i n g s t r e s s was c o n s i d e r e d t h e c i e f c a u s e of f a i l u r e
P
( s h e a r i n g s t r e s s h y p o t h e s i s ) , Mohr t a l k e d a b o u t " s l i d i n g " i n t h e s e a r e a s of i n t e r s e c t i o n and termed them " s l i d i n g l i n s s " , a l t h o u g h t h f s t e r m i n o l o g y a c t u a l l y i n v o l v e s r e s t r i c t i n g t h e p l a s t i c i t y problem t o t h e f i e l d o f a p p l i c a t i o n of t h e s h e a r i n g s t r e s s h y p o t h e s i s , whereas, i n f a c t , normal d i s p l a c e m e n t s o f t h e edges o f t h e a r e a s of i n t e r s e c t i o n a r e p o s s i b l e i n a d d i t i o n t o t,he s l i d i n g s ( i , e ,, :, t a n g e n t i a l d i s p l a c e m e n t s ) , However, t h e c o n v e n t i o n a l t e r m " s l i d i n g l i n e s " w i l l be r e t a i n e d i n t h i s ~ e p o r t , An a r b i t r a r y p l a n o o f i n t e r s e c t i o n i s assumed which makes t h e a n g l eB
w i t h t h e d i r e c t i o n of p r i n c i p a l s t r e s s e s and on which t h e normal s t r e s s a and t h e s h e a r i n g s t r e s s 7 act( F i g , 8 1 , By a p p l i c a t i o n of e q u a t i o n ( 6 7 )
,,
t h i s p l a n e o f i n t e r s e c t i o n r e p l a c e s t h e p l a n e y = c o n s t , i e +,-p
i s s u b - , s t i t u t e d f o r6,
f o r f l y and 7 f o r Txy. s o t h a t Moreover, i f t o b e g i n w i t h , o n l y i s o t r o p i c m a t e r i a l s a r e cone- s i d e r e d , , t h e n a n g l e s p e r t a i n i n g t o ( 2 f f ) and ( 2 7 ) c o r r e s p o n d t o e a c h s y s t e m o f v a l u e s p , q , which s a t i s f i e s t h e i s o t r o p i c p l a s t i c i t y c o n d i t i o n ( a n d which i s g i v e n on t h e p l a s t i c i t y s u r -f ace i n q u e s t i o n ) , These a n g l e s p e r t a i n i n g t o ( 2 6 ) and ( 2 7 ) l i e on a c i r c l e i n a r e c t a n g u l a r system of c o o r d i n a t e s f o ~ m e d by ( 2 a ) and 627)
.
T h i s c i r c l e i s c a l l e d Mohr 1s c i r c l e d i a g r a m f o r s t r e s s ( F i g , 91, I t s c e n t r e i s on t h e u - a x l s a t t h e d i s - t a n c e p and i t s r a d i u s e q u a l s q,, A l l t h e c i r c l e diagrarns f o r s t r e s s which c a n be drawn s o t h a t t h e y c o r r e s p o n d t o t h e i s o - t ~ o p i c p 1 a s t i c i t . g c o n d i t i o n , a c c o r d i n g t o Mohr, t o u c h a common boundary curve IMohr s boundary c u r v e f o r t h e i s o t r o p i c p l a s t i c body, F i g , 9 ) T h i s i s e x p l a i n e d below,Page 2 4
'Tech, T r a n s , TT,-86
The v a l u e s o f t h e s e c t i o n a l s t r e s s e s c o r b p e s p o n d i n g t o t h e boundary c u ~ v s a r e t h e l i m i t i n g v a l u e s whi c h l i e
e x a c t l y a t t h e l i m i t of t h e p l a s t i c behaviourr o f t h e m a t e r i a l , These v a l u e s a r e d e n o t e d by a and T g , , The c o ~ r e s p o n d i n g p l a n e of i n t e r s e c t i o n i s t h e ' s l i d i n g p l a n s ' , and t h e a n g l e t h i s p l a n a makes w i t h t h e d i r e c t i o n of t b ? p ~ i n c i p a l s t r a s s 1 i s d e n o t e d by
P g
( F i g , 8 and 9 ) ; S i n c e U g and r g c o r r e s p o n dt o t h e b o u n d a r y c o n d i t i o n . t h e boundary c u r v e must not be t o u c h e d o r c r o s s e d f o r any o t h e r a r e a o f i n t e r s e c t i o n i , e a t any p o i n t o f t h e p a r t i c u l a r c i r c l e d i a g r a m f o r s t r e s s , (Here o n l y t h e u p p e r b r a n c h o f t h e c u r v e h a s b e e n u s e d ) The b o u n d a r y c u r v e i s t o u c h e d a t t h e p o i n t = 7 i , e , , it i s an enveloping c,uxJve of a l l t h e c i r c l e s wh c h a r e p o s s i b l e f o r t h e i s o t r o p i c p l a s t i c body u n d e r c o n s i d e r a t i o n , T h i s manner o f r e p r e s e n t a t i o n i s n o t s u i t a b l e f o r a n a n i s ~ t ~ r o p i c body, s i n c e f o p f i x e d p t h e r a d i u s q v a r i e s w i t h t h e d i r e c t i o n of t h e p r i n c i p a l s t r e s s , I n t h i s c a s e IvIohrgs t h e o r y must b s s u i t a b l y e x t e n d e d and g e n e r a l i z e d , I t may be c o n c l u d e d from t h e i n f l u e n c e o f t h e a n g l e of i n t e r s e c t i o n on t h e l e n g t h o f t h e r a d i u s t h a t f o r f i x s d q l i d i n g s u r f a c e s t h e s t r e s s v a l u e s p e r t a i n i n g t o t h e f i x e d p - v a l u e s l i e n o t o n a c i r c l e b u t on a c l o s e d c u r v e s i m i l a r t o a n e l l i p s e which must be c o n s t r u c t a d p o i n t .by p o i n t f r o m t h e c o r r e s p o l i d i n g c u r v e p
-
c o n s t o f t h e p l a s t i c i t y s u r f a c e o r i t s p l a n f o r m ( F i g , 7 ) , It i s now e v i d e n t t h a t one b o u n d a r y c u r v e i s no l o n g e r s u f f i c i e n t b u t i n f i n i t s l y many boundary c u r v e s a r e r e q u i r e d , i e , , a boundary a u r f a c e t a k e s t h e p l a c e of t h e boundary c u r v e , I n a d d i t i o n t o t h e two s o o r d i n a t s s a and 7 o f M o h r Q s d i a g r a m f o r u l t i m a t e f a t . i g u e s t r e s s ( O F s l i d i n g l f n e d i a g r a m ) , as a l o g i c a l c o n s e q u e n c e , a t h i r d c o o r d i n a t e , 1 , e , t h e a n g l e a , must be u s e d , T h i s a n g l e c h a r a c t e r i z e s t h e p o s i i f o n o f 'he p l a n e o f i n t e r s e c t i o n on which t h e s t r e s s componen-ts and a c t . I f c y l i n d r i c a l o 0 0 , r d i n a t e s a r e u s e d w i t h ((25) i n a x i a l and ( 2 7 ) % n p a d i a l d i r e c t i o n and t h e a n g l e between 2 1 and2 7 c o s 2 a i n s p a c e , 2a, t h e n a c l e a r p i c t u r e o f t h e a n i s o t r o p i c ~ o u n d a r y s u r f a c e i s o b t a i n e d ( c f , F i g ,
l o ) ,
M a t h e m a t i c a l l y , c ~ n d i t ~ i o n s a r e now a s f o l l o w s , The t h ~ e e q u a n t i t i e s a, 7 , and a which d e t e r m i n e t h e s t r e s s i n t h e
p l a n e o f i n k e m e c t i o n , a r e f u n c t i o n s of t h e q u a n t i t i e s p,, q , and
P ,
which s e r v e a s p a r a m e t e r s , The q u a n t i t i e s p , q and a r e a l s o a s s o c i a t e d a s f u n c t i o n s o f t h e c o n d r t i o n o f p l a s t i c i t y and t h e p l a s t i c i t y s u r f a c e < I n f a c t , i n a d d i t i o n t o t h e c o n d i t i o n o f p l a s t i c i t y t h r e e r e l a t , i o n s between t h e s e q u a n t i t i e s e x i s t , i , e ,, w i t h r e f 3 r e n c e t o e q u a t f o n ( 1 3 2 ) and FJglwe 8: 2 9 = p ; q c o s 28 ,1
2 7 = -q s i n 28 ,, 2cr = 2d 9 28 ;,Page
-
25Tech, T r a n s , TTc.86
I f q ,
6,
a r e g i v e n , a , 7 and a c a n be d e t e r m i n e d , T h i s a p p l i e st o any p l a n e o f i n t e r s e c t i o n , I n each c a s e t h e c o r r e s p o n d i n g s t r e s s c u r v e s form a n a r e a , which may be termed I ' s t r e s s a r e a " , I n F i g , 1 0 t h e t r a c e s of two such s t r e s s a r e a s ( p = 4 and p = 0 ) i n t h e p l a n e s 2 a = 0 and 2 a = +K 2- a r e shown i s o m e t r i c a l l y , It i s now r e q u i r e d t h a t a l l p o s s i b l e groups o f v a l u e s a , 7 , a l i e w i t h - i n t h a boundary a r e a , Then a l l s t r e s s a r e a s w i l l t o u c h t h i s boundary a r e a , i , e , a t h e boundary a r a a i s t h e e n v e l o p i n g s u r f a c e of a l l a r e a s g i v e n b y p o s s i b l e g ~ o u p s o f v a l u e s 0 , r - a,, The group o f v a l u e s , 3 = q g ; T = T g p a = ag, t o which t h e p a r a m e t e r
v a l u e s p , q ,
6 ,
$ belong, t h e r e f o r e r e p r e s e n t s t h e common p o i n t of two i n f i ~ i t e l y c l o s e s t r e s s a r e a s and t h u s may a l s o ber e p r e s e n t e d by t h e p a r a m e t e r v a l u e s p
+
d p , q+
dq,6
+ d 6 , $+
d e , f o e , t h e d e r i v a t i v e s o f e q u a t i o n ( 1 5 3 ) w i t h r e s p e c t t o t h e p a r a m e t e r s v a n i s h ( t ' v a r i a t io n of p a r a m e t e r s " ), v i z , , 8 ( 2 @ 8 5 0 = dp dq c o s 28+
2qdP s i n 28 ,o1
-
b(21r)=
0 =.-
dq s i n 28-
2qd$ c o s 281
( 134 ) From t h e t h i r d e q u a t i o n ( 1 3 4 )i s d e r i v e d , hence from t h e second e q u a t i o n ( 1 3 4 )
By s u b s t i t u t i o n i n t h e f i r s t e q u a t i o n ( 1 3 4 ) t h e f o l l o w i n g r e s u l t f s f i n a l l y o b t a i n e d a f t e r t r a n s f o r m a t i o n : dp c o s 28 = dq ( 1 3 7 ) whence, w i t h r e f e r e n c e t o e q u a t i o n ( 1 3 6 ) , dp s i n 2p = 2qd6
.
(138) By s q u a r i n g and a d d i n g :The folloivfng c o n c l u s i o n i s drawn from t h e r e s u l t s o b t a i n e d f o r t h e c h a r a c t e r i s t i c s :
The " s l i d i n g l i n e s " of t h e g e n e r a l two-.dimensional p l a s t i c s t a t e c o i n c i d e w i t h t h e c h a r a c t e r i s t i c s of t h e
d i f f e r e n t i a l e q u a t i o n s of e q u i l i b r i u m and a t t h e same time r e p r e s e n t t h e 45O s l o p e s of t h e p l a s t i c i t y s u r f a c e ,
Page
-
26Tech, T r a n s , T T - 8 6
It i s noteworthy t h a t the s l i d i n g l i n e t h e o r y a s such c o u l d be d e r i v e d from t h e v a r i a t i o n o f p a r a m e t e r s , a s was done i n t h i s s e c t i o n , without r e f e r r i n g t o t h e d i f f e r e n t i a l e q u a t i o n s of e q u i l f b r i u m , However, o n l y from t h e s e e q u a t i o n s can i t be e s t a b l i s h e d t h a t t h e s l i d i n g l i n e s c o i n c i d e w i t h t h e c h a r a c t e r i s t i c s , which a r e of d e c i s i v e importance f o r t h e i n t e g r a t i o n , The d i f f e r e n t i a l e q u a t i o n s of e q u i l i b r i u m a l s o l e a d t o t h e f o r m u l a f o r s u b s t i t u t i o n of and d o u b l i n g of t h e a n g l e s o f t h e t a n g e n t i n t h e p r o j e c t i o n of t h e x , p p l a n e on t h e (bX
-
b y ) , , (2Txy)-plane ( c f , plo),
Page
-
2 7Tech, T r a n s , TT-86 BIBLIOGRAPHY
1) Review of a l l p r e v i o u s r e s e a r c h on t h e t h e o r y o f p l a s t i c i t y
w i t h numerous r e f e r e n c e s i n Handbuch d e r Physik,, v o l , V I ,
B e r l i n 1928
,
p , 428a l s o i n Nlsmorial d e s S c i e n c a s ~ a t h e ' m a t i q u e s
,
No, LXXXVI and LXXXVII ( 1 9 3 7 ) , / 21H,
T r e s c a , Nlem, p r e s , d i v , s a v a n t s , v o l , 18 ( 1 8 6 8 ) , v o l , 20 (1872 ) , 3 ) B , de S t ; Venant, J o u r n , Math, p u r , e t a p p l , , '2nd s e r i e s , , v o l , 1 6 ( 1 8 7 1 ) , 4 ) 0 , Mohr, 2, d o V D I 44 ( 1 9 0 0 ) pp, 1528-1530.5 )
H,
Henky, 2 , angew,, Math, Mach, 3 ( 1 9 2 3 ) P O 2 4 1 -6 )
L o
P r a n d t l , P r o c , i n t o c o n g r , a p p l , Mec,, D e l f t 1925.7 ) C, C a r a t h e o d o r y and E ~ h a r d Schmidt, Z , angew, M a t h , Mech, 3
( 1 9 2 3 ) p , 468.
8 ) A o Nadai, Dep b i l d s a m e Z u s t a n d d e r W e r k s t o f f e , (The P l a s t i c
F I G . 2 TRANSFORMATION O F CHARACTERISTICS
4
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d X
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F I G . I EFFECT O F FORCES I N CARTESIAN COORDI NATES
FIG. 3 T T - 8 6
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CONST. ON THE PLASTICITYFI G. 4 TRANSFORMATION OF STRESS C O M P O N E N T S
F I G . 5 E F F E C T OF FORCES O N AN ELEMENT C U T O U T A L O N G T H E L I N E S OF PRINCIPAL STRESSES
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F I G . 6 T T - 8 6
q + b ;
= p
ISOMETRIC REPRESENTATION OF AN AN ISOTROPIC
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F I G . 7 T T - 8 6
P L A N OF T H E A N I S O T R O P I C P L A S T I C I T Y SURFACE S H O W N I N F I G U R E 6 W I T H T H E S L O P E S I N D I C A T E D
