The relation of incompatibility and dislocation motion to plastic deformation
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DISLOCATION
motion in p l a s t i c d e f o r m a t i o n is f r e quently a n a l y z e d on the b a s i s of a model in which t h r e e c o n c e p t s a r e d e s c r i b e d : p l a s t i c s t r a i n r a t e is p r o p o r t i o n a l to d i s l o c a t i o n velocity; d i s l o c a t i o n s m u l t i p l y in p r o p o r t i o n to s t r a i n ; and the v e l o c i t y of d i s l o c a t i o n s depends upon s t r e s s in a m a n n e r p e c u l i a r to each m a terial. T h e s e concepts a r e g e n e r a l l y e x p r e s s e d by e l e m e n t a r y m o d e l s , developed on the b a s i s of the b e h a v i o r of i n d i v i d u a l d i s l o c a t i o n s . They a r e r e p r e s e n t e d t y p i c a l l y by the following t h r e e e q u a t i o n s : 1'2 S t r a i n r a t e - v e l o c i t y : ~,p = N b V
[1]
Multiplication:
N
[2]
Velocity-stress:
V = Vof(a), e.g., f ( o ) = e -D/cr
w h e r e ~p
= No + M T p
b a s e d a p p e a r s to be m i s s i n g in the f o r m e r . C o n s i d e r able p r o g r e s s has b e e n m a d e in the m a t h e m a t i c a l a n a l y s i s of a p p l i c a t i o n s of the theory by M u r a , 7 who d e v e l oped f u n d a m e n t a l s o l u t i o n s to the e q u a t i o n s and u s e d them to d e t e r m i n e the s t r e s s fields of s i n g l e d i s l o c a tions. To e m p h a s i z e the p h y s i c a l a s p e c t s of the d i s t r i b u t i o n theory it is the p r e s e n t objective to e x h i b i t a c o n n e c tion between its i n c o m p a t i b i l i t y concept (given below) and the f i r s t two concepts of the e l e m e n t a r y m o d e l above. The c e n t r a l equation of the theory is the i n c o m p a t i bility equation which r e l a t e s i n t e r n a l s t r a i n s to d i s l o cation d e n s i t i e s t h r o u g h g e o m e t r i c c o n s i d e r a t i o n s . In a t i m e - d i f f e r e n t i a t e d f o r m the equation is as follows: s
[3]
is p l a s t i c s t r a i n
N, No a r e c u r r e n t and i n i t i a l n u m b e r of d i s l o c a tions p e r u n i t v o l u m e b
is the B u r g e r s ' v e c t o r of the d i s l o c a t i o n s
V
is the a v e r a g e v e l o c i t y of the d i s l o c a t i o n s
M
is the d i s l o c a t i o n m u l t i p l i c a t i o n f a c t o r (a material property)
Vo, D a r e m a t e r i a l c o n s t a n t s
BRIAN M. LEMPRIERE is Senior Specialist Engineer, The Boeing Co. (Aerospace), Seattle, Wash. Manuscript submitted January 15, 1970. METALLURGICALTRANSACTIONS
+s
dtji,k = 2 s
Clmn P j m , k n
[4]
where Ctjl is the d i s l o c a t i o n d e n s i t y t e n s o r , r e p r e s e n t ing the total B u r g e r s vector in the x j d i r e c t i o n e n c o m p a s s e d by an a r e a dA I. P j m is the p l a s t i c s t r a i n r a t e t e n s o r ; Eij k is the a l t e r n a t i n g t e n s o r which r e p r e s e n t s +1 when the i n d i c e s take a cyclic p e r m u t a t i o n of 123, - 1 when c o u n t e r c y c l i c , and 0 if any two i n d i c e s a r e the
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