Comment on “back-stresses, image stresses, and work-hardening” by L.M. Brown

Scripta METALLURGICA

Vol. 8, pp. I01-I02, 1974 Printed in the United States

Pergamon Press, Inc.

COMMENT ON "BACK-STRESSES, IMAGE STRESSES, AND WORK-HARDENING" BY

L.

M.

BROWN

*

K. T a n a k a Fatigue Testing Division National Research Institute for Metals 2 - 3 - 1 2 N a k a m e g u r o , M e g u r o k u , Tokyo 153, JAPAN (Received In the recent internal

stress

A c t a Met.

in the matrix

S e p t e m b e r 1, 1973)

have appeared

two p a p e r s

of dispersion-hardened

calculating

crystals 1'

2).

the average

However,

the

calculated results i n two p a p e r s h a v e an a p p a r e n t d i s c r e p a n c y i n t h e d e p e n d e n c e o f t h e a v e r a g e s t r e s s on t h e v o l u m e f r a c t i o n of the inclusions, f . Namely, due t o Mori and T a n a k a 1 ) , i t s i m p l y d e p e n d s l i n e a r l y on f , w h i l e due t o Brown 2) i t d e p e n d s n o t o n l y on t h e f i r s t o r d e r o f f b u t a l s o on t h e h i g h e r o r d e r o f f . In t h i s s h o r t n o t e I w o u l d l i k e t o p o i n t o u t t h a t Brown c o m m i t t e d a t r i v i a l error in his calculation, w h i c h l e d t o t h e s p u r i o u s d e p e n d e n c e o f t h e a v e r a g e s t r a i n on f . Now l e t us s t a r t from t h e t h i r d p a r a g r a p h o f p . 8 8 1 i n B r o w n ' s p a p e r . I n a given inclusion, tim he strain, c i j F, c o n s i s t s o f t h e c o n s t r a i n e d s t r a i n , Eij C the image s t r a i n , eij ; and t h e s t r a i n due t o a l l t h e o t h e r i n c l u s i o n s . Brown a s s u m e d t h a t t h e s t r a i n due t o a l l t h e o t h e r i n c l u s i o n s c a n be e q u a t e d t o t h e mean s t r a i n i n the matrix, Thus,

M,

if

t h e number o f t h e i n c l u s i o n s ,

F C im M" eij ~-__eij + eij +

N, i s s u f f i c i e n t l y

large.

(a)

Here we must notice that the mean strain in the matrix, N, originates from the constrained strain and the image strain due to all the other inclusions. Therefore, im the strain, eij , in Eq.(a) should be the image strain of a given inclusion itself and t h e a v e r a g e o f t h e s t r a i n , I , i s t o h a v e t h e o r d e r o f ( V / V 0 ) e i j T ' where V i s . t h e v o l u m e . o f a g i v e n i n c l u s i o n and V0 i s t h e v o l u m e o f t h e b o d y , s i n c e i~__. < ~ i j l m > . I t i s e v i d e n t t h a t i n E q . ( 8 ) i n B r o w n ' s p a p e r the strain, im I c a n be n e g l e c t e d c o m p a r e d t o t h e s t r a i n s , < e i j C > i and <~ijF>M, s i n c e <~ij C >i ~ Eij T ~]p(V/Vo)eij T and < ~ i j F > M ~ ( N V / V o ) e i j T = f~.lj T ~ ( V / V o ) E i j T from the assumption that the number of inclusions is large and any of them does not occupy a large part of the body. Thus, the strain, i, should be omitted from the right hand side of Eq.(10) in Brown's paper. This gives the correct expression, which is

101

i02 COMMENT ON: BACK-STRESSES,

identical

t o Mori and T a n a k a ' s E q . ( 6 ) .

(12) i n B r o w n ' s p a p e r , strain

IMAGE STRESSES AND WORK-HARDENING

due t o a l l

Unfortunately,

he t o o k t h e s t r a i n ,

Vol. 8, No. 2

as e x p r e s s e d in E q s . ( 1 1 )

and

< E i j l m > i , f o r t h e a v e r a g e o f the image

inclusions.

The i n c o n s i s t e n c y i n v o l v e d i n B r o w n ' s p a p e r can a l s o be d e m o n s t r a t e d from a n o t h e r l i n e . From t h e d e f i n i t i o n ,

im>
=

<~. F> C> lj - <~ij

= - fl - (I Comparing Eq.(b) with Eq.(AI)

f)M •

(b)

in Brown's paper, we have

M = -M'

(c)

since = re.. T. Substitution of Eq.(c) into Eq.(b) gives i] ij fl = feij T - f I.

(d)

If we combine Eq.(d) with Eq.(9) in Brown's paper, we will have M = O. Of c o u r s e , t h i s u n r e a l i s t i c image s t r a i n i n E q . ( a ) .

conclusion results

from t h e i n c o r r e c t

treatment

o f the

Acknowledgement

this

I enjoyed discussions matter.

w i t h Dr. T. Mori in Tokyo I n s t i t u t e

o f T e c h n o l o g y on

References 1. T. Mori and K. T a n a k a , Acta M e t . , 21, S 7 1 ( 1 9 7 3 ) . 2. L. M. Brown, Acta M e t . , 21, 8 7 9 ( 1 9 7 3 ) . *Dr. L. M. Brown w r i t e s : "The e x t r e m e r i g h t - h a n d term o f e q u a t i o n s ( 9 ) , ( 1 0 ) , (1]~ and (13) o f my p a p e r s h o u l d be d e l e t e d . This change does n o t a f f e c t the m a j o r conclusions of the paper."