Constructing Ashby Map

7/24/2019 Constructing Ashby Map

1/9

JOURN AL OF MATERI ALS SCIENCE 1 3 1 9 7 8 ) 1 2 8 2 -1 2 9 0

A s i m p l e m e t h o d o f c o n s t ru c t i n g a n

A s h b y t y p e d e fo r m a t io n m e c h a n is m m a p

T E R E N C E G. L A N G D O N F A R G H A L L I A . M O H A M E D

Department o f M aterials Science University o f Southern Cal i fornia Los Angeles

Ca lifornia 9000 7 USA

An Ashb y- type deform at ion mechan ism map m ay be cons iderab ly s impl i f ied by p lo t t ing

in the form of normal ized stress versus the reciprocal of the hom ologous temp erature. In

this fo rm the b oun daries separat ing the var ious f ields appea r as straigh t l ines and the

co ns tant strain rate co ntou rs ma y be ap prox ima ted as straight l ines. Representat ive m aps

are presented for con di t ion s of h igh tem perature creep and a s imple p rocedure is

ou t l ined for cons t ruct ing several maps for the same mater ia l at d i f fere nt grain s izes.

1 . n t r o d u c t i o n

I t h a s b e e n r e c o g n i z e d f o r m a n y y e a rs th a t a p o l y -

c r y s t a l l i n e m a t e r i a l , w h e n s u b j e c t e d t o a n a p p l i e d

s tr es s , m a y d e f o r m b y o n e o r m o r e o f a n u m b e r o f

d i f f e r e n t m e c h a n i s m s . A n i m p o r t a n t c o n c e r n i n

s t u d ie s o f m e c h a n i c a l b e h a v i o u r is t h u s t o i d e n t i f y

t h e d o m i n a n t d e f o r m a t i o n p ro c e ss u n d e r a n y

s e l e c te d c o n d i t i o n s o f s tr e ss , t e m p e r a t u r e , a n d

gra in s ize .

I t w a s f i r s t s u g g e s t e d b y W e e r t m a n a n d

W e e r t m a n [ 1 , 2 ] t h a t i t m a y b e p o s s ib l e t o c o n -

s t r uc t a

creep diagram

i n w h i c h t h e n o r m a l i z e d

st ress ,

o/G

i s p l o t t e d a s a fu n c t i o n o f t h e h o m o l o -

g o u s t e m p e r a t u r e , T / T m whe r e o i s t he a pp l i e d

s t r e s s , G i s t he she a r m odulus , T i s t he a bso lu t e

t e m p e r a t u r e , a n d T rn i s t h e m e l t i n g p o i n t o f t h e

m a te r i a l i n de gr e e s Ke lv in . I n t h i s f o r m , i t wa s

d e m o n s t r a t e d s c h e m a t i c a l l y t h a t t h e d i a g r a m m a y

b e d i v i d e d i n t o f o u r d i s t i n c t re g i o n s: N a b a r r o -

H e r r i n g c r e e p a t v e r y h i g h t e m p e r a t u r e s , h i g h te m -

p e r a t u r e A n d r a d e ) c re e p a n d l o w t e m p e r a t u r e

loga r i t hm ic ) c r e e p a t o / G > 10 - s , a nd a ne l a s t i c

r e c ove r a b l e ) c r e e p a t

o/G

< 10- s . A d i a g r a m o f

t h i s t y p e , w h i c h p l o t s t w o o f t h e v a r ia b l es i n th e

b a s i c r a t e e q u a t i o n w h e n t h e o t h e r v a r ia b l e s r e m a i n

c on s t a n t , i s no w g e ne r a l l y r e f e r r e d t o a s a

defor-

mation mechanism map.

T h e c o n c e p t o f d e f o r m a t io n m e c h a n i s m

m a pping wa s f i r s t de ve lope d f o r r e a l m a te r i a l s i n

t h e d e t a i l e d w o r k o f A s h b y [ 3 ] i n w h i c h t h e b e s t

a va i l a b l e sour c e da t a we r e use d t o c ons t r uc t i nd i -

1282

v i d u a l m a p s o f n o r m a l i z e d s t re s s v e rs u s h o m o l o g o u s

t e m p e r a t u r e f o r s e v er al d i f f e r e n t m e t a l s a n d

c e r a m ic s . S inc e t h i s e a r ly w or k , s im i l a r m a ps ha ve

b e e n p r e s e n t e d f o r a l ar g e n u m b e r o f m a t er i a ls

[ 4 - 2 8 ] , a n d s u ch m a p s a re n o w b e c o m i n g a n

a c c e p t e d fe a t u re o f m a te r ia l s te x t s [ 2 9 - 3 7 ] .

A n i n h e r e n t p r o b l e m w i t h t h e A s h b y - t y p e m a p

i s t h e d i f f ic u l t y o f c o n s t r u c t i o n , s i n ce a m a p m a y

o n l y b e d e v e l o p e d f o r a s p e c if ic m a t e r i a l b y u s i n g

a c o m pu te r t o so lve t he r e l e va n t c ons t i t u t i ve

e qua t ion s f o r a ll t he poss ib l e d e f o r m a t io n p r oc e sse s

a t a v e r y l a rg e n u m b e r o f p o i n t s t y p i c a l l y , ~ 4 0 0 0

t o 6 0 0 0 ) i n s t r e s s - t e m p e r a t u r e s p a c e . F u r t h e r -

m o r e , h a v i n g c o n s t r u c t e d a m a p f o r a m a t e r i a l

ha v ing a se l e c t e d g r a in s i z e , t he r e i s no s im ple

p r o c e d u r e t o o b t a i n a n e w m a p f o r t h e s a m e m a -

te r ia l wi th a di f fe rent gra in s ize . As a resul t , i t i s

n o t e a s y t o u t i li z e d i r e c tl y d e f o r m a t i o n m a p p i n g

t e c hn ique s i n p r a c t i c a l s i t ua t i ons .

O n e m e t h o d o f c i r c u m v e n t i n g t h is d i f f i c u l ty is

t o p l o t a m a p u s i n g t w o o t h e r v a ri a b le s : fo r

e xa m ple , nor m a l i z e d g r a in s i z e ,

d /b

versus

nor m a l i z e d s t r e s s , o/G a t c o n s t a n t t e m p e r a t u r e ,

whe r e d i s t he g r a in si z e a nd b i s t he B ur ge r s ve c to r

[ 38 , 39] , o r nor m a l i z e d g r a in s i z e ,

d /b

ve r sus t he

r e c ip r o c a l o f h o m o l o g o u s t e m p e r a t u r e , T m / T a t

c o n s t a n t n o r m a l i z e d s t re s s [ 4 0 , 4 1 ] . H o w e v e r ,

t he r e a r e m a ny s i t ua t i ons i n whic h i t i s e spe c i a l l y

a t t ra c t i v e t o d i s p la y m e c h a n i c a l d a t a in th e f o r m

of s t r e s s ve r sus t e m pe r a tu r e a t c ons t a n t g r a in s i z e ,

9 19 78 Chapman and Hall Ltd. Printed in Great Brit ain.


2/9

a s , f o r e xa m p le , i n e ng ine e r ing s i t ua t i ons w he r e a

ma ter ia l of a s ingle g ra in size is subjec ted to a wide

r a n g e o f s t r e ss e s a n d t e m p e r a t u r e s . T h e p u r p o s e o f

t h i s p a p e r i s t o r e - e x a m i n e t h e A s h b y - t y p e d e f o r -

m a t i o n m e c h a n i s m m a p a n d t o p r e s e n t a s i m p l e

c o n s t r u c t i o n p r o c e d u r e w h i c h e li m i n a t e s t h e

r e q u i r e m e n t f o r d e t a i l e d c o m p u t a t i o n .

2 . T h e

A s h b y t y p e d e f o r m a t i o n m e c h a n is m

m a p

A s c h e m a t i c i l lu s t r a ti o n o f a n A s h b y - t y p e d e f o r -

m a t i o n m e c h a n i s m m a p i s s h o w n i n F i g . 1 , w h e r e

o/G

c o v e rs e ig h t o r d e r s o f m a g n i t u d e f r o m 1 0 -8

t o 1 . 0) , t h e h o m o l o g o u s t e m p e r a t u r e s c a le e x t e n d s

f r o m a b s o l u t e z e r o t o t h e m e l t i n g p o i n t , a n d t h e

m a p r e l a t e s t o a c on s t a n t va lue o f t he g r a in s i z e.

T h e t h i c k l in e s o n t h e m a p r e p r e s e n t t h e b o u n d -

a r i e s b e t w e e n f i e l d s i n s t r e s s - t e m p e r a t u r e s p a c e

wi th in w hic h a si ng le m e c ha n i sm dom in a t e s the

d e f o r m a t i o n b e h a v i o u r . T h e s e li n es th e r e f o r e t r a c e

o u t t h e l o c i o f a l l p o i n t s w h e r e t h e t w o a d j a c e n t

p r oc e sse s ha ve e q ua l s t r a in r a t e s ; whe r e t h r e e l ine s

m e e t , t h e t h r e e a d j a c e n t p r o c e ss e s m a k e e q u a l

c o n t r i b u t i o n s .

T h e u p p e r h o r i z o n t a l l in e , a t

a/G ~- 4 x 10 -2

r e pr e se n t s t he i de a l s t r e ng th o f t he m a te r i a l . T he

f i e ld m a r ke d d i s loc a t i on g l i de , i n t he nor m a l i z e d

st ress range o f 4 x 103

< o / G < 4

x 10 -2 , cor re-

s p o n d s t o t h e s t r e s s - t e m p e r a t u r e r eg i m e in w h i c h

t h e m a t e r i a l d e f o r m s b y t h e c o n s e r v a ti v e m o t i o n

o r ~ - - V ~ - - ~ r

/ [ C o n s t a n t g r a i n s r z e

L_____ _

~ d ~

e [

Id e I I \

o-/G

O t 0 2 0 3 0 4 0 5 0 6 O 7 0 8 0 9 1 0

T / T m

Figure 1 A s c h e m a t i c A s h b y - t y p e d e f o r m a t i o n m e c h a n i s m

m a p , p l o t t i n g l o g a r i t h m i c n o r m a l i z e d s t r e s s ,

o/G

v e r s u s

h o m o l o g o u s t e m p e r a t u r e ,

TITm

f o r a c o n s t a n t g r a i n s i z e.

of d i s loc a t i ons by g l i de t h r ough the l a t t i c e . A t

va lue s o f

o/G

l es s t ha n - 4 x 10 - 3 , t he de f or -

m a t i o n b e h a v i o u r a r i s e s f r o m d i f f u s i o n - c o n t r o l l e d

c r e e p p r oc e sse s . T he m a te r i a l de f o r m s by a d i s lo -

c a t i on c l im b pr oc e ss a t h igh s t r e s se s a nd t e m pe r a -

t u r e s , b y N a b a r r o - H e r r i n g d i f f u s i o n a l c r e e p

[ 42 , 43] a t l ow s t re s se s a nd h igh t e m pe r a tu r e s ,

a nd by Coble d i f f us iona l c r e e p [ 44] ove r t he

e n t i r e s t r e s s r a nge a t l ow t e m pe r a tu r e s .

T h e t w o t h i n l in e s s u p e r i m p o s e d o n t o t h e f ie l d s

a r e c o n t o u r s o f c o n s t a n t s t ra i n r a t e , f o r s t e a d y - s ta t e

r a t e s o f 10 - 2 a nd 10- 1~ se c - l~ r e spe c t i ve ly . T he se

l i n e s j o i n p o i n t s i n s t r e s s - t e m p e r a t u r e s p a c e

ha v ing t he sa m e pr e d i c t e d s t r a in r a t e s .

A l tho ugh sc he m a t i c in f o r m , F ig . 1 i s i de n t i c a l

t o t h e m a p s o r i g i n al ly d e v e l o p e d b y A s h b y [ 3 ] ,

w i t h t h e e x c e p t i o n t h a t in s o m e e a rl y w o r k t h e

s t r a i n r a t e c o n t o u r f o r 1 0 -8 sec -1 was d rawn as an

a dd i t i ona l bounda r y t o d i s t i ngu i sh a n e l a s t i c

r e g im e a t l ow e r s t re s se s. T h i s is j us t i f i e d w he n

u s in g a t e s t i n g m a c h i n e w h i c h c a n n o t d e t e c t s t ra i n

r a t e s b e l o w 1 0 -8 sec 1 , but in ge nera l i t i s usu a l ly

m o r e c o n v e n i e n t t o s h o w t h e e n t i r e m a p a n d t h e n

t o s u p e r i m p o s e a n u m b e r o f d i f f e r e n t s t r a i n r a t e

c o n t o u r s .

W i thou t e xc e p t ion , a l l o f t he p r a c t i c a l i n t e r e s t

i n t h e u s e o f d e f o r m a t i o n m e c h a n i s m m a p s h a s

c e n t r e d o n t h e i r a p p l i c a t i o n in t h e a n a l y si s o f d a t a

o b t a i n e d f r o m h ig h t e m p e r a t u r e c r e e p e x p e r i m e n t s .

T h i s i n t e re s t a ri se s b e c a u s e o f th e n u m e r o u s p r o -

c e s s e s w h i c h m a y c o n t r i b u t e t o s t e a d y - s t a t e f l o w

u n d e r c r e e p c o n d i t i o n s , s o t h a t t h e u s e o f m a p s i s

pa r t i c u l a r ly a t t r a c t i ve f o r m e c ha n i sm ide n t i f i-

c a t i on . Ac c or d ing ly , t h i s pa pe r i s spe c i f i c a l l y c on-

c e r n e d w i t h t h e c o n s t r u c t i o n o f m a p s f o r t e m p e r a -

tu r e s i n t he r a nge f r o m 0 .4 t o 1 .0 T m a nd f o r

nor m a l i z e d s t r e sse s be low 10 - 3 * : t h i s a r e a i s i nd i -

c a t e d by t he b r oke n l i ne s i n F ig . 1 .

3 . C o n s t i t u t i v e r e l a t i o n s h i p s f o r p l a s t i c

f l o w

For t he pu r pose s o f i l l us t r a t i on , m a p s w il l be

d e v e l o p e d f o r h i g h p u r i t y a l u m i n i u m . T h i s m e t a l

w a s s e l e c te d b e c a u s e o f t h e e x c e l l e n t e x p e r i m e n t a l

d a t a a v a i l a b l e f o r h i g h t e m p e r a t u r e c r e e p , a n d

be c a use t he sa m e m a te r i a l w a s use d f o r i ll us t r a ti ve

pur pose s i n two e a r l i e r a na lyse s [ 38 , 40] .

F o u r d i f f e re n t c r e e p m e c h a n i s m s a r e c o n s i d e re d :

N a b a r r o - H e r r i n g [ 4 2 , 4 3 ] a n d C o N e [ 4 4 ] d i f-

A b r e a k d o w n i n c r e e p b e h a v i o u r g e n e r a l l y o c c u r s a t s t r e s s l e v e ls g r e a t e r t h a n - 2 X 1 0 - 3 , a n d t h e c r e e p r a t e t h e n

i n c r e a s e s e s s e n t i a l l y e x p o n e n t i a l l y w i t h s t r e s s . T h i s b r e a k d o w n i s n o t i n c l u d e d i n F i g . 1 .

1 2 8 3


3/9

f u s i o n a l c r e e p , H a r p e r - D o r n c r ee p [ 4 5 ] , a n d a

d i s l o c a ti o n c l im b p r o c e s s . E a c h o f t h e s e m e c h -

a n i s m s m a y b e r e p r e s e n t e d b y a c o n s t i t u t i v e

r e l a t i onsh ip f o r t he s t e a dy- s t a t e c r e e p r a t e , ~ , o f

t h e f o r m

= A D o e x p ( - - Q / R T ) ~ (1)

whe r e A i s a d im e ns ion l e s s c o ns t a n t , Do i s a f r e -

que nc y f a c to r , Q i s the a c t i va t i on e ne r gy , R i s t he

g as c o n s t a n t ( 8 .3 1 J m o l -x K -1 i f Q i s in J m o l - a ) ,

k i s B o l t z m a n n s c o n s t a n t , a n d p a n d n a r e

c o n s t a n t s .

T he r e l a t i onsh ips f o r the se f ou r p r oc e sse s a r e

s h o w n i n T a b l e I , w h e r e t h e s u b s c r i p ts N H , C o , H D ,

a n d c r e la t e t o t h e N a b a r r o - H e r r i n g , C o b l e ,

H a r p e r - D o r n , a n d d i s c l o c a t io n c li m b p r o c e s se s ,

r e spe c t i ve ly , t he subsc r ip t s 1 a nd gb r e f e r t o l a t t i c e

a n d g r ai n b o u n d a r y d i f fu s i o n , r e s p e c t i v e l y , a n d n e

i s t h e s t re s s e x p o n e n t f o r t h e c l i m b p ro c e s s . T h e

f or m of t he se f our r e l a t i onsh ips wa s d i sc usse d

e a r l ie r [ 38 ] . T he f o l l owing va lue s we r e a s s igned

t o t h e v a r i o u s c o n s t a n t s i n E q u a t i o n s 2 t o 5 :

A N H = 28 ,

A c o = 6 6 . 8 ,

A H D =

1.67

x 1 0 - 1 1 ,

A c = 2.5 X 1 0 6 ,

D0(gb) = Do( l ) = 1.86,

Q1 = 1 4 3 . 4 k J m o l - 1 ,

Q g b = 0.6 Q1,

b = 2 .86 x 10 - s c m ,

n e = 4.4 .

T h e s h e a r m o d u l u s w a s t a k e n a s G = G o - - A G ) T ,

whe r e Go = 3 .02 2 x 104 M Pa a nd AG = 16 .0 M Pa

K -1

T h e f o u r m e c h a n i s m s l i s t e d i n T a b l e I o p e r a t e

i n d e p e n d e n t l y , s o t h a t t h e t o t a l s t r ai n r a te is e q u a l

t o t h e s u m o f t h e r a t e s d u e t o t h e f o u r i n d i v id u a l

p r oc e sse s .

4 C o n s t r u c t io n o f a d e f o r m a t i o n

m e c h a n i s m m a p

T h e p r o b l e m a s s o c i a t e d w i t h t h e c o n s t r u c t i o n o f

F i g . 1 i s t h a t b o t h t h e f i e l d b o u n d a r i e s a n d t h e

s t ra i n r a t e c o n t o u r s a p p e a r a s c u r v e d li n e s, so t h a t

e a c h m a p , f o r a ny se l e c t e d m a te r i a l a nd g r a in s i z e ,

m u s t b e c o n s t r u c t e d i n d i v i d u a l l y b y c o m p u t e r .

284

T h e c o m p u t e r s u b r o u t i n e h a s b e e n p u b l i s h e d

b y F r o s t a n d A s h b y [ 1 2 ] . B r i e f ly , i t in v o l v e s u s i n g

the va r ious c ons t i t u t i ve r e l a t i onsh ips t o e va lua t e

i n c r e m e n t a l l y t h e r a t e -c o n t r o l l in g p r o c e s s a t e a c h

o f t h e 4 0 0 0 p o i n t s d e f i n e d b y 5 0 e q u a l i n c r e m e n t s

o f t e m p e r a t u r e a n d 8 0 l o g a r i t h m i c a l l y s p a c e d

inc r e m e nt s o f s t re s s . T he po s i t i on o f a pa r t i c u l a r

s t r a i n r a t e c o n t o u r i s o b t a i n e d b y s e a r c h i n g

t h r o u g h t h e s tr e ss i n c r e m e n t s a t c o n s t a n t t e m p e r a -

t u r e u n t i l t w o a d j a c e n t i n c r e m e n t s a r e l o c a t e d

whic h y i e ld , r e spe c t i ve ly , s t ra in r a t e s l owe r a nd

h ighe r t ha n t he r e qu i r e d va lue . I n a s im i l a r m a nn e r ,

t h e f ie l d b o u n d a r i e s a r e o b t a i n e d b y l o c a ti n g

a d j a c e n t s t r es s inc r e m e nt s w he r e t he r e i s a c ha nge

i n t h e d o m i n a n t m e c h a n i s m . T h e c o m p l e x i t y o f

th i s p r oc e du r e a r i se s i n pa r t be c a use o f t he ne c e ss i t y

of c on s ide r ing d i s loc a t i on c l im b a nd g l ide a s a l t e r-

n a t iv e m e c h a n i s m s , a n d t h e n c h o o s i n g t h e f a s t e r o f

the se two pr oc e sse s .

M a p s o f th e t y p e s h o w n i n F ig . 1 m a y b e c o n -

s i d e r a b l y s i m p l i f i e d b y p l o t t i n g t h e d a t a i n t h e

f o r m o f n o r m a l i z e d s t r es s , a / G , ve r sus the r e c ip r o -

c al o f t h e h o m o l o g o u s t e m p e r a t u r e , T m / T . A n

e x a m p l e i s s h o w n i n F i g . 2 f o r p u r e a l u m i n i u m

ha ving a g r a in s iz e o f 100 pr o , us ing t he r e l a t i on-

sh ips g ive n i n T a b le I . F ig . 2 c ove r s t e m pe r a tu r e s

f r om 0 .4 t o 1 .0 Tm a n d v a l u e s o f o / G f r om 10 - 8

to 10 3 , so t ha t i t d i r e c t l y c or r e spond s t o t he a r e a

c on ta ine d w i th in t he b r ok e n l i ne s in F ig . 1 . As

ind i c a t e d i n F ig . 2 , a r e c ip r oc a l t e m pe r a tu r e

r e l a ti o n s h i p le a d s t o a m a p i n w h i c h t h e b o u n d a r i e s

se pa r a t i ng t he va r ious f i e lds a ppe a r a s s t r a igh t ,

r a the r t ha n c ur ve d , l i ne s .

T A B L E I C o n s t i t u ti v e r e l a ti o n s hi p s f o r d e f o r m a t i o n

m e c h a n i s m s i n h i g h p u r i ty a l u m i n i u m

M ec h an i sm Co n s t i t u t i v e r e l a t i o n sh ip fo r ~ ( sec -~ )

N a b a r r o - 4 N H = A N H D o ( 1 ) e x p --Q1/RT)

Her r i n g Gb~b~

C o b l e e C o = A C o D 0 ( g b ) e x p ( - -

Qgb/RT)

H ar p e r - ~ I- ID = AI-IDDo 17 x p --Q1/RT)

D o r n

Cl im b d e =

AeDo

1 ) ex p ( - -Q1 /R

T)

G b ( G ) n e

kT

(4)

(s)


4/9

T K )

i d 3 8 0 0 6 0 0 5 0 0 4 0 0

1 T

id 4

1 3

o-/~

I O - 6

/ . I / j

d = l O O f f m ~

. . . . . . . . . . . ? /

r

i i

j

C o b l e

i /

/ [ /

/ i /

- 7 / / i / B o u n d a r y

o f / I m b / H o - p o - O o o

/ / / / / i C o N e / H o r p e r- D o r n

/ / / / i ,

I 0 1 5 2 .0

T m / T

2 . 5

Figure 2 Deformation mechanism map of

normalized stress versu s the reciprocal of

homologous temperature for poly-

crystalline aluminium having a grain size

of 100 m , cove ring conditions of high

temperature creep a / G 0.4 Tin ).

F ig . 2 shows th r e e d i f f e r e n t f i e lds : d i s loc a t ion

c l i m b a t h i g h s t r e s s l e v e l s , N a b a r r o - H e r r i n g c r e e p

a t l o w s t r e s s e s a n d h i g h t e m p e r a t u r e s , a n d C o b l e

c r e e p a t l o w s t r e s s e s a n d l o w t e m p e r a t u r e s .

H a r p e r - D o r n c re e p d o es n o t a p p e a r o n th is m a p ,

b e c a u s e t h e r e i s n o v a l ue o f s tr es s a n d t e m p e r a t u r e

a t wh ic h e I~D i s t he f a s t e s t p r oc e ss . As i nd i c a t e d by

t h e b r o k e n l i n e s i n F i g. 2 , t h e c l i m b / H a r p e r - D o r n

a n d C o b l e / H a r p e r - D o r n b o u n d a r i e s b o t h l i e w i t h in

t h e f ie l d f o r N a b a r r o - H e r r i n g c r e e p , s o t h a t , f o r

d = 1 0 0 /~ m , e N H > e H D u n d e r a ll e x p e r i m e n t a l

c o n d i t i o n s .

F i g. 2 a l so s h o w s t w o c o n t o u r s o f c o n s t a n t

s t ra i n r a t e , f o r 1 0 - P s e c -1 ( ~ 0 . 0 1 % / d a y ) a n d

1 0 - 1 ~ - 1 ( ~ 3 % / 1 0 y e a r ) , c o r r e s p o n d i n g t o a

l o w e r l i m i t i n g l a b o r a t o r y s t r a i n r a t e a n d a n u p p e r

l im i t f o r s t r uc tu r a l de s ign pur pose s , r e spe c t ive ly .

T h e p r e ci s e m e t h o d o f c o n s t r u c t in g a m a p o f

th i s t ype i s i l l us t r a t e d i n F ig . 3 , whe r e t he t a b l e

i n s e t s u m m a r i z e s t h e a p p r o p r i a t e v a l u e s u s e d f o r

t h e d i m e n s i o n l e s s c o n s t a n t s , t h e a c t i v a t i o n

e ne r g i e s , a nd the s t r es s e x po ne n t , n e . T h e

pr oc e dur e f o l l ows f our ba s i c s t e ps :

( 1 ) S i n c e N a b a r r o - H e r r i n g c r e e p a n d t h e c l i m b

p r o ce s s b o t h d e p e n d o n t e m p e r a t u r e th r o u g h t h e

m a g n i t u d e o f e x p - - Q 1 / R T ) , t h e b o u n d a r y

s e p a r a t in g t h e s e t w o p r o c e s s e s i s i n d e p e n d e n t o f

t e m p e r a t u r e . F r o m E q u a t i o n s 2 an d 5 , t h e

b o u n d a r y is g iv e n b y

o [ A N H b ~ 2 ] 1 / n e -1 )

= [ A e \ d ] ( 6 )

T h e c o r r e s p o n d i n g r e l a ti o n s h i p f o r th e c l i m b /

H a r p e r - D o r n b o u n d a r y , o b t a in e d f r o m E q u a ti o n s

4 and 5, i s

a [AHpD_] / nc-t)

T h e a p p e a r a n c e o f H a r p e r - D o r n c r e e p o n a

d e f o r m a t i o n m e c h a n i s m m a p t h u s r e q u ir es t h a t t h e

va lue o f o / G pr e d ic t e d by E qua t ion 7 i s l a r ge r t ha n

the va lue p r e d i c t e d by E qua : t i on 6 . T h i s

m a y b e e x p r e s s e d e x p l i c i t l y i n t h e f o r m o f a

r e q u i re d m i n i m u m g ra in si ze f o r H a r p e r - D o r n

c r e e p :

~ d N I - I ~ 1 / 2

( 2 ) S inc e N a ba r r o - H e r r ing a nd Cob le c re e p a r e

bo th Ne wto n ia n v i sc ous p r oc e sse s ( i . e ., n = 1 ),

t h e b o u n d a r y s e p a r a ti n g t h e s e t w o p r o c e s s es i s

i n d e p e n d e n t o f s t r e s s . F r o m E q u a t i o n s 2 a n d 3 ,

t h e b o u n d a r y i s g i ve n b y

1 2 8 5


5/9

i o 4 ~

td 5

O /G

i o c

i o 7

T ( K )

8 0 0 6 0 0 5 0 0 4 0 0

I } I ) / 1

? . .

/ / j

c l i m b " / / / ~ / ' Q ~ .

F b

2, /,~c-r~

t - : j / ~ } j ~

/ I

I

~ ' - N o b o r r o - H e r r i n g - -

Coble

i

1

Q ~

/ / / L @

/ 1 1 /

/ /

/

/

/ [

1 5

i n A C : 2 5 x 0 6

9 Q ! : 1 4 3 4

k d m o l - I

ANN: 28

Q q b = 0 . 6 Q #

A c o = 6 6 . 8

c = 4 . 4 D o ( g b ) = D o ( ~ )

- Q,~ - Qg b

p o 0 0 ]

-

ids

t . 0 2 . 0

/ T

Figure 3 Me thod o f constructing a defor-

mation mechanism map of normalized

stress versus the reciprocal of homologous

temperature for condit ions o f hig h tem-

pera ture c reep

(e /G


6/9

t he f our c o ns t i t u t i ve r e l a t i onsh ips g ive n by

E q u a t i o n s 2 t o 5 . I n p r a c t i c e , t h i s a p p r o x i m a t i o n

i n t r o d u c e s o n l y a s m a ll e r r o r w h i c h is o f m i n o r

s ign i f i c a nc e i n c om pa r i son w i th t he unc e r t a in t i e s

a ssoc i a t e d w i th the va r ious t e r m s in t h e c ons t i t u t i ve

e q u a t i o n s u s e d t o c o n s t r u c t t h e m a p .

A se c ond e r r o r a r i se s be c a use t he s t r a in r a t e s

due to t he d i f f e r e n t m e c ha n i sm s a r e a dd i t i ve , so

tha t t he c o n tou r s a r e s l i gh t ly c ur ve d in t he v i c in i t y

of t he f i e ld bounda r i e s . Aga in , t he s im pl i f i c a t i on

of d r a wing s t r a igh t l i ne s up to t he bounda r i e s l e a ds

t o a n e r r o r w h i c h is v e r y m u c h l es s t h a n t h e u n c e r -

t a in t i e s i n t he c ons t i t u t i ve r e l a t i onsh ips . T h i s

s e c o n d a p p r o x i m a t i o n w a s a ls o u s e d b y A s h b y [ 3]

i n c o n s t ru c t i n g t h e o r ig i n a l d e f o r m a t i o n m e c h a n i s m

m a p s .

5 C o n s t r u c t io n o f m a p s f o r d i f f e r e n t g ra in

sizes

H a r p e r - D o r n c r ee p is n o t i n c l u d ed i n t h e m a p f o r

a lu m i ni um a t d = 1 0 0 m b ec au se ~ N H > ~ H D

u n d e r a ll e x p e r i m e n t a l c o n d i t i o n s . T h e li m i t in g

g r ai n siz e a t w h ic h H a r p e r - D o r n a n d N a b a r r o -

H e r r i n g c r e e p c o n t r i b u t e e q u a l l y t o t h e o b s e r v e d

c r e e p r a te m a y b e o b t a i n e d f r o m E q u a t i o n 8 , g iv in g

d = 3 7 0 m .

T he m a p f o r t h i s c r i t i c a l g r a in s i z e i s shown in

F ig . 4 , i nd i c a t ing the e qua l c on t r ibu t ions a r i s ing

f r o m t h e H a r p e r - D o r n a n d N a b a r r o - H e r r i n g

p r o c e s se s . A t h i g h e r v a lu e s o f d , t h e C o N e / H a r p e r -

D o r n b o u n d a r y m o v e s to l o w e r t e m p e r a t u r es , b u t

t h e c l i m b / H a r p e r - D o r n b o u n d a r y r e m a in s fi x ed in

p o s i t i o n b e c a u s e b o t h p r o c e s s e s a r e i n d e p e n d e n t

o f g r ai n s iz e . T h u s , t h e i n t e r c e p t o n t h e l e f t h a n d

axis a t o G 2 9 x 10 -6 r e pr e se n t s a l owe r l im i t i ng

n o r m a l i z e d s tr e ss f o r t h e t r a n s i ti o n t o N e w t o n i a n

v i sc ous f l ow . As ind i c a t e d in F ig . 5 f o r d = 0 .1 c m ,

t h e N a b a r r o - H e r r i n g p r o c e s s i s e x c l u d e d w h e n t h e

c l i I n b / N a b a r r o - H e r r i n g a n d C o b l e / N a b a r r o -

H e r r in g b o u n d a r i e s b o t h f a ll w i t h i n t h e H a r p e r -

Dor n f i e ld .

Re f e r e n c e t o F igs . 2 t o 5 i nd i c a t e s t ha t , s inc e

t h e c l im b p r o c e s s d o e s n o t d e p e n d o n g r a in s iz e ,

t h e s t r a in r a t e c o n t o u r s r e m a i n f i x e d i n p o s i t i o n

Figure

Deformation mechanism map of

normalized stress versus the reciprocal of

homologous temperature for polycrys-

talline aluminium having a grain size of

370 m.

lo-3

1 0 4 _ _

1 0 5 _

c r /e

10-6

1138

1.0

K)

8 0 0 6 0 0 5 0 0 4 0 0

/

/

/ / . /

d = 3 7 0 / z m . / j /

. \ / / 1 / ,

c l im b 0 - ? / ~ / ~ /

/ / / / /

/ / /

f / / / - ' / / _

/ / '

/ / / H a r p e r - D o r n

// / + CoNe

/ / / N o b a r r o - H e r r i n g

/

/ / /

1 .5 2 . 0 2 . 5

mm/T

*Using the general relationship for ~ given in Eq uatio n 1, the variation of logarithmic

a G

with

T m T

is precisely

expressed as

n l og = , o g [ ~ t ~ ) ] + 2 . ~ m 14)

Eq uatio n 13 neglects the slight temperature dependence resulting fro m the logarithmic term on the right han d side of

the equality in Eq uation 14.

1 2 8 7


7/9

Id

i o 4

O / G

c~

T ( K )

8 0 0 6 0 0 5 0 0

c l i m b -\ / ~ / "

\~XO

9 /

/ / /

/

/

/ / / / / "

/

/ ' /

. . . . . L - - . / -

/

/

i / /

/ / /

/ /

/ 1 /

H o r p e r - j D o r n

/

I 0 7 / - / B o u n d o r y

/ - - - - - - c l i m b /N o b o r r o -H e r d n g

IO a / / . . . . C o b l e / N o b G r r o - H e r rm g . l

I I I

1 . 0 1 . 5 2 . 0

Tm/T

4O

/ /

/ / -

/ / - /

C o N e

Figure 5 Deformation mechanism map of

normalized stress versus the reciprocal o f

hom ologou s temperature for polycrystal-

l ine aluminium having a grain siz e of

1.0 cm.

2.5

w i th in t he c l im b f i e ld . S im i l a r ly , a t l a r ge g r a in

s iz es w h e n H a r p e r - D o r n c r ee p b e c o m e s i m p o r t a n t ,

t h e c o n t o u r s a l s o r e m a i n f i x e d i n p o s i t i o n w i t h i n

t h e H a r p e r - D o r n f i el d . * T h i s i s a n a t t r a c ti v e

f e a t u r e o f m a p s o f t h i s t y p e , s i n c e, b y u s i n g

E q u a t i o n 1 3 , i t p ro v i d e s a m e t h o d f o r p l o t t i n g

c o n t o u r s o v e r t h e e n t i r e m a p a t a n y s e l e c te d g ra i n

s i z e. A ne w m a p f o r a d i f f e r e n t g r a in s i z e i s t hus

a c h i e v e d b y a v e ry s i m p l e f i v e -s t e p p r o c e d u r e :

( 1 ) U s e E q u a t i o n 8 t o d e t e rm i n e t h e lo w e r

l i m i t i n g g r a i n s i z e f o r H a r p e r - D o r n c r e e p .

( 2 ) D e t e r m i n e t h e i n t e r c e p t o n t h e o/G axis

f r o m E q u a t i o n 6 if e NH > e H D o r f r o m E q u a t i o n

7 i f

6NH < s

( 3 ) D e t e r m i n e t h e i n t e r c e p t o n t h e Tm/T axis

f r o m E q u a t i o n 9 i f 6NH > ~ H D o r f r o m E q u a t i o n

1 1 i f eNH < e I~n-

( 4 ) D r a w t h e f i el d b o u n d a r i e s p a r al l el t o t h o s e

a l r e a dy c a l c u l a t e d f o r t he i n i t i a l g r a in s i z e .

( 5 ) P l a c e t h e s t r ai n r a t e c o n t o u r s a t t h e s a m e

p o s i t i o n s w i t h i n t h e c l i m b f i el d , a n d c o m p l e t e t h e

c o n t o u r s u s in g E q u a t i o n 1 3 .

6 C o n s t r u c ti o n o f a n A s h b y m a p fo r o G

versus T Tm

I f re qu i r e d , i t i s pos s ib l e t o t r a nsp ose t he da t a

f r o m a p l o t o f e/G ve r sus Tm/T t o t h e s t a n d a r d

f o r m o f

e/G

ve r sus

T/Tm. An

e x a m p l e i s s h o w n i n

F i g . 6 , r e p r e s e n t i n g t h e d a t a o f F i g . 2 f o r a l u m i n i u m

w i t h d = 1 0 0 m . I n v i ew o f t h e a d d e d c o m p l e x i t y

i n d r a w i n g a m a p o f t h is t y p e , i t s e e m s p r e f e r a b le

t o c o n s t r u c t t h e m a p s i n t h e f o r m o f a/G ve r sus

Tm/T.

7 Discussion

T h e d e f o r m a t i o n m e c h a n i s m m a p s d e v e l o p e d i n

t h i s r e p o r t r e p r e s e n t a s im p l i f ie d v e r s i o n o f t h e

A s h b y m a p f o r c o n d i t i o n s o f h i g h te m p e r a t u r e

c r e e p

o / G ~

1 0 - 3 a n d T > ~ 0 . 4 T i n ). T h e y h a v e

t w o s i g n i f i c a n t a d v a n t a g e s o v e r t h e s t a n d a r d m a p s :

( 1 ) t h e y a re v e r y e a sy t o c o n s t r u c t w i t h o u t t h e u s e

o f a c o m p u t e r , a n d ( 2 ) t h e r e is a s i m p l e p r o c e d u r e

f o r c o n s t r u c t i n g s e v e ra l m a p s f o r t h e s a m e m a t e r i a l

a t d i f f e r e n t g r a in s iz es . F u r t h e r m o r e , t h e c o n s t r u c -

t i o n p r o c e d u r e p e r m i t s t h e p r e s e n t a t i o n o f t h e

*A comparison of Figs. 4 and 5 reveals a slight difference in the positio n of the two strain rate con tours within the

Ha rper -D orn f ield. This difference ar ises because Fig. 4 re la tes to a grain size of 370 ~m , when Harp er-D orn and

Na bar ro-H errin g creep b oth contribute equally to the strain rate. At larger grain sizes, when

:NIt

is significantly

lower than

~I-ID,

he strain rate co ntours in the H arpe r-D orn f ield are independent of the value ofd .

1 2 8 8


8/9

. . -5 2OO

/ U I ~ x \ I

i \ - \

[ \ . \ . .

i04

iO5

O-/G

i O 6

i O

_ _

i 0 8

( 3 4

T E M P E R A T U R E ~

5 0 0 4 0 0

5 0 0 6 0 0

\,.,

I

0.9 1.0

c l i m b

\ ,

- \

' \ t \

' \ \

\ N o b e r r o - H e r r i n g

C o b l e \ ~ \ ~ '0 ~

\ \ \

\

\ \

\ \ \

\

\ \

, \

\ \ \

\ \ \

\ \ \ \ \

\

\

0.5 06 o .7 o .8

T / T m

Figure 6 The da ta of F ig . 2 t ransposed to

an A shby- ty pe de form at ion mechani sm

map for cond i t ions of high t empera ture

creep cr /G 0.4 Tin) .

m a p s e i t h e r in t h e s i m p l e f o r m a t o f o / G v e r s u s

T m / T o r , i f r e q u i r e d , b y t r a n s p o si n g t o t h e m o r e

u s u a l f o r m o f g / G v e r s u s T / T m .

A s i n d i c a t e d i n e a rl ie r r e p o r t s [ 3 8 - 4 1 ] , t h e

a c c u r a c y o f a ll m a p s i s n e c e s s a r i ly l i m i t e d b y t h e

a c c u r a c y o f t h e c o n s t i t u t i v e r e l a t i o n s h i p s u s e d i n

t h e i r c o n s t r u c t i o n . T h i s l i m i t a t i o n i s o f re l a t i v e l y

m i n o r i m p o r t a n c e f o r p u r e a l u m i n i u m , b u t m a y b e

i m p o r t a n t i n o t h e r m a t e r i a l s w h e r e t h e r e is l es s

g o o d e x p e r i m e n t a l d a t a t o e s ta b l is h t h e v a ri o u s

t e r m s i n t h e c o n s t i t u t i v e e q u a t i o n s . A d e t a i l e d d i s-

c u s si o n o f t h e u se o f d e f o r m a t i o n m e c h a n i s m

m a p s i n t h e p r e d i c t i o n o f c re e p b e h a v i o u r w a s

p r e s e n t e d e a r l i e r [ 4 6 ] .

8 S u m m a r y a n d c o n clu s io n s

( 1 ) I t i s d e m o n s t r a t e d t h a t t h e A s h b y - t y p e d e fo r -

m a t i o n m e c h a n i s m m a p m a y b e c o n s i d e r a b l y

s i m p l i f ie d b y p l o t t i n g i n t h e f o r m o f n o r m a l i z e d

s t r e s s , o / G , v e rs u s t h e r e c i p r o c a l o f t h e h o m o l o g o u s

t e m p e r a t u r e , T m / T . I n t h i s f o r m , t h e f i e ld b o u n d -

a r ie s a p p e a r a s s t r a ig h t l i n e s , a n d t h e c o n s t a n t

s t ra i n ra t e c o n t o u r s m a y b e a p p r o x i m a t e d a s

s t r a i g h t l i n e s .

( 2 ) R e p r e s e n t a t i v e m a p s a r e p r e s e n t e d f o r p u r e

a l u m i n i u m u n d e r c o n d i t i o n s o f h i g h t e m p e r a t u r e

c r e e p a / G < . 1 0 - 3 a n d T ~ > 0 . 4 T i n ) .

( 3 ) A s i m p l e p r o c e d u r e i s o u t l i n e d f o r c o n -

s t r u c t i n g s e v e ra l m a p s f o r t h e s a m e m a t e r i a l a t

d i f f e r e n t g r a i n s iz e s .

( 4 ) I f r e q u i re d , t h e s e m a p s m a y b e t r a n s p o s e d

t o t h e s t a n d a r d f o r m o f o / G v e r s u s T / T m .

A c k n o w l e d g e m e n t

T h i s w o r k w a s s u p p o r t e d b y t h e U n i t e d S t a t e s

E n e r g y R e s e a rc h a n d D e v e l o p m e n t A d m i n i s t r a t i o n

u n d e r C o n t r a c t E Y - 7 6 - S - 0 3 -0 1 1 3 P A - 2 6 .

References

1. J . WEERTMAN and J . R . WEERTMAN , Phys ica l

Meta l lurgy , ed i t ed by R . W. Cahn (Nor th-Hol land ,

Am s te rdam, 1965) p . 793 .

2. J . WEE RTMA N, Trans. Am er. Soc. Metals 61 (1968)

681.

3. M. F . AS HB Y, Ac ta Met . 20 (1972) 887.

4. R. L. STOCKER and M. F. ASHBY, Rev. Geophys .

SpacePhys . 11 (1973) 391.

5 . M. F . ASHBY, The M icros t ruc ture and Design of

Al loy s , Vol . 2 (The Ins t i tu t e of Meta ls and The

I ron and S tee l Ins t i tu t e , Lond on, 1973) p . 8 .

6. J . T. A. R OBERTS and J . G. VOGLEWE DE, j r.

Amer. Ceram. Soc.

56 (1973) 472 .

P . A . U R I C K

and M . R.

N O T I S , i id 5 6

(1973) 570.

J. H. GITTU S, Phi l. Ma g. 30 (1974) 751.

M. R. NOTIS, J. Am er . Ceram. Soc . 57 (1974) 271.

Idem , Powd er Met . In t . 6 (1974) 82.

M . F. A S H B Y and H. J. F R O S T , Cons t i tu t ive

7 .

8.

9 .

10.

11.

1 2 8 9


9/9

E q u a t i o n s i n P l a s t i c i t y , e d i t e d b y A . S . A r g o n ( M I T

P r e s s, C a m b r i d g e , M a s s ., 1 9 7 5 ) p . 1 1 7 .

1 2 . H . J . F R O S T a n d M . F . A S H B Y , R a t e P r o c es s es i n

P l a s t i c D e f o r m a t i o n o f M a t e r i a l s , e d i t e d b y J . C . M .

L i a n d A . K . M u k h e r j e e ( A m e r i c a n S o c i e t y f o r

M e t a l s , M e t a l s P a r k , O h i o , 1 9 7 5 ) p . 7 0 .

1 3 . M . F . A S H B Y a n d R . R A J , T h e M e c h a n i c s a n d

P h y s ic s o f F r a c t u r e ( T h e M e t al s S o c i e ty , L o n d o n ,

1 9 7 5 ) p . 1 4 8 .

1 4 . M. R . NOTIS D e f o r m a t i o n o f C e r a m i c M a t e r i a l s ,

e d i t e d b y R . C . B r a d t a n d R . E . T r e s s l e r ( P l e n u m

P r e s s, N e w Y o r k , 1 9 7 5 ) p . 1 .

1 5 . A . S A M U E L S S O N a n d A . T H O L E N , G r a i n B o u n d -

a r i e s i n E n g i n e e r i n g M a t e r i a l s , e d i t e d b y J . L .

W a l t e r , J . H . W e s t b r o o k a n d D . A . W o o d f o r d ( C l a i t o r ' s

P u b l i s h i n g D i v i s i o n , B a t o n R o u g e , 1 9 7 5 ) p . 1 0 7 .

1 6 . F . W . C R O S S M A N a n d M . F .

A S H B Y A c t a M e t .

2 3

( 1 9 7 5 ) 4 2 5 .

1 7 . M . R . N O T I S , R . H . S M O A K a n d V . K R I S H -

N A M A C H A R I ,

Mater. ScL Res.

1 0 ( 1 9 7 5 ) 4 9 3 .

1 8 . D . B . K N O R R a n d M . R . N O T I S , J . Nu c l ea r M a t er .

5 6 ( 1 9 7 5 ) 1 8 .

1 9 . M. F . A S H B Y , J .

Geol . Soc.

1 3 2 ( 1 9 7 6 ) 5 5 8 .

20. S. WHITE Phil.

T ran s . Ro y . S o c . A 2 8 3 ( 1 9 7 6 ) 6 9 .

2 1 . M . S . P A T E R S O N ,

ib id

2 8 3 ( 1 9 7 6 ) 1 6 3 .

2 2 . E . H . R U T T E R ,

i b m

2 8 3 ( 1 9 7 6 ) 2 0 3 .

2 3 . R . K . B H A R G A V A , J . M O T E F F a n d R . W .

S W l N D E M A N M e t . T ra ns .

7 A ( 1 9 7 6 ) 8 7 9 .

2 4 . B . K . A T K I N S O N ,

Ea r th a n d P la n e ta ry S c i . L e t t .

2 9

( 1 9 7 6 ) 2 1 0 .

2 5 . L . C . A . S A M U E L S S O N , K . N . M E L T O N a n d J . W .

E D I N G T O N A c t a M e t. 2 4 ( 1 9 7 6 ) 1 0 1 7 .

2 6 . R . N . S I N G H , J . Nu clea r M a ter . 6 4 ( 1 9 7 7 ) 1 6 7 .

2 7 . V . K R I S H N A M A C H A R I a n d M . R . N O T I S , M a t e r .

Sci . Eng .

2 7 ( 1 9 7 7 ) 8 3 .

28. R. A. VERRALL R. J. FIELDS a n d M. F . ASHBY

J . Am er . Ceram. S o c .

6 0 ( 1 9 7 7 ) 2 1 1 .

29 . B.

ILSCHNER

H o c h t e m p e r a t u r - P l a s t i z i t ~ i t , R e i n e

u n d a n g e w a n d t e M e t a l l k u n d e i n E i n z e ld a r s t el l u n g e n ,

V o l . 2 3 ( S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 3 ) p . 2 5 8 .

3 0 . J . G I T T U S , C r e e p , V i s c o e l a s ti c i t y a n d C r e e p

F r a c t u r e i n S o l i d s ( J o h n W i l ey , N e w Y o r k , 1 9 7 5 )

p . 2 7 4 .

3 1 . A . G . E V A N S a n d T . G . L A N G D O N ,

Prog. Mater.

Sci . 2 1 ( 1 9 7 6 ) 1 7 1 .

3 2 . A . G . G U Y , E s s e n t i a l s o f M a t e r ia l s S c i e n c e

( M c G r a w - H i l l , N e w Y o r k , 1 9 7 6 ) p. 3 2 4 .

3 3 . R . W . H E R T Z B E R G , D e f o r m a t i o n a n d F r a c t u r e

M e c h a n i c s o f E n g i n e e r in g M a t e r i a l s ( J o h n W i l e y ,

N e w Y o r k , 1 9 7 6 ) p . 1 5 6 .

3 4 . W . D . K I N G E R Y , H . K . B O W E N a n d D . R .

U H L M A N N , I n t r o d u c t i o n t o C e r a m i c s , 2 n d e d it i o n

( J o h n W i l e y , N e w Y o r k , 1 9 7 6 ) p . 7 4 5 .

3 5 . J . -P . P O I R I E R , P l a s t i c i t~ ~ H a u t e T e m p 6 r a t u r e d e s

S o l id e s C r i s t a l l i n s (E y ro l l e s , P a r i s , 1 9 7 6 ) p . 1 5 6 .

3 6 . A . N I C O L A S a n d J . - P .

POIRIER

C r y s t a l l i n e

P l a s t i c i t y a n d S o l i d S t a t e F l o w i n M e t a m o r p h i c

R o c k s ( J o h n W i le y , L o n d o n , 1 9 7 6 ) p. 4 0 3 .

3 7 . S . M . C O P L E Y a n d J . C . W I L L I A M S , A l l o y a n d

M i c r o s t r u c t u r a l D e s i g n , e d i t e d b y J . K . T i e n a n d

G . S . A n s e l l ( A c a d e m i c P r es s , N e w Y o r k , 1 9 7 6 ) p . 3 .

3 8 . F . A . M O H A M E D a n d T . G . L A N G D O N , M e t .

Trans.

5 ( 1 9 7 4 ) 2 3 3 9 .

3 9 . T . G . L A N G D O N a n d F . A . M O H A M E D , j r Mater.

S cL 1 1 ( 1 9 7 6 ) 3 1 7 .

4 0 . l d em M a ter . S cL En g . 3 2 ( 1 9 7 8 ) 1 0 3 .

4 1 . Ide m J. Mater . Sci. 1 3 ( 1 9 7 8 ) 4 7 3 .

4 2 . F . R . N . N A B A R R O , R e p o r t o f a C o n f e r e n c e o n

S t r e n g t h o f S o l i d s ( T h e P h y s i c a l S o c i e t y , L o n d o n ,

1 9 4 8 ) p . 7 5 .

4 3 . C . H E R R I N G , J .

A p p L P h y s .

2 1 ( 1 9 5 0 ) 4 3 7 .

4 4 . R . L . C O B L E ,

ib id

3 4 ( 1 9 6 3 ) 1 6 7 9 .

4 5 . J . H A R P E R a n d J . E . D O R N , A c t a M e t . 5 ( 1 9 5 7 )

6 5 4 .

4 6 . F . A . M O H A M E D a n d T . G . L A N G D O N , j r Eng.

M a ter . T ech .

9 8 ( 1 9 7 6 ) 1 2 5 .

R e c e iv e d 1 2 M a y a n d a c c e p t e d 1 6 S e p t e m b e r 1 9 7 7 .

129

Documents

Constructing Ashby Map