size effect and geometrical effect of solids in micro-indentation test.pdf

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AC TA M EC HA NIC A SINICA (Eng li sh Ser ies ) , Vo i .19, N o. l ,Februa ry 2003

Th e Chinese Society of Theoret ical a nd Appl ied M echanics

Chinese Journal o f M echanics Press, Bei j ing, China

Al lerton Press, INC., New York, U .S.A.

ISSN 0567-7718

S I Z E E F F E C T A N D G E O M E T R I C A L E F F E C T O F S O L I D S

I N M I C R O - I N D E N T A T I O N T E S T *

WEI Yueguang (~f -) t WANG Xuezheng (~! ~) ZHAO Manhong (N N~)

CHENG Che-Min (~ g~ ) BAI Yilong (~t~;~)

(LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China)

A B S T R A C T : Micro-indentation tests at scales of the order of sub-micron show that the measured

hardness increases strongly with decreasing indent depth or indent size, which is frequently referred to

as the size effect. At the same time, at micron or sub-micron scale, another effect, which is referred

to as the geometrical size effects such as crystal grain size effect, thin film thickness effect, etc., also

influences the measured material hardness. However, the trends are at odds with the size-independence

implied by the conventional elastic-plastic theory. In the present research, the strain gradient plasticity

theory (Fleck and Hutchinson) is used to model the composition effects (size effect and geometrical

effect) for polycrystal material and metal thin film/ceramic substrate systems when materials undergo

micro-indenting. The phenomena of the "pile-up" and "sink-in" appeared in the indentation test for

the polycrysta l materials are also discussed. Meanwhile, the micro-indentation experiments for the

polycrystal A1 and for the Ti/SisN4 thin film/substrate system are carried out. By comparing the

theoretical predictions with experimental measurements, the values and the variation trends of the

micro-scale parameter included in the strain gradient plasticity theory are predicted.

KEY WORDS: micro-indentation tests, size effect, geometrical effect, strain gradient plasticity

1 INTRODUCTION

Indentation test is an important and effective

experimental method, and has been used extensively

to estimat e the plastic pro perties of solids undergoing

plastic deformation. Throu gh indentation test, the

loading-unloading relation between hardness and in-

dent depth is measured. Thereby, the material pa-

rameters, such as the yielding stress, strain hard-

ening exponent, Young's modulus, etc. are esti-

mated. Recently, with the advancement of experi-

mental technique and measuring precision, it is pos-sible to carry out the indentation tests at the scale

of one micron or sub-micron for obtaining more de-

tailed material information . Such small-scale indenta-

tion experiments are frequently referred to as micro-

indentation tests (or nano-indentation tests). In the

micro-indentation test, new results, different from the

conventional ones, the size-dependent results, were

obtai ned extensively [1~9]. For metal materials, the

measured hardness may double or even triple the

Received 30 April 2001, revised 29 October 2001

conventional hardness as the indent size (or depth)decreases to a fifth micron. The effect is often re-

ferred to as the size effect. In addition, at the micron

scale, if material is of some micro-geometrical struc-

tures and if the geometrical structure size is compara-

ble to the indent depth, the size-dependent hardness

mentioned above may be influenced additionally by

the geomet rica l size. We call this effect as the geo-

metrical effect in the present research. However, the

trends of both size effect and geometrical effect are

at odds with the size-independence implied by the

conventional elastic-plastic theory. Otherwise, from adimensional consi deration [1~ the hardnes s was only

dependent of macroscopic parameters of material, i.e.,

was size-independent.

In order to predict the size effect phenomena, re-

cently, several strain gradient plasticity theories have

been developed [n~13]. These theories include th e

strain gradient effects and go beyond the conventional

elastic-plastic theo ry frame. In the new constitu tive

* The project supported by the National Natural Science Foundation of China (19891180 and 19925211) and Bai Ren Plan ofCAS

t E-maih [email protected]



6O

r e l a ti o n s , t h e s t r a i n t e r m s a r e m a t c h e d w i t h t h e n e w

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

t e rs . T h u s t h e m a t e r i a l s iz e e f fe c t c a n b e c h a r a c t e r -

i z e d b y t h e l e n g t h - s c a l e p a r a m e t e r s .

O n t h e r e s e a r c h e s o f t h e s i ze e ff e c t i n t h e i n d e n -

t a t i o n t e s t s b y u s i n g st r a i n g r a d i e n t p l a s t i c i ty t h e o -r i es , in R e f .[ 4 ], t h e c a s e w h e n t h e m i c r o - s c a l e p a r a m e -

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

b y a d o p t i n g t h e c o u p l e s t r e s s t h e o r y I n ] u n d e r a n i n -

c o m p r e s s i b l e a s s u m p t i o n . I n R e f. [3 ] , t h e s i m i l a r c a se

t o [ 4] b u t a d o p t i n g t h e F l e c k a n d H u t c h i n s o n ' s s t r a i n

g r a d i e n t p l a s t i c i t y [11] w a s a n a l y z e d . B y a p p l y i n g t h e

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

t e r ia l t u n g s t e n ( W ) I s] , t h e m i c r o - s c a l e p a r a m e t e r f o r

m a t e r i a l W w a s o b t a i n e d w i t h i n 0 . 2 5 ~ 0 .5 2 p m .

A l t e r n a t i v e l y , a d i s lo c a t i o n m o d e l h a s b e e n u s e d

t o s t u d y t h e s i z e e f fe c t f o r t h e m i c r o - i n d e n t a t i o n t e s t

i n R e f. [1 ] . B y u s i n g t h e T a y l o r ' s r e l a t i o n , y o n M i s e s '

p l a s t ic f l o w la w a n d t h e c o n v e n t i o n a l r e l a t i o n o f t h e

h a r d n e s s ( t r i p l e t h e f lo w s t r e s s ) , a n i n v e r s e s q u a r e

r o o t r e la t i o n b e t w e e n h a r d n e ss a n d i n d e n t d e p t h w a s

o b t a i n e d .

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

c o n v e n t i o n a l m e t a l s ( C u , A g , A 1 ) [2,7 '91 w i t h t h a t f o r

t h e u n i q u e l y h i g h m o d u l u s m e t a l W [ s] , i t i s c l e a r t h a t

t h e s e n s i t i v e z o n e s c a l e o f t h e s i ze e f f e ct o f t h e f o r -

m e r is sm a l l e r t h a n t h a t o f t h e l a t t e r f o r a n o r d e r

o f m a g n i t u d e . C o r r e s p o n d i n g l y , t h e s c al e o f t h e c o n -

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

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

i n v i e w o f t h e p r e v i o u s i n v e s t i g a t i o n s [ 3 ,4 ] w h i c h w e r e

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

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

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

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

t a c t r a d i u s f o r t h e c o n v e n t i o n a l m e t a l s ( C u , A g , A 1,

e t c . ). F o l l o w i n g [ 9 ] , i n t h e p r e s e n t r e s e a r c h , t h e F l e c k

a n d H u t c h i n s o n ' s s t r a i n g r a d i e n t p l a s t i c i ty m o d e l [11],

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

c o n s i d er e d . E x p e r i m e n t a l r e se a r c h e s o f t h e m i c r o -

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

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

o u t . F o r t h e p o l y c r y s t a l A1 , t h e e x p e r i m e n t a l p r o -

g r a m m i n g a d o p t e d h e r e is d i f fe r e n t f r o m t h e c o n v e n -

t i o n a l c o n t i n u u m s t i ff n e ss m e t h o d ( C S m e t h o d ) , a n d

t h e m e a s u r e d h a r d n e s s c u r v e is b a s e d o n a u n i q u e

l o a d i n g p o i n t o n t h e s u r f a ce , a n d d e p e n d s o n t h e s t a -

t u s o f t h e l o a d i n g p o i n t . I n t h e p r e s e n t e x p e r i m e n t ,

m a n y l o a d i n g p o i n t s a r e se l e c te d r a n d o m l y , a n d o n e

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

h a r d n e s s / d e p t h r e l a t io n . A l t h o u g h a d a t a s tr i p f o r

A C T A M E C H A N I C A S I N I C A ( E n g li sh S e rie s ) 2 0 03

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

C S m e t h o d , i t s d e f e c ts c a n b e a v o id e d . B y c o m p a r -

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

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

d i f fe r e n t m a t e r i a l c a n b e p r ed i c t e d . F o r c o m p a r i s o n ,

i n t h e i n d e n t a t i o n e x p e r i m e n t f o r t h e T i / S i 3 N 4 , a

t h i n f i l m / s u b s t r a t e s y s t e m , t h e C S m e t h o d w i ll b e

a d o p t e d . T h r o u g h c o m p a r i n g t h e t h e o r e t i c a l p r e d i c -

t i o n s w i t h t h e e x p e r i m e n t a l r e s u l ts , t h e v a l u e s a n d t h e

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

f o r d i f fe r e n t m a t e r i a l , r e l a t e d w i t h m i c r o - g e o m e t r i c a l

s i z e , a r e p r e d i c t e d .

2 D E F O R M A T I O N A L T H E O R Y O F S T R -

A I N G R A D I E N T P L A S T I C I T Y

I n t h e p r e s e n t s e c t i o n , t h e . c o m p r e s s i b l e g e n e r a l

f o r m o f t h e d e f o r m a t i o n a l t h e o r y o f t h e s t r a i n g r a d i -e n t p l a s t i c i t y i s o u t l i n e d a s f o ll o w s .

2 .1 C o n s t i t u t i v e R e l a t i o n s

T h e d e f i n i ti o n s o f t h e s t r a i n a n d s t r a i n g r a d i e n t

a r e1 e e

e i j = ~ (u i , j + u s , i ) = c i j + ~ i ~( 1 )

e ~]p? ] i j k = U k , i j = ? ] i j k ~ - i j k

T h e e x p r e s s io n s r e l a t e d w i t h t h e c o n s t i t u t i v e r e l a t i o n s

a r e l i s t e d a s f o l l o w s

M g e 2 1W e E2 (1 + @ ( 1 - - 2 . ) k k + ~ 2 ( 1/ Q % ' e ' ~ +

-~Le2 e(I) e(1)'~I = 1 I ? ] i j k ~ ] i j k )

= ow~ . = owe/o, 5

2 , , ~ - ' , L 2 ( I ) ( I )= ~ g i j g i j ~ - ~ I ~ l i jk ? ~ i j k

/ = 1

I = 1

7 ( i )~ ( z )

i j k = A i j k l r n n ~ ] lr n n

p _ _ 3 0 J 2 3 ,

r 2 h e O G i j - 2 h e G i j

r i ( I ) = T ( I ) ~ -j k i j k l m n l m n

3p 3 O J e 1 V " r . - 2 ~ , ' ( z )

~ i j k - - 2hP O T ij k - - h P ~ I ~ij klmn TlrnnI = 1

h e -_ Z / ( Z - X / E )

( 2 )

w h e r e X a n d ~ a r e t h e e f f e c ti v e s t r e s s a n d e f f e c ti v e



Vol.19, No.1 W ei Yu egu ang et al.: Size EfFect and Geometr ical Effect of Solids 61

s t r a i n , r e s p e c t i v e l y . L ~ a n d L I ( I = 1 , 4 ) a r e t h e

m i c r o - s c al e p a r a m e t e r s f o r t h e e l a st i c a n d p l a s t i c c a se ,

r e s p e c t i v e l y . F r b m t h e d i s c u s s i o n i n [ 1 1 ] , t h e r e e x i s t s

a g e n e r a l r e l a t i o n a m o n g t h e m i c r o - sc a l e s ( S G t h e o r y )

1 L L 3 = ( 3 )L1 = L L2 = 2

I n a d d i t i o n , t a k e L 4 = L / 2 . S i m i l a r l y , ( 3 ) i s v a l i d

a l s o f o r t h e e l a s t i c i t y s t r a i n g r a d i e n t c a s e , w i t h L re -

p l a c e d b y L ~ i n ( 3 ) . M o r e o v e r , t h e p r e v i o u s r e s e a r c h

h a s s h o w n t h a t t h e s o l u t i o n i s i n s e n s i t i v e t o t h e v a l u e

o f L ~ / L w i t h i n t h e r e g i o n 0 < L ~ / L < 1 [14] . T hu s ,

t a k e L ~ / L = 0 .5 in t h e p r e s e n t r e s e a r c h . I n t h e l a s t

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

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

= S o < _

( 4 )= ~--'0 (~ /C rY ) 1 /N ~ > O'y

w e h a v e

h P : E [ ( ~ / C r y ) l / N - l - 1 ] - i (5 )

T (I) ( f = 1 , 4 ) is t h e p r o j e c t i o nn f o r m u l a ( 2 ) , i jklm~

t e n s o r o f t h e s t r a i n g r a d i e n t, a n d t h e d e t a i l e d e x p r e s -

s i o n s a r e g i v e n i n [ 1 4 ] . T h u s , b y u s i n g t h e f o r m u l a s

( 1 ) a n d ( 2 ) a n d t h e p r o j e c t i o n t e n s o r n o r m a l i t y , t h e

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

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

(7ij : ,:{ gi j@

1 + u + 2 E / h p

3 1 - 2 u l + u + ~ E / h P ](6 )

3

L ~

" I= 1 L ~ /L ~ 2 + 2 E / h P i j m , , ~ "

L e42Ti~21mn ?~lmn

2 . 2 E q u i l i b r i u m a n d V a r i a t i o n a l R e l a t i o n s

F r e q u e n tl y , e q u i l i b r i u m e q u a t i o n s c a n b e d e -

s c r i b e d b y th e d i s p l a c e m e n t - v a r i a t i o n a l r e l a t io n , f o r

t h e f in i te e l e m e n t i m p l e m e n t a t i o n . T h e d i s p l a c e m e n t -

v a r i a t i o n a l r e l a t i o n f o r th e s t r a i n g r a d i e n t p l a s t i c i t y

t h e o r y i s g i v e n b y Ira ]

/ v ( a~ jS c~ j + r ~ jk S ~ j k ) d V = f v f k S u k d V +

f s t k 6 u k d S + / ; r k ( D ~ u k ) d S (7 )

B a s e d o n ( 7 ) , t h e t r a c t i o n o n S i s d e f i n e d b y

0 i ; kt k = n i a i k O X j ] + n i n j w i j k ( D p n p ) -

D j (n i~ - ijk ) (8 )

a n d t h e s u r f a ce t o r q u e b y r k = n i n j T i j k . T h e o p e r a -

t o r s D a n d D j i n (7 ) a n d ( 8 ) a r e d e f i n e d as

D j = O / O x j - n j n k O / O X k D = n k O / X k (9 )

w h e r e n i i n ( 7 ) ~ ( 9 ) i s t h e d i r e c t i o n c o s i n e o n S .

3 P R O B L E M F O R M U L A T I O N S

3 . 1 B o u n d a r y C o n d i t i o n s o f I n d e n t a t i o n T e s t

P r o b l e m

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

c o n e. T h e u s u a l p y r a m i d p r e s su r e h e a d c a n b e re -

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

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

t h e an a l y s i s. A t r a c t i o n c o n d i t i o n o n t h e r e m o t e

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

b o u n d a r y c o n d i t i o n s a s i s u s u a l l y d o n e . T h e a d v a n -

t a g e s o f s u c h a t r e a t m e n t a r e in e i t h e r c a t c h i n g t h e

f u n d a m e n t a l f e a t u r e o f i n d e n t a t i o n t e s t , o r m a k i n g

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

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

a r y a r e g i v e n b y ( s e e F i g . l ) [15]

Pcr~ -- 2 ~ 2 \ 1 - t- s in 0 3 cos 2 0 s in 0

3 P s in 3 0Cr z : - - 271 .~2

3 P ( 1 0 )

- - cos 0 s in 2 0~ r z - - 2 7 r f 2

( 1 - 2 u ) P ( s i n 0 1 )c ry - 2 - -- - 1 + s i n 0

w h e r e f 2 = r 2 + z 2 . T h e d e f i n i t i o n s o f t h e o t h e r s a r es h o w n in F i g .1 . T h e m i x e d b o u n d a r y c o n d i t i o n s o n

t h e m a t e r i a l s u r f a ce ( z = 0 ) fo r t h e s m a l l d e f o r m a t i o n

c a s e a r e

u z = - r t a n / 3 R r = 0

R z < 0 0 < r < R = h / t a n / 3 ( 1 1 )

R r = R z = 0 R < r

w h e r e t h e r e f e r e n c e ( o r i g i n a l ) p o i n t o f v e r t i c a l d i s -

p l a c e m e n t Uz i s c h o s e n a t t h e c o n i c a l a p e x , R i s t h e

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

d e n t e r , R r a n d R z a r e t h e s u r f a ce t r a c t i o n s .



62 A C TA ME C H A N IC A S IN IC A (E ng l i sh S e r i e s )

i N i i i : j , i N-*-~---- d ----~

N , L )

2003

Z ~3

r

Fig .1 S impl i fi ed mod e l fo r po lyc rys t a l ma te r i a l undergoing the m ic ro- indent

F o r t h e c o n v e n i e n c e o f t h e a n a l y s is , d e f in e a

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

R o = ~ / P / ( 3 ~ y ) (12)

R 0 i s t h e c o n v e n t i o n a l c o n t a c t r a d i u s ( w h e n t h e s i z e

e f f ec t i s n e g l e c t e d ) f o r t h e l o w - h a r d e n i n g m e t a l m a t e -

r i a l i n t h e c a s e o f f l a t p r e s s u r e h e a d [ 1 6] . T h e h a r d n e s s

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

r a m e t e r a n d i n d e n t d e p t h a r e

P - - 3c ry = 3Cry t an /3 (13)H -- 7rR~

3 . 2 P o l y c r y s t a l M a t e r i a l U n d e r g o i n g t h e

M i c r o - I n d e n t

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

g r a i n si ze i s c o m p a r a b l e w i t h th e i n d e n t d e p t h . I n

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

b e c o n s id e r e d . W h e n t h e p r e s s u r e h e a d is a p p l i e d

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

t i c s li p z o n e w i ll b e c o n s t r a i n e d t o t h e r e g i o n o f t h e

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

g r a i n b o u n d a r y , d u e t o t h e s h i e l d i n g e f f ec t o f t h e g r a i n

b o u n d a r y . E f f e c t s o f a l l o t h e r c r y s t a l g r a in s c a n b e

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

g e n e o u s ly e q u iv a l e nt b o d y w i t h Y o u n g ' s m o d u l u s a n d

P o i s s o n ' s r a t i o ( E s , u s ) b y u s i n g t h e H i l l' s c o n s i s t e n t

a p p r o a c h [ 1 7 1 . T h e e q u i v a l e n t Y o u n g ' s m o d u l u s v a l u e

E s f o r th e p o l y c r y s t a l m a t e r i a l i s s m a l l e r t h a n E , t h e

e q u i v a l e n t Y o u n g ' s m o d u l u s o f t h e s in g l e c r y s t a l m a -

t e r i a l. T h e s im p l i f ie d m o d e l f o r p o l y c r y s t a l m a t e r i a l

u n d e r g o i n g a m i c r o - i n d e n t i s s h o w n i n F i g . 1 . t i s t h e

c r y s t a l g r a i n s i ze i n t h e i n d e n t d i r e c t io n , d i s a n o t h e r

s i ze o f t h e c r y s t a l g r a i n .

I n a l l r e l a t i o n s g i v e n a b o v e , a l l p a r a m e t e r s o r

v a r i a b l e s w i t h l e n g t h d i m e n s i o n c a n b e n o r m a l i z e d b y

R 0 , a n d a l l q u a n t i t i e s w i t h s t r e s s d i m e n s i o n c a n b e

n o r m a l i z e d b y ~ y , s u c h th a t t h e n o r m a l i ze d h a r d n e s s

c a n b e t a k e n t o b e a f u n c t i o n o f t h e i n d e p e n d e n t a n d

n o n - d i m e n s i o n a l p a r a m e t e r s a s f o l l o w s

H f (E _ _ y E s L t d ) ( 1 4 )~ y ' E ' u , u s , N , / 3 , R 0 R 0 '

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

t h e h a r d n e s s b y ( 1 3) . T h e d e t a i l e d e x p r e s s i o n o f (1 4 )

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

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

t h e o r y .

3 .3 T h i n F i l m / S u b s t r a t e S y s t e m U n d e r g o i n g

t h e M i c r o - I n d e n t

A d d i t i o n a l l y , c o n s i d e r a m e t a l t h i n f i l m / c e r a m i c

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

s u r f a c e o f a t h i n f i lm . W h e n t h e t h i c k n e s s o f t h e t h i nf i lm is c o m p a r a b l e w i t h t h e i n d e n t d e p t h , t h e e f f e ct

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

s i d e r e d . I n t h i s c a s e , t h e h a r d n e s s - p a r a m e t e r r e l a t i o n

c a n b e w r i t t e n a s

g f u , u s , N , / 3 , - - ( 1 5 )c ry E R0

w h e r e t is t h e t h i n f i l m t h i c k n e s s , E s i s Y o u n g ' s

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

f i l m / c e r a m i c s u b s t r a t e s y s t e m , u s u a ll y , E s / E ~ 6 .S i m p l if i ed m o d e l f o r t h i n f i l m / s u b s t r a t e s y s t e m m l -

d e r g o i n g t h e m i c r o - i n d e n t i s s h o w n i n F i g . 2 .



Vol.19, N o. l We i Yueguang et al,: Siz e Effect and Geom etr ical Effect of Solids

: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :::::::::::::::::::::::::::::::::::::::::::::::::::::::::ir

subs t r a te

N , L )

: : : : : : : : : : : : : : : : : : : : : : : : : :

i i ' , i i i i i H i i i i i ! i i i i : i i i i i i i '' ' ' ii ! i i i i i i i i i i i i l

F i g .2 S i m p l if i ed m o d e l f o r t h e m e t a l t h i n f i l m / c e r a m i c s u b -

s t r a t e s y s t e m u n d e r g o i n g th e m i c r o - i n d e n t

63

4 F I N I T E E L E M E N T M E T H O D F O R T H E

D E F O R M A T I O N A L T H E O R Y O F T H E

S T R A I N G R A D I E N T P L A S T I C I T Y

F o r t h e c o n s t i t u t i v e r e l a t i o n t h a t t a k e s a c -

c o u n t o f t h e s t r a i n g r a d i e n t e f f e ct , g e n e r a l l y s p e a k -

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

n o t b e u s ed , a n d a s p e c i a l f i n i te e l e m e n t m e t h o d

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

v a r i a b l e s i s n e e d e d [ 1 3 '1 s ' 1 9 ]. H o w e v e r , f o r t h e

s t r e t c h i n g - d o m i n a t e d s t r a i n g ra d i e n t p l a s t i c i ty p r o b -

l e m , o n e c a n o b t a i n a n e f f ec t iv e r e s u l t b y a d o p t i n g

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

n o d e s [1 4'1 9'2 ~ O b v i o u s l y , t h e i n d e n t a t i o n p r o b l e m

is s t r e t c h i n g - d o m i n a t e d . T h e r e f o r e , i n t h e p r e s e n t

a n a l y s i s , s u c h a n e l e m e n t w i ll b e u s e d . I n a d d i t i o n ,

2 • 2 G a u s s i n t e g r a t i o n p o i n t s a r e a d o p t e d f o r e a c h

e l e m e n t .

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

t e s t , c o n s i d e r t h a t t h e e l a s t i c s t r e s s f i e l d a c t s o n t h e

r e m o t e b o u n d a r y , a n d t h e p r e s s u r e h e a d a p e x i s t a k e n

a s t h e r e f e r e n c e p o i n t ( z e ro p o i n t ) o f v e r t i c a l d i s p l a c e -

m e n t . T h u s , i m a g i n e t h a t u n d e r t h e a c t io n o f t h e

r e m o t e s t r e s s fi el d, m a t e r i a l w o u l d m o v e b a c k t o t h e

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

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

w o u l d b e e q u a l to R . T h e d i s p l a c e m e n t c o n s t r a i n t c o n -

d i t i o n s i n t h e c o n t a c t r e g i o n a r e d e s c r i b e d b y ( 1 1 ) .

U s i n g t h e p l a s t i c d e f o r m a t i o n a l t h e o r y , t h e s o -

l u t i o n p r o c e d u r e s c a n b e o u t l i n e d a s f o ll o w s . F i r s t ly ,

c a l c u l a t e t h e e l a s t ic s o l u t io n . S e c o n d l y , t a k i n g e l a s-

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

t h e h i g h e r - h a r d e n i n g m a t e r i a l b y it e r a t in g . T h i r d l y ,

t a k i n g t h e l a s t s o l u t i o n a s t h e n e w i n i t i a l t r i a l , f i n d

t h e c o n v e r g e n t s o l u t io n f o r t h e l o w e r - h a rd e n i n g m a -

t e r i a l ca s e b y i t e r a t i n g . F o r e x a m p l e , t h e s o l u t i o n

p r o c e d u r e s f o r N - - 0 .1 c a s e a r e d e s c r i b e d a s fo l lo w s .

F i r s t l y , f i n d i n g t h e e l a s t i c s o l u t i o n a n d t h e n t a k i n g i t

a s t h e i n i t i a l s o l u t i o n , c a l c u l a t e t h e c o n v e r g e n t s o l u -

t i o n f o r N = 0 . 3 c a se b y i te r a t i n g . S e c o n d l y , t a k i n g

t h e l a s t s o l u t i o n ( N = 0 . 3 ) a s a n e w i n i t i a l t r i a l , f i n d

t h e c o n v e r g e n t s o l u t i o n f o r t h e c a s e o f N = 0 .2 b y

i t e r a t i n g . F i n a l l y , b a s e d o n t h e s o l u t i o n o f N = 0 . 2 ,f i n d t h e c o n v e r g e n t s o l u t i o n f o r N = 0 .1 b y i t e r a t i n g .

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

p r o b l e m , t a k e t h e r e m o t e b o u n d a r y a s t h e r e f e r e n c e

p o i n t o f v e r t i c a l d is p l a c e m e n t . T h u s , t h e v e r t i c a l d is -

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

v ( r , z ) = u z ( r , z ) - u 7 ( 16 )

w h e r e u ~ i s t h e v e r t i c al d i s p l a c e m e n t a t t h e r e m o t e

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

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

f o r m o f f i n it e e l e m e n t m e s h a d o p t e d i n th e p r e s e n t r e -s e a rc h i s s i m p l y f a n - s h a p e d u s i n g t h e a x i s y m m e t r i c a l

c o n d i t i o n .

5 N U M E R I C A L A N D E X P E R I M E N T A L

R E S U L T S F O R P O L Y C R Y S T A L M A -

T E R I A L

5 .1 N u m e r i c a l R e s u l t s

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

m e t a l m a t e r i a ls , m a t e r i a l p a r a m e t e r s a r e t a k e n a s

( E / ~ y , ~ , N ) = ( 30 0 , 0 .3 , 0 .1 ) a n d th e f ia t p r e s s u r e

h e a d c a s e (/3 = 2 0 ~ i s c o n s i d e r e d . F i r s t l y , t h e r e l a -

t i o n s o f t h e h a r d n e s s a g a i n s t t h e n o r m a l i z e d l e n g t h

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

p r e d i c t e d ( s ee (1 4 ) ) . S e c o n d l y , t h e r e l a t i o n s o f t h e

h a r d n e s s a g a i n s t t h e i n d e n t d e p t h a n d c r y s t a l g r a in

s i ze a re o b t a i n e d . T h e s i z e e f fe c t a n d g e o m e t r i c a l e f-

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

i d e n ti fi e d . N o t e t h a t t h e c o n v e n t i o n a l fl a t p y r a m i d

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

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

a n a l y s i s , a n d t h e e q u i v a l e n t c o n i c a l a n g l e / 3 i s a b o u t

2 0 d e g r e e s . T h e r e f o r e , i n t h e p r e s e n t r e s e a r c h , t h e

p r e s s u r e h e a d a n g l e i s t a k e n a s / 3 = 2 0 ~



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

r a m e t e r s a n d t h e n o r m a l i z e d c r y s t a l g ra i n s iz e s. T h e

n o r m a l i z i n g q u a n t i t y o f h a r d n e s s ( 3 a y ) i s t h e h a r d -

n e s s v al u e f or t h e t y p i c a l m e t a l m a t e r i a l s w h e n t h e

s i ze e f fe c t i s n e g l e c t e d . T h e n o r m a l i z i n g q u a n t i t y ( R 0 )

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

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

t h e c o n t a c t z o n e w h e n t h e s i ze e ff e c t i s n o t c o n s i d -

e r e d . I n F i g .3 , a w i d e re g i o n o f t h e n o r m a l i z e d q u a n -

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F i g .3 V a r i a t i o n o f h a r d n e s s w i t h n o r m a l i z e d p a r a m e t e r s :

l e n g th p a r a m e t e r L/Ro, g r a i n s i ze t / R o , a n d t/L

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t i t y L / R o f r o m 0 t o 3 a n d a w i d e r e g i o n o f t/Ro a r e

c o n s i d e r e d . F r o m F i g . 3 , t h e h a r d n e s s i n c r e a s e s w i t h

L/Ro q u i c k l y , e s p e c i a l l y f o r s m a l l v a l u e o f t /Ro . A s

t/Ro i n c r e a s e s f r o m 0 .3 6 4 ( t a n 2 0 ~ t o 1 0, t h e h a r d -

n e s s c u r v e d e c r e a s e s v e r y q u i c k l y a t s t a r t i n g , a n d t h e n

a p p r o a c h e s a s t a b l e v a l u e , w h i c h c o r re s p o n d s t o a h o -

m o g e n e o u s m a t e r i a l h a r d n e s s c u r v e ( w h e r e t h e g e o -

m e t r i c a l e f fe c t i s v e r y w e a k ) . N o t e t h a t t h e m a t e r i a l

h a r d n e s s s h a r p l y i n c r e a s e s w h e n t h e i n d e n t d e p t h a p -

p r o a c h e s t h e c r y s t a l g r a i n s i z e v a l u e ( h = 0 . 3 6 4 R ) .

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v e n t i o n a l m a t e r i a l ) . T h e e f fe c t o f t h e n o r m a l i z e d p a -

r a m e t e r t/Ro o n t h e h a r d n e s s c h a r a c t e r iz e s e i t h e r t h e

c r y s t a l g r a i n s i z e e f f e c t , o r t h e l o a d e f f e c t . W h e n t h e

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

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

o r a b i g l o a d c a s e w h e n g r a i n s i ze is f i x ed . I n o r d e r

t o e x p l a i n t h e e f f ec t o f t h e c r y s t a l g r a i n s iz e , is o l in e s

o f t h e n o r m a l i z e d p a r a m e t e r t /L a r e p l o t t e d b a s e do n t h e h a r d n e s s r e l a t i o n a g a i n s t L / R o a n d t /Ro, see

d a s h e d l i n e s s h o w n i n F i g . 3 . W e a l s o i n v e s t i g a t e t h e

i n f l u e n c e o f Y o u n g ' s m o d u l u s E ~ o f t h e e q u i v a l e n t

b o d y o n th e m a t e r i a l h a r d n e s s. T h e r e s u l t s a re s h o w n

i n F i g . 4 . F r o m F i g . 4 , c l e a rl y , t h e i n f l u e n c e o f t h e

Y o u n g ' s m o d u l u s R a t i o E ~ / E o n t h e m a t e r i a l h a r d -

n e s s is v e r y w e a k w i t h i n a w i d e r e g i o n o f E~/E = 0 . 67

t o 6 . 7 0 .

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Vol. 19, No .1 Wei Y u e g u a n g et a l.: S ize Effec t and G eom etrical Effec t of Solids 6 5

F u r t h e r m o r e , t h e d a s h e d l in e s in F ig . 3 c a n b e

r e p l o t t e d i n t h e c o o r d i n a t e s: h a r d n e s s / i n d e n t d e p t h ,

s e e F i g . 5 . F r o m F i g . 5 , t h e s i z e e f f e c t a n d t h e c r y s t a l

g r a i n s i z e e f fe c t a r e c l e a r l y s h o w n . W h e n t h e i n d e n t

d e p t h i s s m a l l , i .e ., w h e n t h e i n d e n t t i p i s f a r a w a y

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

n a n t . A s t h e i n d e n t d e p t h i n c r e a s e s , t h e si z e e f f ec t

d e c r e a s e s a n d t h e g e o m e t r i c a l e f fe c t i n c r e a s e s . W h e n

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

g e o m e t r i c a l e ff e ct b e c o m e s d o m i n a n t a n d m a k e s t h e

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

t h e c r y s t a l g r a i n s i z e , b o t h t h e s i z e e f f e c t a n d c r y s t a l

g e o m e t r i c a l s i z e e f fe c t a re e n l a r g e d . T h e e f f e c t s o f t h e

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

r e c t i o n o n t h e m a t e r i a l h a r d n e s s a r e s h o w n i n F ig . 6 .

I n F i g . 6 , t h e g e o m e t r i c a l e f f e c ts f o r t w o v a l u e s o f d,

d = 5 t a n d d = 1 0 0 t , r e s p e c t i v e l y , a r e d i s p l a y e d . T h e

r e s u l t s s h o w t h a t t h e e f f e c t o f t h e c r y s t a l g r a i n s i z e

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

v e r y w e a k . T h e c o n c l u s io n i m p l ie s t h a t a l t h o u g h t h e

a n a l y s i s is fo r th e p o l y c r y s t a l m a t e r i a l , i t c a n a p p r o x -

i m a t e l y b e a p p l i e d fo r t h e t h i n f i l m / s u b s t r a t e s y s t e m

t o b e d i s c u s s e d i n t h e n e x t s u b - s e c t i o n .

T h e p h e n o m e n a o f " p i l e- u p " a n d " s in k - i n" in

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

r i a l a r e a ls o i n v e s t i g a t e d h e r e . T h e r e s u l t s a r e s h o w n

i n F i g .7 . F r o m F i g . 7 , w i t h t h e i n c r e a s e o f t h e m a -

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

a p p l i e d l o a d ( R 0 = V/P/37ray), c o r r e s p o n d i n g l y w i t h

t h e i n c r e a s e o f L/Ro , t h e " p i l e - u p " p h e n o m e n o n d i s -

a p p e a r s a n d t h e " s in k - i n " fe a t u r e g e ts p r o m i n e n t i n

t h e m i c r o - i n d e n t a t i o n t e st . T h e p il e - u p p h e n o m e n o n

i s f a v o r a b l e fo r t h e b i g f o r ce a n d s m a l l l e n g t h p a r a m e -

t e r . T h i s c o n c l u s i o n i s c o n s i s t e n t w i t h t h e s i n g l e c ry s -

t a l m a t e r i a l c a s e [9 ]. W h e n t h e c r y s t a l g r a i n s iz e is

l a r g e , t h e g e o m e t r i c a l e f f e c t i s w e a k , a n d t h e s i z e e f -

f e c t i s d o m i n a n t . A s t h e c r y s t a l g r a i n s i z e d e c r e a s e s ,

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

d e c r e a s e s c o n s i d e r a b l y f o r a f i x e d a p p l i e d l o a d .

10

9

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F i g .5 S iz e e f fe c t a n d g e o m e t r i c a l e f fe c t o n t h e h a r d n e s s / d e p t h r e l a t i o n s

3 0 . . . . i . . . . i . . . . i . . . . i . . . . i , , , . i , . , ,

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F ig . 6 E f f ec t o f the c r ys ta l g r a in size in the h or izon ta l d i r ec t ion on the h a r dness



66 ACTA MECHANICA SINICA (English Series)

0.40

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i i

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Fig.7 The simulation of the phenom ena "pile-up" and "sink-in"

for polycrysta l in micro- indentation test

2003

5 . 2 M i c r o - I n d e n t a t i o n E x p e r i m e n t f o r P o l y -

c r y s t a l A 1

I n o r d e r t o s t u d y th e m ic r o - g e o m e t r i c a l e f f e c t s

e x p e r im e n ta l l y , w e d e s ig n t h e t e s t sp e c im e n s f o r p o ly -

c rys ta l A1 wi th seve ra l d i f fe ren t c rys ta l g ra in s izes .

T h e i n d e n t a t i o n e x p e r i m e n t a l m e t h o d i s s i m i l a r w i t h

R e f. [9 ] fo r t h e s i n g l e c r y s t a l m a te r i a l . T h e e x p e r im e n -

t a l m e th o d c o n s i s t s o f c h o o s in g m a n y lo a d in g p o in t s

a r b i t r a r i l y o n t h e sp e c im e n su r f a c e , a n d f o r o n e l o a d -

in g p o in t , o n ly o n e s e t o f e x p e r im e n ta l d a t a a b o u t

h a r d n e s s a n d i n d e n t d e p th b e in g o b t a in e d . I n su c h am e t h o d w e s h a ll o b t a i n a d a t a s t r i p a b o u t t h e h a r d -

n e s s / i n d e n t d e p t h r e la t io n . T h e s c a t t e r d e g r ee o f t h e

d a t a s t r i p d e p e n d s o n t h e f e a tu r e o f t h e s e l e c t e d l oa d -

in g p o in t o n t h e sp e c im e n su r f a c e . I f a n i n d e n t p o in t

i s j u s t l o c a t e d a t a f a u l t r e g io n o n t h e sp e c im e n su r -

f a c e , a l o w h a r d n e s s w i l l b e o b t a in e d , w h e n th e i n -

dent de pth i s f ixed . I f an indent poin t i s jus t chosen

a t a ha rd pa r t ic le , a h igh ha rdness va lue i s ob ta ined .

F ig u r e 8 sh o w s th e su r f a c e p r o f i l e o f t h e p o ly c r y s t a l

A 1 sp e c im e n f or a SE p h o to g r a p h . F r o m F ig . 8, o n

the spec imen sur face , some fau l t s ex is t inevi tab ly . I f

t h e c o n t in u u m s t i f f n e s s m e th o d i s u se d , t h e o b t a in e d

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

T h e r e f o r e , w e sh a l l u se t h e e x p e r im e n ta l m e th o d h e r e

b y s e l ec t i n g t h e i n d e n t l o a d in g p o in t s o n t h e sp e c im e n

su r f a c e a r b i t r a r i l y a s i n [ 9 ], r a th e r t h a n a d o p t in g t h e

c o n t in u u m s t i f f n e s s m e th o d , so t h a t t h e i n f lu e n c e o f

the fau l t s and the d i f fe ren t loading poin t se lec t ions on

th e e x p e r im e n ta l r e su l t s w i l l b e d i sp l a y e d c l e a r ly .

Fig.8 The prof ile of the spec imen surface for the p olycrysta l A1



Vo l .19 , No .1 W e i Y u e g u a n g e t a l .: S i z e E fF ec t a n d G e o m e t r i c a l E f f e c t o f S o l id s 6 7

I n F ig . 9, t h e e x p e r i m e n t a l h a r d n e s s r e s u l ts f o r

p o l y c r y s t a l A 1 f o r s e v e r a l c r y s t a l g r a i n s i z e s a r e

s h o w n . F r o m F i g s . 9 ( a ) t o ( d ) , t h e s e r ie s o f t h e g r a i n

s iz e s, 0 .6 # m , 1 # m , 2 # m a n d i n f i n i t ly l a r g e s i ze ( s in -

g l e c r y s t a l A 1), a re sh o w n s e p a r a t e l y . T h e r e s u l t s

s h o w t h a t t h e f e a t u r e o f t h e h a r d n e s s v a r i a t i o n w i t h

t h e i n d e n t d e p t h i s i n a " U " f o r m . W h e n t h e i n d e n t

d e p t h i s sm a l l , a s t r o n g s i z e e f fe c t i s c h a r a c t e r i z e d . A s

t h e in d e n t d e p t h i n c r e a s e s , t h e h a r ( b w s s d e c r e a s e s a n d

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

t h e in d e n t t i p a p p r o a c h i n g t h e g r a in b o u n d a r y . C o m -

p a r i n g t h e r e s u l t s s h o w n i n F i g s . 9 ( a ) , ( b ) , ( c ) a n d ( d )

f o r d i f f e r e n t c r y s t a l g r a i n s i z e s , w i t h t h e d e c r e a s e o f

t h e c r y s t a l g r a i n s i z e , t h e h a r d n e s s i n c r e a s e s i n g e n -

e r al . F o r c o m p a r i s o n , t h e t h e o r e t i c a l p r e d i c t i o n s o f

t h e h a r d n e s s u s i n g t h e s t r a i n g r a d i e n t p l a s t i c i t y a r e

a l s o s h o w n i n F i g . 9 . F o r e a c h c a se , t w o c u r v e s c o r -

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

p l o t t e d . T h e s e t w o c u r v es c a n b e t a k e n a s t h e b o u n d s

o f t h e e x p e r i m e n t a l d a t a . A s t h e c ry s t a l g r a in s iz e

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

a p p r o a c h 1 # m a n d 2 # m , c o r r e s p o n d i n g t o t h e l o w e r

b o u n d c u r v e a n d u p p e r b o u n d c u r v e , r e s p e ct i v e ly , fo r

t h e s in g le c r y s t a l c a se . T h e m a t e r i a l h a r d n e s s a p -

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

i n c r e a s e s in f i n it e l y . F o r t h e s i n g l e c r y s t a l c a s e , t h e

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

d e n t d e p t h i s i n a n " L " f o r m .

:n

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400 . . . . .200 400 600 800 1000 12000 0 0 0

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(d )

F i g .9 E x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s o f t h e h a r d n e s s / i n d e n t

dep th r e la t ions f o r d i f f e r en t c r ys ta l g r a in s izes

A s m e n t i o n e d a b o v e , i f o n e a d o p t s t h e m e t h o d

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

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

o b t a i n t h e h a r d n e s s v a r i a ti o n s w i t h t h e i n d e n t d e p t h

i n t h e f o r m o f t h e d a t a s t ri p . F i g u r e 9 s ho w s s u c h

s t ri p s , t h e s c a t t e r e d d a t a s t ri p s . O b v i o u s l y , t h e w i d t h

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

c r y s t a l s p e c i m e n is s o m e w h a t l a r g e r t h a n t h a t f o r t h e

s i n g l e - c r y s t a l s p e c i m e n , c o m p a r i n g F i g s . 9 ( a ), ( b ) , ( c)

w i t h F i g . 9 ( d ) . T h i s r e s u l t i s r e a s o n a b l e a n d s e e m s t o



6 8

b e e x p e c t e d . T h i s i s b e c a u s e f o r a p o l y c r y s t a l s p e c i-

m e n t h e e x p e r i m e n t a l h a r d n e s s i s i n fl u e n ce d b o t h b y

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

e t c . ) a s f o r a s i n g l e - c r y s t a l s p e c i m e n c a s e , a n d b y t h e

g r a i n b o u n d a r i e s a n d t h e g r a i n si z e d i f fe r e n c es . T h e

l a t t e r e f fe c t i s s p e c i f ic a l l y i m p o r t a n t f o r t h e p r e s e n t r e -

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

t h e g r a i n s iz e s. A l t h o u g h t h e p o l y c r y s t a l s p e c i m e n

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

r a r e l y th e c a s e. F r o m t h e S E p h o t o g r a p h o f t h e s u r -

f a c e p r o fi l e fo r a p o l y c r y s t a l A 1 s p e c i m e n w i t h 1 p m

g r a i n s i z e s h o w n i n F i g . 8 , b e s i d e s t h e m a n y f a u l t s b e -

i n g o b s e r v e d o n t h e s u r f a c e , t h e g r a i n s i z e s a r e n o t

u n i f o r m . I n F ig . 8 , t h e l o n g e r b l a c k c u r v e s c h a r a c t e r -

i ze th e g r a i n b o u n d a r i e s a n d t h e s m a l l b l a c k z o n es a r e

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

o b s e r v e d i n t h e f i g u r e .

6 T i F I L M L A I N O N T H E S i3 N 4 C E R A M I C

S U B S T R A T E U N D E R G O I N G M I C R O -

I N D E N T I N G

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

a c e ra m i c s u b s t r a t e . W h e n t h e t h i n f i lm t h i c k n e ss is

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

f i l m / s u b s t r a t e i n t e r f a c e w i l l h a v e a s t r o n g i n f l u e n c e o n

t h e h a r d n e s s ( g e o m e t r i c a l e ff e c t) . I n t h e p r e s e n t c a s e ,

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

m a t e r i a l a n d g e o m e t r i c a l p a r a m e t e r s i s g i v e n i n ( 1 5) .

F r o m d i s c u s s i o n s i n s u b s e c t i o n 4 . 1 , t h e p r e d i c t e d

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

i n F i g . 5 c a n b e a p p l i e d t o t h e t h i n f i l m / s u b s t r a t e s y s -

t e m a p p r o x i m a t e l y w i t h i n a p a r a m e t e r r e g i o n 0 . 6 7 <

E , / E < 6 . 7 0 . I n t h e p r e s e n t c a s e , t h e Y o u n g ' s m o d u -

l u s r a t i o f o r t h e c e r a m i c s u b s t r a t e ( S i a N 4 ) a n d m e t a l

t h i n f i l m ( T i ) i s a b o u t 6 . 0 . I n o r d e r t o i n v e s t i g a t e

t h e h a r d n e s s v a r i a t i o n w i t h t h e t h i n f i l m t h i c k n e s s ,

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

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

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

s t i f f n e s s m e t h o d i n t h e e x p e r i m e n t , i . e . , t h e c u r v e o f

t h e h a r d n e s s v a r i a t i o n w i t h t h e i n d e n t d e p t h i s o b -

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

f a c e . I n s u c h a t e s t m e t h o d , t h e e x p e r i m e n t a l h a r d -

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

F i g u r e i 0 s h o w s t h e e x p e r i m e n t a l h a r d n e s s f o r

t h e T i f i l m l a i n o n t h e S i 3 N 4 s u b s t r a t e f o r t h r e e d i f -

f e r e n t t h i c k n e s s e s o f t h e t h i n f i l m . F i g u r e s 1 0 ( a ) , ( b )

a n d ( c ) c o r r e s p o n d t o t h e t h i n f i l m t h i c k n e s s 1 . 2 p m ,

1 . 8 # m a n d 1 3 # m , r e s p e c t i v e l y . F r o m F i g . 1 0 , t h e f e a -

t u r e o f t h e h a r d n e s s v a r i a t i o n w i t h t h e i n d e n t d e p t h

i s i n a " U " f o r m a n d i t c a n b e d e s c r i b e d a s f o l l o w s :

ACT A ME CH ANICA SINICA (Engl ish Ser ies) 2003

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

s iz e e f fe c t. A s t h e d e p t h i n c r e a s e s , t h e h a r d n e s s d e -

c r e a s e s q u i c k l y u n t i l t h e i n d e n t d e p t h r e a c h e s s o m e

f r a c t i o n o f t h e t h i n f i lm t h i c k n e s s , t h e h a r d n e s s a s -

s u m e s a l o w e s t v a l u e . F r o m t h e l o w e s t v a l u e t h e

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

c r e a s e s f u r t h e r . A l t e r n a t i v e l y , a s t h e t h i n f i l m t h i c k -

n e s s i n c r e a se s ; t h e h a r d n e s s d e c r e a s e s a l l o v e r t h e r e -

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

h a r d n e s s p o i n t i n c r e a s e s. W h e n t h e t h i n fi l m t h i c k -

n e s s i n c r e a s e s to a c e r t a i n e x t e n t ( 1 3 # m ) , t h e i n -

f l u en c e o f t h i n f i lm t h i c k n e s s o n t h e h a r d n e s s c u r v e

c a n b e n e g le c t e d , s o t h a t t h e h a r d n e s s / d e p t h c u r v e

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

i n d e n t d e p t h i n c r e a s e s u n t i l t h e p r e s s u r e h e a d a p -

p r o a c h e s t h e i n t e r f a c e . F o r t h e s m a l l t h i n f i lm th i c k -

n e s s , s e e F i g . 1 0 ( a ) ( c a s e I ) , d u e t o t h e s t r o n g s i z e e f -

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

b u t t h e s i ze ef fe c t i s d o m i n a n t w h e n t h e i n d e n t d e p t h

i s s m a l l . W i t h i n c r e a s i n g d e p t h , t h e s i ze e f fe c t d e -

c r e a s e s a n d t h e g e o m e t r i c a l e f f e c t ( th e e f fe c t o f t h i n

f i lm t h i c k n e s s a n d i n t e r fa c e ) i n c r e a s e s . U n t i l a d e p t h

e q u a ls t o a b o u t 3 00 n m ( a b o u t o n e f o u r t h o f t h i n f i lm

t h i c k n e s s ) , t h e g e o m e t r i c a l e f fe c t is so la r g e t h a t t h e

h a r d n e s s c u r v e t u r n s t o i n cr e a se . T h e h a r d n e s s o b -

t a i n s a lo w e r l im i t v a l u e 5 .6 G P a a t t h e t u r n i n g p o i n t .

F r o m F i g . 1 0 ( b ) , i n c r e a s i n g t h e t h i n f i l m t h i c k n e s s b y

0 . 6 # m , i .e ., t = 1 . 8 # m ( c a s e I I ) , t h e g e o m e t r i c a l el -

f e ct d e c re a s e s r e m a r k a b l y c o m p a r i n g w i t h c a s e I. F o r

e x a m p l e , t h e h a r d n e s s i s a b o u t 1 0 G P a a t h = 1 00 n m

f o r c a s e I , f r o m F i g . 1 0 ( a ) , w h i l e t h e c o r r e s p o n d i n g

h a r d n e s s a t t h e s a m e d e p t h i s a b o u t 8 G P a f o r ca s e I I,

f r o m F ig . 1 0 ( b ). W h e n t h e i n d e n t d e p t h i s s m a l l, th e

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

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

a b o u t h = 5 8 0 n m ( a b o u t o n e t h i r d o f t h e t h i n f i l m

t h i c k n e s s ) is a p p r o a c h e d , f r o m F i g . 1 0 ( b ) . T h e r e a f t e r ,

t h e g e o m e t r i c a l e f fe c t i s d o m i n a n t . C o n s i d e r i n g t h a t

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

l a r g e , s ee F i g . 1 0 ( c ) ( c a se I I I ) , t h e g e o m e t r i c a l e f fe c t

c a n b e n e g l e c te d . W i t h t h e i n c r e a se o f d e p t h , t h e

s iz e e f fe c t d e c r e a s e s . W h e n t h e h a r d n e s s d e c r e a s e s t o

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

n e g l e c t e d .

I n F i g s . lO ( a ) , ( b ) a n d ( c ), t h e t h e o r e t i c a l p r e -

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

s t r a i n g r a d i e n t p l a s t i c i t y t h e o r y a r e a l s o s h o w n . O b -

v i o u s ly , t h e c h a r a c t e r i s t i c s o f th e t h e o r e t i c a l p r e d i c -

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

r e su l ts . T h r o u g h p l o t t i n g t h e t h e o r e t i c a l p re d i c t io n s

a n d t h e e x p e r i m e n t a l r e s u l t s t o g e t h e r i n F i g s . l O ( a ) ,



Vo1.19, No.1 W e i Y u e g u a n g e t a l . : Size Effect and G eometr ica l E ffect of Sol ids 6 9

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

p a r a m e t e r o n t h e t h i n f il m t h i c k n e s s i s o b t a i n e d . T h e

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

a r e a l s o s h o w n i n F i g s . 1 0 ( a ) , ( b ) a n d ( c ) , r e s p e c t iv e l y .

O b v i o u sl y , t h e l e n g t h p a r a m e t e r a ls o d e p e n d s o n t h e

m i c r o - g e o m e t r i c a l s iz e ( t h i n f i lm t h i c k n e s s ) .

10

8

2

0

, I f I , I ~ I ,

o o o o o o

~ exper iment for

"~Ti/Si3N4

theoreticalpredieition

t = l . 2 p m , L = 4 . 5 p mE / c r y = 3 0 0 , u = u ~ = 0 . 3 ,

E ~ / E = 6 . 0 , N = 0 . 1

200 400 600 800 1 000 200 400 600 800

h/nm h/nm

8

7

6

5

4

a

2

1

00

I , I I I ,

E / a z = 3 0 0 , L ' = u ~=0.3 ,

E~/E=6.0,N = 0 . 1t = 1 . 8 p m , L = 2 . 6 p m

booO ~

t h e o r e t i c a l

p r e d i d t i o n e x p e r i m e n t f o r

T i / S i 3 N 4

( a ) ( b )

1 000

8 i I ~ I , I I

E / ~ ry = 3 0 0 , p = ~ = 0 .3 ,

7 E~/E=6.0, = 0 . 16 ~ t = 1 3 p m , L = l . 3 p m

e 5 ~ ~

2

e x p e ri m e n t f o r T i / S ~ '3 ~ 4 . . . . .0 i i i i

0 200 400 600 800 1 000

h/nm

( c )

F i g .1 0 E x p e r i m e n t a l a n d n u m e r i c a l r e s u lt s o f t h e h a r d n e s s / i n d e n t d e p t h f o r

T i /S iaN4 th in f i lm/su bs t ra t e fo r d i ffe ren t th i cknes s of th in f i lm

7 C O N C L U D I N G R E M A R K S

I n t h e p r e s e n t r e s e a r c h , t h e s i z e e f f e c t s a n d g e -

o m e t r i c a l e ff e c ts i n t h e m i c r o - i n d e n t a t i o n t e s t s h a v e

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

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

p l a s t i c i t y t h e o r y . M e a n w h i l e , t h e e x p e r i m e n t a l r e -

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

a l u m i n u m a n d f o r t h e T i / S i 3 N 4 t h i n f i l m / s u b s t r a t e

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

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

f i l m / s u b s t r a t e s y s t e m , w h e n t h e c r y s t a l g r a i n s i z e o r

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

d e p t h , t h e c o m p o s i t i o n e f f e c ts f r o m s i z e ef f e ct a n d

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

c o n s i d e ra b l y s t ro n g . F o r t h e h a r d n e s s / d e p t h c u r ve ,

w h e n t h e d e p t h i s s m a l l, t h e s iz e e ff e c t i s d o m i n a n t .

A s t h e i n d e n t d e p t h i n c r ea s e s, t h e s iz e e ff e c t d e c r e a s e s

a n d t h e g e o m e t r i c a l e f f ec t i n c r ea s e s . B e f o r e a d e p t h( a t u r n i n g p o i n t ) i s r e a c h e d , t h e g e o m e t r i c a l ef f e ct i n -

c r e a se s , d o m i n a t e s a n d m a k e s t h e h a r d n e s s c u r v e g o

u p s h a r p l y .

T h e s i m u l a t i o n r e s u l t s u s i n g t h e s t r a i n g r a d i e n t

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

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

T h r o u g h c o m p a r i n g t h e s i m u l a t io n r e s u lt s w i t h e x p e r -

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

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

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

F r o m e x p e r i m e n t a l r e s u l t s f o r p o l y c r y s t a l A 1 ,

e v e n t h o u g h t h e c r y s t a l g r a i n s i z e i s f i x e d , t h e h a r d -



7O ACTA MECHANICA S INICA

nes s/d ept h relation is still a scatte r strip. The feature

mainly comes from the status of the indenting point

on the specimen surface. Certainly, it is possible that

the prepared specimens always and inevitably include

some faults or reinforced phases and the differences

amon g the crystal sizes and shapes. From experimen-

tal results for Ti/Si3N4 thin film/substrate system

using the continuum stiffness method, the composi-

tion effects of the size effect and geometrical effect are

clearly displayed. The experimen tal results show tha t

the material length parameter has a fixed value for a

uniform material, however, when the micro-structure

is considered and the length parameter increases, de-

pending on the micro-geometrical size.

Ac kn ow le dg em en t A great help for preparing poly-

crys tal A1 speci men from Dr. Wu Xiaolei, is sincerely

acknowledged.

R E F E R E N C E S

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