Tensile, Creep, and ABI Tests on Sn5%Sb Solder.pdf

8/10/2019 Tensile, Creep, and ABI Tests on Sn5%Sb Solder.pdf

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J o u r n a l o f E l e c t r o n i c M a t e r i a l s Vo l . 2 6 No . 7 1 9 9 7

Special ssue Paper

T e n s ile , C r e e p , a n d A B I T e s ts o n S n 5 S b S o l d e r

fo r M e c h a n i c a l P r o p e r t y E v a l u a t io n

K. LINGA MURTY,* FAHMY M. HA GG AG /a nd RAO K. MAHIDHARA*

*North Carolina State University, Raleigh, NC 27695-7909

*Advanced Technology Corporation, 661 Emo ry Valley Road, Suite A,

Oak Ridge, TN 37830

Sn5 Sb is one of the mate ria ls considered for replacing lead containing alloys

for soldering in electronic packaging. We eval uated the tensile properties of the

bulk materi al at varied strain- rates a nd tempe ratu res (to 473K) to determi ne

the underly ing deformation mechanisms. Stress exponents of about three and

seven were observed at low and high stresses, respectively, and very low

activation energies for creep (about 16.7 and 37.7 kJ/mole) were noted. A

maxi mum ductility of about 350 was noted at ambient temper ature . Creep

tests performed in the same temperature regime also showed two distinct

regions, albeit with slightly different exponents (three an d five) and activation

energy (about 54.4 kJ/mole). Ball indentation tests were performed on the

shoulder portions of the creep samples (prior to creep tests) using a Stress-

Strain Microprobe~ (Advanced Technology Corporation) at vari ed ind enta tion

rates (strain-rates). The automated ball indentation (ABI) data were at rela-

tively high strai n-rates; however, they were in excellent agree ment with creep

data, while both these results deviated from the tensile test data. Work is

planned to perform creep at high str esses at ambient and extend ABI tests to

elevated temperatures.

Ke y w ords : Chemical diffusion, creep, deformation mechanisms, dislocation

glide, dislocation pipe diffusion, grain -boundary diffusion, lead-

free, self-diffusion, solder, tensile properties

I N T R O D U T I O N

Sn-Sb alloys are viable candidates as soldering

materi als in electronic packaging and ar e considered

in the development of lead-free alloys to replace

currently used lead-based alloys. Tin alloyed with

5 Sb has a near-perit ecti c composition and a rela-

tively high melt ing point of 245~ This alloy has good

creep resistance and thermal fatigue characteris-

tics. 1,2 The room t empera tur e ductil ity of the alloy is

very high while the contact angle for the alloy on

copper subs tra te is relat ively large (43 + 4 ~ com-

pared to t ha t of 17 + 4 ~ for the Pb-Sn eutectic, albeit

it is quite adequate from the wettability consider-

ation/Mechanical and creep properties have been

reporte d by several investigators using conventional

tensile and creep tests on bulk specimens.4~

It is important to develop techniques to charac-

*Currently with Tessera Inc., 3099 Orchard Dr., San Jose,

CA 95134

(Received November 15, 1996; accepted February 21, 1997)

terize mechanical, creep, and fatigue properties of

solder structures so tha t these charact eristics can be

evaluate d in

near r ea l

situations where the amounts

of material available are relatively small. A funda-

menta l require ment is the development of echniques

to characterize the mechanic al and fracture proper-

ties of components of various joint configurations so

tha t proper a ssessme nt can be made of the degrada-

tion and remaining life of solder joints in various

applications. While destructive tests of specimens

made from materials of which these structures are

constructed provide relevant data, direct measure-

ment, preferably using nondestructive techniques on

the real structures, is desirable for reliable assess-

ment of the safety of these structures. One such

technique is the automated ball indentation (ABI)

which has been demonstrated to yield the stress-

strain behavior of many structural met al s/T he se

tests are relatively simple, rapid, and essentially

nondestructive.

We report here the room-temperature deformation

characteristics obtained on the shoulder portions of


2/8

8 4 0 M u r t y , H a g g a g , a n d M a h i d h a ra

3 0

2

10

S n - S S b / T e n s i l e Tests

j e

. 0 / , 5 1 . 0

p l a s t i c s t r a i n

F i g . 1 . S t r e s s v s s t ra i n a t ~ = I O - 4 s - L

i , 1 1 5 2 . 0

400

300

i 200

100

i i I i

S n - S S b S o l d e r

1041ec'l ~

0 i I t 1 | I i I

275 325 375 425 475

Temperature K

F i g . 2 . T e m p e r a t u r e v a r i a t i o n o f d u c t i li t y a t t w o c o n s t a n t s t r a i n -r a t e s

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

m i c r o p ro b e ( S S M - - d e v e l o p e d b y t h e A d v a n c e d T e c h -

n o lo g y C o r p o r a ti o n b a s e d o n t h e A B I p r i n c i p l e ) t o

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

c r e e p t e s t r e s u l ts . A l t h o u g h t h e S S M c a n b e u s e d o n

s m a l l s o l d e r b u m p s , s u c h t e s t s r e q u i r e f u r t h e r d e v e lo p -

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

s iz e ) n e e d t o b e u s e d a n d v e r y l o w l o a d s ( m i c r o g r a m s )

c a n b e m o n i t o r e d . A m a j o r o b j e c t i v e o f t h e c u r r e n t

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

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

E X P E R I M E N T A L

D E T A I L S

T i n - a n t i m o n y a l l oy w i t h c h e m i c a l c o m p o s i t i o n S n -

5 % S b w a s o b t a i n e d i n t h e f o r m o f i m m t h i c k s h e e t

f r o m A l p h a M e t a l s , I nc . T e n s i l e s p e c i m e n s w i t h a

g a g e l e n g th o f 1 2. 4 m m w e r e m a c h i n e d f r om t h e s h e e t

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

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

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

t e m p e r a t u r e s f r o m a m b i e n t t o 4 7 3 K b o t h a t c o n s t a n t

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

Th e s t r a i n - r a t e s r a n g e d f ro m ~ 5 x 1 0~ s ~1 to - 1 10 2 s -1.

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

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

s t r a i n s w e r e m o n i t o r e d u s i n g a l in e a r v a r i a b l e d i f f e r-

e n t i a l t r a n s f o r m e r ( L V D T ) in t h e l o a d- l in e . T h e

s t r e s s e s r a n g e d f r o m a b o u t 1 . 2 5 M P a t o 30 M P a .

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

n i n g e l e c t r o n m i c r o s c o p y .

A B I t e s t i n g i n v o l v e s m u l t i p l e i n d e n t a t i o n s a t t h e

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

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

t r u e s t r e s s a n d t r u e s t r a i n a r e e v a l u a t e d u s i n g t h e

e l a s t i c a n d p l a s t i c a n a l y s e s . 7 F o r A B I t e s t s , n o s p e c i f i c

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

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

p a r a l le l . W h i l e t h e s h o u l d e r p o r t i o n s o f t h e t e s t e d

s p e c i m e n s c o u l d b e u s e d f o r A B I t e s t i n g , o n e o f th e

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

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

t h a t m a y h a v e t a k e n p l ac e d u r in g l o n g t e r m e x p o s u r e

a t h i g h t e m p e r a t u r e . l ig h t m e c h a n i c a l p o l is h w a s

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

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

p e r f o r m e d o n a t a b l e t o p s y s t e m m o d e l P o r t a F l o w - P 1 @

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

w i t h 1 . 57 5 m m ( 0 .0 6 1 ) d i a m e t e r ( D ), a n d m a x i m u m

d e p t h o f p e n e t r a t i o n (hm ax) o f a b o u t 0 . 1 m m ( 0 . 0 0 4 ) .

T h e i n d e n t e r v e l o c i t y w a s v a r i e d f r o m 5 . 0 8 1 0 -4 to

0 . 5 0 8 m m / s ( 2 x 1 0 - s t o 0 . 0 2 i n / s) a n d t h e i n d e n t a t i o n

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

o n - l i n e l o a d c e l l a n d a n L V D T , r e s p e c t i v e l y ( F i g. l b ) .

A t t h e m a x i m u m i n d e n t a t i o n d e p t h o f 0 .1 m m , t h e

d i a m e t e r s o f i n d e n t a t i o n s w e r e l e ss t h a n 0 .3 7 5 m m

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

v a l u e . A t t h e s a m e t i m e , b e c a u s e o f t h e r e l a t i v e l y

s m a l l ( 1 0 g m ) g r a i n s iz e , a l a r g e n u m b e r o f g r a i n s a r e

c o v e r e d .

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

D I S C U S S I O N

T e n s i l e T e s t R e s u l t s

U n i a x i a l t e n s i l e t e s t s w e r e c o n d u c t e d a t c o n s t a n t

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

1 0-~ s -1 a n d 1 0 -3 s -1 a t t e m p e r a t u r e s f r o m a m b i e n t t o

4 7 3 K . T y p i c a l s t r e s s v s s t r a i n c u r v e s a r e s h o w n i n

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

n o m i n a l s t r a i n - r a t e o f 1 0-4 s -L Th e d u c t i l i t y i s h i g h ,

w i t h a v a l u e a r o u n d 3 50 % a t 3 9 8 K ( 25 ~ M a x i m u m

d u c t i l i t y (3 5 0% ) s e e m s t o c o r r e s p o n d t o a s t r a i n - r a t e

s e n s i ti v i ty o f 0 .3 a n d d e c r e a s e s a s t e s t t e m p e r a t u r e

w a s i n c r e a s e d . O n t h e o t h e r h a n d , a t e = 1 0 ~ s -1,

d u c t i l i t y w a s m u c h l o w e r t h a n a t ~ = 1 0 ~ s -1 a t a l l

t e m p e r a t u r e s ; i t a l so i n c r e a s e d w i t h t e m p e r a t u r e .

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

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

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

t u r e d s p e c i m e n s . S p e c i m e n s d e f o r m e d a t ~ = 1 0-3 s -~

d i d n o t e x h i b i t c o a r s e n i n g o f g r a i n s o r S n S b i n t e r m e -

t a l li c p a r t i c le s a s i n d i c a t e d b y s p e c i m e n s t e s t e d a t

2 9 8 K ( 2 5 ~ a t 1 0-~ s -L H o w e v e r , d e f o r m a t i o n a t a =

1 0 -4 s -~ a t e l e v a t e d t e m p e r a t u r e s s h o w e d c o n s i d e r a b l e

c o a r s e n i n g o f g r a i n s . T h e c o n c e n t r a t i o n o f S n S b i n t e r-

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

d u e t o g r a i n b o u n d a r y s e g r e g a t i o n a n d / o r d i s s o l u t io n

o f t h e S n S b i n te r m e t a l l i c . F i g u r e s 3 a a n d 3 b c o m p i l e

t h e o p t i c al m i c r o g r a p h s t a k e n f r o m t h e g a g e s e c t i o n s

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

4 2 3 K ( 1 5 0~ r e s p e c t i v e l y , a n d c l e a r l y d e p i c t t h e i n -


3/8

Tensile, Creep, and ABI Tests on Sn5%Sb Solder

for Mechanical Property Evaluation

841

situ grain growth and coarsening of he intermetallics

at the elevated temperature. These microstructural

modifications seem to be responsible for the observed

ductility loss at high temper ature s and low strain-

rates.

Deformation enhanced grain-growth has been re-

ported for several alloy systems, including Sn-lBi and

7475 A1, and causes th e stra in-r ate sensitiv ity expo-

nent m and, therefore, resistance to necking, to de-

crease with strain. 6 The SnSb interm etallic s tend to

segregate at grain boundaries, thereby causing a

reduct ion in the overall ductility. It has been sug-

gested s tha t the co ncurrent phase change due to high

activity of dislocations and vacancies ne ar grain bound-

aries during high temperature deformation makes

boundaries preferred locations for anomalous nucle-

ation effects. Therefore, the combined influence of

microstructural instability due to grain and SnSb

particle-coarsening during high temperature defor-

mation reduces the ductility significantly. However,

the s trai n-ra te s ensitivi ty m, indicative of the ductil-

ity at ~ = 10 -4 s -1, rema in ed high er.

Strain-rate change tests were supplemented by

constant cross-head speed (initial strain-rate) tests at

str ain -ra tes of 10 -4 s -1 and 10-3 s -1 to cha ract eriz e the

stra in-ra te depe ndence of the flow stress. The true

(ultimate) tensile stresses were evalua ted at the load

b

Fig. 3. Optical micrographof the Sn-5 Sbsolder deformedat ~ =

-4 1

10 s- to fracture at 298 (a), and 423K (b).

maxim a and true strain-rates were calculated at the

corresponding true uniform strains. These data are

plotted on double-logarithmic scale in Fig. 4a as the

true strain-rate vs the true stress, with low stress

da ta following the power-law,

= A ~ . (la)

Far larger strain-rates are noted at high stresses

where an exponential stress dependence is usually

noted. The stress exponent, n, varied from around

three to slightly less tha n four. Data at high stresses

followed an expon ential stress dependen ce as shown

in Fig. 4b,

= A e B'~, (l b)

with B = 1.33 + 0.29, which is essentially indep ende nt

of the test temp erature .

The activation energy was determined using these

data at a constant strai n-rate of 2 x 10~ s -1 by plot tin g

the normalized stress vs inverse absolute tempera-

ture where the stress is divided by the tempe rature

dependent elastic modulus. Such a plot is shown in

Fig. 5 which yie lded a valu e of 26.8 + 1.6 kJ/m ole for

the activation energy for creep. This value is very

small co mpared to th at for self-diffusion in Sn as well

as t ha t for lattice diffusion of Sb atoms (about 100.5

10 -1

10 -2

lu

9 10 -3

lo 4

1 0 - 5

1 -I

i . . . . .

Sn-5Sb / Tens i le Tests

298

K

A = o D /

o 373 K ~ ~, o [] /

r

423

K

A o o [] /

473K~ n=4

I 0 I 0 0

S t res s MPa

r

S n - 5 S b T e n s i l e D a t a / H i g h S t r e s s

B = 1.325 + 0.285

10 2

t i t D f = 9 4 K 073 K

* 423 K

473 K

10 -3 , I , I I , I ,

0 1 0 2 0 3 0 4 0 5 0

Stress, MPa

b

Fig. 4. a) Log-log plot of strain-rat e vs tensile stress, and b) high

stress data exhibiting exponential stress variation of strain-rate.


4/8


5/8

Tensile, Creep, and ABI Tests on Sn5%Sb Solder

for Mechanical Property Evaluation 843

experiment al results falls in this region, and thus we

believe that the high stress region is due to climb of

edge dislocations. On the other hand, the activation

energy is noted to be 54.4 kJ/mole, which is closer to

tha t for grain boundary diffusion tha n t hat for lattice

or chemical interdiffusion. If his is taken to repre sent

the low-te mperatu re climb mechan ism due to climb of

dislocations by vacancies diffusing through disloca-

tion pipes, one expects a value o f seven for the s tress

exponent.~l,~4

While the tensile and creep data reveal these two

mechani sms at the lower stresses, creep data did not

extend to very high stresses in the power-law break-

down regime. The tensile test results, however, do

exhibit an increasing n value with applied stress at

very high stresses, which is indicative of the power-

law breakdown. Disagreement between the experi-

mental activation energies and model predictions

notwithstanding, the plausible deformation mecha-

nisms a re viscous glide at low stresses, low-tempera-

ture dislocation climb at int erme diat e stresses, and

power-law breakdown at very high stresses. More

experimental work is warranted to determine the

various para met ers in all of the stress ranges. Micro-

struc tural investigations following deformation will

shed light on the un derlyin g creep mechanism. In the

climb creep regime, one expects to find distinct

subgrain formation whereas randomly distributed

dislocations are observed in the viscous glide creep

contro lled region. 15

ABI e s u l t s

As pointed out earlier, a major objective of he study

#3 - 0.005 in/s / ~ ~ t

#2 - 0.001 . / - ~ t

~ . ~ J l . - - ~ Sn5%Sb t

~ r DISPLACEMENT t

t

Indenter Velocity ~ I

#1 - 0.02 ires ..

#2- 0.0001 f_ f

#3 - 0 0000~~ / - f ~

Fig. 7 . Indentat ion load N) vs depth mm) curves at var ied indenter

speeds .

was to use SSM to obtain the stre ss-stra in curves to

demonstrate use of the ABI technique to determine

these mechanical data on small size specimens in a

nondestructive fashion. A numbe r of such tests could

be performed in a limited space on the specimen

shoulder. Typical load vs deformation (depth) curves

are included in Figs. 7a and 7b corresponding to

different indent er speeds. The system, in its cu rren t

configuration allows testing at ambient te mperature,

and a hig h-tem pera ture chamber is being installed.

We report here ABI test results at room tempe ratur e

and compare them with creep and tensile test data

obtained on the same mat erial.

The sys tem software utilizes elastic-plastic analy-

ses to evaluate the true plastic strains and stresses

from the depth and load data; the details are pre-

sented in th e Appendix A. 18,17The strain- rate is evalu-

ated from the indent er speed s from,

= 2 v ~ 3 )

5 d p '

where v i is the indent er velocity and dp is the plastic

inden ter dia mete r at the point of interest. The effect

of flow stress (about 7% true plastic strain) on the

strain-rate is shown on a double log plot in Fig. 8

which reveals that the data set breaks into two dis-

tinct regions.

At low stresses, strain-rate followed a power-law

10 0

.r

101

10 -2

I0 -3

10 -4

105

10

r

E]

~3

Sn-5Sb / ABI

data

oom Temperature

i i i i i i i i

Stress, MPa 100

Fig. 8. Doub le-log plot of s t rain-rate vs s t ress of ABI da ta.

.=

1 0 0 . . . . . . .

10 -1 Sn- 5Sb /

oom Temperature

10 -2 n 9

10 -3

10 -4

10 -s

10 -6 s

I CreepData 1

10 -7 = Tensile

a t a

10-8 . . . . . . . . . . . . . . .

1 1

S t r e s s M P a

Fig. 9. Com pariso n of tens i le, creep, and AB I test resul ts .


6/8


7/8

Tensile, Creep, and ABI Tests on Sn5 Sb Solder

for Mechanical Property Evaluation

where b is the Bur gers vector.

Physically, the activation area may be interpr eted

as the area swept by a dislocation segment in sur-

mounti ng a barri er in its slip plane or the area swept

betw een consecutive events. The ABI data in Fig. 10b

yield a val ue of 96.5b 2, wit h valu es of ~10b 2 to 100b 2

corres ponding to the climb of edge dislocations as the

dominant mechanism vs nonconservative glide of

jogged screw dislocation with a charact erist ic activa-

tion are as in excess of 1000b2.19 One can co mbine t he

two equations [Eq. (5) and Eq. (6)] using a Sinh-

function so that the whole data set may be described

by a single equation,

= A(sinh B(~)n = 1.235 x 10 -5 (si nh 0 . 0 4 ~ ) 4 5 . 8 )

Here, ~ is in per sec and (~ is in MPa. This implies tha t

a doubleqo g plot of str ain- rat e vs sinhB(~ should yield

a stra ight line as indicated in Fig. 11, with the slope

of the line bein g 4.5. Thus, we note th at the ABI

technique is useful in nondestructive evaluation of

the deformation mechanisms in materials.

It is clear tha t by decreasing the size of the in-

denter, say to micron size, ver y small specimens can

be tested using SSM. It is possible to utilize this

machine and technique in characterizing

real

solder

joints in electronic packages. Such development is

now underwa y, especially to use this as a probe to

monitor the degradation of struct ural materialsY

CONCLUSIONS

Tensile and creep tests, perfor med on bulk Sn5Sb

solder at various tempe rat ures from ambi ent to 473K,

exhibited power-law stress dependence of strain- rate

at low stresses and exponential stress variation at

high stresses. In addition, automated ball indenta-

tion (ABI) tests were perf ormed at room temper atu re

at various strain rates, and the results were in excel-

lent agreement with creep tests. Slight deviation

exhibited by the tensile test dat a from the creep and

ABI results was believed to arise from the flat tensile

stress vs strain curves which made an accurate de-

termi natio n of the ul timate tensile stress difficult.

The present study clearly demonstrated the useful-

ness of the ABI technique to charact erize the me-

chanical properties of microelectronic solder joints.

C K N O W L E D G M E N T S

We wish to express our appreciation to the re-

viewers for various constructive suggestions and for

their efforts in correcting the manuscript.

R E F E R E N C E S

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TMS Fall Meeting,

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Cincinnati, 1996.

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P P E N D I X

B I D a t a n a l y s i s

A brief summ ary of the r elevant equations used in

the derivation of the flow properties from ABI tests is

included here, while details can be found in various

references.7,16, '7 As no te d in Fig. 7, the pr im ar y info r-

mation obtained from ABI tests is the indentat ion

load (P) and the depth /hei ght of pene tra tio n (hp). The

system software calculates the plastic strain while

the flow stresses are evaluated using elastic and

plastic analyses. The true plastic strain is given by,

d

s = 0 . 2 D ' (A1)

wher e dp and D are the p las t i c indentation diameter

and diameter of indenter, respectively. The corre-

sponding flow stress is calculated using,

4P

~ i - - - - ~d~5' (A2)

wher e (~i is the tr ue inde ntat ion stress, P is inden-

tation load, and 5 is a param ete r which depends on the

system compliance and indentation stress with a

val ue b etw een 1.12 and 2.87ocm. The fa ctor a mdepends

on strain r ate sensitivity, with a value of unity for

strain- rate insensitive materials. The plastic inden-

tat ion depth, dp, is given by,

l p / I I / D I h 0 2 5 a }

d , = 2.735 ~Ei E-~fi ~h2, ~--~'~ ~-~- hvD (A3)

Here, E i and E~ rep res ent the elastic moduli of the

indent er and t he specimen, respectively. As we note


8/8

8 4 6 M u r t y , Ha g g a g , a n d M a h i d h a ra

h e r e , t h e p a r a m e t e r d D a p p e a r s i n b o t h t h e l e ft a n d

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

s o u g h t w h i c h t h e s o f t w a r e o f t h e s y s t e m h a s t h e

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

c a n b e r e l a t e d b y t h e p o w e r l aw ,

a i = K ~ ' , ( A 4 )

w h e r e K a n d n ' a r e t h e s t r e n g t h c o e f fi c i en t a n d t h e

s t r a i n - h a r d e n i n g p a r a m e t e r , r e s p ec t iv e l y . F r o m t h e

f a c t t h a t t h e s t r a i n - h a r d e n i n g p a r a m e t e r , n ', i s e q u a l

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

t e n s i l e s t r e s s ( az s) ,

Tensile, Creep, and ABI Tests on Sn5%Sb Solder.pdf

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