MEANING OF THE BIAXIAL FLEXURE TESTS OF DISCS FOR STRENGTH MEASUREMENTS

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MEANING OF THE BIAXIAL FLEXURE TESTS OF DISCS FOR STRENGTH MEASUREMENTS

J. Glandus

To cite this version:

J. Glandus. MEANING OF THE BIAXIAL FLEXURE TESTS OF DISCS FOR STRENGTH MEASUREMENTS. Journal de Physique Colloques, 1986, 47 (C1), pp.C1-595-C1-600.

�10.1051/jphyscol:1986190�. �jpa-00225621�

MEANING OF THE B I A X I A L FLEXURE T E S T S OF D I S C S FOR STRENGTH MEASUREMENTS

J . C . GLANDUS

E . N . S . C . I . , U . A . C . N . R . S . 320, F-87065 L i m o g e s ,

France

~esum6

-

La flexion biaxiale de disques e s t un e s s a i couramment pratiqu6 pour l a mesure de l a r6sistance $Z l a rupture des mat6riaux c&amiques. Sa mise en oeuvre r k l a m e toutefois certaines p r k a u t i o n s e t l e s r 6 s u l t a t s awquels il conduit d6pendent 6troitement des conditions exgrimentales. Certains para- m & t r e s e s s e n t i e l s de ce t e s t sont d k a g 6 s & p a r t i r de mesures extensm6triques e t d ' e s s a i s s u r des 6prouvettes en verre sodo calcique.

Alastract

-

The b i a x i a l flexure of discs is a t e s t widely perfonned f o r t h e strength measurement of ceramics. However its use requires some care and it leads t o r e s u l t s strongly dependent on t h e experimental conditions. Some of t h e main parameters of t h i s t e s t are emphasized on the basis of s t r a i n gage measurements and t e s t s performed on s i l i c a soda lime g l a s s .

INl'ROD UCTION

Measurement of t h e mechanical properties of ceramics using samples i n t h e form of discs have been developed f o r the non destructive evaluation of e l a s t i c moduli /1/

and of thermal shock resistance /2/ but chiefly f o r the measure of strength i n biax- i a l flexure /3//4/ because of the s l i g t h grinding needed and of the negligeable edge effects. I f t h e two former t e s t s lead t o values close t o those given by other methods, t h e l a t t e r exhibits a strong dependence on the experimental conditions. On t h e b a s i s of s t r a i n gage measurements and of t e s t s on d i s c s i n s i l i c a soda lime g l a s s , t h e present paper underlines some e f f e c t s of the main experimental parameters.

BIAXIAL FLEXURE OF DISCS

The samples a r e supported on t h e i r lower face and they a r e constrained by the load applied on t h e i r upper face. Many experimental configurations can be obtained depen- ding on whether the supports a r e discretes ( b a l l s ) or continuous ( r i n g ) and on whe- ther t h e load is applied on a point ( b a l l ) , a l i n e ( r i n g ) , o r an area ( p i s t o n ) . Theoretical analyses have shown /5/ t h a t t h e maximum t e n s i l e stress a t t h e center of t h e lower face does not depend on the number In' of bearing b a l l s (n

>,

3 ) so t h a t , from an analytical standpoint, experimental features d i f f e r only i n the manner i n which t h e load is applied. However, i n t h e cases of ring-like supports, s i g n i f i c a n t f r i c t i o n induced s t r e s s e s can a r i s e because t h e supports cannot r o t a t e and it i s rather d i f f i c u l t t o obtain a perfect contact between t h e sample and the ring / 6 / . Piston on n b a l l s (or r i n g ) r i g

The analytical solution f o r t h i s case has been derived by Kirstein and Wolley /5/.The

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986190

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pressure under the piston is assumed t o be uniform, s o the deflection of t h e sample must be small up t o t h e f a i l u r e . The maximum t e n s i l e s t r e s s a t t h e center of the lower face is given by :

P is t h e applied load, e the thickness of t h e sample, R its radius,

4

t h e Poisson r a t i o of the material, a the radius of t h e support c i r c l e and b t h e radius of t h e piston. A development of the previous expression gives the equivalent form /3/ :

Y = ( l + v ) [ l + L n ( a / R ) ] + ( I - V ) 2 ( a / R ) 2

The r i g "piston on 3 b a l l s " has been standardized by t h e A.S.T.M./8/.

B a l l o n n b a l l s ( o r r i n g ) r i g

Theoretically, no exact solution e x i s t s because of t h e tendency of the pressure contact t o reach

an_

i n f i n i t e value, but a semi-empirical equation gives an effective radius of contact, b , between t h e loading b a l l and the sample:

-

b = (1.6b2

+

e 2 ) l / '

-

0.675 e

b is the actual radius of co_ntact, which is a b u t e/10 under typical experimental conditions. Thus one obtains: br\, e/3 and t h e previous equations can be applied.

Ring o n ring r i g

Radial and tangential s t r e s s e s on the lower face a r e uniformely d i s t r i b u t e d on the surface limited by t h e loading ring, and t h e i r magnitude /9//10/ i s expressed by :

b now denotes the radius of the loading ring.

EXPERI MENT AND RESULTS

The loading rod and bearing s e t of t h e device used ( c f . ASTM standards) a r e replaceable and allow us t o carry out measurements with:

continuous support : r i n g 25 mm i n diameter

discontinuous support : 3 t o 12 b a l l s (4.8 mm in diameter) on a support c i r c l e 25 mm in diameter

applied load on a point : b a l l 8 mm in diameter

applied load on a l i n e : rings 5.5 and 10 mm in diameter

applied load on an area : pistons 0.7, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 9 and 10 mm in diameter

Measures of s t r a i n have been performed on s t e e l samples (d 30 x 1.5 mm,

H

30 x 2.5 rnrn,

H

30 x 3.5 mm) on which a s t r a i n gage ( 0 , 6 x 1 mm) was pasted a t t h e centre of t h e face i n tension. The e l a s t i c moduli of the s t e e l samples have been measured by an accurate resonance frequency method /1/ which leads t o :

E=210,87 GPa

4

= 0,312

The s t r a i n measurements have been performed f o r various combinations of the loading rcds and support sets available. Each of t h e following r e s u l t s is the average of 5 measures, a t l e a s t , and when t h e load w a s applied with a piston, the rod was turned on its axis between two measures. The theoretical values of the s t r a i n s a r e cal- culated according t o t h e previous equations (1) and ( 3 ) and generalized Hookets law.

Piston on n b a l l s (3 ( n ( 12 ; 0.7 mm ( 0 piston

<

^{10 mm)}

.

Whatever the experimental conditions a r e , t h e s t r a i n s measured with t h e piston of 0.7 mm in diameter a r e always lower than t h e i r theoretical value (cf Fig. 1 t o 4 ) .

s t a t i c i t y of t h e support set. I n t h e case of a piston on 3 b a l l s r i g ( i s o s t a t i c a l )

.

t h e measured s t r a i n s agree r a t h e r well with t h e i r t h e o r e t i c a l value, a s shown i n Fi- gures 1 and 2 f o r example, but they e x h i b i t s i g n i f i c a n t disagreement i f one b a l l o r more is added on the bearing race. I n these cases, t h e axysymetry of t h e s t r e s s f i e l d vanishes and t h e s t r a i n s measured,cf. Fig. 3, depend on t h e r e l a t i v e o r i e n t a t i o n of t h e gage a x i s and t h e principal axis. However, i t i s worth noting t h a t even with the i s o s t a t i c a l 3 b a l l s s e t , s l i g h t differences are recorded i n t h e measured s t r a i n s according t o t h e gage a x i s o r i e n t a t i o n /7//11/.

Although t h e device used was n o t tooled with very high precision, t h e previous r e s u l t s allow u s t o think t h a t f o r s t r e n g t h measurements of b r i t t l e materials, hyperstatical s e t s must be eliminated t o avoid t h e r i s k of s c a t t e r e d r e s u l t s due t o f a i l u r e s i n i t i a t e d by s u b c r i t i c a l flaws.

I \

PISTON ON 3 BALLS

Diameter: 38 mm Thickness: 1.5 mm Appl ied Load: 78 N

-

THEOREllCAL VALUES

I

^A nEASURED VALUES

PISTON DIAMETER (mm)

0 2 4 6 6 10 1'

PISTON ON 3 BALLS Diameter: 30 mm Ihickness: 2.5 mm Applied Load: 288 N

-

THEORETICAL VALUES A HEISUREO VALUES

F i g .

1

T y p i c & agtreement beAween cdcu- ^{F 4 . 2} T y p i c & ugheement b&een cdcu- lated and meawred

n ~ u i v t s (

bat% 1 U e d and meuw~ed

n.tmim (

b&

)

Piston on r l n g

The disadvantages of t h e r i n g supports have been underlined by Marshall / 6 / . However we have observed t h a t they give b e t t e r r e s u l t s than t h e 12 b a l l s support and t h a t the disagreement between t h e t h e o r e t i c a l and t h e measured values decreases a s t h e thick- ness of t h e sample increases /11/. Figure 4 i l l u s t r a t e s an example of good agreement f o r a sample of 3.5 mm i n thickness.

Ball on n b a l l s

For t h e s i n g l e case of t h e i s o s t a t i c a l 3 b a l l s s e t , t h e measured s t r a i n s are c l o s e t o those measured with a piston e/3 i n radius, a s suggested by Shetty e t al. /7/.

Table I shows t h a t t h e same r e s u l t is observed when the sample is r i n g supported.

Ring o n r i n g

No agreement has been found between t h e t h e o r e t i c a l and the measured values of t h e s t r a i n s f o r t h e r i n g 5.5 mm i n diameter as w e l l a s f o r t h e ring 10 mm in diameter.

Unexpected f r i c t i o n stresses may be responsible f o r such bad r e s u l t s .

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PISTON ON 12 BALL5

Diameter: 30 mm Thickness: 3.5 mm App l i ad Load: 3 0 0 N

-

THEOAETICAL~VALUES A A

A REASURED VALUES A

A

PISTON DIAMETER (mm)

0 0 2 4 6 B 10 12

Fk3.3 Typical dinagheement b&een c d c u - M e d and meam,ted

n.ttrainn .(

b&)

I

PISTON ON RING

I

Diameter: 3 0 mm Thickness: 3.5 mm Applied Load: 300 N

Ln Z u 100 .-:

-

THEORETICAL VALUES A IiEASURED VALUES

PISTON DIAMETER (mm)

I

2 4 6 0 18 12

Fig.4 Typical agtleement b m e e n c d c u - lated and meuuhed

nLttrrainc,

( n i n g )

Table I Meanuhed

n W m z c l h t h e n i g "b& on 3 b&

( o h

hing

I "

E l a s t i c behaviour

For all these experiments, t h e s m a l l loads used lead t o small s t r a i n s

( E

<300 pdef.) which prevent p l a s t i c e f f e c t s t o occur during t h e tests: t h e l i n e a r e l a s t i c b e h a v i o u r of

me

samples has been controlled by recording t h e s t r a i n v s load dependence /11/.

Finally t h e p l o t t i n g of (P/&) vs e 2 f o r d i f f e r e n t piston diameters gives s t r a i g h t l i n e s (Fig. 5 ) which v e r i f y t h e dependence of t h e s t r e s s on t h e square of t h e thickness.

B i a x i a l strength measurements

-

More than 130 samples of s i l i c a soda lime g l a s s of the same batch w e r e used. F i r s t , 30 specimens (

d

25 x 3 -9 mm) w e r e t e s t e d with a p i s t o n on 3 balls r i g ( 0 piston=2.5 mm) and we obtained

6

⁼ 146 Mza and m w 4.5. Due t o t h e r a t h e r small number of samples used t h e mean strength ER, may be considered a s an accurate value, but not Weibulll s modulus m /12/. Second, batches of 10 specimens each ( 0 30 x 3.9 mm) were used f o r strength measurements with various piston on 3 b a l l s r i g s : t h e measured values a r e summarized i n table 11.

.

The measures performed with an i s o s t a t i c set ( 3 b a l l s ) show t h a t t h e strength de- creases when t h e piston diameter increases according t o t h e Weibull e f f e c t . Due t o t h e f a s t decrease of s t r e s s e s along a radius /7/, i t may be assumed t h a t t h e e f f e c t i v e volume in tension is roughly limited by t h e piston diameter, s o t h a t f o r d i f f e r e n t experiments, t h e product

v

&m m s t remain constant. Indeed, t h e p l o t t i n g of LnV versus Lnu,for t h e values of t a b l e I1 g i v e s ( c f Fig. 6 ) a s t r a i g h t l i n e , t h e slope of which i s about

-

5.7. The r a t h e r go& agreement between t h i s value and t h e 4 . 5 value found f o r Weikull's modulus shows t h a t the piston diameter is one of t h e most important parameters of such tests. This point is confirmed by t h e f a c t t h a t s t r e n g t h s measured on samples 25 mm i n diameter and on samples 30 mm in diameter are c l o s e when t h e same piston is used f o r t h e two s e t s of experiment.

143 1 3 2

4 b a l l s 6 b a l l s 1 2 b a l l s

-

I! PISTON 9 mm /

5 4 PISTON 6.5 mm X

4 . - E

+

PISTON 4.5 nm 0 PISTON 3.5 mn + v,

.

A PISTON 2.5 mm

2 @ PISTON 1.5 mm

3 5

THICKNE5SA2 (mmn2)

0 .

8 3 6 9 12 l !

1.5 2.5

FLg.5

Stxttcwo vn t k i c k n u n dependence F i g .6 S ~ % a n vn v o h e dependence

b a l l + 1; 2. 2.6 2.5 2.5 2.5

.

When a hyperstatic bearing s e t is used (4, 6 , 12 b a l l s ) , random s t r e n g t h s are rnea- sured a s previously foreseen.

J 130

139 153 103

CCNCLU SION

Strength measurements i n b i a x i a l f l e x u r e e x h i b i t strong dependence on t h e hyper- s t a t i c i t y of t h e b a r i n g s e t . When t h e c l a s s i c a l p i s t o n on 3 b a l l s r i g is used, t h e p i s t o n diameter is one of t h e most c r i t i c a l parameters of t h e t e s t , t h e main

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d i s a v a n t a g e o f w h i c h is the v e r y l o w r a t i o : volume i n t e n s i o n / t o t a l volume. If t h e p i s t o n i s r e p l a c e d b y a ball a n e f f e c t i v e r a d i u s o f c o n t a c t o f e / 3 may be assumed w i t h g o d a c c u r a c y . I n t h e case o f r o u n d r o b i n t e s t s , t h e strict o b s e r v a n c e o f the A.S.T.M. r e c o m m a n d a t i o n s seems e s s e n t i a l t o a v o i d scattered results.

REFERE NCES

--

/1/ GLANWS, J.C., PLATON, F. a n d BOCH, P . , Mat. i n Eng. ~ p p l i c a t i o n s

1,

1 9 7 9 , 2 4 3 / 2 / GLANDUS,J.C. a n d BOCH, P . , Mat. S c . Monographs

-

E l s e v i e r 6 , 1 9 8 0 , 6 6 1 / 3 / WACHTMAN, J.B. Jr, CAPPS, W. a n d MANDEL, J . , J. o f Mat.

N ,

1 9 7 2 , 1 8 8 /4/ GIOVAN, N.N. a n d SINES, G.J., J. Am. Ceram. S o c .

640,

1 9 8 1 , 6 8

/ 5 / KIRSTEIN, A. F. and WOLLEY, R.M., J. o f . R e s e a r c h N.B.S. 7 1 ~ ( 1 ) , 1 9 6 7 , 1 / 6 / MARSHALL, D.B., Am. Ceram. Soc. B u l . 5 9 ( 5 ) , 1 9 8 0 , 5 5 1

/ 7 / S H E W , D.K. e t al., Am. Ceram. Soc. r 5 9( 1 2 ) , 1 9 8 0 , 1 1 9 3 / 8 / ANSI/ASTM F 394-78

/ 9 / FAYET, A., Rev. F r a n q a i s e de Meca.

37,

1 9 7 1 , 5 3

/lo/

^{FESSLER, H.} a n d TRICKER, D.C., J. Am. Ceram. S o c .

67(9),

1 9 8 4 , 5 8 2 /11/ GLANDUS, J . C . , I n t e r n a l R e p o r t 1 9 8 5

/12/ GLANDUS, J.C. a n d K C 3 P., J. Mat. SC. L e t . 3 , 1 9 8 4 , 7 4