plaster of paris as a model material for brittle porous solids

7/21/2019 Plaster of Paris as a Model Material for Brittle Porous Solids

1/7

JOURNAL OF MATERIALSSCIEN CE 28 (1993) 3221-3227l a s t e r o f a r is a s a m o d e l m a t e r i a l fo r b r i t t l e

p o r o u s s o l i d sG . V E K I N I S , M . F . A S H B Y , P . W . R . B E A U M O N TCam bridge University Engineering Department Trum pington Street Cam bridge CB2 1PZ UKP l a s t e r o f Paris is a b r i tt le p o r o u s s o l id e a s y t o s h a p e w h i c h h a s p o t e n t i a l a s a m o d e l m a t e r ia l fo rt h e s tu d y o f b r it tl e p o r o u s s o l i d s s u c h a s c e r a m i c s r o c k s a n d c e m e n t . T h i s p a p e r d e s c r i b e s t h em e c h a n i c a l p r o p e r t i e s o f p l a s t e r o f P a r is - m o d u l u s s t r e n g t h f r a c t u r e t o u g h n e s s e t c. - asa f u n c t i o n o f p o r o s i ty . T h e m a t e ri a l i s t h e n u s e d t o s t u d y t h e i n i t ia t i o n a n d p r o p a g a t i o n o f c ra c k s i nc o m p r e s s i o n a s a f u n c t i o n o f p o r o s i ty s tr e ss s t a te a n d s t re s s c o n c e n t r a t i o n .

1 I n t r o d u c t i o nP l a s t e r o f P a r is ( h y d r a t e d c a l c i u m s u l p h a t e ) i s a b r i tt l es o l id w i t h f r a c t u r e p r o p e r t i e s w h i c h r e s e m b l e t h o s e o fc e m e n t , s a n d s t o n e , a n d o t h e r p o r o u s c e r a m i c s . I t c a nb e s h a p e d e a s i l y , a n d u s e d a s a m o d e l m a t e r i a l t os t u d y t h e b e h a v i o u r , u n d e r l o a d , o f p o r o u s s o l id sc o n t a i n i n g m a c r o s c o p i c c r a c k s , h o l e s a n d r e i n f o r c e -m e n t . T h e f i r st p a r t o f th i s p a p e r r e p o r t s a s t u d y o f th ep r o p e r t i e s o f p l a s t e r o f P a r i s - t h e m o d u l u s , t e n s i lea n d c o m p r e s s i v e s t r e n g t h s , m o d u l u s o f r u p t u r e , a n df r a c t u r e t o u g h n e s s a s a f u n c t i o n o f t h e d e n s i t y ( o rpo ros i t y ) o f t he p l as t e r .

T h e s e c o n d p a r t d e s c r ib e s a n e x a m p l e o f a n a p p l i c a -t i o n: t h e m e c h a n i s m s o f fa i l u re a t a c y l i n d r i c a l h o l e o rs p h e r i c a l p o r e u n d e r s i m p l e a n d m u l t i a x i a l c o m p r e s -s i o n . S t r e s s i s c o n c e n t r a t e d a t a m a c r o s c o p i c h o l e i na n e l a s t i c s o l i d . I n u n i a x i a l c o m p r e s s i o n , t e n s i o n a p -p e a r s a t o n e p a i r o f p o l e s o f t h e p o r e ; b u t i n m o s tb i a x i a l c o m p r e s s i o n s t a t e s a l l p r in c i p a l s t re s s e s a t a n dn e a r t h e p o r e a r e z e r o o r c o m p r e s s i v e . D e s p i t e t h i s ,f r a c t u r e o c c u r s a t t h e p o r e s u r f a c e i n a p o r o u s o rm i c r o - c r a c k e d m a t e r i a l l i k e p l a s t e r , c a u s i n g i n w a r ds p a l l in g o f m a t e r i a l f r o m t h e p o r e s u r fa c e . T h e p r o b -l e m i s k n o w n i n t h e m i n i n g i n d u s t r y , w h e n g e o s t a t i cl o a d s c a u s e i n w a r d s p a l l i n g o f b o r e h o l e s , a n d i t isp r o b a b l y t h e m e c h a n i s m b y w h i c h f r a c t u r e s t a r t s i np o r o u s c e r a m i c s u n d e r c o m p r e s s i v e s t r e s s s t a t e s . W eh a v e u s e d p l a s t e r o f P a r i s s a m p l e s t o i n v e s t i g a t e th ei n i t i a t i o n a n d p r o g r e s s o f c r a c k i n g f r o m h o l e s a n dpores , un de r a var i e t y o f s t r ess s ta t es . The r esu l t s a r er e l a t e d t o o b s e r v a t i o n s a n d m o d e l s f o r c o m p r e s s i v efa i l u re o f b r i t t l e so l i d s [1 -16 ] .

W h e n p l a s t e r o f P a r is i s m i x e d w i t h w a t e r t h e r e v e r s er e a c t i o n t a k e s p l a c e : w a t e r i s r e a b s o r b e d w i t h t h ef o r m a t i o n o f g y p s u m . T h e r e a c t io n i s e x o th e r m i c a n dr e s u l ts i n a c o h e r e n t m a s s o f in t e r l o c k i n g n e e d l e -s h a p e d g y p s u m c r y s t a l s. T h e c h e m i s t r y o f t h e r e a c t i o nr e q u i r e s o n l y 1 8 .6 w t % w a t e r f o r r e h y d r a t i o n , b u t i np r a c t i c e m u c h m o r e i s u s e d t o g i v e t h e f lu i d i t y n e e d e df o r c a s ti n g . T h e e x c e s s w a t e r e v a p o r a t e s l e a v i n g c o n -s i d e r a b l e p o r o s i ty . T h e t r u e d e n s i t y o f t h e h e m i h y -d r a t e i s a b o u t 2 7 50 k g m - 3 a n d t h a t o f t h e d i h y d r a t ea b o u t 2 3 2 0 k g m - 3 , s o a c o n t r a c t i o n o n s e t t in g w o u l db e e x p e c t e d ; b u t t h e a r r a n g e m e n t o f th e c r y s t a l s iss u c h t h a t s e t t i n g r e s u l t s i n a s l i g h t e x p a n s i o n ( a b o u t0 .5%).T h e m e c h a n i c a l p r o p e r t ie s o f p l a s t e r d e p e n d o np o w d e r - t o - w a t e r r a t i o , c u r i n g t i m e , t e m p e r a t u r e a n dp r e s s u re , a n d o n p o s t - c u r e h e a t t r e a t m e n t . A l l w e r ei n v e s t i g a t e d [ 1 6 ] a n d a s t a n d a r d p r o c e d u r e w a s a d -o p t e d . T h e C a S O 4 " 8 9 s t a r t i n g p o w d e r ( B r it i shD r u g H o u s e ) w a s m i x e d w i t h d i s t i l l e d w a t e r , i n t h era t i o 100 : 62 .5 , r em ov in g a l l a ir t r app ed i n t he suspe n -s i o n, a n d w a s c a s t i n t o s p l it r e c t a n g u l a r ( a p p r o x i m a t es ize 10 m m x 10 m m x 100 mm ) o r cy l i nd r i ca l ( ap -p r o x i m a t e s iz e 1 0 m m d i a m e t e r x 8 0 m m h e i gh t )m o u l d s a n d a l l o w e d t o c u r e f o r a t l e a s t 7 d a y s a t 2 0 ~S p e c i m e n s o f h i g h e r d e n s i t y t h a n t h a t o f t h e a s - c a s ts p e c i m e n s w e r e o b t a i n e d b y f o r c i n g w a t e r o u t o f t h em o u l d u n d e r p r e s s u re , i m m e d i a t e l y a f te r c a s ti n g . T h esamples con t a in smal l spher i ca l po res (F ig . 1 ) , p rob -a b l y d u e t o t r a p p e d w a t e r d u r i n g c a s t i n g , w h i c h d e -c r e a s e i n s i z e w i t h i n c r e a s i n g b u l k d e n s i t y o f t h emater ial (Fig . 2) .

2 . P l a s t e r o f P a ri s : t h e m a t e r i a lP l a s t e r o f P a r i s i s c a l c i u m s u l p h a t e h e m i h y d r a t e ,C a S O 4 - 8 9 I t is m a d e b y h e a t i n g g y p s u m b e t w e e n120 an d 160 ~

C a S O 4 " 2 H 2 0 ~ C a S O 4 . 8 9 + 1 8 9 (1 )

3 . T h e m e c h a n i c a l p r o p e r t i e s o f p l a s t e ro f P a r i s3 . 1 . P l a in s p e c i m e n sA n u m b e r o f m e c h a n i c a l p r o p e rt ie s w e r e m e a s u r e d a sa f u n c t i o n o f d e n s i ty . T h e p r o p e r t i e s i n c l u d e d Y o u n g ' s

* Present address:"Demokritos"Na tional Cen tre for S cientificResearch, Institute of M aterials Science, 153 10 Ag P araskev i Attikis,Athens,Greece.0022-2461 9 1993 Chapman & Hall 3 2 2 1


2/7

20

15o_

gw

5

0 i I q 00 4 0 5 0 6 0 7 O BRelative density

10.0

7.5

s 0oggo

2.5

Figure 3 Young ' s m odulus in unia xial c om pre s s ion a nd m od ulus ofrupture in four-poin t bend ing as a function of rela t ive densi ty.

Figure The s tru c ture of the plas ter : (a ) 50% re la tive densi ty and(b) 70% relative density specimens.

300

2oo

~ 100

00.4 0.85 0 6 0 7Relative density

Figure 2 Macr opo re diame ter as a function of re la t ive densi ty ofhydra ted plas ter of Par is .

modulus (by four-po in t bend ing , Ebend, and underuniaxial compression, Ecomp , he modulus o f rup tu reunder th ree- and four-po in t bend ing (MOR3 andMOR4), the fracture toughness , Ki t , by four-points t raight-edge-notched beam (SENB), the uniaxial ten-s ile s t rength, cyt , and the uniaxial com pressivestrength, ac. In addi t ion, the compressive s t rengthsunder biaxial and hydrostat ic condi t ions were alsomeasured. The resul ts for the as-cast material aresummarized in Table I , which also l is ts data fromearlier published studies.Young s mod ulus , E, was determ ined from the load-ing curves in bending and uniaxial compression, asdescribed below.3 2 2 2

Four-point bend tes ts were used to measure themo dulus of ruptur e (Table I). The bea m specimens(10 mm x 10 m m x 90 mm ) were l ight ly gro und using1000 grade SiC paper prior to tes t ing in order toremove an outer skin which has a different morpho-logy from the bulk of the specimens. M odul us ofrupture values were calculated using the wel l -knownequat ion

P A s dMOR4 (2)81where P is the fracture load (N), As is the differencebetween major and minor spans, I i s the second mo-ment o f a rea o f spec imen bda/12, where b is the widthof specimen), and d is the height of specimen. All thespecimens displayed purely elas t ic behaviour up tofast f rac tu re . The var i a t ion o f modulus and MO R4with relat ive densi ty and associated pore s ize is shownin Fig. 3 . Young s mo dulus for four-po int bending(Table I) was calculated using the elas t ic beam theor yas

dP AS3g b e n d - - d x 4bd 3 (3)

The fracture toughness measurements were carriedou t on rec tangu lar edge-no tch beam spec imens wi ththe same dimensions as those used for the bendingtests. The fracture toughness, Klc (Table I), was cal-culated from the analysis of Strawley and Gross [19]

P As 3 C~ 1/2KIc = b/d d 2(1 - a) 3/2 (4)

where ~ is a numerical constant , o f order uni ty , whichdepends o n the rat io of notch depth, a , to specimenheight, d. The variation of K~c with relative densityand assoc iated pore size is show n in Fig. 4.

Tensi le tests proved difficult : the p roblem is that ofachieving al ignment . Special rubberized spl i t gripswere designed which pro vided effect ive gripping o f thecyl indrical specimens withou t high ho op s t resses [16].Only resul ts of specimens that fractured within the


3/7

T A B L E I Pro perties of plaster of Paris (as cast, 62.5 water)P resen t s tudy P rev ious work

[t7, 183Physical properties 106050

Mechan ica l p roper t ie s

Theoretical density (kg m- 3)Dens i ty (kg m -3 )To ta l po ros ity con ten t ( )Mean diameter of spherical macropores ( lain)Me an grain size (Ixm)Young ' s mod u lus (bend ing) (GPa)You ng's modu lus (uniaxial compression) (GP a)Mod ulus o f rup tu re ( four -po in t bend ing) (MPa)Weibu l l modu lusFrac ture toughnes s (SENB), (MP a m ~/2)Un iaxial compressive streng th (MPa)Hydrostatic compressive strength (MPa)Uniaxial tensile strength (MPa)

23501170 _ 3051+_2212 +_ 183 x 1 5

4.5 +_ 0.14.6 +_ 0.35.8 _+ 0.66.20.14 _+ 0.01514. 6 -t-_0.919.2 _+ 1.4

3.2 _+ 0.612 8

0.20

r0 15

gIE

~, o:og

'.'- 0.05

F r a c t u r etough?r ce '~ 9 0

0 i i I0.4 0.5 0.6 0,7Relative density

8 n ~ Hydrostatic7 c o m p r e s s i o nFracturei n i t i o t i o6 _

/ 9 ~ U n ( o x i o l U n i o x i a l5~ / M - ~ m p r ~ T e / c o m p r e s s i o n

~- Fracture3 initiation2

Figure Fractu re toughness, Kl~, and the tensile fracture strengthas a functio n of the relative density .

g a u g e l e n g t h w e r e a c c e p t e d . T h e t e n s il e s tr e n g t h o f t h ea s - c a s t m a t e r i a l i s g i v e n i n T a b l e I . T h e v a r i a t i o n o f chw i t h r e l a t i v e d e n s i t y a n d a s s o c i a t e d p o r e s i z e i s a l s os how n in F ig . 4 .

C o m p r e s s i o n s p e c i m e n s w e r e p r e p a r e d b y g r i n d i n gt h e e n d s p a r a l l e l t o b e t t e r t h a n 5 Ix m c m - ~ a n d p e r -p e n d i c u l a r w i t h r e s p e c t t o t h e s id e s t o b e t t e r t h a n 0 .2 ~T o a v o i d d a m a g e t o t h e en d s o f th e s p e c i m e n a n dr e d u c e f r i c t i o n b e t w e e n a n v i l s a n d s p e c i m e n s u r f a c e,p a p e r s h i m s w e r e u s e d i n a l l t e s t s . A t y p i c a l c o m p r e s -s i o n c u r v e is s h o w n i n F i g . 5 l o w e r c u r v e ) . T h e i n i t ia -t i o n s t r e s s f o r c o m p r e s s i v e f r a c t u r e a n d t h e u l t i m a t ec o m p r e s s i o n s t r e n g t h a r e s h o w n i n F i g . 6 l o w e r p a i ro f c u rve s ) .

E q u i - b i a x i a l e x p e r i m e n t s w e r e c a r r i e d o u t b o t h i na n I n s t r o n a n d i n t h e S E M c o m p r e s s i o n r i g , u s i n gw e d g e g r i p s [ 1 6 ] . H y d r o s t a t i c t e s ts w e r e c a r r i e d o u to n s h o r t c y l i n d e r s ( 10 m m d i a m e t e r x 1 0 m m h e i g h t )e n c l o s e d i n a l a te x r u b b e r s h e a t h a n d p r e s s u r i z e d i nd i s t i ll e d w a t e r i n a P T F E c a p s u l e i n a h i g h - p r e s s u r ev e s s el (F i g . 7 ); t h e a r r a n g e m e n t c a n b e u s e d u p t o

DisplacementFigure 5 Load contra ction curve for a specimen tested in uniaxialcompression (lower curve) and in hydrostatic compression (uppercurve).

43 -- Hydrostaticcompressionz

u~

g ' 0=4

3Z compression ~ ~o J~

0 l l0 4 0.5 0.6Relativedensity

Figure 6 Fracture init iation stress (f irst

[][3 {3 [3Ultimate hydrostatic /compressive tress / / /

/ ~ 0 0 / 0 1 1 0

D/ / '~;, Fracture. / / / $ initiation stress_ ~o . . . . . . . . . . = /

/ /Oo rUnioxiol ~ ~ oc o m p r e s s i v e strength / / o / /\ / n / ,o ~- z ~ "- Fracture~Uniaxiol i. .I / i o initiationstress

0'.7 0.8non- l inea r i ty o f thestress-stra in curve) and ult im ate strength, as a func tion of relativedensity , for uniaxial com pression (lower curves) and for hy drostaticcomp ression (upper curves).

3223


4/7

Specirnen~Woter E.:-;: :.

P r e s s u r e cel l

~ - ~ i .g si: ji . ,

Figure The PT FE cell used in the hydrostatic compression ex-periments.

p r e s s u re s o f 1 G P a . A l l th e s p e c i m e n s d i s p l a y e d a ne l a s t i c r e s p o n s e u p t o t h e p o i n t a t w h i c h f r a c t u r ei n i t i a t ed , de t ec t ed by smal l s t eps i n t he l oad i ng cu rve .A t y p i c a l p r e s s u r e v o l u m e c u r v e i s s h o w n s c h e m a t i -ca l l y i n F i g . 5 (upper cu rve). The hyd ros t a t i c co l l apsep ressu re i s p l o t t ed aga i ns t dens i t y i n F i g . 6 (uppercu rves) . Spe c i mens r em ove d f rom t he ce ll a f t e r t es t i ngh a d b e e n c o m p a c t e d t o a b o u t t h r e e - q u a r t e r s o f t h e i ro r i g i n a l v o lu m e , a n d w e r e v e r y w e a k , c r u m b l i n g e a s i lyt o a c o a r s e p o w d e r .

S p e c i m e n s w e r e r e m o v e d f r o m t h e p r e s s u r e v e s s e la f t e r v a r i o u s p r e s s u r e s t o e x a m i n e t h e p r o g r e s s iv ec o l la p s e o f t h e c o n t a i n e d p o r o s i t y . D a m a g e s t a r ts a tt he su r f aces o f t he po res . Co l l apse p roc eeds b y f r ac t u r -i n g o f l a r g e s e g m e n t s i m m e d i a t e l y a d j a c e n t t o t h epores wh i ch f a l l i nwards , f i l l i ng t he po re . The f i na ls t a g es o f c o l la p s e a r e c h a r a c t e r i z e d b y a g e n e r a lc h a n g e o f p o r e s h a p e , w i t h g e n e r a l c r u s h i n g o f th es u r r o u n d i n g m a t e r i a l c o m b i n e d w i t h l a rg e - s c a le f ra c -t u r in g . I t a p p e a r s t h a t p o r e c o l l a p s e t a k e s p l a c e p r i o rt o a n y b u l k f r a c t u r i n g o f t h e m a t e r i a l a n d p r o c e e d sex t ens i ve l y befo re l a rge- sca l e f r ac t u re o f spec i mens .

3 . 2 . S p e c i m e n s c o n t a i n i n g c y l i n d r i c a l h o l e sU n i a x i a l a n d b i a x i a l t e s t s w e r e c a r r i e d o u t o n c u b e s(app rox i m at e s i ze 4 .5 mm x 4 .5 mm 4 .5 mm) o f fu l l ycu red p l as t e r o f Par i s wi t h var i ous dens i t i es . A s i ng l ec y l i n d r ic a l h o l e o f d i a m e t e r 0 .5 m m w a s i n t r o d u c e d b yd r i l l i n g a t l o w s p e e d . T h e s p e c i m e n s w e r e p r e p a r e dand t es t ed as befo re , under un i ax i a l and b i ax i a l s t r esss t at e s . T h e o b s e r v a t i o n s o f p o r e c o l l a p s e w e r e m a d eu s i n g a n i n s i t u S E M c o m p r e s s i o n r i g .

T h e s t re s s f o r i n i ti a t i o n o f h o l e c o l l a p s e a n d t h a t f o rb u l k f r a c t u r e i n i t i a t i o n i n s i m p l e c o m p r e s s i o n w e r em e a s u r e d f o r a r a n g e o f d e n si ti e s. T h e r e s u lt s a r es u m m a r i z e d i n F i g . 8 . T h e s e q u e n c e o f e v e n t s t a k i n gp l a c e d u r i n g u n i a x i a l c o m p r e s s i v e f a i l u r e o f a c u b i cspec i men con t a i n i ng a d r i l l ed ho l e i s i l l u s t r a t ed i n t hem i c r o g r a p h s o f F i g . 9 . T h e c o m p r e s s i o n d i r e c t i o n i s3 2 2 4

2 0 [ S t r a i n s .

~ 1 5 [ ~/~- ~'~. /10 ~BUlk r Q c t u r ~ ~ . ~ . . . f ~ . o

oE S ~ S t r e s s e s _ ~ f - - - ~~ ~ , L._

0.5 0.6 0.7Relet ive densi ty

0.z

c

ul

0.1 _~Eo

0.8

Figure 8 Fracture initiation and hole collapse nitiation stress andstrain as a functionof relative density.

v e r ti c a l. S o m e b u l k f r a c t u r e o f t h e s p e c i m e n p r e c e d e sh o l e c o l l a p s e i n s i m p l e c o m p r e s s i o n . T h e h o l e t h e nc o l la p s e s b y t h e f r a c t u r e o f a r c - s h a p e d s e g m e n t sa r o u n d t h e w a l ls o f t h e h o l e , p a r t i c u l a r l y a t a r e a s o fh i g h e s t c o m p r e s s i v e s tr e s s c o n c e n t r a t i o n a t t h e e q u a -t o r o f the ho l e . The co l l apse an d f r ac t u re p rocess i sd i s c o n t i n u o u s . T h e l o a d n e e d e d t o c o n t i n u e t h e c o l -l apse p rocess decreases wi t h i ncreas i ng s t r a i n ( as i nF i g . 5 ), a l t hou gh t he o vera l l spe c i men co l l apse i ss t ab l e up t o l a rge s t r a i n s .

Even a sm al l deg ree o f l a t e r a l cons t r a i n t ( i .e . b i ax i a ll o a d ) r e s u l ts i n t h e s i m u l t a n e o u s i n i t i a t io n o f b u l kf r a c t u r e a n d h o l e c o ll a p s e. W h e n t h e d e g r e e o f c o n -s t r a i n t a p p r o a c h e d e q u i - b ia x i a l l o a d i n g , h o l e c o l la p s ep receded t he i n i t i a t i on o f bu l k f r ac t u re (F i g . 10 ) . Asbefo re , t he co l l apse o f t he ho l e i nvo l ves the i nw ardf r a c t u r i n g o f a r c - s h a p e d s e g m e n t s f r o m t h e w a l ls . T h ec o l l a p s e d a re a s b e c o m e m o r e u n i f o r m l y d i s t r i b u t e da r o u n d t h e w a l l o f t h e h o l e a s a n e q u i b i a x i a l s t re s ss t a t e i s a p p r o a c h e d . H o l e c o l l a p s e r e a c h e s a na d v a n c e d s t a t e b e f o r e s i g n i f i c a n t o v e r a l l s p e c i m e nco l l apse . The co l l apse p rocess i s d i scon t i nuous , bu ts i gn i f i can t l y m ore s t ab l e t ha n i n t he un i ax i a l case , i .e ,t he l oad r equ i r ed t o sus t a i n t he co l l apse p rocess r e -m a i n e d c o n s t a n t o r d e c r e a s e d v e r y s l o w l y w i t h i n -creasing s t rain (as in Fig . 5) .

4 . D i s c u s s i o nM o d e l s f o r t h e c o m p r e s s i v e c r a c k i n g o f b r it t le s o l id s[2 , 14, 15 ] supp ose t ha t c r acks i n i t i a t e a t f l aws ( t hef l aws mi g h t be spher i ca l po res o r sha rp c r acks ) . Th em o d e l s l e a d t o a n e x p r e s s i o n f o r th e s t r e ss e s f o r c r a c ki n i t i a t i on ; i t i s a r e l a t i onsh i p be t ween t he ax i a l s t r ess ,0 .1, and the rad ial s t ress , o3 = (~2, (c~1 > ~3) , w hichc a n b e w r i t t e n a s

O ' 1 = C l 0 3 - - C o ( 5 )w h e r e t h e c o n s t a n t s c l a n d C o d e p e n d o n t h e n a t u r e o ft h e f la w s . W h e n c r a c k s n u c l e a t e f r o m s p h e r i c a l h ol e s ,t h e c o n s t a n t c l i s p r e d i c t e d t o b e a b o u t 3 . 1 -3 . 4 [ 1 5 ] .W h e n , i n s t e a d , t h e y n u c l e a t e f r o m c r a c k s [ 2 , 1 4] i tdepe nds on t he c oef f i c i en t o f fr i c t i on , ~t, be t w een t hes l i d i ng c rack f aces an d r ange s f rom 1 -3 .5 as ~t r angesf r o m 0 - 0 . 7 . ( E x p e r i m e n t a l r e s u l t s o n c r a c k i n i t i a t i o ni n Wes t er l y g ran i t e , quo t ed i n [20 ] , sugges t a va l ue o f


5/7


6/7

0 0, - -

-10

g - 2 0._o= - 3 0

- 4 0 0

< c

10 20 3 0K I C / ro e ) 1 / 2 M P o )

Figure 11 U ni a x i a l f r a c t u r e i n i t i a t i on s t r e s s o f p l a s t e r o f P a ri s :c o m p a r i s o n w i t h m o d e l s [ 1 5 , 2 1 ] .

-GQ _

-I0-203o

' = - 409 - r q = 1,1

-50 2c- - 6 0 50 40 3 0 2 0 I 0

l-f) cFigure 13 T h e h y d r o s t a t i c s t r e s s f o r f r a c t u r e i n i t i a t i o n p l o t t e da g a i n s t (1 - f ) c o , f o l l o w i n g E q u a t i o n 1 1. T h e l i n e c o r r e s p o n d s t o a ni n i t i a t i o n d e p t h r / a = 1.1.

/ \ \// / b \

i P o r e / II - _ ~ / l o ' ~ I\ /\ F l a w /.~,... . , .~\ , --,, /N /\ /

( o )

- #

P1 - f

r = Q t

- C r(b)

Figure 12 ( a) S t r e s se s a r o u n d a s p h e r i c a l p o r e s u b j e c t e d t o h y d r o -s t a t i c p r e s su r e , p , a nd ( b ) t he i r va r i a t i on w i t h d i s t a nc e f r om a po r e ,r ( E qua t i ons 7 a nd 8 ) .

~ o , where [223b 3 ( r 3 - - a 3 )

(3 -- p r 3 ( b 3 _ a 3 ) 7 )(30

b 3 ( a 3 + 2 r 3 )- p 2 r 3 ( b 3 - a 3 ) 8 )

3 2 2 6

F a r f r o m t h e p o r e a l l s t r e s s c o m p o n e n t s a p p r o a c h t h eh y d r o s t a t i c s t r e s s l e v e l, - p , b u t , w h e r e a s t h e r a d i a lc o m p o n e n t , % , d e c r e a s e s f r o m 0 a t t h e p o r e s u r f a ce to- - - p a t r = b , t h e t a n g e n t i a l c o m p o n e n t s i n c r e a s e

f r o m a m i n i m u m v a l ue = - 3 / 2 p / 1 - f ) ( w h e r ef = a 3 / b 3 i s t h e v o l u m e f r a c t i o n o f p o r e s i n t h e m a t e r -i a l ) a t t he por e su r f ace , t o t he va l ue - p a t r = b , a sshown i n F i g . 12 .

A p p l y i n g t h e f r a c t u r e n u c l e a t i o n E q u a t i o n 5 t o t h ec a s e o f h y d r o s t a t i c c o m p r e s s i o n o f p o r o u s b r i tt le m a -t e r ia l s , w e f i n d ( b e c a u s e % i s l e ss c o m p r e s s i v e t h a n q 0 )

~ 0 = c l o t - Co (9 )wh ere c l -~ 3 .1 an d C o ~ - 1 . 6 K l c / O z a p L o ) 1 /2 a s b e f o r ef o r h o l e s . S u b s t i t u t i n g f o r o r a n d c~0 w e o b t a i n

b 3 ( a 3 + 2 r 3 ) b 3 ( r 3 - a 3 )P i n 2 r 3 (b 3 _ a 3 ) + c l p i , r 3 b 3 _ a 3 ) - Co

(10)u s i n g f = a 3 / b 3 a n d s o l v i n g f o r P i , ( t h e h y d r o s t a t i cs t r e s s f o r f r a c t u r e i n i t i a ti o n ) w e f i n a l l y f i n d

2 r3 /a 3 1 - f ) c oP i, = l + 2 c l - 2 ( c t - 1 ) r 3 / a 3 (11)

T h e e q u a t i o n s h o w s t h a t f r a c t u r e i n i t i a t i o n i s e a s i e s ta t t h e s u r f a c e o f t h e h o l e ( r = a ), a n d b e c o m e s r a p i d l ym o r e d i f f i c u l t a s r i n c r e a s e s . T h e i n i t i a t i o n p r e s s u r e ,P i . , b e c o m e s i n f i n i t e a t r / a = [ (1 + 2 c l ) / 2 c l - 2) ] 1/3( -~ 1 .2 fo r c l -~ 3 .1 ) , be yo nd wh i ch po i n t no f r ac t u r ew o u l d b e e x p e c t e d t o i n i t i a t e .

I n o r d e r t o c o m p a r e t h e h y d r o s t a t i c c o m p r e s s i o nm o d e l i n t r o d u c e d a b o v e ( E q u a t i o n 1 0) w i t h t h e re s u l tso b t a i n e d i n t h i s w o r k t h e h y d r o s t a t i c f r a c t u r e i n i t i a -t i o n s t r e s s , p l , , h a s b e e n p l o t t e d v e r s u s ( 1 - f ) K ~ c inF i g . 13 . I t i s c l ea r t ha t t he r e su l t s can be desc r i bedq u i t e w e l l b y a l in e a r r e l a t i o n t h r o u g h t h e o r i g i n a ss u g g e s t e d b y th e m o d e l . F r o m t h is g r a p h t h e b e s t - fi tv a l u e o f r / a i s -~ 1 .1 wh i ch l ie s w i t h i n t he c r i t i ca l r / av a l u e p r e d i c te d b y t h e m o d e l a n d w i t h t h e v a l u e o f L 0f o u n d f o r t h e u n i a x i a l c o m p r e s s i o n c a s e a b o v e . T h u sf r a c t u r e i n i t i a t i o n i n t h i s m a t e r i a l o c c u r s v e r y c l o s e t ot h e p o r e s u r f a c e a s p r e d i c t e d b y t h e m o d e l .

F i n a l ly , t h e o v e r a l l b e h a v i o u r o f th e p l a s t e r o f P a r i si s s u m m a r i z e d b y t h e s u r f a c e s o f F i g . 1 4, w h i c h s h o wt h e c o m b i n a t i o n o f s t re s s e s re q u i r e d t o c a u s e c r a c k


7/7

100

- 2 0

9" - 4 0

- 6 0

T e n s i o n

Compress ionI I- 6 0

/ / o -~ 50oA/ - ~ 6 ~/

. . ) 10

l o~ 2o

u ~

4 o

- 6 0

- 8 0 2 - 8 0- 8 0 - 4 0 - 0 1 0 - 8 0 - 6 0( G ) A x i a l s t r e ss ( M P a ) ( b )

T e n s i o nI

/ / / , /~/ / . . . . . . > / // / ~ - ~ 5 0 o / o / /

// / J - I 7 0

Co m p r e ss i o nI i I i I- 4 0 - 2 0A x i a l s t r e s s ( M P a ) 0 10

Figure 14 ( a) F r a c t u r e i n i t i a t i on , a n d ( b ) f ina l c o l l a pse su r f a c e s f o r p l a s t e r o f P a r i s , f o r t h r e e r e l a t i ve de ns i t i e s .

in i t i a t ion , a t a ), an d f ina l fa i lu re , a t b ) , fo r th reere la t ive dens i t i e s .

5 C o n c l u s i o nP l a s t e r o f P a r i s c a n b e u s e d a s a m o d e l m a t e r i a l t os t u d y f r a c t u r e p h e n o m e n a i n b r it t le p o r o u s s o li d s. T od o t h is , th e p r o p e r t i e s o f t h e p l a s t e r a r e n e e d e d : t h e ya r e d e t e r m i n e d a n d l i st e d in t h is p a p e r . T h e m a t e r i a l i st h e n u s e d t o s t u d y s p a l l i n g a t h o l e s , i n m u l t i a x i a ls t r es s s ta t e s. T h e e a s e o f f a b r i c a t i o n a n d m a n i p u l a t i o nm a k e p l a s t e r a n a t t r a c t iv e m o d e l m a t e r i a l f o r s u chs tud ies .

A c k n o w l e d g e m e n t sW e t h a n k t h e t e c h n i c al s t a ff o f t h e M a t e r i al s G r o u p ,R i c h a r d B r a n d , B r i a n B u t l e r , A l a n H e a r e r , S i m o nM a r s h a l l a n d O s c a r S k u l sk y j , w h o p r o v i d e d i n v a l u -a b l e t e c h n i c a l s u p p o r t . T h e r e s e a r c h w a s m a d e p o s -s ib l e t h r o u g h t h e f i n a nc i a l s u p p o r t o f th e U n i t e dS t a t e s A i r F o r c e O f f ic e o f S c i e n ti f ic R e s e a r c h u n d e rg r a n t n u m b e r A F O S R - 8 7 -0 3 0 7 .

R e f e r e n c e s1 . A . A . G R I F F I T H S , i n P r o c e e d i n g s o f t h e 1s t I n t e r n a ti o n a l

C o n g r e s s o n A p p l i e d M e c h a n i c s , D e l f t ( D e l ft U n i v e r s i t yPress , 1924) p. 55.

2 . S . N E M A T - N A S S E R a n d H . H O R I I , J. Geophys. Res. 8(1982) 6805.

3 . D . G R I G G S a n d H . H A N D I N , (e ds ), R o c k D e f o r m a t i o n ,( G e o l o g i c a l S o c i e ty o f A m e r i c a , W a v e d P r e s s , B a l t im o r e ,1960) Ch. 13.

4. dem, Geol. Soc. Amer. Mem. 79 (1960) 347.5 . F . A . M c C L I N T O C K a n d J . B . W A L S H , i n P r o c e e d i n g s o f

t h e 4 t h U S C o n g r e s s o n A p p l i e d M e c h a n i c s ( A S T M , I n t e r -sc ience , 1962) p. 1015.

6 . S . A . F . M U R R E L L , i n R o c k M e c h a n i c s , e d i t e d b yC . F a i rh u r s t , P r o c e e d in g s o f th e 5 t h S y m p o s i u m o n R o c kM e c h a n i c s ( P e r g a m o n P r e s s , O x f o r d , 1 9 63 ) p . 56 3 .

7. Idem, Geophys. J. R. Astron. Soc. 10 (1965) 231.8 . S . A . F . M U R R E L L an d P. J . D I G B Y , ibid. 19 (1970) 309.9. Idem, ibid. 19 (1970) 499.

1 0. J . C . J A E G E R a n d N . G . W . C O O K , F u n d a m e n t a l s o f R o c kM e c h a n i c s ( M e t h u e n , L o n d o n , 19 6 9) .

I 1. S . K O B A Y A S H I , J. Soc. Mater. ScL Jpn 20 (1971) 164.1 2. J . B . N E W M A N , i n D e v e l o p m e n t s In C o n c r e t e T e c h n o -

l ogy- I , e d i t e d by F . D . L ydon ( A pp l i e d S c i e nc e , 1979) .13. Idem, Appl. Sci. 5 (1978) 151.1 4. M . F . A S H B Y a n d S . D . H A L L A M , Acta Metall. 34 (1986)

497.1 5. C . G . S A M M I S a n d M . F . A S H B Y , ibid. 34 (1986) 511.1 6. G . V E K I N I S , M . F . A S H B Y a n d P . W . R . B E A U M O N T ,

C a m b r i d g e U n i v e r s i t y E n g i n e e r i n g D e p a r t m e n t R e p o r t sC U E D / C - M A T S / T R 1 48 (1 98 9), C U E D / C - M A T S / T R 1 74( M a r c h 1990) ,1 7. A . D I N S D A L E , P o t t e r y S c i en c e : M a t e r i a l s , p r o c e s s e s a n dp r o d u c t s , ( H o r w o o d , L o n d o n , 1 9 8 6 ) .

1 8. F . S I N G E R a n d S . S . S I N G E R , I n d u s t r i a l C e r a m i c s ( C h a p -m a n a n d H a l l , L o n d o n , 1 9 6 2 ) .

1 9. J . E . S T R A W L E Y a n d B. G R O S S , i n C r a c k s a n d F ra c t u re ,A m e r i c a n S o c i e t y f o r T e s t i n g a n d M a t e r i a l s S p e c i a l T e c h n i c a lP ub l i c a t i on 601 ( A S T M , P h i l a de l ph i a , P A , 1976) pp . 559 79 .

2 0. M . L . K A C H A N O V , Mech. Mater. 1 (1982) 19.2 1. M . F . A S H B Y a n d C . G . S A M M I S , PAGEOPH 133 (1990)

481.2 2. R . J . R O A R K a n d W . C . Y O U N G , F o r m u l a s fo r s t re s s a n d

s t r a in , 5 t h E d n ( M c G r a w - H i l l , N e w Y o r k , 1 9 75 ).

Received 7 Januaryand accepted 4 February 992

3 7

plaster of paris as a model material for brittle porous solids

Documents