Stress intensity factor for ceramics toughened by microcracking caused by dilatant second phase particles

STRESS INTENSITY FACTOR FOR CERAMICS TOUGHENED BY MICROCRACKING CAUSED BY DILATANT SECOND PHASE PARTICLES KAI D&AN, BRIAN COTI-ERELL and YIU-WISG MAI Centrc Iw Advmctxi K~tcrials T~hnology. Department of Mechanical Enginwring. Unitcrsity of Sydney. Sydney. NW 2006. Austrrllia

Abstract-Touphening of many ceramicscan be accomplishedby creating dilaiaticw in the seccmci phase partida that C‘;LUKthe mutrit to crack. In this paper the stress intensity factors for annular cracks uhoul Jilatunt particles in a matrix under a normal stress are calculatd.

I. lNTRODUCTlON

can be toughened by second phase particles that produce a residual stress system during cooling ;IS iI result ofditl&nccs in the coefficient of thermal expansion (Evuns

Ceramics either

and Cannon. 1986; Porter c*f crl., 1979; Gupta TV al.. 197X; Riihit (*I trf.. 1986, 19x7; Davirig,c. I974 ; Davidgc and Green. 1968 ; Lange. 1974 ; Mujata t*t rd.. 19x3 ; Mcchoisky. i0~y.1) or rfuc to ;I stress-induced phase ~r~lnsf~?rr~la~i~)n.In both cast’s the rcsiduai stresses may tcatt to rl~i~r~~~r;l~killg ticpcnding upon the partictc size (Clausscn. trl.. 1977; Riitiiccr crl., 1986, 1957; IIavidgcnnd Mujata

(*I uf.. IO82 ; Mc~~holsky,

if Ihc suuontl

tlcllcct Ihc frac~urc Daviitgc

Grew.

illltt

pilth antI cause

shrinks

IMX). Circumfcrcn~iai higgcr

than

2nd matrix

if the partictcs

196X). Thcsc circumfercntiat

the ceramic

away from the matrix

loughcning

and Green. importance

1976; Cktusson c*t 1974; Lange, 1074;

19X3).

phusc parliutc

iIK

Green, 1968; Davidge,

( BWIIS

microcracks il

tcnsitc. radial slrcsscs can

;III~ C’ittItlott,

1986

; Davittgc.

1973;

will occur bctwccn the pi1rtiCtCS

ccrlain critical size (Davidgc.

1074: Davitlgc

microcracks do not grcatty al1’ect the slrcngth of

provicicrl the particics arc not too large. This method of toughening in many ceramics ot’commcrciat

is ot

significance such as eicctricat porcelain con-

taining quartz fitter particles. However,

this paper is aimed

;II ceramics

where second phase p;lrti&s

reialive to the matrix and cause radial microcracks 01.. 1979; Gupra PI ~1.. 1978; Claussen, 1987; Mujata

rr ui., 1983; Mecholsky,

(Evans and Cannon,

increase in sj~e 1986; Porter er

1976; Ctaussen r~ ttl.. 1977; Riihie el al., 1986. 1983). Providing

that they readily coalesce, the dilatation

the microcracks are not so large

caused by them can produce a significant crack

growth resistance (Evans and Faber, 1984). A secondary much smaller

increase

in toughness

results from the decrease in elastic modulus in the fracture process zone due to the microcracks (Evans and Faber,

1981, 1984). The relative increase in the size of the particles

can result from differences in the coe%cient Mecholsky. (Claussen.

of thermat expansion

1953) or from phase transformation 1976; Ctaussen L*F cd., 1977;

due to the volume expansion microcracks.

Riihte CF cd., 1986,

is an important

(Mujata

ef ~1.. 1983:

as in the zirconia-toughened

atuminas

1957). The residual stress

factor which affects the formations

In fhc former ceramics. stress-induced

microcracking

size is less than a critical value, or existing microcracks

propagate

of the

occurs if rhe particle if the residual stresses

alone are sufficient to Cause microcracking (Mujata c*Fal., 1983). With zirconia-toughened alumina, radial microcracking does not usually accompany the stress-induced transformation--a given particle cithcr transforms under the stress field near the tip of a crack or if already transformed causes microcracking undor the combined action of the residual and appticd stresses (Riihte cl cd., 1956). Existing calculations of the stress intensity factors at the tips of radial cracks emanating from dilatation

particles (Riihie <‘I nt.. I986 ; Krstic and Vtajic, 1953 ; Krstic. 1954) assume

that Sncddon’s classic solution )Iowetcr.

(Sneddon.

esccpt under very high q-Aicrl

1046) for the penny-shaped crack can be ustd. stress, the crack will not propag:ltc far into the

rosidtui comprcssivc stress rcpime of tho second not

prohlcm of 311annular crack surrounding six

PililSC

partick. A penny-sh;tpcd

accurately model the hchaviour of the actual itnnl~l:tr crack.

method (COokC, 1963 ; ‘I‘Sili. iW-1; Scivxltir~~i stress rcginic.

iii sifiw

C;ISCS

c~mpic

of thih type

trxk

tiocs

using the triple integral equation

and Singh,

iOS4. iW5.

19X7; Scivxiurai.

crack will prop;lgatc iuto the rCSitllliIi comprcssivo

whcrc fiie scconct phase m;ileri:rl

very \vtA hortdctl the attniii;tr

UiKk

this paper wc solve the

3 Second phaSC particlc which untkrgocs ;i WliltiVC

incrcxsc riuc to thcrm;li cxp;insion or triltlsfitrf~latiotl

1085). At high applied btrcss lhc anniii:ir

in

will

piX)pilg;ttt2

is similar

to the matri?t aid

into the second phase particks.

An

of crackit~g

is SI~OWII by Mujat;l c’f (11.(1983). I I owcvcr, in other casts whcrc the particlc is not so well hontlcd, any prop:gation into (tic oomprcssive region will take pixc by the crack rttnIli!i~

along the p~~rti~l~~r~l~ltrix intcrfncr: (Iliihie

<*Ird., 19x7). The

present anitlysis only ctculs with the former type of crack growth whcrc the annular crack may pcnctratc into the pilrtiClC.

’ ‘TIIE ANNlJLAK _,

CRACK

I’Koftt.E?v~

A system consisting of ;I sphcricnl particic cmbcddcd in itn inlinirc having ;I surrounding

brittle matrix and

annular crack is considered (Fig. I ). The USC whcrc the crack oxtencls

into the particle is also considcrcd (Fig. 7). It is assumcd that the chtstic constants for the particle and the matrix arc itlcntic;ll so that the principle Thcrc

arc ~H’O ioatl systems:

ofsuperposition

can bc appiid.

(a) the residual strcssss due to the mismatch bctwccn the

particlc and matrix ;md (b) ;i uniform tcnsilc stress 6. The pressure f’ hctwccn the partictc and the matrix is given by

p

=

3,.’ E ._.,- :... -. _

3(l -V)

(1)

whcrc for phase transformation 1:’ is the stress-free strain and for thcrm;ll expansion E is the Young’s modulus: I’ is Poisson’s ratio: x is the mismatches 6’ = (1,” - r,)AT; cocflicicnt of thcrmai cxpitilSiOll ; and the subscripts III itnd p rcfcr to the matrix and particlc. In the itbstncc of any crack the residual stress field on the plant z = 0 is given by al(r.O) and

= -P

for r < K,

(2)

233

Stress intensity factor for ceramics

Fig. 1. “particle

penetrating”

annular crack.

for r > R.

(3)

whcrc R is the radius of the particle. In the prcsencc of an annular crack this residual stress hold is superimposed by the stress field a:(r,O)

for C, c r < R,

= P

P

o,(r.O) = -

I.

R’

0 r

(4)

for c,, > r > H.

where (; and C, arc the radii of the inner and outer edges of the annular crack, them are also the added conditions

that on the plane : = 0 the displacement

II, is zero outside the

crack and the shear stress is zero. The stress field due to an applied uniform stress u superimposed

on the crack system

is -u

CT,=

for c, < r -c c,,

(6)

with the conditions that in the plane c = 0, ur is zero outside the crack and the shear stress is zero. The solution to this problem of the annular crack under uniform stress has already been given by Selvadurai and Singh (1985), but only for q/c, < 0.7. The stress intensity factors K, and K, at the inner and outer edges of the annular crack are given by

K, = lim a,(r,O)J2n(r-c,).

(8)

r-r,:

3. THE

SOLUTION

OF

THE

ANNULAR

CRACK

PROBLEM

Hankel transforms can be used in axisymmetric problems to reduce the two independent variables (r. z) to a single variable z (Harding and Sneddon. 1945: Sneddon 1946.

K. DL.AS cr al.

234

1951 ; Sneddon and Lowengrub, 1969). The biharmonic equation for the stress function Q1 then becomes

where

Jo(+) is a Bessel function of zero order and < is a parameter. iongitudinal stress and displacement can be written as

On the plane : = 0. the

Inscrting the boundary conditions given in Section 2 into eqns (I I) and (12). WCobtain the following triplo intcgfill cquolions,

sOL

.f(v)J&w)dq = 0 (1 < P

<

(16)

a),

where

g(P)=(~+v)(I-~v)c~d(p E t

)i

0)

r
1*

(17)

and a:(p,O) is given by eqns (4) and (5) for the residual stress and eqn (6) for the applied

stress and where IX= c,/c,,. Let g,(p) = g(P)

(0 < P (: a).

g?(P) =.9(P)

(1 < P c a).

(18)

and

Then. we have.

(1%

235

Stress intensity factor for ceramics

(‘0)

and

K, = lim ‘$ y-1+

Jm,

(21)

where C=

(l+v)(l-2v)c,J

E It is seen that one only needs to find g,(p)

(23

’

and g?(p) in order to determine the stress

intensity factors. We make a note that

(0 s s < a).

(23)

and

’ ~t~z(u)du

G,(s) =

($2 _ l,?) ‘12

Then the triple integral cqns (14-16)

(I < s c cc).

can bc simplified

as a pair of simultaneous

equations forg,(u) and gr(u) that is written as (Cooke. 1963; Tsai. Singh. 1984. 198.5, 1987; Sclvadurai. 198s).

i’[G~(.~)+I’y(u)du-l-r--= 2 (qz_U?)li?

(sz

Let G,,(s) and G?,(S), GIz(.r) and G*&) tions :

Then

$)I/’

(24)

-l$‘$

satisfy the following

integral

1984; Sclvadurai and

(I
simultaneous

(26)

integral equa-

K. DCAN rr al.

236 G

(s)

=

I

G:(S)

=

2v,c:

! I + v)( ’ E

--[G,,(.s)+G,:(.s)]

(3’)

(0
--E-‘--(‘+P)(‘--2\)(~‘:[G:,(.s)+G?~(.s)] (I < .s< xl.

gives the solution of integral eqns (25) and (76). Function y(rt) is written as

c;,g

g(u) = -

(/I <

If

<

(33)

I).

for the case when the crack does not penetrate into the particle and

2-P

$,T(ll)=

[I)

14 <

y,$

-

i

(x <

(/I <

II

<

I)

(31)

’

if the crack penetrates into the particle, where /I = R/c,,. In order to get the approximate

solutions

of eqns (27 -JO), WC WC a perturbation

method and express the solutions in scrics form.

G,,(s) =

(I

r”A

i n-

(;:l(.s) =

c

r7

n-

G,:(s)

,”

0

r”/l,,,(.s)

(35)

(0 < .s < x),

.y

x

I

(I < .s < ^I,),

(36)

I

=

(0 < s < cl),

G?>(.Y) = y

f

(37)

(I < .v < m).

#fIJ,(.S)

(38)

- ,,.- I From Abel’s integral equation

(Cooke.

2c Y,(P)

= -

d

np dp

1963; Sclvadurai

“.s[G,,(.s)+G~~(.s)J (I’!

_s?)l

2

and Singh, 1985). WC have,

-ds

(I
The stress intensity factors for the residual stress system K’(‘. Kl and those for the applied stress KY, Kz can bc obtained

from eqns (20-21)

as a scrics.

4,TfIE STRESS INTENSITY FACTORS FOR ANNULAR DILATATION PARTICLES The non-dimensional crack (P

under

the

CRACKS

SURROUNDING

stress intensity factors at the inner and outer edges of an annular

influence

of

the

residual

stresses around

a mismatched

particle

= h+/P J TIC --;, are g’lven in Table I and are shown in Figs 3 and 4. To achicvc sufficient

Stress mtensity factor for ceramics

: P!

?

7

r01003030-

‘?

9

-.

19 ?

?

(41)

Ew

I

i

: I

I

: 8

:

:

,

:

I,

: I.

0.2

0.3

;

:

I

:

0.1

0

;

: ,

:

li)r

tlic stress

intctisily

0.4

up to ;I hudrcd 0.2S%).

l‘licrc

:

solutiori

icIcnlic;iI to ;i two-dimciisioti~il ~IIC limiting

I’xtor

,

;

: ,

0.8

0.7

0.9

*

i

,

0.9

1 .o

Q

liar x < 0.6 it 15 ~~iiiy ncccssary

for cc1115 (35 3s). I lon0w

terms to cnsurc an xcuratc is ;i limiting

;

:

,

I.

0.5

t‘wlors

terms in tiic scrics c~p;iilsioiis

I

: ,

,

;

4’

:

I

Ratio.

accuracy

;

:

I

lo rctaln aboiit

result (I’or ^I < 0.95 tlic acctiracy is bcttcr tiun

liar 2 close to unity since ilr this c;isc tlic problcni is

cr;icli 01’lcrigtli (l,,, .- (‘,I ulidcr ;I slalc 0I’ plant

condition a11 hc ohtainctl by intcgrxtion

I;)r ;I two-dinicnsioiial

~t’thc txprcssim

crack with point loads on tlic crack fxcs

slrxin.

(Ibris

and Sill,

(42)

_.-;<_“‘‘6s.-‘d? ;I::-g-_

_.=

”

““Q?

”

”

”

”

*’

”

.’

,“’

”

”

”

0’

‘0

.“Q. k4

“,”

“‘BO

”

I’

’

”

”

“’

“’

”

I

0

0.1

”

0.2

”

” I

0.3

0.4

I

I

I

1

I

0.5

0.6

0.7

0.9

0.9

Ratio.a

ticncc,

fiw tlw stress intensity

z’_zz;_;LEqn ,a-’

lice

. ;IS x -+ I it is ncccswry to take

1

1.0

1905)

Stress density

factor for ceramics

and is given by

The non-dimensional

forms of these limiting solutions are shown in Figs 3 and 4. The stress

intensity factors are given by the empirical expressions k,P = 0.334fl” T?“(I -p,o

J’9,

(43)

k,P = 0.33582 ?5( 1 _ppJy which are accurate to 0.25%

(4)

over the entire range p = O-1.

The stress intensity factors for an annular crack under a uniform already been given by Selvadurai

tensile stress have

and Singh (1985) for u up to 0.7. In their solution they

take only five terms in the expression for eqns (42-45).

We have extended the range up to

r = I which again requires up to a hundred terms-.-in eqns (42-45). The results for the nondimensional stress intensity factors (A” = K/~,,/IKc~) are given in Table 2 and Fig. 5. Once again a limiting solution can be obtained for z close to unity and is given by k:

= kn = (I

-r)“‘/J2.

(45)

Thcsc stress intensity factors arc given by the empirical cxprcssions

(46)

A:,= which again are accurate to 0.25%

0.644( I --r)”

4Sh,

(47)

over ths entire range CL= O-l.

Figure 5 also shows the non-dimensional

stress intensity factor obtained

by super-

position for a dilatant particle with uniform stress applied. The effect of a crack penetrating the dilatant particle is shown in Fig. 6 for P/o = 2.

T;rblc 2. The non-dimensional stress intonsicy factors at Ihc inner (k:) and ouw edge (k:) of the crack 1

I.0 0.833

0.714 0.625 0.556 0.500 0.333 0.250 0.200 0.100 0.050 0.020

k: 0.0395 0.2963 0.3984 0.4685 0.5231 0.5686 0.7324 0.8510 0.9499 1.3230 I.8455 2.8881

0.0388 0.2x30 0.3657 0.4149 0.4484 0.4729 OS369 0.5648 OS804 0.6097 0.6235 0.6314

K. DCA~ et ul.

%

0

0.1

0.2

0.3

5.

The stress intensity obtained by use of Hankcl

fxtors

0.4

0.6 0.6 Ratio, a

0.7

0.8

0.9

1.0

CONCLUSIONS

for annular

cracks around

dilatant

particles have been

transforms after the method of Sclvadurai itnd Singh (1985) for

the complete range of inner to outer radii. These stress intensity fixtors ;1rc accurate to 0.25%. Previous calculations of the stress intensity (Riihle t’t 01.. 1957 ; Krstic CI rrl.. \9Y3 ; Krstic. 1953) made using Sncddon’s (1946) classic solution for it penny-shaped crack xx only ~ppro~irn~~tcly correct if the annular crack is very large compared with the difarant particle---for small annular cracks the stress intensity fxtors are grossly overestimated.

Stress intensity

factor for ceramics

241

The stress intensity factor at the inner edge of an annular crack formed outside a dilatant particle is always greater than that at the outer edge. Thus there is a strong tendency for a crack to penetrate the dilatant particle. if the particle is well bonded to the matrix. However, if there is no applied stress, the stress intensity factor decreases rapidly as the crack penetrates the compressive stress zone in the dilatant particle (Fig. 3). If the compressive stress due to the dilatant particle is greater than the applied stress, initial crack propagation into the dilatant particle is always stable. Crack propagation into the particle becomes unstable only when the penetration is large.

.-I~,knr,~~k~,tfy~~Jt,nrs-TThe authors wish to thank the Australian Research Council for the support of this work whtch is part of a larger project on “Structural Reliability of Tough Zirconia Ceramics”. One of us (D.K.) is supported by the CSIRO, Sydney L’niversity Research Scholarship.

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