STRESS INTENSITY FACTOR FOR CERAMICS TOUGHENED BY MICROCRACKING CAUSED BY DILATANT SECOND PHASE PARTICLES KAI D&AN, BRIAN COTIERELL and YIUWISG MAI Centrc Iw Advmctxi K~tcrials T~hnology. Department of Mechanical Enginwring. Unitcrsity of Sydney. Sydney. NW 2006. Austrrllia
AbstractTouphening 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 stressinduced 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 zirconiatoughened
atuminas
1957). The residual stress
factor which affects the formations
In fhc former ceramics. stressinduced
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 zirconiatoughened alumina, radial microcracking does not usually accompany the stressinduced transformationa 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 qAicrl
1046) for the pennyshaped 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 pennysh;tpcd
accurately model the hchaviour of the actual itnnl~l:tr crack.
method (COokC, 1963 ; ‘I‘Sili. iW1; 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 stressfree 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(rc,).
(8)
rr,:
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 ‘$ y1+
Jm,
(21)
where C=
(l+v)(l2v)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 (1416)
(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)dulr= 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
2P
$,T(ll)=
[I)
14 <
y,$

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 (2021)
as a scrics.
4,TfIE STRESS INTENSITY FACTORS FOR ANNULAR DILATATION PARTICLES The nondimensional 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 twodimciisioti~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 twodinicnsioiial
~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 nondimensional
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 = O1.
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 (4245).
We have extended the range up to
r = I which again requires up to a hundred terms.in eqns (4245). 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= Ol.
Figure 5 also shows the nondimensional
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 nondimensional 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 pennyshaped crack xx only ~ppro~irn~~tcly correct if the annular crack is very large compared with the difarant particlefor 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,nrsTThe 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.
REFERENCES CI,~u. A. (;. .~ntl I:;lhcr. K. T. (IYXJ). Cr;lckgrowth resistance of microcr:lcking hrittlc matcri:ds. 1. rfnr. C’c*n,m. .%I,. 67. 255 200. (iupt.~. ‘I’. K . I.;~npc. I:. I:. ;~ntl Itcchtoltl. J. I I. (lY7H). LQTcct of stressinduced phase tr:insfortn;ltions on the propcrtic* ol’l~olycry~t;~llu~c lirconia containing mctastablc tcrragonal phase. J. illtrrt,r. Sci. 13. I464 1470. I I.lrthnp. J. W. and Swtldon. I. N. ( IYJS). ‘I’hc elastic stresses produced by the indentations of the plane surface 111.1scnuintinitc cktstic solid by ;I rigid punch. I’roc. C’trrrrh. Phil. Sot. 41. I6 26. Kr\tlc. V. I). (IYXJ). Fracture ol’ brlttlr solids in the prescncc of thcrmoelastic stresses. J. Am. Ci,ru~~. Sot. 67, 5x0 5u.I. Kr\tIc. V. I). and Vlajic. M. D. (IYS3). Conditions for spontaneous cracking ofa brittle matrix due to the presence of thcrmocl;tstlc stressss. :lcro ,\/zrlrl/. 31. I3Y 1U. I.;lnyc. I:. 1:. ( lY7l). Criteria Ibr crack cxtsnsion and arrest in residual. localircd rtrcss fields assosiatcd with \econd ph;lss particles. In Frtrcrrrre Mrthtrnics oJ’ C’crcrtnics 2 (Edited by R. C. Bradt. D. P. II. llasssclman and I’. f:. Lange). pp. 55Y 6OY. Plenum Press. New York. hlccholshy, 1. J. (19X3).Toughening in glursceramics through microstructural design. In Frucvttrc Mrchtmicv of (‘mrnrics h (Edited by R. C. Bradt. A. G. Evans. D. P. H. Classclman and F. F. Lange). pp. 1651X0. Plenum I’re>s. Nrw York. X!uJata. N., L;lmg;lwa. K. ;md Ginno. Il. (IYX3). Fracture behavior of glass matrix/glass particle composites. In Fru<.rtrn* .If~clr~~rtic.~d’Ccrwttic.r 5 (Editrd hy R. G. BradI. A. G. Evans, D. P. tl. Ilassclman and 1:. F. Lange). pp. MIY 634. Plenum Press. New York. I’:tria, P. C. and Sih. A. G. (1965). Stress analysis of cracks. In Frrlcrurr Routjme.w Ttwiry trd Applicurions, .4Sl’.U .‘XP 3X1. 30 82. Philadslphm. Porter. I). L.. Evans. A. G. and f lcucr. Il. fl. (1979). Transformations toughening in partially stabilized zirconia (I’S%). :l~./rr. .I/<*rrr//. 27. l64Y 1654. Riihlc. M.. Claus.scn. N. and Ilcusr. A. II. (19x6). Transformations and microcracking toughening as complementary process in Z,O,toughrncd AIzO,. J. Anr. Ccsrtrnr. .%c. 69, I95 197. Riihlc. M.. Evans. A. Ci,. Mcblceking. R. M.. Charalamidss. I’. G. and flutchinson. .I. W. (IYX7). Microcrack toughening in alumim~irconia. .~lrra. .\/rrcr/l. 35. 2701 2710. Scl\adurai. A. P. S. (1985). On integral equations governing an internally indcntcd Fnnyshaped crack. ,%f&. Rcr. C‘onr. 12.34735 I. Sclvadumi. A. P. S. and Singh. 0. M. (19X4). On the expansion of a pennyshaped crack by a rigid circular disc inclusion. Irfr. /. Fnrr. 25. 59 77. Scl\adurai. A. P. S. and Singh. 0. M. (1985). The annular crack prohlcm for an isotropic elastic solid. @uurt. J. .Ifcch. ..lppl. Mtrrh. 38. 233 243. Scivadurat. A. P. S. and Singh. f% hl. (19x7). Axisymmctric problems for an externally cracked elastic solid. 1. Effect of a pennyshaped crack. Inr. /. 01qf1.q Sri. 25. 10491057. Sneddon. I. N. (IYJ6). The distribution of stress in the ncighbourhood of a crack in an elastic solid, Proc. Roy. so<.. ( Lf~nrlw~ A 1x7. 2’9 3io. Sncddon. I. k. (I YSI 1. Fvrrricr Trcmrfivms. McGrawHill. New York.
Sncddon. 1. S. and Lowengrub. 51. ( IVhV).Crud Prr~idmw m rhr C‘luwcul Sew York. Timoshcnko. S. and Good~cr. J. N. ( IV%). Thwrv ~~/EILLsIK~[L hlcCruuHill.
Throrv
ot Elusrrc~f_v. John W&y.
Sew York.
Tsar. Y. M. (I9Nt).Indentation of a Penn!shaped crack by nn ablate spheroldril rigid inclusion in a transversely isotropic medium. /. .4ppl.,Wech. Truns. (.4SJfE) 51. 8 I IX 15.