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.
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
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
pilth antI cause
IMX). Circumfcrcn~iai higgcr
if the partictcs
196X). Thcsc circumfercntiat
away from the matrix
and Green. importance
1976; Cktusson c*t 1974; Lange, 1074;
Green, 1968; Davidge,
tcnsitc. radial slrcsscs can
will occur bctwccn the pi1rtiCtCS
ccrlain critical size (Davidgc.
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
significance such as eicctricat porcelain con-
taining quartz fitter particles. However,
this paper is aimed
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
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
ef ~1.. 1983:
as in the zirconia-toughened
1957). The residual stress
factor which affects the formations
In fhc former ceramics. stress-induced
size is less than a critical value, or existing microcracks
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.
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
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.
of thih type
using the triple integral equation
crack will prop;lgatc iuto the rCSitllliIi comprcssivo
whcrc fiie scconct phase m;ileri:rl
very \vtA hortdctl the attniii;tr
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
to the matri?t aid
into the second phase particks.
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 _,
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:
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
3,.’ E ._.,- :... -. _
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
for r < K,
Stress intensity factor for ceramics
Fig. 1. “particle
for r > R.
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,
o,(r.O) = -
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
for c, < r -c c,,
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,).
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.
1951 ; Sneddon and Lowengrub, 1969). The biharmonic equation for the stress function Q1 then becomes
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,
.f(v)J&w)dq = 0 (1 < P
g(P)=(~+v)(I-~v)c~d(p E t
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).
(1 < P c a).
Then. we have.
Stress intensity factor for ceramics
K, = lim ‘$ y-1+
E It is seen that one only needs to find g,(p)
and g?(p) in order to determine the stress
intensity factors. We make a note that
(0 s s < a).
($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?
Let G,,(s) and G?,(S), GIz(.r) and G*&) tions :
satisfy the following
1984; Sclvadurai and
K. DCAN rr al.
! I + v)( ’ E
--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
g(u) = -
for the case when the crack does not penetrate into the particle and
if the crack penetrates into the particle, where /I = R/c,,. In order to get the approximate
of eqns (27 -JO), WC WC a perturbation
method and express the solutions in scrics form.
(0 < .s < x),
(I < .s < ^I,),
(0 < s < cl),
G?>(.Y) = y
(I < .v < m).
- ,,.- I From Abel’s integral equation
and Singh, 1985). WC have,
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
stress intensity factors at the inner and outer edges of an annular
= 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
up to ;I hudrcd 0.2S%).
icIcnlic;iI to ;i two-dimciisioti~il ~IIC limiting
liar x < 0.6 it 15 ~~iiiy ncccssary
for cc1115 (35 3s). I lon0w
terms to cnsurc an xcuratc is ;i limiting
terms in tiic scrics c~p;iilsioiis
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
crack with point loads on tlic crack fxcs
fiw tlw stress intensity
. ;IS x -+ I it is ncccswry to take
factor for ceramics
and is given by
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
k,P = 0.33582 ?5( 1 _ppJy which are accurate to 0.25%
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
Thcsc stress intensity factors arc given by the empirical cxprcssions
A:,= which again are accurate to 0.25%
0.644( I --r)”
over ths entire range CL= O-l.
Figure 5 also shows the non-dimensional
stress intensity factor obtained
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
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.
The stress intensity obtained by use of Hankcl
0.6 0.6 Ratio, a
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.
factor for ceramics
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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