CRACKGROWTH RESISTANCE IN TRANSFORMATIONTOUGHENED CERAMICS DAVID
M. STUMP
and BERNARD BUDIANSKY
Dibislon of Applied Sciences. Harvard
University.
Cambridge.
MA 02138. U.S.A.
AbstractCrackgrowth rcsistcrnce in transformationtoughened ceramics is studled by modehng the reglon surroundmg cln advancing crack tip 9s a zone which has undergone a uniform dilatational ph;w transformation. This zone is allowed to evolve around the advancing crack tip under conditions of incrcaslng farticld loud whtlc the tip is maintained at I critical stress intensity necessary for fracture. This procedure loads to the surprising conclusion that maximum toughening occurs for tinite amounts ofcrxk ndvancc.
INTRODUCTION
The discovery of enhanced fracture toughness in zirconiaenriched
ceramics has Icd to a
Hurry of cspcrimcntal and theoretical analyses. It has been well established that the high strcssos near ;I crack tip can cause small
zirconia particles. typically I /ml or less in diamotcr,
to undergo a phase transl,nllation
from ;I tctragonal to a monoclinic crystal structure. The
unconstrained
bc docomposcd into a 4% dilatation
strain.
transformation
tlolvcvcr,
can
due to the elastic constraint
composite gcncrally transform transformation
of the matrix
and a l6”/; shear
phase. the particles in the
with twin bands of altcrnatc character resulting in an overall
strain considerably less than 16%. Conscqucntly,
the strain transformation
of particles embcddcd in the composite is usually assumed to bc dilatant. The transformed
region has the rcmarkablc property of allowing stable crack growth
to occur in the composite material,
whcrcas the unreinforced
ceramic exhibits
behavior. Stable crack growth on the order of scvcral millimeters Hcurves (i.c. stress intensity
no such
has been observed. and
versus crack extension) for a variety of zirconiareinforced
composites have been mcasurcd. Theoretical
studies of transformation
toughening including those of McMceking and
Evans (15X32). Iludiansky el rrl. (1983). Amazigo and Budiansky Lambropoulos
planar crack is surroundctl Using ;I primitive transformation
(19%).
Rose (1986a) and
(1986) have consitlcretl only steadystate toughening. wherein a semiinfinite by ;I semiintinitr
zone of dilatation,
as shown in Fig.
l(a).
analysis, Rose (1986b) also examined the e!Tcct of allowing an initial
Lone to grow with an advancing crack tip. The objective of this study is to
present a complctc analysis of the growing crack by solving for the initial transformation zone and studying
its growth
Model loading. This
around a crack tip advancing under increasing. far field,
analysis will rcvcal the surprising
result that maximum toughening
occurs for a finite amount of crack advance.
INITIAL ZONES
The modcling approach adopted in this study, consistent with previous work, is to assume that a transformation
zone surrounding the initial crack tip has undcrgonc an irrcvcrsiblc transformation dilatation of strength ctl’p. as shown in Fig. I(b), whcrc c is the zirconia particle volume fraction and (I,!’ is the unconstrained particle dilatation. The crack will be taken to be semiinfinite and planar. The transformed and untransformed will bc assumed to have the same elastic moduli. and to be under plane strain.
regions
Since typical zone sizes arc of the order of 20 /lrn or less. small compared to specimen dimensions. the small scale zonesize approximation will be invoked. The strcsscs at distances far from the zone, but small compared to the size of the body, will be assumed to be dominated by the classic Modo 1 Kfield. For r 4 ‘so. the stresses Q,~ are given by
D. M. STUMPand B BCDIANSKY
636
(b)
where the &($) ttre well known trigonometric functions and K is the applied f;trfield stress intensity. Similtlrly, in the vicinity of the crrtck tip, r , 0, the stresses are given by
where K,,, is the stress intensity at the cmck tip. The full transformation will be assumed to occur when the mean stress, am = a,,/3, attains ;L critical value a:,. This corresponds to the “supercritical” transformation ctlse considered by Budiansky er al. (1983). The value of ai, ;1sdiscussed by Evans and Cannon (1984). wiii dcpcnd on the stress and tcmperaturc history of the composite. Other transform~ltion criteriil h;tvr rtlso bocn proposed (L~~mbropou~os, 1986). As shown by Bud&sky er (II., the applic;ltion of the Jintegral to the sttltionary crack results in the conclusion that K,,, = K. Accordingly, if crack growth occurs at a critical intensity, Kc,*= K,,. where A:, is independent of the particle concentration, the initial transformation region induces no toughening. This neglects the possible effects of other toughening mechanisms such as microcracking. To determine the boundary of the initial transformation zone (Fig. 1(b)), it is necessary to insist that the mean stress a,,, attain the value 02 as the boundary is approached from the cxtcrior of the zone. By the superposition of stresses. the mean stress exterior to the zone boundary is the sum of contributions from the applied farfield and from dilatations in the transformed region. The mean stress due to the far&id stresses of eqn (1) is
Crackgrowth
resistance in transformationtou~hrnrd
Fig. 2. Symmetricnlly
WI+\‘)
D =3 The mean stress contribution
631
located spots of dilatation
?rr ’
0
? cos
5
m
crrrtmin
(4+)
i.
(3)
of the transformed zone can be found by first considering
that due to the two small circular spots of dilatation
shown in Fig. 2. For spots of area
drl,,. strength cdl;. and located at z,, = _r,,+i_~*~and Z,, = .I,) i,.(), the mean stress as found by Hutchinson
(1974) is
(4)
l‘hc
spots
at I(, ;111rl f,,
induct ;I change in tip stress intensity
mu;
Ah,,,, = 6J(2,)(
Rewriting
cqn (5) in terms
ofpolar
Kc
I r)
[z,,’ 2+ :,, ’ ‘1 11.4 ,,.
(5)
coordimltcs r and (/J.yields the rqui\alent
form,
(6)
Transformations
in the region (/I < n/3 increase A’,,,,, while those in the region 4 > n/3
dccrtxse A&. The effect of the entire zone on the mean stress can be calculated by integrating eqn (4) over the upper half A of the transformed
zone (Fig.
l(b)).
The equation for the zone
boundary is then obtained by adding the far field mean stress to the zone contribution equating the sum to a:;,. Thus,
and
the cquution governing points z = R(c,f~)e” on the boundary
is
(7)
whcrc F( z.:“)
is given by cqn (4).
Carrying
out the integration
with respect to dx, and setting
following condition governing R(g) when crack growth is imminent:
’ =(4 L
I ’
(0s
R(4)
where
($$,‘Z) _ ;
A’ = A’, leads to the
D. M. STMP
638
and B. BCDUMCI
with
: = K(fl,) lcb. 1 lcrc the p~~ranictcr (II, dclincd
is ;I nondimensional length
mtxsurt’
Iqfp’)
c”‘)
by
of the strength
of the transformation,
stress
The
that induced
intensity
intercept
for K,,r.
by the prcscncc
Integrating
with
at J/I = 0 oT the a:;, boundary
at the tip is equal
half of the zone and adding
to the sum
or the transformation
the applied
farfield
rcspcct to d.~,, results
zone.
for
the initial
crack. = K/K,,,
K,,, =
OJ =
Integrating results
0.
stress
intensity
eqn (5) over
and upper
in the equation
in the equation z
dY,,(~P) L .
cos (fP/‘,
(I 3) with &_/A:,,
for
of the applied
intensity
I
But.
and the characteristic
f. is
that L is the frontal
Note
z,, =
= A’. and for crack
growth
to occur
I scrvcs as ;I cheek on the solution
of(8)
(13) K,,,, = K,,,.
Hcncc,
for the initial
eqn
boundary
curve R(~/I). The results
procedure for
various
used to solve values
on the relative
of
incrcascd
CJ
distances
have btcn normalized
(IJ
eqn (8)
ranging
shapes
for from
of the zones
by the Frontal
R(rp)/L
is described
0 to 30 arc shown can be seen in Fig. intercept
R(0).
in Appendix in Fig.
3. The
4, whcrc
A and the effect of
for each
w
all
Crackgrowth
resistance in transformationtoughened
Fig. 4. Inltisl transformed
zones normalircd
639
ceramics
by frontal intercept length.
1.5
H/L 1.0
Figure
5 shows
thr
initial
zoneheights
of Fig.
3, together with the zoneheights
corresponding to steadystate crack growth calculated by Amazigo and Budiansky For o z 30, the steady state zone height becomes infinite. Up”
ix. infinite tougheningdiscovcrcd
GROWING
corresponding
(1988).
to the “lock
by Rose (19Y6b).
CRACKS
Once the initial zone has been found, it is possible to contemplate its growth around an advancing crack tip. As a growing crack moves into the body, material in the vicinity of the tip attains the critical moan stress ilnd transforms, the transformation
while duo to the irreversibility
iI wake region is Icft behind. Along ;L frontal portion
of
of
the transformed
zoneboundary
the mean stress criterion is satisfied, while on the wake portion of the in bound;lry the mean stress wiil have dropped below a:,. According to cqn (6) miltcriill the transformed region which lies to the left of the radial lint running through the tip at the angle n/3 rcduccs the stress intensity at the tip. To continue driving the crack forward. the applied stress intensity must be incremented. Consequently to solve the growing crack
problem. both the stress intensity
and zone shape must be found as functions
of crack
extension. The upper half of the instantnncous zone around a growing crack is shown in Fig. 6(a). The boundary is modclcd by three segments ; active. passive. and residual. The active
D. Xi.
STLMP
and
B. BL’DMNSK\
Y
scgmcnt AI% is the portion of the hountl;~ry whcrc the mc;~n stress has just rcxhcd man stress on the rcniaindcr ol‘thc hound;~ry has droppcd Mow (‘I)
is the p
growth. The intcrnictliatc active
arid
01’previous
hounil;~ry
p;tssivc portion
Ml
Mind
n:,,. The
rr:,,. The residual scgmcnt
with the first
incrcnicnt 01‘ crack
fK’ is ;I growth dcpcndcnt pica connccling the
rcsidu;iI scgnicnts. The p;issivc portion is conipriscd cntircly ol‘thc end points I3 xtivc
scvcral 01’ which ;irt’ shown in the sketch.
s~giiiciits.
‘I‘hc solution ol‘lhc growing crack prohlcm involves adapting the three scgmcnl approxh to ;I 5crics of’ linitc crack incrcmcnts. In the limit scgmcnt provide
:I
scgmcnt. t lowcvcr f’or ;I scrics ol’linitc to the passive zone
C;III
bc
active scgmcnt with ;I xrics active picctx
III
ol’ continuous crack advance. Lhc passive
smooth connection from the residual portion ol’thc instantaneous active
the limit
crack incrcmcnts. ;i piecewix linear ~tpprouinixtion
constructed by connecting the residuaI scgmcnt to the currently ot’ strxight
lines running
01’ inlinitcsimal
through the end points ut’ previous
crack increments
the approximate
boundary
should coincide with the actual passive scgmcnt. The setup t’or ;I typical growth increment All is shown in Fig. 6(b). The active segment Al3 is dcscribcd with rcspcct to moving crack tip coordinatr’s r and 0. where 9 is assumed to span the angular interval I‘rom 0 to an unknown angle I. For the growth incrcrnent AU the passive
scgrn~~~t
cntcnsion is modclcd by using ;I strxight
line to connect the end of the
instantaneous active scgmcnt, f3. to R’, the end of the previous active scgmont. Tangency bctwccn the active and passive scgmcnts at R is cnlixccd.
For the first crack incrcmcnt, the
passive scgmcnt connection with the initial /WC shape must LIISO be round. As shown in Fig. 6(b), the passive residual boundary at c’will bc dcscribcd by the unknown angle/I. mcasurcd with
rcspcct to the initi;Il
crack position.
segments will also bc unlbrccd at point C. The analysis prcscntcd lor the initial crack configuration
Tangency
bctwccn
the residual and passive
crack problem can bc applied to the growing
with some slight modilications.
Tho initial zone will be ailowcd to grow
with an advancing crack tip under the dual requircmcnts of satisfying the critical mean stress condition on the active zone boundary and maintaining the tip stress intensity at K”;,,. The mean stress criterion. cqn (7). is enforced with the understanding that the angle $I is rcstrictcd to the intcrvnl 0 < C/J< z. Introducing the nondimensional toughening parametcr I\ = K K,,,, carrying out the integration with respect to d.r,,. and regrouping eqn (7) results in the cxprcssion
Crackgrowth
resistance in transformationtoughened
ceramics
0.75
Y/L 0.
so
o)
IS
5
2s
IO
35
X/L Fig. 7. Growmg
zones for w = 5. IO.
(14)
governing the active zone boundary the crack tip and the integration
R(c/J)/L. All distances in eqn (14) are measured from
extends over the entire boundary 0 < 4’ < n.
The tip stressintensityfactor
can be maintained
at K,,, by enforcing eqn (l3),
which
now rcquircs
The system (14) and (15) constitutes a nonlinear integral equation and a scalar equation for R($)/L and A. For the initial crack increment. eqns (14). (15) and tangency conditions at a and /?, can be solved for R(a)/L in (0 < (b < a), A, z and j?. For a series of subsequent crack increments R($)/L. A. and a are found repeatedly to generate the zone shape and the crackgrowth in Appendix B.
resistance. The solution procedure
Growing zone shapes for
OJ =
for each crack increment is outlined
5 and IO are shown in Fig. 7. The dashed curves indicate
the positions of active segments for various amounts
of crack extension;
the innermost
D. M.
STLMP
B.
and
BLDL\SSKY
A = KjKm
1.00
‘.‘..I. 0. 0
~‘~“‘L”“““~““~“’ 2. s
5. 0
7. s
10.0
12. 5
1s. 0
AaiL
A  K/Km
curve is the initial boundary. The Kcurves, plots their respective steadystate tough&q (19Y8). arc shown in Fig. 8. The results
of A
asymptote.
vr’rsus AU/L, for (II = 5. IO along with as found by Amazigo and Budiansky
of Figs 7 and Y arc quite unexpected. The zone
height H/L and the toughening A both overshoot their steadystate lrvcls for finite amounts of crack advance bcfort: approaching them asymptotically The Rcurves
of Fig. 8 are qualitatively
mcasurcmcnts. A number of investigators,
consistent
from above. with some availublc experimental
including Swain (1983). and Swain and Hannink
(1983). have reported itcurves which exhibit peaks in toughness. However, the corresponding peak in zone height has not been rcportcd. Even though the actual zone boundary bctwccn transformed and untransformed regions occurs over ;I dilt‘usc region, the issue of zone widening should bc cxplorcd cxpcrimcntally. Rcurves and transformationzones
hove been calculated for various values of w. and
the results for the pciik toughening A, and peak zone height rl,, arc shown in Table I. Also shown art: the nondimensional crack extensions Atr(c\,)/L and Arr( t/,)/L at which A and tl.
respectively. are maximized.
Roth A,, and If,, art seen to incrcasc dramatically
in the
interval I9 < OJ < 20. Denoting the steitdy stiltc toughening ratio K,//C,,,. as calculated by Amazigo and Budiansky (I9KS) by A,. we compare the toughening predictions for the steady state and growing crack ci~lcul:~tions in Fi g. 9, Lvhich shows (A,) ’ and (A,,) ’ versus CIJ.
The implication
for ;I growing crack is “lockup”
for
UJ
gcater than X.2.
The steady
Crackgrowth
resistance in transformationtoughened Table
5 IO I5 17.5 19.5 20.0
1.29
1.03
1.80 3.07 5.06 lb.3 71.3
1.91 5.17 13.6 351 27hJ
643
ceramics
1.
5.5 10.25 29.4 81.0 2400 I!0200
2.1 5.6 IS.4 55.0 1875 1.900
1.2
0.9
dcK/Km) d(Aa/L) 0.6
Fig. 10. Initi;ll crackgrowth
rcsisranco vc’rsus w.
state configuration (Fig. l(a)) cannot bc rcachcd from the growth of an initial zone for c) > 20.2. It may be noted from Fig. 8 that a tiny secondary maximum appears in the Rcurve for w = IO. For values of 01 grcatcr than IO (but less than 70.2) additional peaks of decreasing amplitude wcrc found to appear in the Rcurves as the toughness decayed in an oscillatory fashion to its steadystate magnitude. By considering the initial slope of the Rcurves, resistance parameter
it is possible to define an initial tearing
D.
Fig.
tlcrc
I
I I. Comparison
M.
SXMP
with Hutchinson’s
B. BUDIANSKY
and
( 1987)
results for initial crackgrowth
reststance.
rcprcscnts the r;ltc d toughening incrcasc i1s the crack begins its initial growth
and is plottcrf versus w in Izig. 10. A cornprison
can hc mu& with the approximate results
of’ tlutchinson
(19X6) who ncglcctcti the ctli’ct of the transformation on the zonk boundary in his study of initial tearing rcsistancc. Figure I I shows ;t comparison of both sots of
~~~l~ul~~ti~~ns for small ($1.In the limit (1)+ 0, the pr~~i~~ti~ns coincide.
In the presence of transforming
particles, the resistance to crack growth. as mcasurcrf
by the applied stress ii~t~nsity K, reaches a mi~ximltm at a finite amount of crack growth. This
maximum
tWnSform:itiOn
~xcccds the steadystate toughness by an amount inta’Isity
p:Wamctcr
occurs ift a critical value of 01 = X.2,
(II.
Furthermore,
“lockup”
that varies with
(i.e. inlinitc
the
toughening).
which is subst~tnti~llly sm;tflcr than thr v;tIuc (I) = 30
corresponding to steadystate toughening. The potrntial
benefits of transformation
tough
ening must be asszsscd on the basis of tmnsicnt crack growth since the steadystate condition seriously undcrcstimatcs the toughrning
efl&t
of tr~nsforn~in~
particles.
~fclo,orrh~~~~rrot~.sThig work was supportal in part hy the DARI’A linivcrsity Research Initiative (Suhagreement P.O. No. V~3~6391~ with the University of Calilixni~, Santa Barbara, ONR Prime Contract NOfXf1b 86K0753). the Oll’icc of Naval Research (Contract NfWOI4X4K0510). and the Division of Applial Sciences, llarvard University. The work of DMS was partially supported by a National !Xmce Found&m Grxlu;ttr: Fetluwchip.
REFERENCES Amaziyo, J. C. and Budiansky, U. (198R). Steadystate crack growth in supercritically transforming materials. [I larvxd Univcniry report M ECt l107, IOX7f far. J. .%li& Struc?ures 24. 75 1 7%. Budiansky. 8.. Hutchinson. J. W. and ~lmbropoulos. J. C. (1983). Continuum theory ofdilatant transformation toughening in ceramics. Iptr. 1. So/i& S~ruclurec 19, 337355. Evans, A. G. and Cannon. R. M. (19X6). Toughcninp of brittle solids by martcnsitic transformations. /Icm i21cfall. 34.761800. ~lut~hinson. J. W. (1974). On steady state quasistatic crack growth. Harvard University Report, Division of Applied Sciences. DEAP S8. April 1974.
Crxkgrowth Lambropoulos. Srrucrurrs McMceking.
J. C. (1986).
resistance in transformationtoughened
Shear. shape and orientation
22. IOU1 106. R. M. and Evans. A. C. (1981).
effects in transformation
Mechanics of transformation
645
ceramics
toughening
toughening.
Inr. /. So/i&
in brittle materials. 1. .4m.
Crum. SC. 69.W336. Rose, L. R. F. (1986a). A kinematical
model for stressinduced transfotmation toughening in brittle materials. /. /(m. Ceram. Sot. 34.20821 I. Rose. L. R. F. (I986b). The size of the transformed zone during steadystate cracking in transformationtoughened materials. 1. Mcch. Phg. Solids 34, 60%616. Swain. Xl. V. ( 1983). Rcurve behavior of magnesiapartiallystabilized zirconia and its significance for thermal shock. In Frtrcrure .\lrckunics of Czrmmcs (Edlted by R. C. Bradt rr (II.). Vol. 6. pp. 355370. Plenum. New York. Swam. M. V. and Hannink. R. H. J. (IWI). Rcurve behavior of zirconia ceramics. In .4th~r~s in Ccrumics “CZZY. Amencan Ccrsmics Society. Columbus. OH. (Edited by N. Cluussen and A. Heter). Vol. 12. pp. ___
APPENDIX
A
The initial zone boundary. R(fb). can be found by solving the nonlinear integral equation (8). However before proceeding. it is instructive to examine the solution for w = 0 given by eqn (3) and shown in Fig. 3 :
/7((b)= L[I,‘2+
I,? cos (f$)].
(Al)
Due to Model symmetry. the boundary at 4 = 0 intersects the axis ahead of the tip normally. However. at cb = n. the houndxy runs through the crack tip tangentially to the crack face. For (1) # 0. the possibility that the boundary at lb = I dctachcs from the crack t~p must bc admitted. An an;llyG of the inlcgr4 in cqn (R) shows th;lt tf R(n) # 0. the boundary intcrqccts the crack faces norm;llly. An cxp;ln4~‘n for K(f/‘) meeting the boundnry rcquircmenls at 0 and n is Y
far/J) = 1 I

0”
LX’S
(m/l).
(A?)
0
In this study. the solution Ii’r the (.Y~tI ) unknowns cocllicicnts was xcomplishcd by colltu;~ting cqn (A3) ilt ,V t I cqu;illy S~WXI puints. C/J= n,~:( .S + I ) whcrc ,I = (0. I. 2. , lV), in the interval 0 < t/’ CI n. A Newton 1(;1ph~ou ilcrxtivc tcchniquc wxs then u~d to solve fur the IV + I unknown cosllicicnts. The integral in cqn (A3) w:is cv;~lu~~lrd hy (;;ulssi;tn quatlraturc. C;~nvcrgcncc W;IJ :~ssumcd when the rcl;ttivc changr in successive itcrntions uf cxh k)1’the unknowns was less th;m 0.001. I:or ths initial zones shown in Fig. 3. it was found that :I IOterm crp:uGon ws sullicient to obtain ;I highly accur:ttc solution. As an additional chcwk on the accuracy of the overall sh.lpc. the constraint on K(rj~)/f.. cqn (13) with &,,/A’ = I. w;is evaluntcd and found to hold to within 10 ’ for cxh (‘I’ ths Lone’s 01’ I:iy. 3.
Ths solution for ths unknowns X(g). ‘1, and A clftcr a typical crack incrcmcnt (Fig. 6(b)) the intsgr;d cquatlon (ll). the scal;lr equation (I 5). and mortlng ths tangcnsy condition
involves solving
(81) For the initi:ll crxk cyu;ition
increment tho system must be supplemented
dl
,:;(o,“,., An appropriate conditions is
with the unknown
/! and the additional
dl
= d:;(C)I_d.
cnpansion for the mdius of the instxttancous
R(b) = i %TJdh)
active scymcnt which meets ModeI
tangency
(W symmetry
(83)
”_ 0
where the T, are the even Tchcbyshev polynomials. Substituting this expression into (lt). (IS), and (RI) results in a set of equations for the unknowns ((I,,. (I,. , uy). z. and A. The solution procedure will ix dcscribcd for the gcncral growth incrcmcnt as shown in Fig. 6(b). For the initial incrcmcnt. the process is the same with fi and (B2) added to the list of unknowns and equations. For a
646
D.
M.
STUMP and B. BUD~ANSKY
given Au. a system of .V+3 equations was generated by enforcing equations (15). (81). and collocating eqn (II) at.VsIequallyspacudpoints~=n~I.V+I~rn=O.I.~..... Nt I ). The equations were written in residual form and SewtonRaphson method was used to tind the solution. Gaussian quadrature was used to evaluate the integrals in eqns (III and (15). Con\rrpnce to a solution was specified by a reIdrIve change of less than 0.001 between Iterations. .A btrrm expansion for R(4) was found sutficirnt to obtain a solution at all stages of growth. Initial prouth mcremrnts had to be small. ~0.0.5 L. to capture the earl> Lone shape. As crack extension proceed&. It was found that the increment sze could bc Increased. The number of Increments necessary to grow the crack to peaks In hclght and toughening varied greatly For small values of (I) less than 50 increments were sulfcient. However for values of c) between I7 and 20. several hundred increments were necessary to reach H, and A,. During the growth process. the mean stress exterior to the zonr was chcxkrd to confirm that it was less than 0;. The tinal results shown herein were checked by calculations with the increment steps halved.