Crack-growth resistance in transformation-toughened ceramics

Crack-growth resistance in transformation-toughened ceramics

CRACK-GROWTH RESISTANCE IN TRANSFORMATION-TOUGHENED CERAMICS DAVID M. STUMP and BERNARD BUDIANSKY Dibislon of Applied Sciences. Harvard University...

720KB Sizes 0 Downloads 54 Views

CRACK-GROWTH RESISTANCE IN TRANSFORMATION-TOUGHENED CERAMICS DAVID

M. STUMP

and BERNARD BUDIANSKY

Dibislon of Applied Sciences. Harvard

University.

Cambridge.

MA 02138. U.S.A.

Abstract-Crack-growth rcsistcrnce in transformation-toughened 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 far-ticld 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 zirconia-enriched

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 H-curves (i.c. stress intensity

no such

has been observed. and

versus crack extension) for a variety of zirconia-reinforced

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 steady-state toughening. wherein a semi-infinite by ;I semi-intinitr

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

Mode-l 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 semi-infinite 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 zone-size 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 K--field. 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;tr-field 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 J-integral 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 far-field and from dilatations in the transformed region. The mean stress due to the far&id stresses of eqn (1) is

Crack-growth

resistance in transformation-tou~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 non-dimensional 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

far-field

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

Crack-growth

resistance in transformation-toughened

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 zone-heights

corresponding to steady-state crack growth calculated by Amazigo and Budiansky For o z 30, the steady state zone height becomes infinite. Up”-

ix. infinite toughening-discovcrcd

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

zone-boundary

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

Crack-growth

resistance in transformation-toughened

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 stress-intensity-factor

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 crack-growth 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 K-curves, plots their respective steady-state 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 steady-state lrvcls for finite amounts of crack advance bcfort: approaching them asymptotically The R-curves

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 it-curves 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. R-curves and transformation-zones

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 non-dimensional 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

g-cater than X.2.

The steady-

Crack-growth

resistance in transformation-toughened 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 crack-growth

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 R-curve 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 R-curves as the toughness decayed in an oscillatory fashion to its steady-state magnitude. By considering the initial slope of the R-curves, 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 crack-growth

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 steady-state toughness by an amount inta’Isity

p:Wamctcr

occurs ift a critical value of 01 = X.2,

(II.

Furthermore,

“lock-up”

that varies with

(i.e. inlinitc

the

toughening).

which is subst~tnti~llly sm;tflcr than thr v;tIuc (I) = 30

corresponding to steady-state toughening. The potrntial

benefits of transformation

tough-

ening must be asszsscd on the basis of tmnsicnt crack growth since the steady-state condition seriously undcrcstimatcs the toughrning

efl&t

of tr~nsforn~in~

particles.

~fclo,orrh~~~~rrot~.s-Thig work was supportal in part hy the DARI’A linivcrsity Research Initiative (Suhagreement P.O. No. V~3~639-1~ with the University of Calilixni~, Santa Barbara, ONR Prime Contract NOfXf1-b 86-K-0753). the Oll’icc of Naval Research (Contract NfWOI4-X4-K-0510). 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). Steady-state crack growth in supercritically transforming materials. [I larvxd Univcniry report M ECt l-107, 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, 337-355. Evans, A. G. and Cannon. R. M. (19X6). Toughcninp of brittle solids by martcnsitic transformations. /Icm i21cfall. 34.761-800. ~lut~hinson. J. W. (1974). On steady state quasi-static crack growth. Harvard University Report, Division of Applied Sciences. DEAP S-8. April 1974.

Crxk-growth Lambropoulos. Srrucrurrs McMceking.

J. C. (1986).

resistance in transformation-toughened

Shear. shape and orientation

22. IOU-1 106. 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.

C-rum. SC. 69.-W-336. Rose, L. R. F. (1986a). A kinematical

model for stress-induced transfotmation toughening in brittle materials. /. /(m. Ceram. Sot. 34.208-21 I. Rose. L. R. F. (I986b). The size of the transformed zone during steady-state cracking in transformation-toughened materials. 1. Mcch. Phg. Solids 34, 60%616. Swain. Xl. V. ( 1983). R-curve behavior of magnesia-partially-stabilized zirconia and its significance for thermal shock. In Frtrcrure .\lrckunics of Czrmmcs (Edlted by R. C. Bradt rr (II.). Vol. 6. pp. 355-370. Plenum. New York. Swam. M. V. and Hannink. R. H. J. (IW-I). R-curve 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 Mode-l 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 IO-term 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 (l-l). 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 Mode-I

tangency

(W symmetry

(83)

”_ 0

where the T, are the even Tchcbyshev polynomials. Substituting this expression into (l-t). (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 Sewton-Raphson 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.