Effect of Water Penetration on the Strength and Toughnessof Silica Glass

Sheldon M. Wiederhorn,w,z Theo Fett,y Gabriele Rizzi,z Stefan Funfschilling,y Michael J. Hoffmann,y andJean-Pierre GuinJ

zMaterials Science and Engineering Laboratory, National Institute of Standards and Technology,Gaithersburg, Maryland

yKarlsruhe Institut fur Technologie, Institut fur Keramik im Maschinenbau, 76131 Karlsruhe, Germany

zKarlsruhe Institut fur Technologie, Institut fur Materialforschung II, 76344 Karlsruhe, Germany

JLARMAUR ERL-CNRS 6274, University of Rennes 1, Rennes, France

In this paper, we discuss the effect of water on the strength andstatic fatigue of silica glass. When a crack is formed in silicaglass, the surrounding environment rushes into the crack; waterthen diffuses from the environment into the newly formed frac-ture surfaces to generate a zone of swelling around the crack tip.Because the swollen material is constrained from expanding bythe surrounding glass, a zone of compressive stress is generatedat the fracture surface around the crack tip. The results aresimilar to those found for transformation toughened zirconiumoxide, with the exception that the transformation zone in silicaglass grows with time, so that the effect gets progressivelystronger. Using diffusion data from the literature, we showthat the diffusion of water into silica glass can explain severalsignificant experimental observations: the reported strengthen-ing of silica glass by soaking in water at 881C; an increase in theslope of dynamic fatigue curve by prior exposure to water at881C; the observation of a static fatigue limit in silica glass atvery low values of the applied stress-intensity factor; and theobservation of crack face displacements caused by water pene-tration into the glass at the crack tip.

I. Introduction

THIS paper discusses the role played by water in determiningthe strength of silica glass. The detrimental effect of water

on the strength of glass is well known, but not everyone agreesas to the causes of its effect. It is, however, universally believedthat the strength of silica glass is controlled by cracks containedin the surface of the glass. Removal of these cracks enhances thestrength of the glass, and if the glass can be made completelyfree of surface cracks then the strength approaches the theoret-ical strength of glass, �14 GPa, as it does in optical fiber glass.1

Even slight mechanical contact with the glass surface reduces itsstrength substantially.2,3

Water is found to be deleterious to the strength of glass. Withwater present, the breaking stress under load is time dependent,the higher the load applied to the glass, the more rapidly itbreaks. For very low loads, silica glass can support the load for avery long time before breaking4; some glass scientists believethat silica glass possesses a static fatigue limit, a stress belowwhich the silica glass will not break no matter how long the load

is applied. A fatigue limit of this sort has been measured on silicaglass by Sglavo and Green.5

This effect of time on the strength of glass is believed by manyto be caused by subcritical crack growth, resulting from a stress-enhanced interaction between water and glass. Crack growth insilica glass has been very well characterized by fracture mechan-ics techniques, in which the crack velocity is measured as afunction of the applied stress-intensity factor, relative humidity,and temperature.6–10 Chemical reaction rate theory provides asound basis for understanding crack growth in glass.11,12 Fur-thermore, a chemical reaction path for crack growth has beenidentified, as has the molecular structure of the water that isresponsible for crack growth.13 Based on this discovery, otherchemicals that cause crack growth in glass have also been iden-tified, for example, ammonia.13 The growth of cracks in largefracture mechanics specimens and in glass fibers is similarenough that the belief that the strength of glass is controlledby subcritical crack growth is almost unquestioned.

Areas where scientists disagree on the causes of strength de-gradation and static fatigue are centered on the nature of thecrack tip and how the crack tip controls the growth of cracks insilica glass. Some glass scientists believe that cracks in glass arebasically elastic, i.e., the stresses, strains, and displacementswithin the silica glass follow the elastic solution almost to thecrack tip, differing from the elastic solution only where the mo-lecular structure of the glass becomes apparent. Their view issupported by studies on the atomic force microscope (AFM),which demonstrate that displacements near the crack tip followthe elastic solution to within 10 nm of the crack tip.14 Experi-ments using spatially and spectrally resolved cathode-lumines-cence show that stresses follow the predicted stress level towithin 6 nm of the crack tip.15 Finally, transmission electronmicrographs demonstrated that the crack tip ‘‘radius of curva-ture’’ in silica glass was o1.5 nm.16 These measurements limitthe size of any possible nonlinear zone near the crack tip.

In apparent contradiction to the idea that cracks are ‘‘sharp’’and that strength and lifetime are limited by crack length, someglass scientists argue that crack tip blunting, either by precipi-tation of silica at the crack tip, or by plastic flow, is crucial to themeasured time to failure.17–20 Blunted cracks cannot propagateuntil they become atomically sharp; hence, the kinetics of blunt-ing and sharpening is important in establishing the strength anddynamic fatigue behavior of silica glass.17–20 In support of theirargument, Hirao and Tomozawa20 show that the strength ofsilica glass can be increased by annealing the silica glass at hightemperature, 9101C, while Ito and Tomozawa show thatstrength can be increased by soaking the glass in water at881C.18 In either case the strength of silica glass increases by

A. Heuer—contributing editor

wAuthor to whom correspondence should be addressed. e-mail: smwied@aol.comManuscript No. 29002. Received December 3, 2010; approved February 25, 2011.

Journal

J. Am. Ceram. Soc., 94 [S1] S196–S203 (2011)

DOI: 10.1111/j.1551-2916.2011.04530.x

r 2011 The American Ceramic Society

S196

about 10%–20%, the increase being attributed to blunting ofthe crack by the annealing process.

Other strengthening mechanisms have been suggested to ex-plain changes in the strength of glass as a consequence of ageingor annealing without invoking crack tip blunting as a part of themechanism. For example, most cracks in glass are formed as aresult of mechanical impact that also leaves residual damagebehind in the form of plastic deformation and densification inthe surface of the glass. Such damage raises the stresses at thetips of cracks and is the main driving force for crack growth.21

Annealing the glass at high temperature relieves these stressesand thus increases the strength of the glass. Alternatively, agingglass in water results in the formation of an interfacial layer thatacts as a ‘‘glue’’ to bind the two fracture surfaces together, thus,increasing the strength of the glass.22

In this paper, we discuss an alternative explanation of theobservations of the strength increase reported by Hirao andTomozawa.19,20 We assume that the cracks are always sharp,and that the observed strengthening occurs by the penetration ofwater into the glass surrounding the crack tip and into the ad-jacent walls of the crack. Such penetration induces a compres-sive stress within the walls of the crack, which in turn results in anegative stress-intensity factor at the crack tip. A mechanismsuch as this has been reported for soda lime silicate glass as aconsequence of ion exchange near the crack tip, hydronium ionsfor alkali ions in the glass.23,24 The stresses set up by the ionexchange process, �2000MPa,25 can result in a substantial timedelay in restarting crack growth in soda lime silicate glass.23,26,27

Similar levels of stress can be achieved when water penetratesinto silica glass.

II. Water Diffusion and Volume Expansion in Silica

A substantial literature exists documenting the diffusion of wa-ter into silica glass.28–31 Water diffuses by a diffusion-reactionprocess,28 in which the water molecules can react with the Si–Obonds to form –SiOH groups. At low temperatures, o2501C,the –SiOH groups are immobile, whereas, at high temperatures,�10001C, they exhibit a limited mobility and can take part inthe equilibrium reaction that establishes the relationship be-tween the concentration of –SiOH and the molecular water. Partof the water is present in molecular form and part is present inreacted form and both parts have to be taken into account whenwater diffuses though silica glass. The transport of waterthrough the glass structure, however, is by motion of the mo-lecular water alone.

The diffusion coefficient, Dw, of the molecular water is givenby the following equation:

Dw ¼ Dw0 expð�Qw=RYÞ (1)

whereQw is the activation energy,Y is the absolute temperature,and R is the gas constant. As reported in Zouine et al.,30

Qw 5 71.271 kJ/mol and log10 Do5�7.7870.06, where Do isin m2/s.

For a constant diffusion coefficient the water concentrationprofile, C(z,t), as a function of depth z and time t is given by

Cwðz; tÞ ¼ Cw0 � erfcz

2ffiffiffiffiffiffiDtp

� �(2)

where Cw0 is the surface concentration of the water.32 As shownby Nogami and Tomozawa,32 diffusion is enhanced in the pres-ence of stresses.

The diffusion coefficient as a function of the hydrostaticstress, sh, can be written

D ¼ Dð0Þ expðDVwsh=RYÞ (3)

where, DVw is the activation volume for water diffusion in glassand D(0) is the diffusivity at zero pressure, i.e., sh5 0 atm. The

value of the diffusivity at 1 atm is almost the same as at 0 atm, sofor practical purposes the diffusivity at 1 atm is used in thispaper. At room temperature, D(0)D10�21 m2/s.30 The activa-tion volume for water at room temperature has not yet beenevaluated, but is believed to be approximately equal to the molarvolume of water, 18� 10�6 m3/mol (M. Tomozawa, privatecommunication). This paper suggests a new method of calcu-lating the activation volume of water. The value obtained,10� 10�6 m3/mol, is reasonably close to the molar volume ofwater. Clear evidence has been reported in the literature forvolume swelling, e0, measured directly via dimension changes inthe silica glass, and indirectly via stress generation within theglass. The swelling behavior has been determined by length33

and curvature measurements on silica glass bars34 and by X-raydiffraction strain measurements35 on silica glass. Gorbachevaand Zaoints33 soaked fused silica prisms (25� 5� 5 mm) in wa-ter at 801C for 20 months. They found the measured length tohave increased by 0.17% by this treatment. Because only a thinsurface layer of water penetrated glass, �5 mm,ww could havebeen generated during the water storage, the suppressed linearstrain in the surface must be clearly larger than 0.17%.

An estimate of the surface stress caused by water penetrationis relevant to the present study, because it is this surface stressthat affects the stress-intensity factor driving the crack growth.Knowing the effect of water penetration into the glass on thevolume of the glass, permits us to determine that surface stressby first estimating the volumetric strain of the glass as a conse-quence of the water penetration. The volumetric strain, e(z,t), isa function of time, t, and distance, z, from the free surface:

eðz; tÞ ¼ e0 � erfcz

2ffiffiffiffiffiffiDtp

� �(4)

with e0 being the time-dependent volumetric strain at the sur-face. The stresses caused by swelling are equi-biaxial (sz 5 0)and given by

sx ¼ sy ¼ �eðz; tÞE3ð1� nÞ (5)

where E is Young’s modulus and n is Poisson’s ratio. Conse-quently, the hydrostatic stress is

sh ¼1

3ðsx þ syÞ ¼ �

2eðz; tÞE9ð1� nÞ (6)

Because the diffusion coefficient depends on the hydrostaticstress component according to Eq. (3), the hydrostatic stress dueto swelling must affect the concentration profile. Hence, Eqs. (2)and (4) are no longer correct with the consequence thatthe diffusion differential equation has to be solved numerically.For reasons of clarity and simplicity, such higher-order effectson concentration profiles will be neglected in the followingdiscussion.

III. An Approximate One-Dimensional Treatment of theSwelling Zones

A static crack undergoing water diffusion shows a diffusionzone such as that represented in Fig. 1(a). Water entering theglass through the plane of the crack can be described by a one-dimensional diffusion problem, the solution of which is given byEq. (4). Near the crack tip and ahead of the crack a much morecomplicated two-dimensional nonaxial solution to the diffusionequations is required.

The following considerations were made to allow the repre-sentation of the general trends by approximate analytical ex-pressions from the one-dimensional analysis. First, we ignore

wwThis distance is calculated from the known diffusivity30 and the time of exposure usingthe definition of the diffusion distance, x5O(Dt).

June 2011 Effect of Water Penetration on Strength and Toughness of Silica Glass S197

the dilution of water concentration due to the increase of thearea elements dAprdr in the two-dimensional problem and usedApdz from the one-dimensional model. Also, the continu-ously varying swelling strain e(r) of Eq. (4) is replaced by a step-shaped dependency with a change at the characteristic contourheight beff (Fig. 1(b)) resulting in the same shielding stress-in-tensity factor.

(1) Swelling Zone for Unloaded Cracks

The volume expansion at crack surfaces must result in an in-trinsic shielding stress-intensity factor Ksh. In the case of an un-loaded crack, the hydrostatic stress term in Eq. (3) disappears.Because of the Arrhenius dependence of the diffusivity on tem-perature, Eq. (1), the diffusion process becomes more importantas the temperature is increased, even though no stress-enhance-ment takes place. The shielding stress-intensity factor, Ksh, aftera time, t, of water contact is given by36

Ksh ¼ �0:25e0E1� n

ffiffiffiffiffiffiffibeff

p(7)

with an effective water layer thickness, beff, of

beff ffiffiffiffiffiffiffiffiffiffiffiffiffiDð0Þt

p(8)

which is roughly the distance from the surface at which thestrain has dropped to one-half of the surface value (Fig. 1(b)).Finally, it holds that

Ksh ffi �0:25e0E1� n

½Dð0Þt�1=4 (9)

(2) Swelling Zone Near the Tip of a Nonpropagating CrackUnder Load

The hydrostatic stress is given by the trace of the stress tensor,

sh ¼1

3ðsrr þ sjj þ szzÞ (10)

Because very high tensile stresses occur in the vicinity ofa crack tip, a water containing zone must rapidly extend fromthe tip to the surrounding material during crack growth. As thecrack moves, the zone extends along the sides of the crack at thesame velocity relative to the crack tip as the crack is moving inthe test material.

If KI denotes the stress-intensity factor, the near-tip stressfield for the stationary crack in the absence of a T-stress term36

readszz

sii ¼Ktipffiffiffiffiffiffiffi2prp giiðjÞ (11)

where r is the crack-tip distance. In Eq. (11), gii are well-knowngeometric functions depending on the polar angle j.37 The hy-drostatic stress under plane strain conditions is given by

sh ¼2

3ð1þ nÞ Ktipffiffiffiffiffiffiffi

2prp cosðj=2Þ (12)

Introducing Eq. (12) into Eq. (3) yields location-dependentdiffusion coefficients. The introduction of such diffusion coeffi-cients into the problem makes the resulting equations highlyintractable, because Eqs. (2) and (4) are correct only forconstant diffusion coefficients. The authors are unaware of anexact analytical solution for this diffusion problem. Therefore,to simplify the problem, we assumed that the two-dimensionaldiffusion problem at the crack tip can be represented by the one-dimensional problem. Therefore, the erfc-relation, Eq. (4), wasused for the near-tip zones in the presence of stresses.

Using Eqs. (12), (4), (3) and (1), an equation for the swellingstrain can be derived:

eðr;j; tÞ ¼e0 � erfcr

2ffiffiffiffiffiffiffiffiffiffiffiffiDð0Þt

p"

� exp � 1

3ð1þ nÞKtip

cosðj=2Þffiffiffiffiffiffiffi2prp DVw

RY

� �� (13)

The contour for an effective swelling strain of e(r,j,t)5 0.5e0results in

r

2ffiffiffiffiffiffiffiffiffiffiffiffiDð0Þt

p exp � 1

3ð1þ nÞKtip

cosðj=2Þffiffiffiffiffiffiffi2prp DVw

RY

� �

¼ 0:477 ffi 0:5 (14)

with a solution of

Fig. 1. (a) Diffusion zone in the absence of an external load, (b) swelling profile under assumption of a one-dimensional (solid curve) and axial-symmetric two-dimensional water diffusion (dashed curve); effective zone thickness beff in the one-dimensional case.

zzThe stress distribution around a crack tip is usually expressed in the form of a powerseries in rn/2, where r is the distance from the crack tip. The series starts with n5�1. Thecoefficient of the first term is the stress-intensity factor; the coefficient of the second term ofthe series is the T-stress. See Fett36 for further discussion of this subject.

S198 Journal of the American Ceramic Society—Wiederhorn et al. Vol. 94, No. S1

r ¼ k2 cos2 ðj=2Þ

4 PLogk cosðj=2Þ2½Dð0Þt�1=4

" #20@

1A

(15)

with the abbreviation for k:

k ¼ ð1þ nÞ3

Ktipffiffiffiffiffiffi2pp DVw

RY(16)

‘‘PLog’’ stands for the Lambert W function or product logfunction, i.e., the solution x5 f(y) of the equation y5x exp(x)(in Mathematica38 called ProductLog).z Contour radii for con-stant swelling are plotted in Fig. 2(a) with k/(D(0)t)1/4 as theparameter.

In the case of strongly stress-enhanced diffusion, i.e.,k/(D(0)t)1/4 � 1, Eq. (15) tends to

r

rmax¼ cos2 ðj=2Þ; rmax

¼k2 2Plogk

2½Dð0Þt�1=4

" # !�2 (17)

This can be seen from Fig. 2(a) where the thick curve repre-sents the limit case for k/(D(0)t)1/4-N in Ksh. The shape of therelated swelling zone is shown in Fig. 2(b). The result is similarto that for the phase-transformation zones in zirconia ceramics.But in contrast to those zones, the size of the swelling zones istime-dependent.

The height of the swelling contours for an arbitrary k/(D(0)t)1/4 results in very complicated expressions. For reasonsof simplicity, the contour radius for j5 0, rmax, may thereforebe used as a characteristic size parameter. According to theanalysis by McMeeking and Evans39 this type of zone must re-sult for k/(D(0)t)1/4-N in Ksh5 0.

IV. Shielding Term for Growing Cracks

A crack growing at a constant subcritical velocity, v, develops anextended zone of biaxial compression on the newly formed crackfaces. The zone forms from the crack tip and as the crack movesforward, it passes through the compressive zone leaving half thecompressive zone along each fracture surface. Figure 3(a) showsthe swelling zone after a crack increment of Da. For a very sim-

plified analysis, the continuously varying swelling strain e(r) maybe replaced by a step-shaped dependency with a change at thecharacteristic contour height o5 b (Fig. 3(b)), defined by thesame shielding stress-intensity factor.

In this normalized representation, Ksh,max denotes the shield-ing for an infinitely long zone, Da/o-N. Without placing greatdemands on accuracy we can represent the Ksh�Da dependency,numerically given by McMeeking and Evans,39 by the approx-imation:

Ksh

Ksh;maxffi tanh 0:757

Daoeff

� �3=4" #2=3

(18)

with an effective zone height oeff�rmax, Eq. (17) (Fig. 3(b)).The time available for the formation of the swelling zone is

roughly t5 rmax/v. Instead of Eq. (17), this now results in theimplicit equation:

rmax ¼ k2 2Plogk

2½Dð0Þrmax=v�1=4

" # !�2(19)

The solution in terms of the power law of subcritical crackgrowth, which holds far from the thermodynamic threshold forcrack growth12

v ¼ AKntip (20)

is then

rmax ¼k2

PLog2 kffiffiffiffiffiffiffiffiffiffiv

Dð0Þ

r" # ;

k ¼ ð1þ nÞ3

ðv=AÞ1=nffiffiffiffiffiffi2pp DVw

RY

(21)

It has to be reemphasized that this maximum zone size is onlyreached if a crack has grown a sufficiently large increment ofDa4reff at a constant rate.

The maximum shielding stress-intensity factor increases withdecreasing crack growth rate

Ksh;max ffi �0:22e0E1� n

k

PLog kffiffiffiffiffiffiffiffiffiffiv

Dð0Þ

r" # (22)

Fig. 2. (a) Diffusion zone radius normalized on the maximum value as a function of the polar angle j, (b) shape of the swelling zone.

zThe use of commercial names is for identification only and does not imply endorsementby the National Institute of Standards and Technology.

June 2011 Effect of Water Penetration on Strength and Toughness of Silica Glass S199

with the coefficient 0.22 derived in.39 This equation implies thatKsh,max-�N for v-0. Consequently, the Ktip versus Kappl

curve is given by

Ktip ¼ Kappl þ Ksh; Ksh < 0 (23)

For fully developed zones according to Fig. 3(a), the crack tipstress-intensity factor Ktip is plotted in Fig. 3(c) as a function ofthe applied stress-intensity factor Kappl. The minimum appliedstress-intensity factor, Kappl,min, is indicated by the circle. Thiscurve together with Eq. (23) establishes a threshold value forsubcritical crack growth as shown in Fig. 3(d). The experimen-tally nonaccessible region of the v(Kappl)-relation is indicated bythe dashed curve in Fig. 3(d).

V. Interpretation of Experimental Results from Literature

(1) Inert Strength

The parameter e0 can be estimated from strength measurementsby Hirao and Tomozawa19 on hot-water soaked and freshlyabraded silica glass specimens. Hirao and Tomozawa deter-mined strength of high-silica glass for freshly abraded samplesand specimens that were hot-water soaked for 112 h at 881C.The results of strength tests in liquid N2 as the inert mediumunder two loading rates are shown in Fig. 4 by the circles; closedcircles for freshly abraded glass; open circles for abraded glass

aged in hot water. Whereas the freshly produced samples hadstrength of sc 5 180 MPa, an increase of �12% was caused bythe water soaking procedure. Subcritical crack growth wasabsent in liquid N2 as can be seen from the independence ofstrength on loading rate.

Because all specimens were prepared in the same way, theirinitial flaw population must have been the same. This makes itclear that the apparent toughness must have been increasedduring soaking by about 12% according to

Kappl ¼ scYffiffiffiap¼ KIc þ jKshj ¼ 1:12KIc (24)

(Y5 geometric function). As a consequence of these inertstrength results, the difference in stress-intensity factor|Ksh|5 0.12 KIc must have been generated during water storagewith KIcD0.8 MPa �Om, KshD�0.096 MPa �Om results.

The intrinsic stress-intensity factor, Ksh, caused by the com-pressive stresses in this layer is given by Eq. (9) where the surfacestrain was assumed to be a constant value e0, independent oftime. Because Ksh is negative for volume expansion, the com-pressive zone of expansion partially shields the crack tip fromthe applied load.

With E5 73.3 GPa and n5 0.17 from40 it follows from Eq.(7) that

e0ffiffiffiffiffiffiffibeff

p¼ 4� 10�6

ffiffiffiffimp

(25)

The diffusion coefficient for the interesting temperature of881C could be taken from Zouine et al.30 to be D0D3� 10�19

m2/s. With E5 73 GPa and n5 0.17 from40 and Ksh5�0.096MPaOm; from Eq. (8) we obtain,

beff ffi 0:3mm (26)

and

e0 � 0:7% (27)

Introducing this value in (5) gives at the surface (where e5 e0)a biaxial compression stress of sx5sy 5�200 MPa.

Fig. 4. Strength measurements on silica glass in two modifications;solid symbols: freshly abraded glass rods, open symbols: measurementafter hot-water storage for 112 h at 881C (circles: inert strength sc

measured in liquid nitrogen, squares: strength sf affected by subcriticalcrack growth in water) from Hirao and Tomozawa.19

Fig. 3. (a) Shape of the swelling zone for a crack growing with aconstant crack rate v, (b) shielding stress-intensity factor K computedwith the method proposed by McMeeking and Evans39 normalized onthe limit value for Da/o-N(solid curve), approximation by Eq. (18)(dashed curve), (c) related Ktip versus Kappl curve (solid curve); thestraight dashed curve is an extension of the linear portion of the curve,extended from high Ktip, (d) related log (v) versus Kappl curve. BelowKappl, min the crack stops growing; the dashed curve represents acontinuation of the log (v) versus Kappl relationship; this portion of thecurve is not accessible to real crack growth.

S200 Journal of the American Ceramic Society—Wiederhorn et al. Vol. 94, No. S1

In context with the result of (27), it has to be mentioned thatvolumetric strains may not be fully developed in the case ofthe strength measurements. In the first part of the soaking time,the crack is free of stresses and water can freely diffuse into thecrack faces. With increasing time, a negative stress-intensityfactor develops, so that the crack faces are pressed on eachother and water diffusion is suppressed, resulting in a thinnerzone compared with that in specimens under a superimposedpositive stress-intensity factor. Nevertheless, we expect the gen-eral conclusions of this section to be correct: the differences instrength of the two types of specimens, freshly abraded andaged, occur as a consequence of the zones of swelling around thetips of cracks contained in the aged glass, but not in the freshlyabraded glass.

(2) Static Lifetime Tests

Figure 5 shows a schematic v–K curve for tests on fresh cracks ofinitial length a0. Under constant load, the curve increasesmonotonically with Kapplp(a01Da)1/2. The circle at the originof the dashed curve applies for the cardioid shape of the swellingzone developing at Da5 0. Because the shielding term for thiszone is very small, Ksh-0, this point is also located on the v–Ktip

curve represented by the light dash-dotted line. Experimentally,this curve is difficult to determine, because all crack growth dataare affected more or less by a shielding stress-intensity factorterm. The solid curve represents the v–Kappl curve for a test witha fully developed zone length, Ksh5Ksh,max. The upper limitcurve for Kappl5Ktip1|Ksh,max| will be reached for a swellingzone length of Da 3o (Fig. 3(a)). The presence of a minimumin Kappl, dKappl/d(log(v))5 0, suggests a static fatigue limit forthe v–Kappl curve, see Fig. 3(d).

The dashed curve in Fig. 5(a) describes the v–Kappl curve for alifetime test, which uses fresh cracks previously exposed to anaqueous environment, but not yet propagated in the presence ofwater. These cracks have no swelling along the fracture surfacethat would result in toughening. The water that penetrates intothe tensile stress region around the crack tip has a cardioid shapeindicated in the figure. On propagation in a moist environment,a zone of swelling sweeps from the crack tip to the surface of thecrack and develops into a shielding zone that toughens thecrack, Section IV. The crack growth curve starts at (Da5 0,Ksh5 0) and asymptotically approaches the solid curve as thelength of the swelling zone increases. The zone thickness is lessfor more rapidly moving cracks, and the slope of the v–Kappl

curve increases as the curve approaches the solid curve. Theseresults suggest that different slopes can be obtained for v–Kappl

curves depending on how the data are collected. Curves ob-tained on fresh cracks, are expected to be shallower than curvesfrom cracks with well-developed zones of swelling. Data thatdistinguishes between these two kinds of behaviour are discussedin the next section.

(3) A Possible Explanation of Results from DynamicStrength Tests

In dynamic bending tests carried out in water, the slope of log(sf) versus the log ( _s) was found by Hirao and Tomozawa19 todepend on test history. Curves obtained on freshly abraded glasswere steeper than those obtained on glass soaked in hot water(Fig. 4). In the power-law representation of subcritical crackgrowth data,12,37 the value of n for the data in Fig. 4 was foundto be greater for data collected on hot water soaked specimensthan for data collected on freshly abraded specimens. Theseresults can be explained by the ideas presented in the previoussection. For freshly abraded specimens, shielding zones developduring the test and resistance to crack growth gradually buildsup as the crack progresses, as illustrated in Fig. 6 by the thickdashed curve. By contrast, specimens that have been exposed towater at elevated temperatures will contain cracks that havefully developed shielding zones and for each value of log(v),crack growth will require a higher value of Kappl, solid line inFig. 6. The slope of the v–Kappl curves from such data will besteeper than curves from freshly abraded specimens. As theslope on log–log representation of a v–Kappl curve is n, the valueof n for the specimens exposed to hot water is expected to begreater than the value of n obtained on fresh specimens, as isobserved experimentally19 (Fig. 4).

We expect as the general conclusions of this section: thedifferences in slopes of the two types of specimens, freshlyabraded and aged, occur as a consequence of differences inthe zones of swelling around the tips of cracks contained in theirsurfaces (preexisting from hot-water soaking or to be generatedin the dynamic strength test).

(4) Crack-Face Measurements by AFM

A discussion of crack-surface inspections by atomic forcemicroscopy (AFM) is given in Wiederhorn et al.41 In particular,data was obtained on a crack in silica glass that was first arrested

Fig. 5. Development of the swelling zone during crack growth. Thesolid curve is expected for tests carried out with v5 constant over longcrack extensions Da (e.g., in a DCB-test) reaching saturation in theshielding term. The heavy dashed curve is for a crack that develops azone of swelling along the fracture surface as it propagates. The lightdash-dotted line represents the v–Ktip curve.

Fig. 6. v–K curve for silica; solid line: crack with a fully developedshielding zone present already at the beginning of the test, dashed line:test with a shielding zone to be created during a test. The thin dashedcurve indicates the possibility of fatigue limit at low values of Kappl, as inFigs. 3 and 5.

June 2011 Effect of Water Penetration on Strength and Toughness of Silica Glass S201

and held under load and then repropagated under a muchhigher load. The stress-intensity factor applied to the crack tipat the point of arrest was 0.245 MPa �Om; the load wasmaintained for 80 days in water at room temperature, thenincreased to 0.48 MPa �Om to restart the crack. Crack motionwas immediate upon increasing the load on the specimen.

Examination of a crack profile by comparing the twofracture surfaces after completing the fracture (Fig. 7(a))indicates that with the exception of a small area near thecrack tip, a, the two surfaces matched over the length of thecrack that had been opened and exposed to the water. Aninteresting observation in Fig. 7(a) is the overlap of the twofracture surfaces near the crack tip, a. One possible explana-tion of the overlap might be the occurrence of silica precipitatefrom the crack tip solution and its deposition at the crack tipbefore final fracture.16 This was the interpretation given forthe observation in Wiederhorn et al..41 We now discount thispossibility for two reasons. First, the experiment was carriedout in water, and the specimen had to be dried before anystudies could be carried out on the fracture surface. In dryingthe fracture surface, it was wiped with high-quality tissuepaper, which would have removed any precipitation depositedon the surface during the experiment. In our previous experi-ence, gels that precipitate on the fracture surface are easilyremoved by such a procedure.

Second, an AFM scan of the apparent excess material at thecrack arrest point using the tapping modet indicated no shift inthe phase image, which suggests that the viscoelastic/elasticproperties of the excess material were close to that of the originalglass. If the excess material represented a dried precipitate, theelastic constants would have been lower because of internalporosity and a phase shift would have been observed in thephase image.

Based on these considerations, we interpret the excess mate-rial ahead of the actual crack tip as a water diffusion zone whereswelling by the volumetric strain, e0, took place. After completecracking of the specimen, each half of this region can expand inthe direction normal to the new crack face. This results in avolume expansion and in an overlapping of the surface profiles.Figure 7(b) gives measurements of the overlapping displace-ments with a maximum value of about 6 nm.

In order to study this effect, a finite element (FE) modelling ofthe heart-shaped zone according to Fig. 2(b) was performed.The case of a half-space was realized by a plate 60o wide and30o high including 1400 elements and 4300 nodes. Solid con-tinuum elements (8-node bi-quadratic) in plane strain werechosen and the computations carried out with ABAQUS version6.8. The volume strain was replaced by the equivalent thermalproblem by heating the inner of the zone with a temperature ofDT keeping the temperature outside the zone constant (e.g., at

Fig. 7. (a) Overlapping of crack faces for an arrested crack after final fracture.41 The arrow gives the direction of crack growth, (b) crack–facedisplacement,Vz, from (a), (c) finite element result for the surface displacements due to a volumetric swelling strain ev, (d) the cardioid shape of the regionof the glass penetrated by water. The width of the cardioid is 2o and the distance from the crack tip in front of the crack is x (geometric data are definedin the bottom schematic).

S202 Journal of the American Ceramic Society—Wiederhorn et al. Vol. 94, No. S1

T5 0). This results in the volume strain

ev ¼ 3aDT (28)

The displacements are shown in normalized representation inFig. 7(c). The maximum displacement from FE-modelling wasfound at 0.6o ahead of the crack tip to be vy/(oev)D0.55. FromFigs. 7(b) and (c) we can conclude that oe0D11 nm.

The width at half of maximum displacement, x1/2/o, is (seeFig. 7(c)) x1/2/o5 1.8. The corresponding measured value of x1/2is x1/2�140 nm (see Fig. 7(b)). From these two results weobtain: oD80 nm ) e05 13.7%, or, a linear strain of about4.6%, which is equivalent to a biaxial compression stress ofsx5sy5�4000 MPa, Eq. (5).

The height of the zone (identical with r� sinj for jDp/3)and an estimate of DVw can be determined from Eq. (17).Inserting the values of D0 5 10�21 m2/s 25 into Eq. (17) yieldsoD140 nm for DVw 5 30� 10�6 m3/mol, oD110 nm forDVw 5 20� 10�6 m3/mol and oD80 nm for DVw 5 10� 10�6

m3/mol. The latter value of o is in good agreement with thevalue resulting from the evaluation of Fig. 7(b). This calculationsuggests that DVw 5 10� 10�6 m3/mol, a value not that far fromthe molar volume of water, DVm 5 18� 10�6 m3/mol. Thus, asystematic investigation of the zone size at different holdingloads, K, holding times t, and temperatures, Y, provides anindependent tool for the determination of the activation volume,DVw, for the diffusion of water in silica glass.

VI. Summary/Conclusions

In this paper, we investigated the effect of water diffusion intosilica glass fracture surfaces on the strength and static fatigue ofthis glass. The approach used is identical to that used byMcMeeking and Evans39 in their calculation of the effect of aphase transformation on the strength and toughening of zirco-nium oxide. The toughness of zirconium oxide was enhancedsubstantially by a volume expansion of the material around thecrack tip, caused by a stress-induced phase transformation in thezirconium oxide. In the case of silica glass, the volume expansionis the result of swelling when water diffuses into the fracturesurface of the glass and the glass around the crack tip. Theswelling is time-dependent and so the toughening of the glassincreases with time. The theory is able to predict an increase ofstrength when testing is carried out in liquid nitrogen; a smallchange in the slope of the static fatigue curve as a result of thezone of swelling around the crack tip, and an overlapping of thefracture surfaces by about 6 nm in the near vicinity of the cracktip due to water diffusion into the highly strained regionsurrounding the crack tip. A further benefit of this study wasthe development of a new method of measuring the activationvolume for the diffusion of water through silica glass.

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