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NBSIR 80-2023

Micromechanisms of Crack

Growth in Ceramics and

Glasses in Corrosive Environments

S. M. Wiederhorn, E. R. Fuller, Jr. and R. M. Thomson

Fracture and Deformation Division

Center for Materials Science

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

May 1 980

Interim Report

Prepared for

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NBSIR 80-2023

MATIONAL BUREAU OF STANDARDS

LIBRARY MICROMECHANISMS OF CRACK

GROWTH IN CERAMICS AND

GLASSES IN CORROSIVE ENVIRONMENTS

JUL 1 ^ '980

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S. M. Wiederhorn, E. R. Fuller, Jr. and R. M. Thomson

Fracture and Deformation Division

Center for Materials Science

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

May 1 980 *

Interim Report

Prepared for Office of Naval Research

Department of the Navy

Arlington, VA

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J, Baruch. Assistant Secretary for Productivity, Technology, and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Micromechanisms of Crack Growth in Ceramics and Glasses

in Corrosive Environments

S.M. Wiederhorn, E.R. Fuller, Jr., and R. Thomson

, National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

At normal temperatures and pressures, water is known to have a strong

influence on the strength of ceramics and glasses. Behaving as a stresscorrosion agent, water causes these materials to fail prematurely as a

consequence of subcritical crack growth. A basic premise of this paper

is that stress-corrosion cracking of ceramics is a chemical process that

involves a stress-enhanced chemical reaction between the water and the

highly stressed ceramic near the crack tip. Plastic deformation is

believed to play no role in this fracture process. After a brief survey

of chemical reaction rate theory, the basic rate equation from this theory

is modified to reflect physical and chemical processes that occur at crack

tips. Modification of the rate equation is based on the assumption that

the crack tip can be modelled as an elastic continuum, an assumption that

is supported by a simple atomistic model of crack growth. When tested

against experimental data collected on glass, the theory was found to be

consistent with measurements of the crack-growth dependence on temperature,

applied stress intensity factor, and concentration of reactive species in

the environment.

1
1. Introduction

In the presence of corrosive environments, structural materials often experience!

delayed failure when subjected to mechanical loads. These environments react

'

I chemically or physically with the stressed material causing subcritical crack *

growth that results in structural failure when the crack has grown to a critical

| size. The time to failure is the total time required for the nucleation and 1

growth of a crack to a critical size. Known as stress corrosion cracking, | this process is superficially the same for ceramic, metallic, and polymeric |

I materials, since all three kinds of material exhibit similar types of delayed

i failure curves. Failure occurs most rapidly at high loads. Below a critical

j value of the load knov/n as the stress corrosion limit, failure does not occur

j regardless of the time that the load is supported.

j

j

J Despite these similarities in fracture behavior, the mechanisms of fracture j:

ii for the three types of materials differ greatly. Polymers and metals have ^

plastic zones at their crack tips, so that stress corrosion cracking of these i

j materials involve the generation, propagation, and fracture of plastic zones '

I in these materials. By contrast brittle materials such as AI

2

O

2

, SiC or Si do

;j not have plastic zones at their crack tips, so that the fracture of these

'

materials must be explained by mechanisms that preclude th& occurrence of '

plastic deformation [1,2]. It is these types of materials that are discussed ini

this paper. Since these materials are treated as ideally brittle solids, a 1

short description of recent experiments examining the crack-tip structure is [

resented to siionort this proposed model of fracture. 1

1 Evidence for the absence of a plastic zone at the tips of cracks in brittle, I

crystal 1 i macerials nas been obtained by the use of transmission electron |

2
microscopy [1,2]. Examination of thin foils of Al^O^, SiC, Ge and Si that

contain cracks has shown the crack tips to be devoid of mobile dislocations

when the cracks are introduced into the foils at low temperatures.

Furthermore, removal of the driving force for fracture results in spontaneous

crack healing of cracks in these materials, which suggests that in the absence

of an opening mode force, the fracture surfaces can become reattached by strong

molecular forces. At elevated temperatures, however, where plastic deformation

is expected, mobile dislocations are observed at crack tips.

Although inorganic silicate glasses are an important class of brittle material,

the technique of transmission electron microscopy cannot be applied because of

the lack of long-range order in these materials. However, recent transmission

electron microscopy studies on crystalline quartz, which has the same type of

atoms and atomic bonding as glass, also indicate an absence of plastic deformation during fracture [3]. In addition, since crack healing has been observed

in glass [4], and since the fracture toughness of glass [5] is similar to that

of silicon [6] (which has the same hardness and exhibits no plastic deformation

during fracture) it is probable that inorganic silicate glasses also fracture in

a completely brittle manner.

The consequence of completely brittle behavior is the development high strains

at crack tips prior to crack propagation. Bonds at the crack tip in a completely

brittle solid are strained until they reach a state of instability at which point

fracture occurs. The level of elastic strains that can be achieved in brittle

materials can be estimated from strength measurements on silica glass fibers and

rods [7,8]. Because test specimens of silica glass can be made by fusion

3
I

i'

I techniques, their surface are highly perfect, lacking the small flaws and cracks

that act as stress concen rators in normal silica glass artifacts. Consequently,

silica glass can be subjected to stresses as high as '^^14 GPa (2 x 10 psi) in vacuum

or in other inert environments. Based on a Young's modulus of 75 GPa, a stress of:

this magnitude corresponds to an elastic strain before fracture of approximately

20 percent. Because test specimens used in these measurements undoubtedly contain ;

structural defects that are several times the size of the holes in the glass net- j!

work structure, strains in the vicinity of fracture origins must be even greater

j than 20 percent. |

r

I strained bonds at crack tips of brittle materials can break for two reasons:

|

thermal fluctuations that are normally present in materials can cause i:

individual bonds to stretch beyond their cohesive limit so that they |

rupture; or reactive environments can preferential ly cleave highly '

strained bonds by chemical attack. Subcritical crack growth resulting

| from either process can be described in terms of reaction rate theory,

j

] and theoretical approaches of this type have been taken by a number of '

authors [9-18]. Bond rupture can also be rationalized in terms of chemical

j

j reactions, which for certain types of reactions have a great deal in

i common with fracture. Although the chemical approach to fracture is ;

I also based on reaction rate theory (so that the two are compatible), i

expressions for the rupture process can be related to the structure of i

brittle materials in a simple manner. The purpose of this paper is to

j

! treat the rupture process at crack tips as a chemical reaction and to

| show how parameters normally obtained from fracture mechanics experiments '

can be related to .parameters that are normally used to describe chemical '

4
reactions. In particular it will b shown that the slope of the stress

corrosion curve can be related to ie activation volume obtained from

chemical kinetics studies. Since ctivatior volumes are particularly

amenable to structural interpretation, this relation permits crack

growth data to be discussed directly in terms of the atomic structure of

the crack tip. The chemical approach to fracture is consistent with the

statistical mechanics models developed by other authors [ 16 ].

5
2 . Chemical Reaction Rate Theory

A brief review of the chemical approach to reaction rate theory is presented

for the reader who is unfamiliar with this subject. A complete description

of reaction rate theory can be found in reference 19.

The thermodynamic formulation of reaction rate theory is based on the assumption that reactants are in a state of equilibrium with an activated complex

that forms during the reaction. The activated complex then decomposes to '

form the reaction products. Based on this assumption it can be shown that

the rate coefficient, k^, for a single step reaction is given by [20];

k = [<^exp (-fiGyRT)l [(.1 f.)/fT, (1)

i where k is Boltzmann's constant, h is Planck's constant, R is the gas constant

and f. and f"^ are the activity coefficients of the reactants and the activated

complex, respectively. The parameter k is the so-called transmission coefficient

which accounts for the fact that for some reactions, not all activated complexes

decompose into reaction products. For most reactions, is equal to unity.

The free energy of activation can be expressed in terms of chemical potentials

for the reaction referred to their standard state:

.G* = y - (2)

Hence, is the difference in partial molar free energy between the initial i

and activated state of reaction, all referred to their standard state. As

such, can be expressed in terms of other partial molar quantities:

sG^ = + PaV^ (3)

^ f f

where cS is the activation entropy, aE ts the activation energy, and aV is the

activation volume, each of these quantities being determined from the change

in state trom the initial to the activated state of reaction.

i

6
The temperature dependence of a chemical reaction is characterized by the

enthalpy of reaction (aH^ = ae"^ + Pav"^):

3(ln k^)/3T = 1/T + aH*/RT^, (4)

provided the activity coefficients are independent of temperature. Usually,

the temperature dependence of the rate constant is determined from a plot of

In versus 1/RT, and the slope of the line is reported as the experimental

activation energy,

Eexp

= ^ <5)

Since activation energies for chemical reactions range from ^>10 to 'vlOO

kcal/mol [19], RT is usually small relative to ah"*" and can be neglected.

The pressure dependence of chemical reactions is characterized by the

activation volume, AV .

:(ln k^)yaP = aV*/RT (6)

Since the activation volume is the difference in partial molar volume

between the reactants and the activation complex, it can be modelled by

changes in molecular dimensions as the reaction goes through the activated

state.

Two important effects are usually considered in estimating the activation

volume [21]. The first effect has to do with structural changes that occur

during the reaction (i.e., whether bonds are being formed or broken) and

with the number of molecules taking place in the reaction. In liquids,

unimolecular bond rupture always leads to a positive activation volume,

whereas bimolecular bond formation always leads to a negative activation

volume [20,22,23]. ^he second effect has to do with the formation.

7
destruction or alteration of charge during chemical reactions. Charges alter

the volume of the solvent through the process of electrostriction, and the

change in volume of the solvent must be added to the change in volume of

the reacting species as they become activated. Quantitative approximations

developed to account for electrostriction indicate large negative volume

changes (-10 cm /mole) if charge formation occurs during the activation

step. As a general rule, the portion of the activation volume due to

electrostriction is larger than that due to structural changes when

reactions take place in liquid solvents [22]:

For a wide variety of chemical reactions, activation volumes are found to

3 3

range from '\^-20 cm /mole to ^-lO cm /mole [20,22,23]. An interesting

observation made by Laidler [19] is that the activation entropy, aS^,

appears to be related linearly to the activation volume, which can be

used to obtain an estimate of the activation entropy. Activation volumes

are also used to obtain information on the type of reaction occurring in

a liquid system. As noted by Laidler [19], reactions can be classified

into three types, "slow", "normal" and "fast", according to their volume

of activation. Slow reactions often involve the formation of opposite

charges, or the approach of like charges and have large negative volumes

of activation. In normal reactions, electrostatic effects are unimportant,

and volumes of activation are usually small and negative. Fast reactions

often involve the approach of opposite charges or the spread of like

charges and have positive volumes of activation. Since chemical bonds are

stretched and broken during fracture, fast reactions with positive activationvolumes are clearly most applicable for characteri zi ng crack-growth rates.

8
3. Application q

-^ Chemical Theory to Fracture

The chemical reaction rate theory discussed in the previous section was

developed for reactions occurring in a liquid environment. To apply these

ideas to fracture, two modifications to the above equations are necessary.

First, since reactions at a crack tip occur in the presence of high stress

fields, stresses rather than pressures have to be incorporated into the

external work term {'P'V'^ of eqn. 3). Second, since the crack tip can be

modelled as a highly curved surface, effects of surface curvature on

chemical potential must be considered.

The two modifications discussed in this section are developed in the same

spirit as that used in the previous section. This development is supported,

however, by a more rigorous statistical mechanics formulism for the advance

of a crack by thermal fluctuaf’ons in a solid il6j. The activation energy

barrier for ihe advancing crack involves the reaction of the chemically

interacting species with the atoms of the solid whose bonds are about to

break at the tip. The normal chemical reaction of the gas with the solid

atoms at tne crack tip will be drastically altered by the fact that the

bonds at the crack tip are violently stretched. Therefore, the activation

energy will depend upon the stress in a complex manner which can only be

calculated in terms of specific interatomic forces and configurations.

Nevertheless, in section 6 we show that an activation volume can be

derived for the crack-tip bond rupture event in a discrete model of a

solid, which supports the development in this section. Accordingly, here

we follow the lead of Charles and Hillig [24] and write an expression for

which is constructed from considerations based on continuum stress and

liquid surface tension arguments, and from this expression derive an activation volume ^or crack growth. The expression tnus derived will then be

9
compared to experiments. In section 6 the expression is shown to be

equivalent to the discrete atomistic model analysis.

Accordingly, the crack tip is assumed to have an elliptical shape, with J

a curvature equal to p. The stress at this crack tip, a, can be evaluated

from the continuum elasticity solution of an elliptically shaped hole subj

jected to an applied stress normal to the crack plane. In fracture mechanics '

terms the crack-tip stress is related to the applied stress intensity factor, i

:l Kj, and the crack-tip radius, p, by the following equation [25]:

j = 2Kj /(Tfe)'^^ (7)

Since the crack-tip stress lies normal to the fracture plane, it also lies

normal to the chemical bonds that are broken during crack propagation, and

hence, can be considered as a negative pressure applied to the atoms in the

vicinity of the crack tip. By substituting for P in eqn. 3, the reaction

rate equation can be expressed in terms of the driving force for fracture,

Kj. As will be shown below, the volume of activation can be determined

from crack propagation curves obtained by fracture mechanics techniques.

Activation volumes determined in this manner are equal to the partial molar

volume difference between the activated complex and the stressed solid that I

is undergoing reaction. Because the corrosive environment is not subjected

to stress, it does not contribute to the activation volume. This is a

significant difference in behavior between chemical reactions in a pressurized

fluid, and reactions at the surface of a stressed solid.

Surface curvature can alsO' modify the rate of chemical reaction by modifying

the chemical potential of reactants at the surface. The dependence of chemical

potential on surface curvature is given by the following equation [26]:

10
u = Uq + yV/p ( 8 )

where u is the chemical potential at a surface of curvature, p, and is

the chemical potential at a flat surface. The surface free energy of the

solid is Y and the partial molar volume of the material undergoing reaction

is V. For material undergoing reaction at a crack tip, the surface curvature is negative. Surface curvature is expected to modify the chemical

potential of the activated complex (since it lies on the surface) and the

chemical .potential of the initial reactants that form part of the solid,

but not the chemical potential of the corrosive species that are reacting

with the solid because they are not part of the surface. The effect of

curvature on the activation free energy is obtained by substituting eqn. 8

into eqn. 2:

= :"o

- yV)/p : 9 )

The fact that p is negative is taken into account in eqn. 9 by use of a

minus sign in front of the term in parentheses.

By modifying eqn. 3 to account for crack-tip curvature and crack-tip stress,

the following equation is obtained for the free energy of activation:

iG" = -TiS^" + -[2 Kj/(ttd)^/^]aV'' - (y^v"^-yV)/p . (10)

From eqn. 10, we see that the application of a load to a sotid containing

a crack accelerates the reaction provided the activation volume for the

reaction is positive. If the activation volume is negative, the reaction

at the crack tip will De inhibited by the application of an applied load,

and subcritical crack growth will probably not occur. The effect of curvature on the reaction will depend on the relative values of the surface free

energies anc of molar volumes in eqn. 10.

11
4. Experimental Data

A considerable body of experimental data has been collected to characterize

subcritical crack growth in glasses and ceramics. By far, the largest

amount of data has been collected on commercial glasses, for which the

crack growth rate has been characterized as a function of applied stress

intensity factor, temperature and environment. Data have also been obtained

on single crystal aluminum oxide [27] and on alkaline earth fluorides [28],

however these data are not as reliable, nor as extensive as those obtained

on glass and will not be discussed in this paper. Data obtained on polycrystalline ceramic materials will also not be discussed because of

complications introduced by the polycrystal 1 ine nature of these materials.

Crack velocity studies on soda-lime-silica glass in nitrogen gas containing

water vapor indicate three regions of crack growth, Figure 1, that depend

on stress intensity factor and on percent relative humidity of the environment [29]. In region I, the crack growth rate is proportional to the partial

pressure of the water in the gas for a given stress intensity factor, and

the slope of the curves is approximately equal to 0.1/RT in MKS units. In

region II the crack growth rate is weakly dependent on the stress intensity

factor, and is linearly proportional to the relative humidity of the water

in the nitrogen gas. In region III the crack growth rate depends only on

the applied stress intensity factor; the slope of the curve is approximately

equal to 0.9/RT. Within experimental error, data obtained on soda-lime-silica

glass in vacuum at room temperature [30] are identical to the region III data

shown in Figure 1, which suggests that region III crack growth represents a

fracture process that is environment independent.

12
Studies conducted on a variety of commercial glasses in water indicate

that the data obtained are similar to those obtained for region I crack

propagation in nitrogen gas [24]. The position of the crack propagation curve is sensitive to temperature, crack propagation occurring more

easily as the temperature is increased. Figure 2. Experimental crack

growth data have been fit by the method of least squares to the following

empirical equation;

U = Uq exp [(-E* + bKj)/RT] (11)

where u^, E* and b are empirical constants of the fit. A summary of these

constants is given in Table 1.

Data collected on other commercial glasses as a function of temperature in

vacuum could also be fit to eqn. 11 [30]. Subcritical crack growth was

observed for glasses that had normal elastic properties, Figure 3. Generally,

the slopes, b/RT, obtained for these glasses were higher than those obtained

in water (Table 1). High silica content glasses such as silica, chemical

borosil icate, and low thermal expansion titania-sil ica glasses have anomalous

elastic properties. These glasses did not exhibit subcritical crack growth

in vacuum, even though crack growth was observed when the glasses were

tested in water [30,34].

The importance of water to crack propagation in glass was emphasized by

recent studies of crack growth in alcohols [31,32]. As is illustrated in

Figure 4 for butyl alcohol, the curves obtained in alcohols are similar

to those obtained in nitrogen gas containing water. In region I the

shift in the crack velocity curves is proportional to the partial pressure

of wacer in the alcohol. Within experimental error, the curve obtained

13
in fully saturated butyl alcohol is identical to that obtained in water.

Similar findings have been obtained for silica glass [33] and for a 7.5;o

Ti02 silica glass tested in water saturated air [34]. However, the crack

velocities measured on scda-1 ime-si 1 ica glass appears to be slightly

higher in water than in water saturated gas (Figure 1).

^14
5. Comparison of Theory and Experiment

Consistency between theory and experiment require predictions from reaction

rate theory to agree with experimental measurements of the dependence of

crack growth rate on temperature, stress intensity factor and concentration

of reactive species in the environment. As will be shown, reasonable predictions of behavior are obtained in all three cases.

5. 1 Dependence of Crack Propagation Rate on Concentration

Under the influence of chemical environment the rate of crack propagation

is predicted to be proportional to the concentration of reactive species

times the reaction rate constant. For water attack on glass in a liquid

or gaseous environment, the crack velocity is given by the following

equation

;

o = [H

2

O] k^ (12)

where [H^O] is the concentration of water in the environment, and is a

proporti onal i ty constant that incorporates the frequency factor and the

concentration of active sites at the crack tip. Since [H.,0] = a ,/f ,

where a is the chemical activity of the water in the environment, and

f is the activity coefficient, the crack velocity is predicted to be wproportional to the chemical activity of the reacting species. In gases,

the chemical activity is proportional to the partial pressure of water

vapor so that the crack growth rate should be proportional to the partial

pressure of water in the gas. This prediction is confirmed by crack growth

studies in soda-lime-silica glass [29], in which the crack velocity is

found to be proportional to the partial pressure, p, of water vapor in

the gas for a ^ide range of partial pressures. At low partial pressures

1 /2 of water, however, the crack growth rate is proportional to p , suggesting

15
a possible change in fracture mechanism. Freiman [31] has recently obtained

similar results for crack growth in alcohols containing varying amounts of

water in solution.

The finding that water saturated liquid or gaseous environments give the

same crack propagation curve for region I crack growth data is also consistent with reaction rate theory. To demonstrate this point, we note that

[H^O] = 1 for water. In addition, k == exp(u /RT), where u is the

chemical potential of water in its standard state. From eqn. 12:

For the water saturated environment [HoO] = a /f , where a and f are,

2 s s s s s respectively, the activity and the activity coefficient of water in the

saturated environment. Since f^ of the water concentration term in eqn. 12

cancels with f^ in eqn. 1, the crack velocity is given by:

u - u^a^ exp(uQ^/RT), (14)

where u is the chemical potential of water in the saturated solution

os referred to its standard state.

At saturation, the chemical potential of the water and the water saturation

solution are identical:

' + RT tn a^ os s 15) ow s By substituting eqn. 15 into eqn. 14, we obtain eqn. 13, which shows that

the crack velocity in water saturated environments should be identical to

that in water for a given applied stress intensity factor.

16
Lest the impression be left that the curves for saturated environments are

the same over the entire range of crack growth behavior, we note that

region II in different environments occurs at different crack velocities.

Region II appears to be more dependent on the absolute concentration of

the water in the environment rather than on its chemical potential [32].

-4

Thus, the plateau for water saturated nitrogen occurs at x 10 m/s;

-4

the plateau for water saturated butanol occurs at "-10 m/s; and

the plateau for water saturated decanol occurs at -^^lO’^m/s. The plateau

for pure water occurs at ^10 ^m/s, but is the result of viscous drag on

the crack surfaces rather than a limit on the rate of transport of water

to the crack tip [35]

.

5.2 Dependence of Crack Propagation Rate on Stress Intensity Factor

For chemical reaction rate theory to be consistent with empirical observations

of crack propagation in glass, the term containing the stress intensity factor

in eqn. 10 must be set equal to the term containing the stress intensity

factor in eqn. 11. This equality leads directly to an expression for the

volume of activation in terms of the crack-tip radius and the slope of the

crack propagation curve:

iV* = (b/2) (16)

As shown in Table 1 values of b for glass determined from crack propagation

studies in water range from "-0.110 to ''-0.216 m^^'^/mol, which corresponds to

3 activation volumes ranging from '^2.3 cm /mol for soda-lime-silica glass to

3

'^'4.3 cm /mol for silica glass. This calculation assumes p = 0.5 nm which

corresponds roughly to the size holes expected in the glass network structure.

The degree of bond stretching can be estimated from the activa.tion volume

by assuming that stretching occurs along an axis of a cylinder of constant

17
cross section. The cross section of the Si-0 bond in glass can be taken

to be the mean of the cross sections of the Si and 0 atoms which are

calculated from the radius r of these atoms in glass (r^^ = 0.04 nm; Tq = ,

.1 0.14 nm [36]). The equation used by Hamann [22] to calculate the bond ;

elongation occurring as the bond passes through the activated state is:

(r^. + r^)ol/2 (17)

where is Avogadro's number. Using this formula, values for 52, of '^^0.1 nm

and -^0.2 nm are calculated for soda-lime-silica glass and for silica glass,

respectively . Thus, the degree of strain based on the unstressed bond length

of 0.16 nm ranges from -^^63 to 131 percent. This degree of stretching is

similar to that estimated for organic molecules undergoing free radical or

molecular dissociations [22].

Based on crack propagation data collected in vacuum, the activation volume

for crack growth and the degree of stretching in vacuum are much larger

than that just calculated for crack growth in a moist environm.ent . For sodalime-silica glass, for which the only network former is Si 02 , b = 0.88

/mol, and the activation volume determined from eqn. 16 is '^16 cm /mol,

which is 8 times that found for crack propagation in v;ater. Although this

value is at the high range of that reported in the literature for activation

volumes, it is not excessively high. The degree of bond stretching that is

calculated from eqn. 17 is 0.8 nm, which is approximately 5 times the normal

Si-0 bond length in the unstretched state. This high degree of stretching

is probably unrealistic and results from the assumption that only the crack-ti

Si-0 bond participates in the bond rupture. If interatomic interations are

such that the bond rupture is a cooperative process (i.e., -a wide cohesive

region around the crack tip), then the activation volume corresponds to

i nteraztions t''r-;u:.no jt the cohesive region as well as to the di spl aceir.ent

of the crack-tip bona.

18
One possible explanation for the large activation volume for the vacuum

data results from the fact that the Si-0 bond has a partial ionic character,

37 percent [37], so that long range coulombic forces have to be overcome

for fracture to occur. Water molecules, by reacting with the Si-0 bonds

at the crack tip, partially cancel coulombic attraction since the OH

bond is also partially ionic, 29 percent. Thus, in water only short

range valence bonds would have to be overcome for bond rupture to occur,

and the degree of bond stretching in water would be less than that in

vacuum.

Another explanation for the large activation volume may be electrostriction

effects, which are known to accompany chemical reactions in liquids. These

effects arise as a result of an electrostatic interaction between charged

chemical species and the dielectric medium in which the reaction takes

place. Using the Born [33] approximation for the change in Gibbs free

energy requireo to transfer a charge of magnitude Ze from a vacuum to a

medium having a dielectric constant D, the molar volume change,

resulting from electrostatic interactions between the charged species

and the medium is given by [20]:

={N^Z^e^/2r)(|A + (l-O''') (18)

where P is the applied pressure and r is the radius of the charge. For

reactions in liquids 3in r/aP is always negative, and aD’VsP is

usually negative [20]. Consequently, has a negative value in

liquids if charges are created, but a positive value if charges are

destroyed during the reaction. The electrostatic contribut-ion to the

activation volume is found to range from ''^-10 to -30 cm^/mol in most

sol vents

.

19
Although electrostrictive forces are also expected to occur in solids, the

effect will be quite different from that occurring in solution. First,

the bulk modulus of solids is approximately 100 to 1000 times greater

than that of liquids, so that the second term of eqn, 18, which depends

on the compressibility of the medium, will be very much smaller in solids

than it is in solutions and can probably be ignored in the calculation of

AVg^. Second, the dielectric constant of solids decreases with increasing

pressure [39], which means that the first term in eqn. 18 will be positive,

instead of negative as it is for liquids. Consequently, charge formation

in solids results in a positive activation volume. The magnitude of the

effect can be estimated from eqn. 18 for the scission of Si-0 bonds. From

-1 15 2 work by Reitzel [40] on silica glass, jD /3P is approximately 4 x 10 m /N.

Assuming that unit electric charges (i.e., Z = 1) are formed on both the

silicon and oxygen atoms as the bond between these atoms breaks, , is

equal to ^^2 cm'^/mol for oxygen and 7 cm'^/mol for silicon. The combined

3

volume change "-9 cm /mol is approximately 56 percent of the total activation

volume for fracture.

The effect of el ectrostri ction is to partition the volume of activation

between the crack-tip bond and the rest of the solid surrounding the crack

tip. For Si-0, the activation volume associated with the crack-tip bond

3 is approximately 7 cm /mol which corresponds to a o'.’ of '^0.35 nm. This

value is 2.2 times the normal Si-0 spacing and is more reasonable than

the value quoted earlier. Consequently, el ectrostriction seems to play

an important role in the fracture of brittle materials, as it does in

chemical 'reactions carried out in solutions.

20
5.3 Zero Stress Activation Energy Term

When stresses are completely removed from a specimen containing a crack,

chemical reactions can still occur at the crack tip. The activation energy

for crack motion is then given by the zero stress intensity factor portion

of the activation free energy for crack growth, eqn. 10. If the concentration of reactant at the crack tip also depends on temperature, its temperature

dependence will have to be added to the activation free energy for crack growth.

The sum of these two terms should be equal to E* (eqn. 11), which is an

empirical measurement of the zero stress activation energy for crack growth:

aE^ - - yV)/p] + n aH^ = E* . (19)

The term nAH.|is the temperature dependence of the concentration of reactive

species, and is determined by chemical reactions within the solution, or

between the solution and the crack surface, or on the crack surface near

the crack tip. The temperature dependence of the concentration of reactive

species will therefore be determined by the equilibrium constants of these

reactions, which have an Arrehenius dependence on temperature. Measurements

of the oh” ion concentration of slurries of ground glass in water [41], for

example, indicate a temperature dependence given by the following relation:

[0H”l = A exp (aH^/RT) (20)

t h If the crack tip reaction is n^' order with regard to the active species in

solution, then the term to be added tc the zero stress activation energy is

nAH-j , as indicated in eqn. 19.

Equation 19 can be used to compare crack propagation data with corrosion rates

that occur at normally unstressed surfaces. Based on the theory presented

earlier, ae'*' is the activation energy for corrosion at a free surface.

Consequently, by evaluating E*, aH.j and the surface curvature terms of

21
eqn. 19, aE* can be evaluated from crack growth data. The value of ae"^ so

obtained can then be compared with independent measurements of corrosion

rates in the reactive environment of interest. Since all of the necessary

data is available, comparison of this sort will be made for the attack of

hydroxyl ions on the network structure of glass in alkaline environments.

Experiments have shown that at a pH greater than 9, hydroxyl ions directly

attack the network structure of glass to cause corrosion [42,43]. Hubbard

and Hamilton [42] measured the activation energy for the corrosion of a

variety of glass in a 5 percent solution of NaOH in water, and obtained

values of the activation energy that ranged from 15.3 to 19.7 kcal/mol.

The order of the reaction was not studied specifically. For purposes of

this paper, a first order reaction, n = 1, will be assumed.

In order to determine the probable pH at crack tips in glass, measurements

were made on slurries of water and glass [41]. The pH determined on four

of the glasses shown in Table 1 were high enough for direct attack of

hydroxyl ions on the network structure of the glass. Values of aH-j , obtained

from temperature studies on these glasses are shown in Table 2.

The surface tension term can be evaluated directly from crack propagation

data. At the stress corrosion limit, the corrosion rate of the glass at

the crack tip will be identical to that occurring at a free surface, so

that the crack will not increase in length relative to the surface, and

catastrophic fracture will not occur. The condition for the stress

corrosion limit is obtained by equating the surface curvature term and

22
( 21 )

the stress intensity factor term in eqn. 10;

aV* = -[ (y*V*' -yV)/p]

or using the equality between the stress intensity factor terms in eqns. 10

anci 1 1

‘>'^1500 '

( 22 )

Soda-lime-silica glass is observed to undergo a rapid decrease in crack

velocity at stress intensity factors that are slightly greater than 0.25

MPa-m

1/2 and it is believed that this decrease in velocity indicates an 1/2 approach to a fatigue limit for this glass. Assuming ~ MPa-m

bKiscc = 2.75 X 10^ J/mol (6.6 kcal/mol). This value can be used to

estimate the change of surface energy during the fracture process. Since

4 3 3

aV = 2 cm /mol, and since V = 25 cm /mol for soda-lime-silica glass, an

estimate for the right hand sice of eqn. 22 can be obtained by using a mean

value for the molar volume terms. Assuming = V = 25 cm^/mol , and p = 0.5

4 2 nm, it follows that y-y =0.55 J/m .

Estimates of the corrosion rate of glass in basic solutions are presented

in Table 2, using measured values of aH-j , and E*. In the absence of

additional data on the stress corrosion limit in the glasses studied, it

is assumed that the surface tension term of eqn. 19 is the same for the

four glasses shown in Table 2. The activation energies for corrosion

calculated from crack propagation data fall within the range measured by

Hubbard and Hamilton [43]. This favorable comparison supports the arguments

presented in this paper that crack growth in brittle materials can be

/ explained in terms of chemical reaction rate theory.

23
6. An Atomic Model of Crack Growth

In previous sections of this paper, the material surrounding the crack tip

was treated as if it were a continuum, a procedure that may be questioned

when dimensions of the order of the atomic spacing are being considered.

In this section, a somewhat simplified atomistic model of fracture is

applied to the problem of chemical reactions at crack tips. It is shown

that the assumptions used in previous sections of the paper are valid for

estimating the activation volume for crack growth. In addition, the

partitioning of energy between the internal energy of the crack-tip bond,

and work terms that involve the remainder of the solid is shown to be

appl icable.

The atomistic model of the crack used in this section of the paper (Figure 5)

was proposed first by Fuller and Thomson [13,14]. This quasi -one-dimensional

model of a crack consists of two semi-i nf i ni te chains of atoms that maintain

their rigidity by means of an interatomic interaction which is modelled as a

flexual force with a force constant 6. The chains of atoms are bonded

together by a series of cohesive interactions that are modelled as stretchable spring elements between the atom pairs. A crack is formed by assuming

that the first n stretchable elements are ruptured, or stretched beyond

their range of interaction. With the exception of the crack-tip element,

the remaining cohesi ve-spri ng elements are assumed to be in the linear

regime of their force-displacement response with a spring constant a. The

crack-tip element is assumed to be a non-linear element, which reflects

the inherent cohesive strength of the solid. A transverse opening force,

F, is applied to the end atoms of each chain to force the crack open.

24
The total potential energy of this crack system is given by [14]:

U = -Fo + 7T 6 Z • 0 4 j=i

* " ^ ^ h "j = n+l (23)

where the first term on the right-hand side is the potential energy of the

loading system; the second term is the strain energy stored in the flexural

bonds; and the remaining terms are the contributions from the cohesive bonds

across the crack. These cohesive contributions are, respectively, the total

bond energy from the n ruptured bonds, the bond energy of the nonli nearly

strained bond at the crack tip, and the linear-elastic strain energy of the

remaining cohesive bonds. From a fracture mechanics viewpoint, this potential

energy provides both the driving force for fracture and the resistance force

to fracture. The crack driving force is derived from the stored elastic

strain energy which is released upon crack advance; whereas, the resistance

to fracture is contained in the cohesive-bond energy of the crack-tip bond

that must be ruptured to advance the crack.

In contrast to the Griffith continuum analysis, where the fracture resistance

is provided by the average cohesive energy required to form a new element of

surface, the atomistic treatment reveals a fine structure in this process

through the discrete nature of the bond rupture event. To illustrate this

atomistic influence, the variation of the system potential energy is examined

as the crack-tip bond ruptures. For each crack-tip bond displacement, 6^,

the remaining atoms of the solid are required to be in an equilibrium

configuration with respect to this displacement. Necessary conditions

for these equilibrium configurations are:
(24)

!

( 3U/36j ) = 0, for j / n (24)

which correspond to an infinite set of fourth-order difference equations.

Analytical solutions of these equations are obtainable [14] in the functional

form n, 6^). Substitution of these equilibrium-displacement solutions

into eqn. 23 reduces the system potential energy to an explicit function of

both the applied fracture mechanics parameters (load and crack length) and

the crack-tip bond displacement 6 : n

U (F, n, = n Ugg (F^/6b) (2n^ + 3n^%+ n)

, 2/_2

(25)

where ^ is an elastic coefficient defined by c (^ -1) = (23/ot), and K

=

(n+c)F is an effective one-dimensional "stress intensity factor".

This expression is particularly useful for demonstrating the origin of the

atomistic energy barrier to intrinsic bond rupture and crack advance.

Focusing attention on the n^' bond, the first two terms on the right-hand

side of eqn. 25 are constant for a given applied force and will not be

considered further. The next term, corresponds to the reduction

in potential energy that results from the applied fracture driving force

as the crack-tip bond ruptures. The two remaining terms represent the

resistance of the system to the rupture of the crack-tip bond. The first

of these resistance terms, Uj^(6^)', corresponds simply to the interatomic

1 "-1 2 interaction between the crack-tip atoms; whereas the second term, 2”

is more subtle and corresponds to the harmonic restoring potential exerted

on the crack-tip atoms by the remainder of the (linear elastic) structure.

It is the balance between these last three terms that determines the atomistic

energy barrier, as is illustrated graphically in Figure 6. The individual
terms are plotted in Figure 6a and their sum is shown in Figure 6b. Clearly,

the stress intensity factor term, -K6^/c, biases the bond rupture to make

the final state the more stable configuration in this example.

An important part of the above construct is that the interatomic potential

between the crack-tip atoms is decoupled from the interations of the rest

of the solid. This construct provides a model whereby an arbitrary bondrupture reaction can be studied in an equivalent linear-elastic continuum

of crack-tip stiffness In particular, the construct can be used

to study the effect of chemical reactions on crack propagation, in which

case, a modified cohesive force law is substituted for the interatomic

potential of the crack-tip atoms. The initial and final equilibrium

configuration in Figure 6b now correspond to unreacted and reacted states

of the bond respectively. The unstable configuration corresponds to the

activated complex of the reaction. The exact influence of the chemical

reaction on the rupture of the crack-tip bond depends on the details of

the chemical reaction and its activation barrier. Various possibilities

have been considered in a recent manuscript [18]. The main effect of

the chemical reaction is to further bias the intrinsic activation energy

barrier and to allow crack growth to occur at a lower applied stress. An

important conclusion of this work is that the thermodynamic surface energy,

which determines the thermodynamic condition for crack propagation, is

lowered to that of the chemically reacted surface.

The construct can also be used to estimate the activation volume for fracture.

Consider the two equilibrium configurations corresponding to the initial

/
stable configuration (unruptured, crack-tip bond) and to the saddle-point

configuration in the potential energy curve (the activated state for the

crack-tip bond). The bond displacement at these two configurations are

denoted by 6^^ and respecti vely (Figure 6b). The energy barrier, aU_^,

required to reach the activated complex is given by;

aU, = U(F, n, ) - U(F, n, 5 ) no'

_ rii ^ i dU, 1 /C-1 ^ . 2, , '^n

( 26 : d6. no

where the energies in each equilibrium configuration, U(F, n, 6^ ) and

U(F, n, 5^^), are obtained from eqn. 25. The dependence of this activation

energy barrier on the fracture driving force, K, is obtained by taking the

derivative of eqn. 26 with respect to K (utilizing the stability condition of

each state to obtain the K dependence of 5 and 9 *):

^ no n

d(AUJ

dK -(6^-5 )C ^ n no

1

(27)

Since K has units of force in the simple atomistic model used in the present

construct, an applied crack-tip stress can be defined by dividing K/^ by the

effective area. A, of the bond at the crack tip, Figure 7:

- = K/(i'A) (28)

Substituting this relation into eqn. 27, the following relation is obtained:

d(AU^)

da

- A (5 - 6 ) = -uV ^ n no

(29)

where aV is the activation volume for the crack-tip reaction. This

definition of aV^ is identical to that used in the literature of chemical

reaction rate theory. Thus, it has been shown that an atomistic model of

crack growth is consistent with a model of crack growth derived by

28
treating the crack tip as an elastic continuum. This finding supports the

assumption that crack-tip stresses can be treated by a continuum approach

to fracture.
REFERENCE

1. B. J. Hockey and B. R. Lawn, J. Mater. Sci . 1_0, 1275 (1975).

2. B. R. Lawn, B. J. Hockey and S. M. Wiederhorn, J. Mat. Sci., in press (1980),^

3. J. Dunning and D. Dunn. Geol . Soc. of Amer., Annual Meeting, Toronto,

Abstract with Programs, 1_0 [7], 395 (1978).

4. S. M. Wiederhorn and P. R. Townsend, J. Am. Ceram. Soc. 486 (1970).

j;

5. S. M. Wiederhorn, J. Am. Ceram. Soc. 99 (1 969).

[ 6. R. J. Jaccodine, J. Electrochem. Soc. 1 1 0 , 524 (1963). li

7. W. B. Hillig, "The Factors Affecting the Ultimate Strength of Bulk Fused

Silica," Symposium sur la Resistance Mecanique du Verre et les Moyens de

I'Amelioree, Union Scientifique Conti nentale du Verre, Chari eroix, Belgium,

(1962).

8. B. A. Proctor, I, Whitney and J. W. Johnson, Proc. Rog. Soc. A 297 ,

9. D. A. Stuart and 0. L. Anderson, J. Am. Ceram. Soc. 41 6 (1953).

10. S. M. Cox, J. Glass. Tech. 127T (1948).

11. B. R. Lawn, J. Mater. Sci. 10, 469 (1975).

12. G. M. Bartenev and I. V. Razumovskaya , pp 1871-80, in Proc. 1st Int. Conf. |li on Fracture, Vol . 3, Sendai, Japanese Society for Strength and Fracture f

of Materials (1966) .

1

13. E. R. Fuller, Jr. and R. M. Thomson, "Nonlinear Lattice Theory of Fracture,":

pp 387-394 in Fracture 1977, Vol. 3, edited by D. M. R. Taplin, Univ. of

Waterloo Press, Canada, (1977).

14. E. R. Fuller, Jr. and R. M. Thomson, "Lattice Theories of Fracture,"

pp 507-548 in Fracture Mechanics of Ceramics, Vol. 4, edited by R. C. Bradt,,

D. P. H. Hasselman, and F. F. Lange, Plenum Publishing Corporation, (1978). ;

15. E. R. Fuller, Jr. and R. M. Thomson, "Theory of Chemically Assisted

Fracture," pp 485-494, in Proceedings of the Third International Conference

on Mechanical Behaviour of Materials, Vol. 2, edited by K. J. Miller and

R. F. Smith, Pergamon Press, (1979).

16. R. Thomson, J. Mat. Sci., in press (1980).

17. E. R. Fuller, Jr. and R. M. Thomson, J. Mat. Sci. in press (1980).

i

30
18. E. R. Fuller, Jr,, B. R. Lawn, and R. M. Thomson, "Atomic Modelling

of Crack-Tip Chemistry," submitted to Acta Met.

19. K. J. Laidler, Chemical Kinetics, McGraw-Hill Book Co., New York (1965).

20. G. Kohnstam, Kinetic Effects of Pressure, pp 355-408 in Progress in

Reaction Kinetics, G. Porter, Editor, Pergamon Press, London (1970).

21. M. G. Evans and M. Polanyi, Trans. Faraday Soc. 4^, 875 (1935).

22. S. D. Hamann, "Chemical Kinetics" pp 163-207 in High Pressure Physics

and Chemistry, R. S. Bradley ed. , Academic Press, London (1963).

/* 23. C. A. Eckert, Ann. Rev. Phys. Chem. 239 (1972).

24. R. J. Charles and W. B. Hillig, pp 511-527 in ref. 7.

25. S. M. Wiederhorn and L. H. Bolz, J. Am. Ceram. Soc. 53, 543 (1970).

26. A. W. Adamson, Physical Chemistry of Surfaces, Interscience Publishers

New York (1967).

27. S. M. Wiederhorn, Int. J. Fract. Mech. 171 (1968).

28. P. F. Becker and S. W. Freiman, J. Appl . Phys. 3779 (1978).

29. S. M. Wiederhorn, J. Am. Ceram. Soc. 407 (1967).

30. S. M. Wiederhorn, H. Johnson, A. M. Diness and A. H. Heuer, J. Am. Ceram.

Soc. 336 (1 974).

31. S. W. Freiman, J. Am. Ceram. Soc. 350 (1974).

32. S. W. Freiman, S. M. Wiederhorn and C. J. Simmons, J, Am. Ceram. Soc.,

submitted for publication.

33. S. M. Wiederhorn, unpublished data.

34. S. M. Wiederhorn, A. G. Evans, E. R. Fuller and H. Johnson, J. Am. Ceram.

Soc. 57, 319 (1974).

35. T. A. Michalske and V. D. Frechette, "Dynamic Effects of Liquids on Crack

Growth Leading to Catastrophic Failure in Glass", J. Am. Ceram. Soc. to

be published.

36. B. A. Proctor, Appl. Mat. Res. 28 (1964).

37. M. F. C. Ladd, Structure and Bonding in Solid State Chemistry, Ellis Harv/ood

Limited Chichester (1979).

38. M. Born, Z. Physik., 1_, 45 (1920).

31
I

39. R. S. Bradley, "Other Miscellaneous Effects of Pressure^' pp 325-37

in High Pressure Physics and Chemistry, Vol . 2, R. S. Bradley, ed.

Academic Press, London (1963).

40. J. Reitzel, Nature, Lond. 1 78 , 940 (1956).

41. S. M. Wiederhorn, J. Am. Ceram. Soc. 81 (1972).

42. R. W. Douglas and T. M. M. El-Shamy, J. Am. Ceram. Soc., (1967).

43. D. Hubbard and E. H. Hamilton, J. Res. Nat. Bur. Stand., ^ 143 (1941 ).

44. S. M. Wiederhorn, "Subcri tical Crack Growth in Ceramics," pp 613-646

in Fracture Mechanics of Ceramics, Vol. 2, R. C. Bradt, D. P. H. Hasselman

and F. F. Lange eds. Plenum Publishing Corp. New York (1974).
Table 1 - Summary of Stress Corrosion Data

Envi ronment Glass E* kcal/mol

( kJ/mol

)

b (m^^^/mol

)

In Ujj

Water [25] Si 1 ica 33.1

(139)

0.216 -1 .32

Aluminosilicate I 29.0

(121)

0.138 5.5

Aluminosilicate II 30.1

(126)

0.164 7.9

Borosil icate 30.8

(129)

0.200 3.5

Lead A1 kal

i

25.2

(106)

0.144 6.7

Soda-Lime-Si 1 ica 26.0

(109)

0.110 10.3

Vacuum [30] 61^^ Lead 83.1

(348)

0.51 5.1

Aluminosilicate III 176

(705)

0.77 14.1

Borosil icate Crown 65.5

(275)

0.26 6.6

Soda-Lime-Si 1 ica 144

(605)

'0.88 -5.7

I

33
Table 2 - Calculated Activation Energy for Corrosion,

o m kcal/mol

Glass E* - Yv)/p AH -, aE

^

Aluminosilicate I 29.0 6.6 4.7 17.7

Aluminosilicate II 30.1 6.6 2.4 21.1

Lead A1 kal

i

25.2 6.6 2.1 16.5

Soda-Lime-Silica 26.0 6.6 1.0 18.4

Hubbard and Hamilton [42] E^ = 15.3 to 19.7

34
1. Dependence of crack velocity on applied stress intensity factor,

soda-lime-silica glass. The percent relative humidity for each set

of runs is given on the right-hand side of the diagram. Roman numerals

identify the different regions of crack propagation. (Wiederhorn, ref.

43; original data from ref. 29).

35

1
II

I

I Influence of temperature on fracture behavior of glass in v/ater. Some 'I data, such as that given above for soda-lime-silica glass, v;ere curved at j low values of indicating an approach to a static fatigue limit. Other '

data, such as that for silica, were straight for the entire range of Kj.

j Curved data were fit by the method of least squares only over the linear

portion of the data to obtain the summary given in Table 3. (Wiederhorn,

. I rer. aDj.

36
Influence of temperature on the fracture behavior of soda-lime-silica glass

-4 -2

tested in a vacuum of 10" torr (10 Pa). (Wiederhorn, Johnson, Diness

and Heuer, ref. 30)

.
Stress Intensity Factor, k|, MN3/2

Fracture behavior of soda-1 ime-si 1 ica glass tested in butyl alcohol

solutions that contained different quantities of dissolved water. (Freiman,

Vii ederhorn and Simrr.ons, ref. 32.)

38
F

5. Atomistic model of crack. (Fuller and Thomson, ref. 13, 14).

39
6. Schematic diagram of the potential energy terms involved in the atomistic

model of fracture shown in figure 5; (a) plot of the three individual

potential ene**gy terms; (b) plot of the sum of the three energy terms

for a potenrial barrier for fracture. (Fuller and Thomson, ref. 14).

40
Resistance

From Solid

7. Schematic diagram depicting the applied force for fracture K/c, the

cross-sectional area, A, of the strained bond and the bond displacement,

5^^ - 5 , during activation. The figure also depicts the resistance force

n no from the crack-tip bond, dUj^/d6^, and the hanrionic restoring force,

a(c-l)(5^/2, exerted on the crack-tip bond by the remainder of the linearelastic structure.

41
NBS>114A (REV. 9-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-2023

3. Acfie$sS9>t Mo* -m

4. TITLE AND SUBTITLE

"Micromechanisms of Crack Growth in Ceramics and Glasses in

Corrosive Environments"

5. Publication Date MAY 1980

Code

7. AUTHOR(S)

S.M. Wiederhorn, E.R. Fuller, Jr., and R. Thomson

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fStreer. City, state, ZIP) 13. Type of Report & Period Covered

Same

15. SUPPLEMENTARY NOTES

I

I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200~word or less factual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.) At normal temperatures and pressures, water is known to have a strong influence on the

strength of ceramics and glasses. Behaving as a stress-corrosion agent, water causes

these materials to fail prematurely as a consequence of subcritical crack growth. A

basic premise of this paper is that stress-corrosion cracking of ceramics is a chemical

process that Involves a stress-enhanced chemical reaction between the water and the

highly stressed ceramic near the crack tip. Plastic deformation is believed to play

no role in this fracture process. After a brief survey of chemical reaction rate

theory, the basic rate equation from this theory is modified to reflect physical and

chemical processes that occur at crack tips. Modification of the rate equation is

based on the assumption that the crack tip can be modelled as an elastic continuum,

an assumption that is supported by a simple atomistic model of crack growth. When

tested against experimental data collected on glass, the theory was found to be

consistent with measurements of the crack-growth dependence on temperature, applied

stress intensity factor, and concentration of reactive species in the environment.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first tetter of the first key word unless a proper name;

separated by semicolons)

Ceramics; Glass; Fracture; Strength; Stress-corrosion cracking; Sub-critical crack

growth

.

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