NBSIR 74-442

High Temperature Slow Crack

Growth in Ceramic {Materials

A. G. Evans

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, 'D. C. 20234

February 1974

Interim Report for Period September 1, 1973 - January 31, 1974

Prepared for United States Air Force

Air Force Systems Command

HQ 4950th Test Wing. 4950/PMMA

Wright 4>atterson AFB, Ohio 45433

Delivery Order F 33615 73 ^Vl ^501
V
NBSIR 74-442

HIGH TEMPERATURE SLOW CRACK

GROWTH IN CERAMIC MATERIALS

A. G. Evans

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

February 1974

Interim Report for Period September 1, 1973 - January 31, 1974

Also to be published in Proceedings of Conference on Ceramics for High Performance Applications, Hyannis, Mass., Nov. 13-16, 1973.

Prepared for United States Air Force

Air Force Systems Command

HQ 4950th Test Wing, 4950/PMMA

Wright-Patterson AFB, Ohio 45433

Delivery Order F 33615-73-M-6501

Jul.

U. S. DEPARTMENT OF COMMERCE. Frederick B. Dent. Secretary

NATIONAL BUREAU OF STANDARDS. Richard W. Roberts. Director

ABSTRACT

High temperature slow crack growth processes in several cercimic

materials are examined under static and cyclic loading conditions. Data

obtained at temperatures up to 1400 "C are used for purposes of failure

prediction and for analysis of the slow crack growth phenomena. It is

shown that purity plays a major role in slow crack growth resistance,

particularly in the hot pressed materials, and that cycling in the low

frequency regime does not significantly increase the rate of slow crack

growth. The slow crack growth mechanisms appear to be primarily

plasticity related. Two semi-quantitative mechanisms are presented,

one due to dislocation motion and the other due to grain boundary

sliding.

i

- 1 -

1. INTRODUCTION

Slow crack growth is a widespread phenomenon and several reviews

of this extensive topic have appeared in the recent literature .

'

The content of this paper is thus restricted primarily to the more

recent developments; especially in slow crack growth at elevated

temperatures. High temperature slow crack growth occurs in a number

r3 41 of ceramic systems; e.g., silicon nitride above 1000°C, ' alumina

above 800 °C;

^''^ and since this temperature regime is a critical one for

the structural applications of primary interest at this symposium,

it seems appropriate to address this problem in some detail.

First, the fracture mechanics techniques for evaluating slow crack

growth at high temperatures are described, for both static and cyclic

loading conditions. Then, data obtained for a variety of materials are

described. Since purity is one of the most important microstructural

variables affecting high temperature slow crack growth, impurity effects

are examined in a controlled way. Then some conclusions are reached

concerning the requisite types and distributions of impurity for good

slow crack growth resistance.

The primary role of slow crack growth studies is the application of

[ 6 ] the slow crack growth data to failure prediction. Hence, the

analytical procedures for developing the time dependent failure

parameters from slow crack growth data are examined. Then the materials

with the greatest potential for effective structural use at very high

temperatures are identified and the possibilities for further
- 2 -

improvements discussed.

Finally, the data are used to discuss the processes that lead to

slow crack growth in ceramic systems at high temperatures. We identify

two fundamental processes; crack propagation due to grain boundary sliding,

and crack propagation due to dislocation activity. Semi-quantitative

models for both processes are developed and correlated with data for

silicon nitride, silicon carbide, and alumina.

2. TECHNIQUES ...

The fracture mechanics specimen which has the greatest versatility for

effective slow crack growth rate measurements at elevated temperatures is

the double torsion specimen ^"^^ (Fig. 1). This specimen can be most easily

[8,9" used for slow crack growth rate measurements under fixed grip conditions; '

where the crack growth rate is determined directly by the rate of load

relaxation. This approach is only effective when crack growth is not

accompanied by gross plasticity, and unfortunately, several of the important

structural ceramic materials exhibit gross plasticity under typical

[4] operating conditions. But, the specimen can be used in a different mode,

fixed displacement rate, to obtain valid slow crack growth rate measurements

when gross plasticity, of the linear viscoelastic type , is occurring. In

this mode a load plateau is observed when crack propagation commences, as

shown in Fig. 2, and the crack growth rate, V, is related to the plateau

[4] load, Pp, by:

y , V = (1) Bp^
- 3

-

where y^j^ is the elastic displacement rate at the loading positions

(given by the applied displacement rate if the plastic displacements are

[2] small) and B is a constant which can be determined either empirically

or analytically . ^^ The corresponding value for the stress intensity

[8] factor is:

Kt = pw,m

3(l+v)

WdM

n _

(2)

where v is Poisson' s ratio and W , W, d and d are specimen dimensions, m n shown in Fig. 1. When non-plateau behavior is observed, substantial plastic

[4] displacements are generally occurring, and valid parameters can no longer

be obtained from Eq. (1) , because y^-j^ is indeterminate. This limits the

velocity range which gives valid fracture mechanics parameters; for example,

in silicon nitride at 1400°C, plateau behavior is only observed at velocities

larger than 10"^ cm/sec. But the velocity regime below 10 cm/sec is the

crucial one for failure prediction, and it is clear that further modifications

are required to extend the capabilities of the technique.

A suitable modification entails crack growth measurements at constant load,

where the crack growth rate is obtained from direct measurement of the

extent of the crack tip displacement, Aa, in a predetermined time interval. At,

using a fluorescent dye penetrant to indicate the position of the crack
- 4

-

tip.* Since the crack moves essentially at constant velocity in this mode,

the velocity is determined directly by, V = Aa/At. The stress intensity

factor must still, however, be determined from the load. There is some

concern, therefore, that the substantial plastic displacements may affect

the relation between load and stress intensity factor, given in Eq. (2)

,

even for linearly viscoelastic materials. Analysis and experimentation

have shown that the stress intensity factor, after plastic deformation of

the loading arms, is less than the value given by Eq. (2) : but that the

diminution is rather small, even for plastic displacements several tj.mes

the specimen thickness; e.g., a maximum correction of 20% is required for

a plastic displacement three times the specimen thickness. The effect of

plastic displacements on the stress intensity factor, load relation is not

therefore a crucial consideration for fracture mechanics measurements at

low crack velocity, and we feel that the detailed evaluation of a correction

factor is not merited at this time. We shall thus use the standard relation

(Eq. 2) to evaluate the stress intensity factor. (We also assess the validity of

* For side-grooved double torsion specimens, this is achieved by inserting

the specimen in the test fixture with the side groove along the top. Then

the furthest extension of the crack is clearly indicated on the bottom

face of the specimen. This inversion of the specimen from normal

[6 7 81

usage ^

' ' has no effect on the crack propagation parameters.
- 5 -

this approximation for each spscimen by obtaining the critical load for

crack propagation at room temperature—which can be equated to K-^q—after

the high temperature measurements have been completed.)

The same specimen configuration can be used to determine crack propagation

rates under cyclic loading conditions . Cycling is conducted between two

fixed loads, p-j^ and p^, and the amount of crack growth that accompanies the

imposed cycle is measured. Again the crack growth rate is determined from

direct measurement of the crack tip displacement in a predetermined time

interval, and the loads p-^ and P2 give the corresponding stress intensity

factor range, 2AKj. The crack growth rate can then be plotted as a

function of the average stress intensity factor (K-j-)^^. for various AKj,

and compared with the "static" slow crack growth rates, using an analysis

developed by Evans and Fuller. ^^"^^

3. SLOW CRACK GROWTH RATE MEASUREMENTS

Measurements of slow crack growth rate have been obtained for a range

of silicon nitrides , "^^^ silicon carbides, ^^'^^ and aluminas. The

most extensive work has been performed on the silicon nitrides and this is

examined first.

3. 1 Silicon Nitride

The temperature dependence of the slow crack growth rate obtained

using the constant displacement rate approach is shown in Fig. 3. We

note two distinct regions of behavior, denoted regions A and B. In region

A (at low velocity) the crack growth rate increases with increase in

temperature (for a given stress intensity factor); whereas in region B,
- 6

-

the inverse occurs. The region A behavior can be analyzed in terms of a

conventional rate process to obtain an apparent activation energy for slow crack

[41 growth, 200 Kcal/mol.

More extensive data obtained at 1400°C,* using the constant load

approach for low velocity measurements , are shown in Figs. 4 and 5.

Interestingly enough, when the data are plotted using a linear stress

intensity axis (Fig. 4) , there is a suggestion of a slow crack growth limit;

but when plotted on a logarithmic K axis (Fig. 5), a good linear fit is

obtained over the entire range. A good representation of the slow crack

growth data is thus;

V = AK-j."^ (3)

where A and n are constants for a given temperature. This we will find later

is very convenient analytically; but, as yet, we find no fundamental reason

for predicting this type of relation for a rate-controlled process. It

is also noted that the incorporation of the lower velocity data (compared

The most critical temperature for high temperature applications, such

as in gas turbines.
- 7 -

to Fig. 3) permits more precise evaluation of the exponent, n. A least

squares fit of the data for the same 'standard' material indicates that n

(at 1400°C) is 6 ± 1.

The orientation dependence of the crack growth rate has also

been evaluated at 1400°C as shown in Fig. 6. It is apparent that the

crack propagation rates on planes containing the hot-pressing direction are

marginally larger than on the planes orthogonal to the hot-pressing direction,

[3] consistent with the orientation dependence of the strength noted by Lange.

All subsequent crack propagation rate measurements performed on research

grade silicon nitrides* are conducted on the "strdng" (orthogonal)

planes because specimens of the other orientation cannot be prepared from

the relatively small compacts.

Crack propagation rates obtained for a range of research grade silicon

nitrides are also plotted in Figs. 4 and 5. These materials are characterized

by different total impurity contents and/or different spatial and size

distributions of essentially the same impurities. The classification is

summarized in Table I. It is obvious that the higher purity materials

(characteristically those with a low calcium content) exhibit substantially

superior slow crack growth behavior. For example, the slow crack

growth rate in the critical low velocity regime can be reduced by more

* Supplied by the Norton Company.
- 8 -

than two orders of magnitude, compared to the "standard" (HS130) material.

This has a profound effect on the life expectancy under stress at high

temperatures, as we shall demonstrate later. Conversely, the size and spatial

distribution of impurity has no significant effect on the slow crack growth

rate. The particulate composites of silicon nitride and silicon carbide

exhibit inferior slow crack growth behavior.*

The effects of stress cycling on the crack growth rates are examined

next. Slow crack growth rates have been determined for a range of AK

and (K ) and are plotted as a function of (K ) at 1400°C in Fig. 7.

It is first noted that the cyclic slow crack growth rates are only

marginally larger than the static rates. This is exactly what we would

expect if the cyclic slow crack growth mechanism is the same as the

static mechanism. Quantitative cyclic and static slow crack growth rate

correlations, obtained using the Evans and Fuller analysis, ^^^^ are

shown in Table II. The predicted crack growth rates are in good

agreement with the measured rates. This is a rather unexpected result

* However, there may still be advantages to using silicon carbide and silicon

nitride composites if the slow crack growth in these materials is accompanied

by acoustic emission. ^^^^ This aspect of the failure prediction problem

is not dealt with extensively in this paper, but nevertheless, it is an

interesting possibility which will be examined and reported in a future

publication.
- 9 -

because it is generally found, in metals, that when there is extensive

crack tip plasticity, cycling yields crack growth rates substantially

[13] larger than the rates expected from the static process. It is noted

however that measurements have only been performed at relatively low

frequencies, approximately 1 cycle per second or less, and that enhanced

effects may exist at larger frequencies. (But the enhanced cycling effects

observed in ductile materials show up primarily at the low frequencies

;

and if an enhanced effect is found for silicon nitride at the higher frequencies

it will be due to some process other than the conventional cyclic fatigue

process,

)

3. 2 Alumina

Slow crack growth rate measurements in aluminas at high temperatures

are less interesting. Only a relatively narrow (stress intensity factor) range

of slow crack growth is observed (Fig. 8) , and in contrast to the observations

in silicon nitride in region B the critical stress intensity factor diminishes

as the temperature is increased.

Acoustic emission measurements have been performed on these materials

during high temperature crack propagation (Fig. 8) . Only limited information

is available at present, but there does appear to be a good correlation between

the acoustic emission rate and the slow crack growth rate, consistent with

similar measurements performed at room temperature.

The very interesting phenomenon of crack healing has also been examined

in alumina at elevated temperatures. ^^^^ Relatively low loads are imposed

on precracked double torsion specimens and the rate at which the crack
- 10 -

heals is determined by direct measurement of the crack healing increment

in a certain time interval. Only limited information (at 1400°C) is presently

available (Fig. 9) , but some very interesting features are emerging.

Note, especially, that the crack healing rate diminishes very rapidly at

quite small stress intensity factors.

^ .;3.3 Silicon Carbide

Some very preliminary measurements for slow crack growth have been

performed using a recrystallized silicon carbide. For this material, only

a limited range of slow crack growth (K^ > 0.9 K^.^) is observed even at

1400°C; and both the range of slow crack growth and the critical stress

intensity factor are essentially temperature independent. Presumably,

rather more interesting slow crack growth behavior will be observed in the

hot-pressed silicon carbides. We have not yet attempted to perform measurements on these materials, but studies will be initiated presently.

4. FAILURE PREDICTIONS

4.1 General Failure Time Analysis

The crack growth, and hence the failure time,* can be predicted

* These predictions are based on the two premises that failure is primarily

propagation controlled (i.e., the flaw initiation time is insignificant)

and that the fracture mechanics parameters apply to the relatively small

pre-existing flaws that lead to fracture in a typical component. This

appears to be justified for silicon nitride, although more work is

needed to substantiate this assumption.
- 11 -

from the slow crack growth data, by combining the Kj,v relation (Eq. (3))

with the fracture mechanics relation, = aY r^. (a is the stress and a

is the crack length) , to give;

da n n n/ 2 , V = — = AO Y a , (4)

and then rearranging and integrating to obtain;

^-2 t

(5) = A { a^dt

^1 -

Normally, the geometric parameter Y is considered to be constant '• and taken

outside the integral. However, in silicon nitride where slow crack growth

occurs over a wide range of stress intensity factors, the crack length at

failure is often a significant fraction of the component dimensions.

Then it is not valid to consider Y as constant. We shall therefore perform

the time-to-failure integration in general form, to evaluate the importance

of a variable Y in failure- time predictions. Initially, the integration is

conducted at constant stress, and then for a cyclic stress at constant Y.

The complete case of variable Y and variable stress can be performed, but

only by using numerical methods.
- 12 -

(a) Constant Stress

For constant stress, a , Eq. (5) becomes

t = da

n I _,n n/2 (6)

where a^ is the initial crack length and a^ is the critical crack length

for rapid fracture. Inserting the appropriate Y for the test configuration

into Eq. (6) enables t to be computed. For example, if we consider a tensile

[7] specimen, Y is approximately constant {"^ 2.0) in the range 0 < a/W < 0.2

(where W is the specimen thickness) ; but increases for larger a/W, with

Y ~ 10 a/W. Hence the time to failure becomes.

0.2W

_ , . n.,n / n/2 ACa^) 2 J

a.

'

Integration gives;

W^ da

- , . n, _n

A(a-,) 10 3n/2

0.2W

(7)

'-{-^)°^"['-(3K)J-(«)fef (t)° (8)
- 13 -

where is the time to failure for constant Y, given by;

to = — (9)

-3 o n- 2. A(n-2)a ^y^K^.

a Ii

where K-j.^^ is the maximum stress intensity factor on initial loading. It

is apparent from Eq. (8) that the crack length at rapid fracture, a^-, does

not significantly affect the time to failure; in fact, the larger is a^

the closer t comes to t^ (because an increase in a^ im.plies a decrease in

the stress and more of the time is expended at small crack lengths) . The

corrections due to variable Y only become important when a^ is relatively

large (> 0. 3W) , and this is not the case for typical silicon nitride

components. Hence, the simple relation for failure time given by Eq. (9)

can be used for failure prediction at constant stress without incurring

significant error, even when extensive slow crack growth occurs,

(b) Variable Stress

For a variable stress, a(t), and constant Y, Eq. (5) gives;

a (t)dt (10)
- 14 -

where t is the failure time, or.

_2 ( oA

r a(t)dt = —^

I — / AY^(n-2) ^Kj^

J

n-2

(11)

The left hand integral can be solved using series solutions for various

o (t) , ^^^^ and the results can be expressed in the form;

where t^ is the static time to failure at the equivalent average stress

and g is a dimensionless quantity which depends on the type of stress cycle,

the amplitude of the cycle and n. For example, values of g for a sinusoidal

cycle are shown in Fig. 10.

4.2 Failure Prediction Techniques

The relations in Eqs. (9) and (11) can be used directly for failure

prediction by inserting values for K^^. This can be achieved either

statistically or absolutely.

The statistical result, based on a Weibull distribution function is,

for low probabilities (< 0.1);
- 15 -

log t = - ^^ log P - n log oa + <{> (13)

m

where m is the shape parameter for the distribution, P is the fracture

probability and 0 is a constant.

For absolute failure predictions some flaw detection procedure must

be used. One of the most effective "flaw detection" techniques for ceramic

materials is overload proof testing. ^' "'"^^ A proof test to a stress, o^,

permits a minimum in-service failure time, t^-i_j^/ to be predicted, given by;

^min (14)

where K^^ is the critical stress intensity factor under the proof test

conditions

.

The minimum failure time after proof testing is determined

directly from the slow crack growth data, and the statistical distribution

of times is obtained from the slow crack growth data in conjunction with

strength data (determined in the absence of slow crack growth, i.e., at

room temperature for silicon nitride) . Both failure time solutions can

be plotted on the same diagram to obtain a complete picture of the

structural capabilities of the material.
- 16 -

4.3 Application

The failure prediction analysis developed above is now applied

to the silicon nitride data to evaluate the performance capabilities of the

various compositions. First, the proof stress diagrams for room temperature

proof testing and a 1400 °C operating temperature are evaluated for two

proof stress ratios; 2 to 1 and 6 to 1. These are shown in Fig. 11.

First, it is noted that a substantial discrepancy emerges when comparing

failure predictions (for the standard material) obtained using the old data

(Fig. 3) with the predictions from the new data (Fig. 5) . The discrepancy

is due, of course, to the lack of low velocity data in the former and reemphasizes the crucial importance of low velocity data for failure prediction.

The higher purity materials exhibit substantially longer failure times,

by about two orders of magnitude; for example, a proof stress ratio of

_2

6 to 1 ensures failure times > 1 yr for stresses < 10 MNm

The statistically related failure times are estimated for the higher

purity materials from four-point flexural strength data obtained at room

temperature. ^-^^^ The strength data relate to the standard material and

it is assumed that comparable room temperature strengths can be obtained

for the higher purity materials. The strength data exhibit a reasonable

fit to an assumed VJeibull distribution, giving a zero location parameter

and a value for m = 3. The corresponding failure times at 1400°C computed

using Eq. (13) are shown in Fig. 12 for the probabilities lO''^, 10""^, and

10~^. Also shown in the diagram are approximate failure times in tension.
- 17 -

for P = 10~ , assToming that the tensile strength is approximately half the

flexural strength.

Let us now consider the ir.aximum operational stresses that this

material can be subjected to, at 1400°C. If we take 1 yr {'^ 10,000 hrs)

as a useful life expectancy, end consider that the component has a median

strength about half the mediai;. flexural strength, then we see from Fig. 12

-5

that the stress levels corresponding to fracture probabilities of 10 , -3 -1 -2

10 and 10 without proof testing are 12, 20 and 30 MNm , respectively.

_2 Alternatively, a failure time of 1 yr is assured at 30 MNm by proof

_'> testing at a 9 to 1 proof stress ratio, i.e., 270 MNm . (This is, of course,

a very large proof stress ratio which may well be impractical.) The

maxim'jjn stress capability of this material may thus be taken as 30 MNm

at 1400"C. This may suffice in some applications, but additional improvements

are evidently desirable. For example, an improvement comparable to that

achieved from the standard to the higher purity material discussed here

would be sufficient for many applications.

Finally, it is noted that the material exhibits substantially improved

strength capabilities at lower temperatures. For example, decreasing the

temperature by 100°C to 1300°C gives a 10,000 hr strength more than 5 times

the strength at 1400°C, i.e., > 150 MNm~^.

5. MECHANISMS

Elevated temperature slow crack growth processes in ceramic materials

are primarily independent of environment. The slow crack growth mechanisms
- 18 -

are mostly related to crack tip plasticity,* (which occurs in these materials

at high temperatures) . Two distinct plasticity-related slow crack growth

processes can occur in ceramic materials (although it is noted that

additional mechanisms are possible in ductile systems), due to two different

modes of plasticity; dislocation motion and grain boundary sliding. Models

describing slow crack growth under these conditions are developed, but only

in a semi-quantitative fashion, noting that fracture is predominantly

intcrgranular. .- -

. 5.1 Dislocation Assisted Slow Crack Propagation

The model for the dislocation-related process is depicted schematically in Figure 13- In polycrystalline ceramics, grain boundaries are

frequent sources of dislocations, particularly in the oxide ceramics and

alkali halides. Hence, we envisage that when a stress is applied to a

cracked body, the stresses in the vicinity of the crack tip will cause

dislocations to be emitted from nearby grain boundary sources (S in

Fig. 13) . These dislocations will glide along their slip planes

and pile up at the opposite grain boundary (X in Fig. 13) . The details

of the dislocation transmission across the grains are rather complex, due

to the undulations of the crack tip shear stress field. However, detailed

* A possible alternative mechanism for the alumina is vacancy condensation

n 7

1

at the crack; but this process is not examined in this paper for

conciseness of presentation.
- 19 -

examination of these stress fields shows that the process depicted in

Fig. 13 is expected for most grain orientations (depending on the slip

systems) . The dislocation pile up initiates a grain boundary crack (in

the Stroh sense), this links with the primary crack, and hence produces the

crack length increment which constitutes the slow crack growth. A very simple

analysis which identifies the important crack propagation parameters for

this process is conceived as follows:

If we consider that the dislocations move across the grain at a

stress equivalent to the average shear stress acting along the plane, t,

[19] then the number of dislocations that form the pile up, n » is given by;

n = o_

yb

(15)

where I is the length of the pile-up ( G, the grain size) , is the

friction force opposing dislocation motion, u is the shear modulus and b

the Burgers' vector for the dislocations. The number of dislocations in

[19]

the pile up determines the tensile stress at the tip of the pile up

(o^j^p = n (x-Xq) ) . Once this exceeds the theoretical strength of the material,

a grain boundary crack will form at the end of the pile up. Putting the

condition, therefore, that the stress at the tip of the pile up is equivalent

to the theoretical strength ^^^^ (a . « /ey/C ) gives;
- 20 -

(16)

where is the lattice parameter, E is Young's modulus and y is the grain

boundary fracture energy. The shear stress along the slip plane is clearly

[211 dependent on the stress intensity factor and is given by; '

T = f (9) , (17)

/2T7r

where r is the distance from the crack tip and f(6) is the angular dependence

of T. Combining Eqs. (16) and (17) gives the stress intensity factor

required to initiate a secondary crack, due to the dislocation pile up;

Kj = BT^/2r+B' ' (18)

where B and B' are constants. Generally, when a crack is initiated at the

tip of a pile up, the crack will propagate to a size substantially larger

[221 than the pile up length. ^ ^ We expect therefore that the stress intensity

given by Eq. (18) is also sufficient for the secondary crack to link with
- 21 -

the primary crack; hence, the condition for propagation of the primary

crack along one grain boundary has been obtained.

We can further extend the model by noting that the lattice resistance

to dislocation motion is a function of both the dislocation velocity, v,

[23] and the temperature;

V = A'T^*f (T) (19)

where A* is a constant and m* is the strain rate sensitivity. If we now

combine this relation with Eq. (18) , we obtain a relation between K and

crack velocity which suggests that the slow crack growth rate is controlled

by the dislocation strain-rate sensitivity exponent, i.e., n « m* - Hence,

a good test of the model is a comparison of measured values of n and m*

.

The existing data on alumina (which might exhibit this mode of slow crack

growth) is unfortunately rather limited and it is not useful to make

direct quantitative comparison, but the trends are quite consistent with

[241 this model; i.e., both n and m* are large and the Kj range in which slow crac

growth occurs is small. We await more detailed empirical evaluation of

slow crack growth rates for quantitative comparison with the model.

5.2 Grain Boundary Sliding Induced Slow Crack Growth

Slow crack growth can also occur due to grain boundary sliding

near the crack tip. A model for this process that applies to materials
- 22 -

with rigid grains and a narrow viscous grain boundary phase (typifying

the grain boundary sliding process in silicon nitride) is depicted

schematically in Fig. 14. Analysis of this process presents several

formidable problems and a complete analysis has not yet been developed.

The sequence of events is as follows:

Sliding occurs along the grain boundary (AB) adjacent to the crack

tip. Using Eq. (17) in conjunction with the stress dependence of the

N sliding rate i^aa • where N ^ 1) , the displacement, 5, that occurs in

• A [25] time, t, can be obtained;

6 = k

N

t (20)

where k is the proportionality in the sliding rate relation. This displacement

is transferred by the rigid grains to the triple point at B, and a grain

size crack will form from B, along the grain boundary (BC) , when 6 reaches

a critical value, 6^, The crack at BC enhances the stress at the primary

crack tip, but only by a small amount* (< 10%) . Hence m the range

* A detailed analytical solution for the stress intensification is not

available, but the solution for two equivalent size cracks can be used

to estimate the order of the effect. ^^^^ Also a simplified analysis

assuming non-interacting cracks can be used and this gives a similar

result.
- 23 -

0.9 K-j.^ ^IC formation of this secondary crack may induce

separation along AB and the primary crack will move forward, due to this

event, by 3/2 G. The V,Kj relation in this range is thus obtained from

Eq. (20) and is given by;

V

Gk

6_

2-'

f (6) (21)

It is also noted at this stage that K^^ corresponds to fracture by grain

separation (opening) at A and will occur when 5, at A, is 6^. The

increase in K^^ in silicon nitride with increase in temperature is thus

accounted for by an increase in 5^ due to a decrease in the viscosity of the

grain boundary phase

-

At lower Kj , separation along AB can no longer occur and sliding now

commences along CD due to the formation of the secondary crack. The rate

of sliding here can be obtained by summing the shear stresses along CD

due to the primary crack and the wedge opening of the secondary crack

(due to the sliding along AB) . The stress intensity factor at C due to

[2' the wedge opening at B can be obtained from Barenblatt's wedging analysis,

and in its simplest form the solution is;

e6

(22)
- 24 -

The sliding displacement alone; CD due to the total Kj at C can thus be

found in the exact manner used to derive Eq. (20) . Putting the displacement

equal to 6^ then gives the condition for forming another secondary crack

along DE, which further enhances the stress at the primary crack tip. This

process can be continued until a large zone of subsidiary microcracks is

developed, and the corresponding Kj at the primary crack evaluated. The

analysis has not yet been completed, but the important trends are emerging.

Notably, the crack growth rate will be a function of Kj and the grain boundary

sliding rate-constant, and hence qualitatively consistent with the slow

crack growth observations.

6. SUMMARY

Slow crack growth has been examined in several systems at elevated

temperatures. The double torsion specimen has been identified as an

appropriate specimen for fracture mechanics measurements at high temperatures,

but with the use mode (fixed grips, constant displacement rate, fixed

load) depending on the extent of the accompanying plasticity.

. Data obtained for silicon nitride, silicon carbide and alumina have

been examined, and these show that purity plays a major role in controlling

the slow crack growth resistance. Crack growth rate data evaluated for

silicon nitride under cyclic loading conditions show the unexpected result,

that cycling does not enhance the crack growth rate at low frequencies,

even though substantial crack tip plasticity is occurring.
- 25 -

Failure prediction analyses have been presented which enable failure

times to be evaluated from the fracture mechanics data. Two types of

failure prediction have been emphasized; a statistical prediction based

on the initial strength distribution, and an absolute prediction for the

minimum failure time after proof testing. The silicon nitride data have

been transformed into failure diagrams (containing these predictions) which

may be used for design purposes or for materials evaluationFinally, models for the plasticity related slow crack growth process

which may occur in ceramic materials at elevated temperatures have been

described. The models are not fully quantitative, as yet, but they

demonstrate the important parameters controlling the rates of slow crack

growth.
- 26 -

REFERENCES

, .1. S. M. VJiederhom, Fracture Mechanics of Ceramics, to be pu±;lished.

2. A. G, Evans and T. G. Lapgdon, Structural Ceramics, Progress in Materials

Science, to be published.

3. F. F. Lange, J. Am. Ceran. Soc, to be published.

4. A. G. Evans and S. M. Wiederhorn, J. Mater. Sci., to be published.

5. A. G. Evans, M. Linzer and L. R. Russell, Mater. Sci. Eng., to be

published,

6. A. G. Evans and S. M. Wiederhorn, ITBSIR 73-147, March 1973; Int.

J. Frac, to be published.

7. A. G. Evans, Fracture Mechanics of Ceramics, to be pviblished.

8. D. P. Williams and A. G. Evans, J. of Testing and Evaluation, 1_,

264 (1973).

9. A. G. Evans, J. Mater. Sci. 7_' 1137 (1972).

10. A. G. Evans, L. R. Russell and D. W. Fdcherson, to be published,

11. A. G. Evans and E. R. Fuller, Met. Trans., January 1974, to be published.

12. F. F. Lange, unpublished work performed at Westinghouse Research

Laboratories.

13. See for exairple: Corrosion Faticrue , NACE Pi:iblication (Ed. O. F. Deveraux,

A. J. McEvily, R. W. Staehle) , 1972.

14. A. S. Tetelman and A. G. Evans, Fractiire Mechanics of Ceramics, to be

published.
- 27 -

15. A. G. Evans and L. R. Russell, to be published.

16. A. G. Evans, J. Mat. Sci., to be published.

17. R. M. Stephens and R. Button, Mater. Sci. Eng. 8_, 220 (1971).

18. R. M. Thor.son and A. G. Evans, to be pi±>lished.

19. J. P. Hirth and J. Lothe, Theory of Dislocations (McGraw Hill, New York),

1968.

20. A. Kelly, Strong Solids , (Clarendon, Oxford) 1966.

21. P. C. Paris and G. C. Sih, ASTM STP 381 (1967) p. 30.

22. A.N. Stroh, Advances in Physics 6_, 418 (1957) .

23. A. G. Evans and R. D. Rawlings, Phys. Stat. Sol., 34_, 9 (1969).

24. H. Conrad, G. Stone and K. Janowski, Met. Trans., 233 , 889 (1965).

25. A. G. Evans, to be published.

26. M. Isida, Methods of Analysis and Solutions of Crack Prcbleras (Ed.

G. C. Sih, Noordhoff, 1973), p. 121.

27. G. I. Barenblatt, Advances in Applied Mechanics, 7, (1962)

.
- 28 -

TABLE I

SILICON NITRIDES CLASSIFICATION

Materials

Relative

Deleterious

Irpurity

Content

Impurity

Distribution

Type I 1 Wide Inclusion

Size Distribution

Standard

Type II 1 Narrow Inclusion

Size Distribution

High Purity

Type I 0.2

Type II 0.3

Composites

:

SiC/Si,N . 3 4

30/70 > 3
- 29 -

TABLE II

CYCLIC/STATIC ilLO':! CP.'CK GRO-.TH COMPARISON

Velocity Ratio (g)

AK/(K ) Predicted Measured

I av

0.2 1.8 1.5

0.3 3.2 3.0

0.4 6.5 7.0
- 30 -

FIGURE CAPTIONS

The double torsion specimen.

The load, displacement behavior observed in a double torsion

specimen loaded at constant displacement rate, y.

Slow crack grov/th measurements in "standard" (HS 130 type) hot

pressed silicon nitride at various temperatures, using the

constant y approach.

Slow crack growth in several silicon nitrides at 1400°C; C is a

SiC/Si^N^ particulate composite, S is "standard" (HS 130 type)

material and HP is a research grade high purity material.

The same data as in Fig. 4, but plotted using a logarithmic Kj

axis, indicating the good fit of all the data to the relation:

V = AKj".

Crack growth rates in "standard" silicon nitride in the "strong"

(containing hot pressing direction) and "weak" (orthogonal to

hot pressing direction) planes, at 1400°C.

Slow crack growth rates for "standard" material under cyclic

loading conditions, plotted against the average Kj for the cycle

(Kj)^^, for various ratios of AK/(Kj)^y at 1400°C.
- 31 -

Fig. 8. Slow crack growth rates and acoustic emission rates in "lucalox"

alvunina at elevated temperatures.

Fig. 9. Crack healing rates, vs. , in lucalox alumina at 1400°C.

Fig. 10. The g parameter for a sinusoidal stress cycle, for various n

and stress arplituaes.

Fig. 11. A proof stress diagram^ for "standard" and high purity silicon

nitrides for proof testing at room temperature and in-service at

1400°C.

Fig. 12. Probability related failure timies for failure in flexure and

in tension for various probability levels for high purity silicon

nitride at 1400°C; also shown are the proof test predictions from

Fig. 11.

Fig. 13. A schematic of dislocation assisted slow crack growth in polycrystalline ceramics.

Fig. 14. A schematic of grain boundary sliding induced slow crack growth.
P/2

Fig. 1. The doiible torsion specimen.

- 32 -
SHARP CRACK

INITIATION

LOAD 7CRACK

PROPAGATION

Fig. 2. The load, displacement behavior observed in a double torsion

specimen loaded at constant displacement rate, y.

i

- 33 -
10r2

'ow

E

-JlU

>

<

u

10-

lOr8

ISSO^'C

I300°C

1250 *'C

B

5 6 7

Ki(MN m~3/2)

8 9 10 12

Fig. 3. Slow crack growth measurements in "standard" (HS 130 type) hot

pressed silicon nitride at various temperatures, using the

constant y approach.

- 34 -
Fig. 4. Slow crack growth in several silicon nitrides at 1400°C; C is a

SiC/Si^N^ particulate composite, S is "standard" (HS 130 type)

material and HP is a research grade high purity material.

- 35 -
- 36 -
0.4 0.6 0.8 I 1.5 2 3 4 5 6 7 8 10

Kj (MN nr^^"^)

Fig. 6. Crack growth rates in "standard" silicon nitride in the "strong"

(containing hot pressing direction) and "weak" (orthogonal to

hot pressing direction) planes, at 1400°C.

- 37 -
I IO"'Q| I \ I \ \ \ \

I I I I 0.4 0.6 0-8 1 1.5 2 3 4 5 6 7 8 10

KidVlNm"^^^)

Fig, 7. Slow crack growth rates for "standard" material under cyclic

r\:,r.''-y;.; loading conditions, plotted against the average Kj for the cycle,

(Kj)^^, for various ratios of AK/(Kj)^^ at 1400°C.

- 38 -
'E

k_a>Q.

ouni

LiJ

cr2

O

CO

t/)

llJ 10'

u

1-

coo

u

<

10

-I

10

c-2

o

to .-3 [- J 10

UJ

O

q:

<

O

10'^

10

-7

10

.-8

1 6 'o

'

I300°C^ O O

1 •

600'^C

/ / laoo^'c / looo^'c

-IP T

w —

• —

i

j

• CRACK GROWTH RATE •

—

O ACOUSTIC EMISSION RATE

1 1 1 1 1 2.0 2f» 2.8 3.2 3.6 4.0

8. Slow crack growth rates and acoustic emission rates in "lucalox"

alxomina at elevated temperatures.

- 39 -
Fig. 9. Crack healing rates, vs. K . in lucalox alvcnina at 1400°C.

- 40 -
0 .05 .1 .15 .2 .25 .3

Fig. 10. The g parameter for a sinusoidal stress cycle, for various n

(11) and stress amplitudes.

- 41 -
STRESS (MN m^)

Fig. 11. A proof stress diagram for "standard" and high purity silicon

nitrides for proof testing at room temperature and in-service at

1400°C.

- 42 -
Pig. 12. Probability related failure tdLmes

in tension for various probability

nitride at 1400*C; also shown are

Fig. 11.

- 43 -

for failure in flexure and

levels for high purity silicon

the proof test predictions from
SECONDARY CRACK

Fig. 13. A schematic of dislocation assisted slow crack growth in polycrystalline ceramics.

- 44 -


NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 74-442

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE 5. Publication Date February 1974

6. Performing Organization Code

7. AUTHOR(S)

A. G. Evans

8. Performing Organ. Report No. NBSIR 74-442

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 3130459

11. Contract/Grant No. F 33615-73-M-6501

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

United States Air Force

Air Force Systems Command

HQ 4950th Test Wing, 4950/PMMA

Wright-Patterson AFB, Ohio 45433

13. Type of Report & Period Covered t.Interim

14. Sponsoring Agency Code

^* IH.^^^^¥)^'^|)^ii$tf3S?ed in the Proceedings of Conference on Ceramics for High

Performance Applications, Hyannis, Mass., 11/13-16/73.

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.)

High temperature slow crack growth processes in several ceramic materials are examined under static and cyclic loading conditions. Data obtained at temperatures

up to 1400° C are used for purposes of failure prediction and for analysis of the

slow crack growth phenomena. It is shown that purity plays a major role in slow

crack growth resistance, particularly in the hot pressed materials, and that cycling

in the low frequency regime does not significantly increase the rate of slow crack

growth. The slow crack growth mechanisms appear to be primarily plasticity related.

Two semi-quantitative mechanisms are presented, one due to dislocation motion and the

other due to grain boundary sliding.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Ceramics; crack healing; crack propagation; cyclic fatigue; failure prediction;

high temperature; static fatigue.

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