NBSIR 75-661

Strength Degradation of Brittle

Surfaces: Sharp indenters

B. R. Lawn, E. R. Fuller, and S. M. Wiederhorn

|RUi1!J^Jg!=-- e^, lmT ^ r

._ m

_

Inorga

Institi Natior

Washi

May 1

Interir

Lawn, B. R

.

, Fuller, E. R. , Wiederhorn, S. M.

,

Strength degradation of brittle surfaces:

i Sharp indenters. Am. Ceram. Soc. 58, No. 5-6,

193-197 (May-June 1976). \ NBSl I 7<>

313

Prepared for Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

NBSIR 75-661

STRENGTH DEGRADATION OF BRITTLE

SURFACES: SHARP INDENTERS

B. R. Lawn, E. R. Fuller, and S. M. Wiederhorn

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

May 1975

Interim Report for Period July 1, 1974 through June 30, 1975

Prepared for

Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

STRENGTH DEGRADATION OF BRITTLE SURFACES:

SHARP INDENTERS

B. R. Lawn*, E. R. Fuller, and S. M. Wiederhorn

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234, U.S.A.

*On study leave, from School of Physics, University of New South Wales,

Kensington, N.S.W. 2033, Australia

Abstract

A theory of strength loss for brittle surfaces in contact situations

,

developed in a previous paper for "blunt" indenters , is here extended

to the case of "sharp" indenters. A prior fracture mechanics analysis

of crack growth beneath ideal cone indenters serves as the basis for

predetermining the prospective surface degradation of ceramic components in service. Compared to blunt indenters, severe degradation

can occur at lower contact loads. However, at high loads the extent

of degradation becomes remarkably insensitive to details in the

indenter geometry. Essential theoretical predictions are verified

by bend tests on glass slabs. The effect of indenter "sharpness"

and initial specimen surface flaw state are investigated systematically,

along with some secondary rate effects in the contact process. The

possibility of minimizing degradation via adjustment of material

parameters (including hardness) or surface condition (e.g., residual

stresses, frictional properties) is briefly discussed.

1
I . Introduction

In a previous paper"'" an analysis was presented of the degradation

incurred by brittle surfaces under circumstances in which indenting

particles could be considered "blunt". The classical Hertzian cone

crack produced beneath a hard sphere was adopted as a model representation

Of this situation. "Real contact" situations involving irregular

particles fall somewhere between this and a second limiting type of

indentation configuration, namely configurations in which the particles

may be viewed as "sharp" (i.e., possessing an ideally sharp point).

The contact conditions, and hence the ensuing indentation fracture

patterns, are somewhat different in this second case, and are not as

well understood as in the cone crack configuration. Nevertheless, recent

2-4

fracture mechanics studies of sharp- indenter systems provide us

with an adequate basis for predicting degradation behavior. The present

study is directed toward this end, and accordingly serves to complement

the earlier study."'"

Because the deformation beneath a perfectly sharp indenter cannot

be completely elastic (as it can beneath a blunt indenter) , we must

expect some new features in the degradation properties of a given

brittle material. We shall indeed show this to be the case, particularly

in the initiation stages of indentation fracture at low loads. At the

same time, once the crack system develops well beyond the contact zone

it might be anticipated that the influence of indenter geometry will

become less marked. Again, we shall demonstrate this to be so.

Considerations along these lines are useful in any discussion of damage

in real contact processes.

2
As before, it will be assumed that quasistatic conditions prevail

throughout the indentation process. Ideally rigid cones will be taken

as typifying the sharp-indenter configuration.

II. Point Indenters and Median Penny Cracks

Let us examine the essential features of the crack pattern produced

in a brittle surface loaded with a hard, sharp point indenter . The

solution for the linear elastic stress field in the indented specimen

prior to cracking contains a singularity about the contact point, and has

2 3 an inverse-square fall-off with the radial distance. ' Removal of this

singularity is accounted for physically by the formation of an irreversible,

Ktress-relievinq zone of deformation (plastic or viscous flow, structural

demsification, etc.) around the contact; the intensity of the "local

stresses (characterized by the mean contact pressure) at which this deformation

occurs determines the hardness of the material. The stress concentration

immediately below the indenter point nevertheless remains high, and

downward-extending cracks can be nucleated at very low loads , depending

on the degree of brittleness. This represents a major departure from

the blunt-indenter case, where pre-present surface flaws control the

crack initiation process. Indeed, if the tip region of the indenter is

somehow bluntened, or if the pre-present flaws are sufficiently large,

the formation of a small cone crack may precede the development of the

deformation-induced cracks: this is a complication which we take up

again later (Section IV) , but which for the present may be ignored.

Once the cracks have been nucleated, they extend downward on median

planes in accordance with the scheme of Fig. 1. Several mutually

intersecting median cracks (depending, for example, on the symmetry

3
of the indenter or the anisotropy of the specimen) may propagate

simultaneously. However, we shall focus attention on a single crack

system, noting from empirical observation that the depth of cracking

4 is not too sensitive to the number of cracks formed. We describe the

variation of median-crack depth with load for the following stages in

4

the growth:

(1) Formation of "contained" median penny crack

Whereas in reality the nucleation of median cracks involves a threshold

in the loading, for the most brittle ceramic materials this threshold is

insignificantly small (typically less than one newtron, as is readily evident

from hardness test observations) . The cracks assume the form of

well-defined pennies, wholly contained below the contact zone, and grow

downward stably as the load is applied. This configuration constitutes

a complex elastic-plastic problem, for which an exact solution is out

2 of the question. A simplified fracture mechanics analysis indicates

that in addition to the usual fracture parameters, namely the fracture

surface energy r_ and the Young's modulus E_, the hardness H_ should enter

into the relationship between indenter load and crack size; the scale

of the deformation zone determines the degree of stress relief that occurs

about the singular point in the ideally elastic field. A second analysis,

based on a more or less dimensional argument for general penny-shaped

4

cracks, predicts a slightly different relationship between the parameters.

Although less explicit in the dependence on hardness, the second treatment

provides a better empirical fit to crack growth data, and will thus be

cited here.

4
Accordingly, we find, for smooth indenters,

P

2 /D

3 = 2rE/K^(V, H/E, r\» (1)

where P_ is the load on the indenter, p_ is the depth of the crack (the penny

daimeter for the contained median penny crack) , V_ is Poisson's ratio, is

the cone half-angle and is a dimensionless factor (f_ signifying formation

stage of crack growth) whose value needs to be determined by experimental

calibration. It may be noted that this equation does not involve the state

of the specimen surface (flaw population, etc.) . However, the angle of the

cone, through its influence on the indentation field, does enter as a

variable

.

Up to this point the cracks cannot be considered to be "well-developed,"

for they still remain within the sphere of influence of the nucleation

zone immediately below the contact. However, there is a critical

configuration, albeit ill-defined, above which the contact zone can no longer contain the expanding penny (D>>a, Fig. 1)*. At this point the

crack "breaks through" to the specimen surface, and thereby becomes

fully propagating.

(2) Propagation of "center- loaded" median half-penny crack

Beyond the breakthrough load the median crack assumes the form

of a center-loaded half-penny crack (radius D, Fig. 1) . Details of

events at the contact zone no longer exert a strong influence on the

fracture mechanics: the crack has escaped the nucleation forces, and

*This in turn implies a critical load, since, taken together with the

2 definition of hardness, H = P/ott (cx a geometrical factor, which is unity for cone indenters) , Eqn. (1) predicts, for a fixed value of H, D/a « P

1^6

.

5
is now controlled by the component of the applied load which acts to

wedge open the mouth region. Nevertheless, the crack geometry retains

the essential features of a (somewhat modified) penny configuration,

in which case we may predict for smooth indenters , in analogy with

Eqn. (1) , the relation

P

2 /D

3 = 2rE/K-£(v,ijj) (2)

with another dimensionless factor (p signifying propagation stage of

crack growth) ; we note that the hardness now disappears from the

fracture mechanics calculation. A straightforward analysis, based on

4 a center-loaded, full penny in an infinite solid, gives

k^(v,i|j) = (1-v

2

)/tt 3 tanV (3)

Generally, the contact will not be smooth. It can be shown

that the effect of including frictional tractions at the indenterspecimen interface is simply to replace x^J_ by ip_' = \p_ + arctan y_

(y = coefficient of friction), i.e., effectively to "blunt" the

4 indenter tip. The effect of ignoring friction is therefore to

overestimate in Eqn. (3) , and hence D in Eqn. (2) . In other words,

we deal with a "worst case" situation.

In practice, the transition from stage (1) to stage (2) in

the growth of a given median crack can be detected, either by

direct observation of the indentation process (in transparent solids)

,

by monitoring the acoustic emissions as the contained penny pops

through to the specimen surface, or by recording load drops in a

4

fixed-grips loading arrangement. While such discontinuities in

growth occur at generally lower critical loads as the cone halfangle diminishes, the critical loads themselves vary considerably
under apparently constant indentation conditions. Moreover, mutually

intersecting median cracks may "pop-in" at different stages in the

loading, and accordingly be retarded in further growth by their

less-developed neighbors . Multiple-cracking complications are

particularly noticeable at the lower cone angles. This aspect of

the indentation cracking remains poorly understood.

(3) The residual median crack

Unlike the elastic-brittle contact of the ideally blunt indenter

situation , the present case is complicated by further developments

in the crack pattern upon unloading . Incompatibility between the

stress-strain responses of nonlinear material within the irreversible

deformation zone and linear material without gives rise to mismatch

tractions at the contracting zone boundary. The net result is a

residual stress field which begins to dominate the field of the applied

loading just prior to complete removal of the indenter. This residual

field contains a component of tension sufficiently intense to generate

2 5 an entirely new, laterally-extending system of cracks. ' However,

such lateral cracks are relatively shallow, and thus do not contribute

significantly to degradation. A more important influence of the

residual stresses here is evident in the behavior of contained

penny cracks already formed during the preceding, loading half-cycle.

These cracks progressively grow toward the specimen surface as

the indenter is withdrawn; an alternative route to the fully-developed

form thereby exists , without the need for the critical pop-in load

4 ever to be exceeded.

7
Thus the final state of the strength-controlling median crack

system inevitably approaches the half-penny configuration, regardless

of the loading history. This is most commonly evidenced by the

appearance of characteristic radial traces of the median cracks on

the indented surfaces. The dimensions of these traces tend to

reflect the depth of cracking one would anticipate if the half-penny

4 configuration were to be realized prior to maximum loading. With

this information we may sensibly make use of Eqn. (3) for fullydeveloped median cracks to determine the effect of indenter "sharpness"

(all else being constant) on the resultant crack size. Fig. 2 is

an appropriate plot of [Kjjjj ] ^ f°r cone indenters on glass, with

V = 0.25. Experimental data from indentation observations, for

tungsten carbide indenters on glass surfaces in air environments,

fall somewhat below the calculated curve, suggesting that the effects

of friction are not insignificant here. Nevertheless, we shall

proceed for the present on the basis of a smooth contact, mindful

that the resulting upper limit K^(jp) to be used in Eqn. (2) will lead

to an overestimate in D, thus providinq a conservative value for the

strength of the indented surface.

III. Strength Degradation Tests

(1) Test procedure

The standard Griffith formula for the rupture stress Q in a

homogeneous tensile field once again provides the basis for the

strength tests

a = [2rE/7T(l-v

2

)c

f

] 1/2

(4)

3
where c, is the effective flaw size of an equivalent through crack. —

r

Of the variables in Eqn . (4), it is the flaw size which is most

widely susceptible to extraneous influences (mechanical, thermal,

chemical , etc . )

.

Soda-lime silicate glass laths for four-point bend testing were

prepared in much the same manner as in the previous study, ^ with

one or two minor differences. In the present case, the 10 ym

level of pre-present flaws typical of carefully handled ceramic

surfaces is eclipsed by the extent of median cracking produced at

indentation loads as low as IN. As-received laths hence provided

perfectly adequate test surfaces for the bulk of the testing,

although the test areas of one set of specimens were pre-abraded

by a grit-blast treatment"'" to determine whether severe pre-present

surface damage produces any side-effects in the indentation process.

For each specimen the prospective indentation site was covered

with a drop of paraffin oil to prevent moisture-assisted slow crack

growth during subsequent testing. (In the case of grit-blasted

specimens, the abraded area was covered immediately after the treatment.)

Each specimen surface was then indented with a tungsten carbide

conical indenter of given half-angle ip, at a prescribed load P_.

(The tip radius of the cones used never exeeeded 5 ym throughout the

experiments.) The specimens were then loaded to rupture with the

indented surface in tension. Unless specified otherwise, the bend

test was effected within four hours of indenting in the experiments

to be reported here.

9
Examination of the laths after breaking indicated the indentation

site to be the origin of fracture in the vast majority of tests.

Isolated edge failures did occur, and these were rejected from the

data accumulation . Reproducibility in strength values was limited

mainly by variations in the indentation crack pattern under any

specified set of test conditions; the scatter in data points in the

figures to follow therefore serve to indicate the experimental accuracy

(2) Strength of indented test pieces

Strength degradation was investigated as a function of indentation

load, for different cone half-angles. Up to a (very low) limiting

load, P_' say, the rupture stress maintained the level characteristic

of the unindented surface (depending on the prior history of the

glass specimens) . In this region of behavior the effective length

of the dominant flaw is presumably determined by the size of prepresent microcracks, i.e., = c_° say, (PjCP_' ) . Above the limiting

load the rupture stress declined steadily. The effective flaw

length here becomes a function of the size of median cracks, i.e.,

c_ = c_(D), (P>P ' ) : an analysis of penny cracks in a tensile field

~~f —f — — —

2 gives the explicit relation c^«(4/tt )D (Appendix). With Equation (2)

this result allows us to re\*rite the strength equation (4) in

terms of indentation load, for a given cone, under fixed test

conditions

,

a = [2rE/7T(l-v

2 )c°] 1/2 (P<P') (5a)

O = {(2rE ) 2/3

7T 1/2/2(l-V2

) 1/2 [lcP(v, 1i;)] 1/6 }p- :L/3 (P>P). (5b)

An evaluation of these equations may be carried out in terms

of readily measurable fracture mechanics parameters. For tungsten

10
cone indenters on soda-lime glass we take the following values:

TOO *5 C

E = 7.0 x 10 Nm" , V = 0.25 and T_ = 3.9 Jm~ (glass data );

k£ (30°) = 9.0 x 10~ 2

, Kg (40°) = 4.3 x 10~ 2, Kg (50°) = 2.0 x 10~ 2

,

f£ (60°) = 1.1 x 10~ 2

, Kg (70°) = 5.0 x 10" 3, teg (80°) = 1.5 x

10~ 3

(as per the curve in Fig. 2). This then leaves c° and as test variables,

of which the second has the more interesting implications.

(a) Effect of flaw size . A preliminary indentation/strength test

run was conducted to check the effect of initial surface state on

degradation. Fig. 3 compares results for surfaces as-received and

grit-blasted (No. 100 SiC grit, corresponding to c° = (23 ± 5)ym)^, —

i

for a cone of half-angle 70°. Also included in the figure are the

theoretical predictions (full lines) of Eqn. 5a and 5b. Other than at

low loads, where the abrasion flaws dominate the indentation damage

in the case of grit-biased surfaces , the results are indistinguishable

.

Consequently, for the remainder of the tests it suffices to work

simply with as-received surfaces.

(b) Effect of cone angle . The most extensive runs were directed to

an investigation of the role of indenter sharpness in surface

degradation. Fig. 4 illustrates the experimental data (points)

,

along with the theoretical curves (full lines) from Eqn. (5)

.

Agreement here is reasonable, taking into account the spread in the

data, although some discrepancy is apparent at the largest cone angle,

o

= 80 . We should note that the strength behavior is not highly sensitive

to indenter sharpness, except perhaps again at the larger angles, notably at ^ = 80°. We may note further that the strength fall-off is remarkably

slight with increasing indenter load. That the loads attained in

11
these experiments are substantial is borne out by the fact that the

sharper cones quickly became rounded (in which case their tips

were reground) , and the glass beneath the contact began to crush,

at levels not much higher than those represented in Fig. 4.

(c) Some secondary, rate effects . Finally, two interesting time7 dependent effects observed by Holland and Turner in analogous

strength-loss tests on scratched glass surfaces, were briefly examined.

In the first, Holland and Turner noted a tendency for specimens to

slowly recover their strength as the delay between indentation and

rupture in bending was prolonged. Fig. 5 confirms this effect, for

the specific case of normal indentation at ^ = 70°, P_ = 100 N. However,

the recovery is significant only for aging times of a day or more,

and therefore does not enter as a factor in the results reported

earlier in this study.

In the second effect, which relates to load rate effects, Holland

and Turner reported a drop of about twenty per cent in strength values

as the scratch velocity was increased over a range 0-5 ms ^ , at normal

load P = 5 N. Results of a parallel experiment, in which the crosshead

speed to a peak (quasistatic) normal indentation load P_ = 100 N was

varied over four orders of magnitude, are shown in Fig. 6. This time

there is no evidence of a rate effect.

IV. Discussion

The present model establishes a basis for predicting a priori

the prospective strength degradation in contact situations where

the indenting particles can be considered "sharp". Comparing the

12
strength equation (5) with its counterpart for blunt indenters in

the earlier study,"'" we find again that degradation is minimized for

surfaces of large characteristic r_ (tough) , E_ (stiff) and V (tensioninhibiting, in basically compressive loading) . Further, we might

anticipate, following the description of Section II (1) , low

values of H_ to be of advantage in suppressing the formation of

indentation-induced cracks at low loads; we shall say a little more

about the role of hardness again later.

A major distinction between the cases of sharp and blunt

indenters arises in connection with the state of the brittle surface

prior to indentation. Whereas with blunt indenters pre-existing surface

flaws exert an important influence through control of fracture

initiation processes, the same is not true with sharp indenters,

which induce their own flaws. On the other hand, there are other

aspects of the surface condition which may have significant effects

in crack initiation independently of indenter geometry. An example

is the fracture-inhibiting effect of residual compressive stresses

in a toughened surface. All such effects will be manifest at lower

indentation loads , where the indentation cracks remain within the

sphere of influence of the near-contact region. Generally, however,

it is found that the more pointed indenters, by virtue of their

greater capacity to concentrate stresses;, always produce the most

severe damage in this domain.

At higher loads the differences due to details in the nearcontact conditions tend to "wash out." In their well-developed form,

all indentation cracks inevitably assume some modified penny configuration,

1.3
the growth of which becomes determined by the stresses of "point loading".

The field in this limit is radial (inverse-square fall-off) about

the contact point, which explains the parallel behavior, toward the

upper end of the indentation load scale, between the strength-loss

curves in Fig. 4 of the present paper and Figs. 3 and 4 of Ref. 1.

There is accordingly less need to take into account specifics of

indenter geometry when the contact damage is severe. Moreover,

while sharper indenters do tend to produce deeper cracks at a given

4

load, these same indenters tend also to become flattened at the

tip, or to crush severely the supporting surface.

Further, to produce these deeper cracks the sharper indenter must

penetrate further between the crack walls, i.e., it must do more

work. The implication here is that in situations in which the indentation

event is governed by a fixed energy input (e.g., in a fixed-velocity

impact situation) the effects of indenter sharpness may well disappear

g altogether

.

Nevertheless, taking a broad view of the strength degradation

curves obtained in this and the earlier study, the degree of indenter

sharpness, as signified by the cone half-angle (or sphere radius)

,

remains an important factor to be considered. This is particularly

so in relation to the interpretation of degradation processes in

"real contact situations"; here it may be feasible to characterize

a sharp indenting particle by some "effective half-angle" (we may

note, for instance, that a Vickers pyramid, whose opposing diagonals

subtend an angle of 136° , produces cracks in glass of very much the

4 same depth as does a cone of ^ = 70° ) . However, in most cases the

14
assignment of a simple index of sharpness will not be so straightforward.

Even the sharpest particles must in reality have a finite non-zero

tip radius, so the possibility exists of a gradual "blunt-sharp

transition" in the strength response with increasing severity of

* indentation. An opposite, "sharp-blunt transition", corresponding

to an increasing argument in the tangent term of Eqn. (3) , may be

** realized in cases where frictional effects become pronounced.

The most general real contact situation is clearly very complex

indeed.

Friction is a further factor which warrants a good deal more

attention than we have been able to give here. It should be reemphasized that the present tests were conducted under oil. In a

different environment the frictional properties of the indenter/

specimen contact interface could be quite different. In most instances

it would be wise to base any strength design criterion on the

conservative assumption that frictional tractions are entirely absent

(thereby overestimating the depth of indentation cracking) . On the

*The corresponding change in fracture pattern in such a case has been

well demonstrated by k. Phillips, whose results are illustrated in

Fig. 21 of Ref. 3.

**As mentioned in Section II (2) , the effective cone angle ^ when friction

is present is ^ + arctan ]J. Thus for the indentations corresponding

to lj) = 80° in the present experiments, a friction coefficient y_ » 0.2

gives ^'«5 90°, a "perfectly blunt" indenter. In this one case examination of the cone-indented specimens revealed Hertzian cone fracture,

typical of ideally blunt indenters, to form at the expense of the median

fracture system. 4 The data points in the plot for ^ = 80° in Fig. 4

do in fact fit reasonably well to the degradation curve predicted for

well-developed cone cracks.^- In this description the cone-crack curve provides an upper limit to the strength resulting from any sharp-point

contact situation.

15
other hand, the possibility is raised of fabricating surfaces in

such a way that the operative frictional forces are deliberately

made large, in order to suppress the relatively dangerous median

fracture system.

Finally, there are a number of secondary effects associated

with non-equilibrium, and history-dependent, crack configurations.

Several of these effects have been discussed in the earlier work,'''

and we need only briefly consider one new aspect here. We have

already indicated the possible role of the irreversible deformation

properties of the indented solid, as characterized by the hardness,

in controlling the initial stages of crack growth. The hardness of

9 10 glasses and many other brittle materials is strongly ratedependent, so we might expect to observe additional kinetic effects in

the strength degradation curves, particularly at lower loads where

the inelastic processes appear to have some influence on the crack

formation (Sections II (1), II (3)). The null effect observed in

Fig. 6 may arise from the fact that our tests were conducted under

oil, an environment which tends to minimize the time dependence of the

9 hardness of glass. The analogous experiments by Holland and Turner

(Section III);, on the other hand, were conducted in an ordinary

laboratory atmosphere: their positive effect, in which the strength

was observed to diminish with velocity of the scratch indenter, correlates

with a corresponding increase in hardness of glass with decreasing

9

contact duration in air. This is one interesting aspect which

invites further, special attention, particularly in connection with

less brittle ceramic materials

.

16
Acknowledgements

The authors are grateful to A. G. Evans for stimulating

discussions relating to this work. Sponsorship through the Offi

of Naval Research under Contract No. NR-032-535 is acknowledged.

17
Appendix

Let us examine the mode of failure of a well-developed median

half-penny crack in a uniform tensile stress field. In the most

general case, the median plane of the crack will be inclined at

some angle, 3_ say, to the tensile axis, as in Fig. Al . We anticipate

that full-scale fracture will initiate at points on the crack perimeter

coincident with the specimen surface (point A) . For completeness, however, a general point P will be considered.

The procedure follows that outlined in the Appendix of the

earlier paper for cone cracks.^" For inclined penny cracks in uniaxial

tensile fields the standard stress-intensity factors at a prospective

extension point, P, are"^

K = 2Q (D/7T)

1/2 sin2

(6)

K = 20 ( DA) 1/2 sin(23)cos (C)/(2-V) (Al)

K

III = 20 (D/7T)

1/2 (l-v)sin(23)sin(i;)/(2-v)

where the subscripts denote I, "opening", II, "sliding", and III,

"tearing", modes of fracture. (These solutions omit a small correction

factor due to the presence of the free specimen surface in the halfpenny configuration.) The action of the shearing stresses in the general

crack orientation is to direct the crack-plane extension through some

energetically favorable tilt angle 0_ and twist angle At point A,

this extension will only consist of a tilt; whereas at point B, it will

only consist of a twist (Fig. Al) . Considering just an incremental

extension, we identify opening, sliding, and tearing types of local

stresses in the prospective crack plane:

18
cryIyl =

[K

l

/(2TTr) 1/2

] + g* 1 [K^/ (27Tr)

1/2

] + g* 11 [K^/ (27Tr)

1/2

]

= K^/(2Trr) 1/2

,

a

x'y'

= g

il tV27Tr)1/2] + g

il

ZI

fKn/^ 2^) 172] + gn

IH

[K

Ii:[

/27rr)

1/2

]

- K'^r) 1/2

, . (A2)

a

y'z' = g^1 [K

l

/(27rr) 1/2

] + gzlx

1J

[^TJ/ (2TTr)

1/2

] HI r„ /,„ ,1/2,

= K^

I:[ /(27Tr) 1/2

,

where x', y', z' are the coordinates in a system of the prospective

crack plane and the "transformed stress-intensity factors" K.J., K 'jjf an^

K

1 define the field for the modified crack. The g terms are calculable

—III

by transforming the angular-dependent components of the standard crack12 tip stress expressions. For plane strain we obtain

I ,0 W 2 ,0. 2 , . , _ . 2 , , . , g

l = cos (j) [cos (j)cos (cp) + 2Vsm ((p)]

Vjj

1 = sin (j) cos2

(^-) cos ((p)

g-j.-j.-j. 1 = - cos (j) [cos

2

(j) -2V] sin (<J>) cos (cp)

II . ,0,,, 2 ,0, 2 . „ . 2 , , g^. = - sm(j)[3cos ( —) cos (cp) + 2vsin (cp)]

gjj

11 = cos (j) [3cos2 (j)-2]cos ((p)

gIII

11 = sin(|-) [3cos

2 (j)-2v]sin((p)cos((p)

HI A • g

x = cos (j) sm(2cp)

HI • A • ,A» g^ = sxn(j) sxn((p)

) (mode IU) (A5)

HI A /oan

gIIX

= cos (j)cos (2<p) .

19
Combining (A2) and (Al) gives

1/2 T 2 tt 1 K'(6,C,G,4>) = 2a(D/7r) X/Z

[g';sin (3) + g"~ (~) sin(2B) cos (?)

+ g^

11

(~^) sin(26)sin(?)] , (A6)

where a represents I, II, or III.

From the stress-intensity factors the mechanical-energy-release

rate G, per unit width of crack front, follows:

G(B,C,0,<f>) = [d-V2

)/E] [K^

2 + + (i^) K

iiT.]- (A7)

For a specific crack angle 3_, we postulate that the crack at point P on

the crack perimeter will extend at the tilt angle 0* and twist angle <£* 12 for which the energy release G_* £J is greatest , i.e.,

3G/30 = 0 and 3G/3cf> = 0. (A8)

From Eqns . (A3) - (A8) we may compute the required energy term in the

form

G*(B,C) = [4(l-V

2 )a2D/7TE]C0(B,i;) (A9)

where to (B_, O is a dimensionless constant that attains a limiting value of

unity for 3_ = tt/2 (independent of . It remains to determine the point

P on the crack perimeter from which the crack will first extend. As

before, this is determined by the angle which maximizes the energy

release rate G*(B,jC*).

By introducing the Griffith energy-balance condition for

equilibrium cracks,

G* = 2I\ (A10)

and writing

c f = (4/tt 2 )d , (All)

20
we can combine with Eqn. (A9) to reproduce the strength equation

(4) of the text. The quantity

c

f

/D = (4/tt

2

) co(B,C*) E fl(B,e*) (A12)

accordingly determines the effectiveness of the inclined half-penny

crack as a critical flaw in the strength test. The quantity

fi(|3,£), is plotted for several values of £ in Fig. A2 for V_ = 0.25

(glass value) . The corresponding tilt angle 0* and twist angle are

plotted in Figs. A3 and A4, respectively. [At point A on the crack

perimeter (£ = 0°) there is no twist (i.e., <j)* = 0°) and at point

B (£=90°) there is no tilt (i.e., 0j*=0°).] It is seen from Fig. A2

that £*= 0, and as speculated, full-scale fracture commences at point

A on the crack perimeter (assuming the correction for the free specimen

surface is approximately constant along the perimeter) . However, the

variation of £_(j3_,£) with £_ is small, when the median crack plane is close

to normal coincidence with the uniform tensile stress field.

In practice, it is generally found that the sharp indenter

crack pattern contains several intersecting median vents. This is

certainly true of the cone indentation of glass, where a minimum of

3 six "arms" in the surface trace pattern is usual. (In certain

circumstances, where anisotropic effects in the indentation process

are very strong, a single median crack may be produced, but this* is

very much the exception.) Typically, then, the probability of

finding a median crack lying within 30° or so of the most favorable,

normal orientation is sufficiently high that the value ft(g =90°) =

4/tt provides a reasonably accurate assessment of the effective flaw

size. ni
References

1. B. R. Lawn, S. M. Wiederhorn and H. H. Johnson, "Strength

Degradation of Brittle Surfaces: Blunt Indenters," J. Am. Ceram.

Soc. , submitted for publication.

2. B. R. Lawn and M. V. Swain, "Microfracture Beneath Point Indentations

in Brittle Solids," J. Mater. Sci. , in press.

3. B. R. Lawn and T. R. Wilshaw, "Indentation Fracture: Principles

and Applications," J. Mater. Sci . , in press.

4. E. R. Fuller and B. R. Lawn, "Fracture Mechanics of Penny-like

Cracks in Indentation Cracking," in preparation.

5. B. R. Lawn, M. V. Swain and K. Phillips, "On the Mode of Chipping

Fracture in Brittle Solids," J. Mater. Sci ., in press.

6. S. M. Wiederhorn, "Fracture Surface Energy of Glass," J . Am .

Ceram. Soc , 52 [2] 99-105 (1969).

7. A. J. Holland and W. E. S. Turner, "The Effect of Transverse

Scratches on the Strength of Sheet Glass , " J. Soc. Glass Tech .

,

21_ 383-394 (1937) .

8. S. M. Wiederhorn and Brian Lawn, unpublished work.

9. S. P. Gunasekera and D. G. Holloway, "Effect of Loading Time

and Environment on the Indentation Hardness of Glass," Phys . and

Chem. Glasses , 14 [2] 45-52 (1973) .

10. R. E. Hanneman and J. H. Westbrook, "Effects of Adsorption on the

Indentation Deformation of Non-metallic Solids," Phil . Mag . , 18

[151] 73-88 (1968)

.

22
11. M. K. Kassir and G. C. Sih, "Three-dimensional Stress Distribution

Around an Elliptical Crack Under Arbitrary Loadings," Trans .

A.5.M.E.: J. Appl. Mech ., 33 1-11 (1966).

12. B. R. Lawn and T. R. Wilshaw, Fracture of Brittle Solids,

Cambridge University Press, Cambridge, 19.75, Ch. 3.

23
Figure Captions

1. Median crack parameters. Crack nucleates at extremity of deformation

zone below indenter point, and forms into a penny wholly contained

beneath contact surface (broken circle) . At some critical load

the crack breaks through spontaneously to the free surface

("pop-in") , usually with an attendant increase in crack

depth p_, and transforms into a well-developed half-penny

configuration (full semicircle) . This transformation may also

be effected in an alternative, stable manner, via the action of

residual stresses about the deformation zone, upon unloading the

indenter prior to pop-in.

2. Plot of K^OjO from Eqn. (3) (full line) for glass. Data points

(circles) represent measurements, using tungsten carbide indenters

,

made in air.^

3. Strength of soda-lime glass surfaces as function of indentation

load, for given WC cone, = 70°. Surfaces both as-received and

abraded with No. 100 SiC grit. Oil environment. Crosshead speed

of 0.5 mm min ^ for indentation tests and 50 mm min ^ for bend tests.

4. Strength of as-received soda-lime glass surfaces as function of

indentation load, for WC cones of different half-angles

(indicated). Oil environment. Crosshead speed: 0.5 mm min for

indentation tests; 50 mm min 1 for bend tests.

5. Strength of as-received soda-lime glass surfaces as function of

aging time between indentation and bend testing. Data for WC

cone = 70°, P = 100 N, oil environment. Crosshead speed: 0.5 mm min

24
6. Strength of as-received soda-lime glass surfaces, as function of

crosshead speed in indentation test. Data for WC cone =

70°, P = 100 N, oil environment. Crosshead speed for bend

. -1

tests 50 mm min

Al. Half-penny configuration of median crack, (a) plane view,

(b) side view. One seeks the crack increment orientation 0_ and £

which optimizes the energy release.

A2 . Angular terms for median-crack extension in tensile field at

several points (£_ = 0°, 45°, and 90°) on the crack perimeter.

A3. Tilt angle, 0*, of median-crack extension in tensile field.

A4 . Twist angle, , of median-crack extension in tensile field.

25


CONE INDENTER HALF-ANGLE,^
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Z-WIMIAI/H19N1U1S QN39
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CM

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

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 75-661

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Strength Degradation of Brittle Surfaces:

Sharp Indenters

5. Publication Date April 1975

6. Performing Organization Code

7. AUTHOR(S)

B. R. Lawn, E. R. Fuller, and S. M. Wiederhorn

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 3130453

11. Contract/Grant No.

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

Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

13. Type of Report & Period Covered Interim

7/74 - 6/30/75

14. Sponsoring Agency Code

ONR

15. SUPPLEMENTARY NOTES

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.)

A theory of strength loss for brittle surfaces in contact situations,

developed in a previous paper for "blunt" indenters, is here extended to the case of "sharp" indenters. A prior fracture mechanics analysis of crack

growth beneath ideal cone indenters serves as the basis for predetermining

the prospective surface degradation of ceramic components in service.

Compared to blunt indenters, severe degradation can occur at the lower contact

loads. However, at high loads the extent of degradation becomes remarkably

insensitive to indenter geometry. Essential theoretical predictions are verified by bend tests on glass slabs. The effect of indenter "sharpness"

and initial specimen surface flaw state are investigated systematically,

along with some secondary rate effects in the contact process. The

possibility of minimizing degradation via adjustment of material parameters

(including hardness) or surface condition (e.g., residual stresses, frictional

properties) is briefly discussed.

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)

Brittle solids; ceramic surfaces; degradation; fracture; indentation; strength

18. AVAILABILITY [^Unlimited

I For Official Distribution. Do Not Release to NTIS

1 Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. CJJ

[X 1 Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

37

22. Price

$3-75

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