NBSIR 75-664

Strength Degradation of Brittle

Surfaces: Blunt Indenters

B. R. Lawn, S. M. Wiederhorn, and H. H. Johnson

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

February 1975

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

LSvm^ B. R.^ Wiederhorn^ S. M. , Johnson^ H.

Strength degradation of brittle surfaces:

Blunt indenters, J. Am. Ceram. Soc , 58,

No. 9-10, 42S-432 (Sept. -Oct. 1975).

Prepared for Department of the Navy

Office of Naval Research

^ Oi

0

Arlington, Virginia 22217

313.00

NBSIR 75-664

STRENGTH DEGRADATION OF BRITTLE

SURFACES: BLUNT INDENTERS

B. R. Lawn, S. M. Wiederhorn, and H. H. Johnson

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

February 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. Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

STRENGTH DEGRADATION OF BRITTLE SURFACES:

BLUNT INDENTERS

B. R. Lawn*, S. M. Wiederhorn and H. H. Johnson

Institute for Materials Research

National Bureau of Standards

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

*0n study leave, from School of Physics, University of New South

Wales, Kensington, N.S.W. 2033, Australia.

Abstract

Indentation fracture mechanics is used to develop a theoretical

basis for predetermining the strength properties of brittle surfaces

in prospective contact situations. Indenters are classified as

"blunt" or "sharp", of which only the first is considered in the

present work. The classical Hertzian cone crack conveniently models

the fracture damage incurred by the surface in this class of indentation event. Significant degradation is predicted at a critical contact

load; however, with increasing load beyond this critical level,

further degradation occurs at a relatively slight rate. Bend tests

on abraded glass slabs confirm the essential features of the

theoretical predictions. The role of controlling variables in the

degradation process, notably starting flaw size and indenter

radius, is systematically investigated. An indication is also

given as to optimisation of material parameters.

I . Introduction

The possible strength degradation of a brittle surface due to

contact with a hard particle, especially in an impact situation,

is an important consideration in ceramics engineering. A typical

contact event may not in itself cause the failure of a structural

component, but may nevertheless generate highly localized stresses

of sufficient intensity to develop potentially dangerous cracks.

With variables relating to particle geometry, loading conditions

(e.g., load rate), mechanical properties of the materials,

state of the brittle surface, etc. complicating the general issue,

it is little wonder that a fundamental description of the problem

has been slow in evolving.

However, the emergence within the past decade of the field of

"indentation fracture mechanics " recently reviewed by Lawn and

Wilshaw^, establishes the necessary foundation for a theoretical

2 analysis of contact-induced degradation. Evans was the first to

propose a theory along these lines, but his treatment addressed

only one specific situation, that of impact loading with a hard

spherical indenter, and did not explore the role of many important

indentation variables. The present work was initiated in an effort

to generalise the Evans analysis. The philosophy adopted here

was that the strength properties of a material should be predictable

from basic fracture parameters as determined from standard indentation

tests
To this end it is convenient, following the scheme of Ref. 1,

to distinguish two extremes of indenter type: (i) "blunt indenters"

(typified by a hard sphere) , characterised by a perfectly elastic

contact such that crack initiation is controlled by pre-existing

flaws (usually at the specimen surface) ; (ii) "sharp indenters"

(e.g., cone or pyramid), characterized by a partially plastic contact

such that the starting flaws are produced by th<t actual indentation

process itself. "Real contact situations" may then be viewed as

lying somewhere between these two limits. We accordingly deal with

only blunt indenters in this paper, and defer investigation of sharp

indenters till a subsequent publication. For simplicity, crack

growth is considered to occur under quasistatic conditions throughout,

an assumption that should hold good for contact rates up to an

appreciable fraction of sonic velocities.

II. The Spherical Indenter and Hertzian Cone Fracture

We consider the nature of the cracking that results from the

contact between an elastic sphere (indenter) and an elastic-brittle

surface (specimen) . This is the most widely used test configuration

of the classical Hertzian contact^. The elastic field, although

complex, remains well-defined up to a critical load, at which point

a cone-shaped crack suddenly develops in the specimen. Initiation of

the cone crack invariably takes place et a pre-existing surface flaw

located just outside the contact circle, where the tension is greatest.

Fig. 1 indicates the basic test parameters.

2
The mechanics of cone extension through the inhomogeneous

Hertzian field has been investigated in great detail, both theoretically

3-10

and experimentally . For the purpose of strength evaluation we

need to know how the size of the cone crack varies with indenter load.

It is convenient to consider the behavior prior and subsequent to

critical loading in^ separate parts.

(1) Subcritical loading (P^P^)

As the indenter load steadily increases from its initial zero

value the ultimate starting flaw will experience a tensile stress

field of growing intensity. In the absence of kinetic effects in the

crack growth the condition for sudden development of the surface

flaw into a propagating cone may be readily formulated in terms of

the fundamental, equilibrium energy-balance criterion of Griffith'^''':

essentially, the cone develops when the crack-extension force associated

with the applied loading exceeds the resistance term associated with

the creation of new surface area. Because of the strong inhomogeneity

of the Hertzian field, it is no simple matter to derive a single,

exact analytical relation for the critical load as a function of

starting flaw size. However, one can obtain working expressions for the

1 3 limiting cases of "small" and "large" flaws '

:

(a) Small flaws . For surface flaws of effective initial length c° very small

in comparison with the scale of the elastic field (typically, ^<<0.01 a_,

3 where a^ is the contact radius ) the gradient of stress along the crack

trajectory may be considered to be negligibly small. In other words,

the flaw will tend to respond as if it were located in a field of

uniform tension. Thus, when the tensile strength of the material is

exceeded at the site of the flaw unstable crack propagation ensues,

3
and the cone accordingly develops spontaneously until a position

of stability is once more attained remote from the contact area.

On the assumption that initiation is favored at the surface location

of maximum tensile stress, i.e, at the contact circle, the Hertzian

equations predict^' ^' ^' ^

P = r^/W/x(V)El/2c° (c°«0.01a); (1) c f ^ here T_ is the fracture surface energy, k = (9/16)[(l-v ) + (l-v' )E/E'

]

is a dimensionless constant with v Poisson's latio and E Young's

modulus (the primes refering to the indenter material) , r is the sphere

radius and x(v) = { (3/4) [3 (1-V^) (1-2V) ^/327t] ^}-^/^ is another

dimensionless constant. We note in particular the dependence of the

critical load on flaw size and ball radius

(b) Large flaws . For flaws sufficiently large that the (negative)

gradient of tensile stress along their length becomes appreciable

(typically, c^O.Ola^) the approximation of a uniform field is no

longer valid. Instead, there is a tendency to limited initial growth

3 7

of the flaw as a stable surface ring just prior to cone formation '

That is, the crack must first surmount an energy barrier before it can

become unstable, and the condition for this to occur may again be

1 3—6

evaluated from the Hertzian equations '

F^ = 2Vkr/(p*{V) = Ar (c°>p.01a) , (2)

where ^* (V) is another dimensionless constant whose value can be

estimated (but only to within an order of magnitude) by numerical

» analysis, and A is a material constant known as Auerbach's constant

12 after the discover of the empirical "law" P^°^£. in 1891 . The

dependence of the critical load on ball size is, therefore, less

marked in the large-flaw region. The same is true of the dependence
on flaw size; indeed, c° has no direct influence on P at all in

Eqn. (2)

The predictions of Eqns . (1) and (2) are represented in Fig. 2.

In reality, the transition between the two regions of behavior is not as

sharply defines as the analysis above would suggest. This is indicated

in Fig. 2 by means of an experimental curve, evaluated from some

9

data on glass . (These data, and indeed the data described in the

present work, will be seen to fall more within the realm of Eqn. (2)

than of Eqn. (1).) Accordingly, the theory will generally underestimate

the true value of P , so that our ultimate estimates of strength will

be on the conservative side.

Throughout the subcritical stage^P^<P^, therefore, we may expect

little or no degradation of the brittle surface. It is true that limited

stable extension of surface flaws might occur just prior to critical

loading, but any such extension is predicted to be imperceptibly small

and will tend to be obscured in the general variation of flaw sizes

sampled in the typical strength test.

(2) Supercritical loading (P>P )

Once the critical load is exceeded the crack grows substantially

(typically, R>2a_, Fig. 1) . For the limiting case of a true cone

(i.e, R "^0) the mechanics of fracture become independent of affairs —

within the contact zone (i.e., of £ and £,£ ) » seen in the simple

equilibrium relation obtained by Roesler^*^,

P^R^ = 2Te/k.(.v) (r»r ) (3) K O

where k (v) is a dimensionless term which can be computed (to within a R

1 2 factor of about two) by numerical methods . Evans and others have

analyzed more general cone configurations, but the expressions

derived tend to be unwieldy. Moreover, the Roesler fomula underestimates the indenter load necessary to maintain the cone at a
prescribed depth, so that our strength estimates will once again be

conservative ones.

III. Strength Degradation Tests

The strength of a brittle solid is defined in terms of the

applied stress required to break a test piece in uniform tension.

From the Griffith energy-balance condition one obtains the standard

expression , ,

; "

r 2 T 1/2

a = [2rEA(l-v^)c^j

^ (4)

where a is the applied stress and c^ is the effective length of the

dominant flaw. This an equilibrium expression involving material

properties and surface state. The strength will accordingly be

reflected to some extent in the microstructural influence

on T_ (possibly also on E) , but to a much greater degree in the

influence of a wide range of past mechanical, thermal and chemical

events on c_. Most ceramic surfaces contain a diversity of surface flaws, —

up to 10pm deep in even carefully handled test pieces; of these it is the

weakest which determines the strength.

Since the prime concern in the present work was a systematic

investigation of the factors affecting the mechanics of degradation,

all strength tests were conducted under controlled surface conditions

13 14 after the manner of Mould and Southwick ' . Soda-lime glass was

selected as a test material, mainly because of the availability of

1 15 reliable data on its fracture properties ' (but also for reasons

of economy, mechanical isotropy and absence of microstructural

complications). The test pieces were laths 250x37.5x5.65 mm, suitable

for fracture in four-point bending with major span 204mm and minor

6
span 37.5 mm. A center spot approximately 15mm in diameter was grit

blasted with a given grade of silicon carbide abrasive on each

tension face prior to testing, thereby introducing a more or less

uniform density of starting flaws for the ensuing fracture. By

covering the abraded spot with a drop of paraffin oil immediately

after the blast treatment any effects of moisture-assisted slow crack

growth during subsequent testing could be largely eliminated.

The bulk of the specimens were indented within the abrasion area

with a tungsten carbide sphere of prescribed radius, at a preselected

load, before rupture in bending. The delay between grit blasting

and ultimate rupture never exceeded four hours.

(1) Strength of abraded test pieces

Strength tests were conducted on some non-indented specimens

to establish the reproducibility of the grit-blast treatment, and to

determine the effective length of the flaws resulting from this

treatment. In most cases the fracture could be unambiguously traced

to an origin within the central abrasion spot, but a few isolated

examples of premature edge failures were encountered; these latter

were rejected from the data accumulation. Table I summarizes

the results. The scatter in measured strengths for each grit size,

viz. less than ten percent, will be seen to be small compared to the

magnitude of the degradation effect to be investigated in the following

o subsection. The effective lengths of the abrasion flaws, c^=c_ —f —

10 -2

(grit size) , then follow from Eqn. (4) , using E = 7. 0 x 10 Nm , -2 15

V_ = 0.25 and r_ = 3.9 Jm for glass . Microscopic examination of

the grit-blasted areas showed individual microcrack damage centers

14 of dimensions consistent with those listed in Table I.

7
(2) Strength of indented test pieces

The effect of indentation load on strength was systematically

investigated in a series of runs on glass specimens under conditions

of given pre-abrasion treatment and sphere radius. In all such runs

the strength remained essentially unaffected up to a critical indentation load, beyond which it showed a steady decline. This picture is

consistent with the theoretical description outlined earlier: for

P^P^ (Section II (1) ) , we would predict the eff 'ictive length of the

dominant flaw to hold constant at the characteristic size of the

abrasion damage, i.e., c_ = c^ ; similarly, for P>P (Section II —f —T C

(2)), we would predict the effective flaw length to be governed by

the geometry of the newly propagating cone crack, i.e, £^ = c^(R,

(Fig. 1) . A more detailed analysis of the effectiveness of the

conical crack as an initiating center for catastrophic failure in a

tensile field can be given in terms of standard fracture mechanics

formulae (Appendix); we obtain, at R»R , c =R Q,(a) . where fi(a) is — —o —r — — — —

an angular-dependent dimensionless function (evaluated in the

Appendix) . Taken in conjunction with Eqn. (3) , this term then gives

the effective flaw size as a function of identation load. The

strength equation (4) accordingly becomes

a = [2rEA(l-V^) c°]^^^ (P<P^) (5a)

f

' C f

a = {(2rE)^-/Vf^(a)^/2 [TT(l-v2)]l/2 K^(V)l/6}p-Y3 (p^p^j^ ^^^^

These relations, together with Eqns. (1) and (2), provide, in principle,

the basis for predetermining the degradation behavior without resort

8
to extensive (and, for all but the most common ceramic materials,

expensive) failure testing under simulated service conditions.

Apart from the quantities IE, V and r_ already specified, the

information necessary for a complete evaluation of our expressions

may be obtained using straightforward indentation testing methods'*"

Typical values for a tungsten carbide sphere on a soda-lime glass

specimen are as follows: = 0.55 and X

= 5.1 x 10 ' (calculable

directly from the elastic constants) ; a = 22±1° (direct observation

3 -5

of cone angle ), giving Q= 0.25±0.02 (Appendix); ^= (1.7±0.2) x 10

(from tests on No. 320 grit-blasted glass in oil, A = (390±35) N/1 . 58mm,

-3

Eqn. (2)); and l<^= (1.3±0.2) x 10 (vacuum observations of fully

1 * developed cone crack ) . This leaves the abrasion flaw size

£° and sphere radius t_ as convenient test variables for investigation

here:

(a) Effect of flaw size . The first set of indentation/strength

tests was conducted to investigate the role of initial flaw size

on the ultimate degradation. The experimental results (data points)

are compared with the theoretical predictions of Eqn. (5) (full lines) in

Fig. 3, for a sphere of radius 1.58mm. We may note the following

points of interest concerning the correlation in the different

regions of behavior: (i) At p<P , there is no detectable degradation,

c as predicted (other than the usual deterioration associated with

*As indicated in Section II it is possible to compute values of

these dimensionless constants directly from a knowledge of the

Hertzian elastic stress field using basic fracture mechanics.

Hov/ever, the degree of uncertainty in absolute values thus computed

is currently such that direct experimental calibration affords by

far the more reliaible data^.

9
increasing severity of the preabrasion treatment) . (ii) At P-P , the StrengtK cut-off point is observed to be reasonably independent of the

abrasion flaw size^ (although small, systematic deviations from

constancy in P are apparent at the larger flaw sizes) . This is ~c consistent with behavior in the Auerbach domain (Table II)

(iii) At P.-*P^f the strength data asymptotically approaches the

(flaw-size independent) limiting curve for true cones. The degradation

is seen to be remarkably slight considering the intensity of loading

supported by the glass during indentation. However, above loads

of 3kN the experiment tended to end abruptly due to sudden failure

of either indenter or specimen. /- ' r^ ';

(b) Effect of sphere radius . A second set of tests was run to

investigate the influence of indenter size. Data and theory are

represented in Fig. 4, for an abrasion flaw size lOym. Considering

the different regions of behavior as before, we have: (i) At

P<P , negligibl© degradation, (ii) At P"P , the fall-off point

scales with sphere radius, once more in accord with Auerbach

behavior (Table III) . (iii) At P>P , an asymptotic approach to the

limiting curve is again observed (this occurring more rapidly with

diminishing indenter scale, with the smallest spheres producing a

slight tendency to overshoot below the calculated curve) . Again,

the limits to data accumulation were determined by an abrupt failure

during indentation.

IV. Discussion . -["

Within the limits of the assumptions embodied in the Hertzian

fracture model, our analysis establishes a conservative basis for

10
predetermining the strength properties of brittle ceramics in

contact situations involving blunt indenters . In addition, it

provides an indication as to which materials might be expected to

suffer the least degradation. Thus, according to Eqns . (1) and

(2) , the onset of strength loss many be suppressed to a certain

extent by choosing materials with large r_ and v (larger leading

9

to smaller y in Eqn. (1) , ^ in Eqn. (2) ) . Again, according to

Eqn. (5) , the strength loss once degradation does occur may be

minimized at large r_ being influenced by micros tructure , e.g.,

grain size) and (larger v_ diminishing £ in Equation (5)),

and also large E_. That is, the material should be tough and stiff,

with a large Poisson's ratio to restrain the development of a high

level of tensile stress in the indentation field.*

However, for a given indenter the most practical means

of controlling the prospective degradation is via the state

of the brittle surface. As seen in Fig, 3, the size

of the starting flaw, once above the lOym level typical of ceramic

surfaces, does not enter as a sensitive factor in the reckoning of

strength degradation. Indeed, we may note that, if anything, the

critical load for the onset of significant degradation is greater for

the more severely sibraded surfaces . This is in agreement with previous

observations of the function (£° ) , as indicated by the experimental

curve in Fig. 2; indeed, it seems that a ten-fold increase in flaw

size can lead to a two-fold increase in critical load in certain

*At the limiting value V=0.5, the tensile component in the general

indentation field disappears altogether

11
favorable casts''. While such behavior may be contrary to intuitive

expectation, a logical explanation is to be found in the very factor

responsible for the Auerbach relation Eqn. (2) , namely, the inhomogenious nature of the Hertzian field. Basically, for very large flaws

(typically, c° >0.1a) the stress gradients in the indentation field

can reach the point where the component of normal stress along the

flaw length actually changes from tensile alt the surface to compressive

9 below the surface ; such flaws will accordingly pi opagate less easily

than smaller flaws contained entirely within the region of complete

tension near the indented surface. This result is interesting in

that it suggests deliberate , severe pre-abrasion treatment as a

potential means of inhibiting the onset of degradation. (At the same

time, however, such treatment would, of course, weaken the material

in flexure, etc.)

The present analysis, although quasistatic, may be usefully extrapolated to certain time-dependent contact situations. Impact-induced

damage is an important practical case in point. Here, the quasistatic

approximation should hold provided the rate of contact does not approach

the velocity of elastic waves. The impact conditions under which such

extrapolation may be made are currently under investigation. The

advantages of using characteristic indentation fracture parameters

obtained in simple, hardness testing routines to predict more

complex phenomena involving in-service damage events are selfevident. Once the velocity of projectile impingement approaches

sonic values, the problem becomes a dynamic one, and our analysis

becomes invalid.

There are other time-dependent features in the cone-crack

growth which cause further complication in the general description.

12
In particular, we have specifically avoided consideration of complex, nonequilibrium crack configurations. Kinetic crack extension can

occur in a wide range of ceramic materials, notably as a result of

crack-tip interactions with hostile environments. This is especially

true of silicate glasses, where ambient moisture causes subcritical

cracks to propagate at substantial velocities (typically, up to

1mm s

"'")'''. Our procedure of administering a thin film of oil to

the abraded glass surfaces to restrict access of moisture to the

cracks therefore limits the range of applicability of the calculations

in their present state. Nevertheless, some of the most important

practical degradation situations would appear to be sufficiently

well simulated in the present experimental arrangement: the case

of impact already mentioned, where the duration of contact is too

brief for environmental effects to manifest themselves, and operations

under inert or vacuum condition (e.g., spacecraft), are just two

examples

In relating to practical problems one must also give consideration

to the history of a ceramic component subsequent to any contact

event. Indentation-induced cracks are susceptible to mechanical or

thermal shock; stepwise repropagations can occur, thereby enlarging

18 apparently innocuous cracks to potentially dangerous dimensions.

On the other hand, some cracks may actually close up and heal,

either spontaneously if sufficiently fast running that contaminants

do not penetrate along the interface, or via m.ass transport at

18 19 elevated temperatures '

13
One further point needs to be brought out here. As mentioned in

Section III (2) , the glass test pieces were prone to sudden failure

at high indentation loads (that is, if the spherical indenter did not

itself fail first) . An examination of indented surfaces invariably

revealed a pattern of radial cracks extending outward from the contact

center just prior to the failure event. This was especially noticeable

for the smaller indenters; indeed, in the case of the sphere of

radius 0.40mm in Fig. 4 radial cracking was evident at loads not

much in excess of the critical value for cone formation, and

appeared to be associated with the slight tendency for the data

points to overshoot the predicted degradation curve. Such radial

crack patterns are more typical of sharp indenters. Evidently, at

suitably high loads small spheres are capable of producing irreversible

flow in the material, and thus of penetrating the surface"*"; this is

aided by the fact that the expanding contact area tends to

encompass the surface trace of the cone crack at higher loads,

and thereby to suppress further cone extension. Since this alternative

crack system is potentially more degrading than the cone crack

system it demands closer examination. This provides the topic

for the second paper in this study.

Finally, brief comment may be made on a certain similarity between

the strength degradation curves described here for indentation damage

20

and those described by Hasselman for thermal shock. In both cases

there exists a range of "loading" (i.e., a load-range limit, AP_, or

temperature range, AT) over which little if any degradation is apparent.

14
At a critical load, both descriptions imply an abrupt drop-off in

strength, followed by a more gradual decrease with further load

increase. A common feature of the indentation and thermal shock

situations at subcritical loading is the availability of an amount

of strain energy insufficient for unstable crack propagation . The

condition for ultimate crack arrest once a supercritical condition

has been exceeded is, however, somewhat different for the two cases,

being due to a rapid (approximately inverse-square) fall-off in the

stress field about the contact point in the indentation operation,

and to the "fixed grips" nature of the loading in thermal shock.

Further differences also arise from a somewhat different role of

initiating surface flaws in the two cases.

Acknowledgements

The authors are indebted to A. G. Evans for motivating

discussions in the early stages of this work, and to E. R. Fuller

for assistance throughout. The sponsorship by the Office of

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

15
Appendix

The mode of failure of a cone-shaped crack in a uniform tensile

Stress field has been given only a cursory consideration.'^

Full-scale fracture expected to initiate from one of

two diametrically opposite positions on the base rim of the cone

coincident with the symmetry plane containing the axes of intial

contact loading and subsequent flexural tension. We may obtain an

approximate solution for this configuration by disregarding the crack

curvature on either side of any such favored position, considering

instead the representative case of a plane crack of half-length

C, width infinity (parallel to the eraak f:font) , inclination £

to the tensile axis. The scheme is shown in Fig, hi.

We are now in a position to apply standard fracture mechanics

21

procedures to the problem . The stress-intensity factors for an

inclined plane crack in uniform tension are

= a(Trc)"'"'^^ sin^ a . ,.,.

. V 1/2 . ^11 ~ ^^"^^^ sma cosa

(Al)

where the subscripts denote 1, "opening", and II, "sliding",

modes of fracture. (Actually, since our representative configuration

strictly involves an edge crack, an additional multiplying factor

20

?»1.12 should appear on the right side of the expressions in Eqn. (Al)

However, this is nullified to some extent by a further factor,

smaller than unity, arising from the curvature in the cone crack

situation . ) The present problem is complicated by the fact that

the crack will not, in general, extend in its own plane; rather,

it will tend to extend at some energetically favorable tilt angle 9,

16
as indicated in Fig. Al, This complication can be handled

by regarding the critical incremental extension as a reinitiation

22

process in the near field of the original crack . We note that

the incremental crack experiences both opening and sliding types

of local stress:

= [K^/(2T^r)^/^]fQQ + [K^^/(2TTr)l/2JfII ^ •/ (27rr)

O^Q = [K^/(27Tr)^/^]fjQ + [K^j/(27Tr)^/^]f^Q = K^^ '/ (2TTr )

-

(A2)

where the _f terms are the angular-dependent components of the standard

21,22 crack-tip stress formulae

I 3 fgg = cos (9/2)

f^g = sin (0/2) cos^ (9/2)

fgg = -3 sin (9/2) cos^ (9/2)

f'^l = cos (9/2) [1-3 sin^ (9/2)] rc

(mode I) (A3)

(mode II) (A4)

and the "transformed stress-intensity factors" K^' and Kjj' define

the field for the modified crack. From (A2) and (Al) we have

1/2 I 2 II K^'(9,a) = a(7TC) [fgg sin a + fgg sina cosa] . ^^^^ 1/2 I 2 II K^^'(9,a) = a(7rc) [f ^ sin a + f „ sina cosa]

II re re Now to determine the energetically most favorable path from

the original crack tip we compute the mechanical-energy-release

rate G per unit width of crack front as a function of angular

variation,

G(9, a) = [(l-v^)/E] tKj'^(9, a) + a)]. (a6)

For a given inclination angle a, we may assume that for isotropic

solids the crack will extend at that angle e_* which gives a

maximum G* (a) in the energy release, i.e.

17
0G/8C =0. (A7)

This equation, in conjunction with Eqns. (A5) , (A4) and (A3) reduces

Eqn. (A6) to the form

G*(a) = [TT(i-v'^)a^c/E]w(a) ,

' (a8)

where w(a) is a dimensionless constant whose value attains a

limiting value unity for normal cracks.

Equation (AS) then leads directly to the strength equation

(4) of the text if the Griffith energy-balance condition

G* = 2T "

(A9)

is satisfied, and if we write '

= C oj(a) . (AlO)

as the effective length of the flaw. In the limit of well-developed

cones (R>>R , Fig. 1) we have — —

C = R/cosa. (All)

Thus, from (AlO) and (All),

c^/R = a)(a)/cosa E f2(a). ' (A12)

The quantity fi(a) therefore defines our required angular term

in Eqn. (5) of the text. Both Q. evaluated numerically,

are plotted in Fig. A2, For the special case of a=22° (glass)

,

we compute 0=0.25, 0_*=-63°. An independent, empirical evaluation

2 from strength data by Evans yields ^^=0.20. Since Q_ ultimately

appears in the degradation formula Ban. (5b) under a square root,

uncertainties in the above analysis will not be of major

significance .

18
References

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

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

2. A. G. Evans, "Strength Degradation by Projectile Impacts,"

J. Am. Ceram Soc . , 56 [8] 405-409 (1973).

3. F, C. Frank and'B. R. Lawn, "On the Theory of Hertzian Fracture,"

Proc. Roy. Soc. Lond . , A299 [1458] 291-306 (1967).

4. B. R. Lawn, "Hertzian Fracture in Single Crystals with the

Diamond Structure," J. Appl. Phys ., 39 [10] 4828-4836 (1968).

5. F. B. Langitan and E. R. Lawn, "Hertzian Fracture Experiments

on Abraded Glass Surfaces as Definitive Evidence for an Energy

Balance Explanation of Auerbach's Law," J. Appl. Phys . , 40

[10] 4009-4017 (1969)

6. T. R. Wilshaw, "The Hertzian Fracture Test," J . Phys . D; Appl. Phys . , £ [10] 1567-1581 (1971).

7. A. G. Mikosza and B. R. Lawn, "Section-and-Etch Study of

Hertzian Fracture Mechanics," J. Appl. Phys . , 42 [13] 5540-5545

(1971)

8. ' J. S. Nadeau, "Hertzian Fracture of Vitreous Carbon," J. Am.

Ceram. Soc , 56_ [9] 467-472 (1973).

9. B. R. Lawn, T. R. Wilshaw and N. E. W. Hartley, "A Computer

Simulation Study of Hertzian Cone Crack Growth," Intl. J.

Fract. Mech ., 10_ [1] 1-16 (1974).

10. F. C. Roesler, "Brittle Fractures Near Equilibrium," Proc. Phys . Soc, B69 [10] 981-992 (1956).

11. A. A. Griffith, "The Phenomena of Rupture and Flow in Solids,"

Phil. Trans. Roy. Soc. Lond ., A221 , 163-198 (1920).

12. F. Auerbach/ "Measurement of Hardness," Ann. Phys. Chem . , 43 , 61 (1891)

13. R. E. Mould and R. D. Southwick, "Strength and Static Fatigue

of Abraded Glass Under Controlled Ambient Conditions: I. General

Concepts and Apparatus," J. Am.. Ceram. Soc , 42 [11] 542-547

(1959) .

14. R. E. Mould and R. D. Southwick, "Strength and Static Fatigue

of Abraded Glass Under Controlled Ambient Conditions: II. Effect

of Various Abrasions and the Universal Fatigue Curve,"

J. Am.. Ceram. Soc, 42 [11] 582-592 (1959).

19
15. S. M. Wiederhorn, "Fracture Surface Energy of Glass," J . Am . Ceram. Soc , 52 [2] 99-105 (1969).

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

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

17. S. M. Wiederhorn, "Influence of Water Vapor on Crack Propagation

in Soda-lime Glass," J. Am. Ceram. Soc , 50 [8] 407-414 (1967).

18. S. M. Wiederhorn, B. J. Hockey and D. E. Roberts, "Effect of

Temperature on the Fracture of Sapphire," Phil . Mag . , 28 [4]

783-795 (1973)

19. B. J. Hockey and B. R. Lawn, "Electron Microscopic Observations of

Microcracking About Indentations in Aluminum Oxide and Silicon

Carbide," J . Mater. Sci . , in press.

20. D. P. H. Hasselman, "Unified Theory of Thermal Shock Fracture

Initiation and Crack Propagation in Brittle Ceramics," J. Am.

Ceram. Soc, 52 [11] 600-604 (1969).

21. P. C, Paris and G. C. Sih, "Stress Analysis of Cracks," Am. Soc . Test. Mater . , Spec. Tech. Publ. 381, 30-83 (1965).

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

Cambridge University Press, Cambridge, 1975, Ch. 3.

20
Table I. Strength and effective flaw size (mean value ± standard

deviations for at least ten tests per run) of glass test

pieces preabraded in SiC grit blast at air pressure 0.275

MNm . Bend tests in oil environment at Instron crosshead

speed soTM min-\

Grade SiC grit

-2

Strength, a /MNm Flaw size,£°/ym

100 79±7 2 3±5

150 86±4 19±3

200 94±7 16±3

^40 107±9 12±2

320 123±5 10±1

21
Table II. Critical leads to cone fracture, computed from Eqns . (1)

and (2) for soda-lime glass for given indenter r^=1.58mm,

as function of flaw size. Note that P (Eqn, (2))>P (Eqn. —c —

(1)) for all £° ; i.e., data lie in "Auerbach region"

(see Fig. 2)

19 16 12 10

0.5 0.7 1.0 1.4

390 390 390 390

Flaw size, c. /ym 23

—t

P /N (Eqn. (D) 0.4 —

P /N (Ean. (2)) 390

22
Table III. Critical loads to cone fracture, computed from Eqns . (1)

and (2) for soda-lime glass for given flaw size £^ = lOym,

as function of indenter radius. Note that P >(Eqn. (2))> —

P (Eqn. (1)) for all r, i.e., data lie in "Auerbach region".

Indenter radius , r/mm 6.34 3.17 1.58 0.79 0 .40

P /N (Eqn. (1)) —

22 5.5 1 .

0.3 0 .1

P /N (Eqn. (2)

1560 780 390 195 98 .5

23
Figure Captions

1. Hertzian cone crack parameters.

2. Critical load to cone fracture as function of starting flaw size.

Limiting cases of Eqns. (1) and (2) shown as full, straight lines.

(Note, controlling equation is determined by which predicts the

larger value of P . ) Real behavior shown as curve. (Note —c

"tail" at large £° in real curve.)

3. Strength degradation as function of indentation load, for given

VJC sphere, £ = 1.58mm, on soda-lime glass surfaces containing

different abrasion flaw sizes (indicated) . Oil environment.

Crosshead speed for indentation tests 0.5mm min ''' , for bend tests

50mm min .

4. Strength degradation as function of indentation load, for sodalime glass surfaces containing given abrasion flaws c° - lOym,

using WC spheres of different radii (indicated) . Oil environment.

Crosshead speed for indentation tests 0.5mm min ^, for bend tests

50mm min ^.

Al. Plane-crack representation of Hertzian cone configuration (side

view) . One seeks the crack increment at {r_, 6) , with respect to

origin at tip of representative crack, which optimizes energy

release conditions.

A2. Angular terms for cone-crack extension in tensile field.
2R

1. Hertzian cone crack parameters.
10'

u

Q.d

O

c

0)c

8

•c

<DU

3

o

10^

10"

10-

1 1 1 1

\ Eq. (1) 1

^ \ Expt. \ ^

\^ Eq. (2)

" \

\ .. ... . .. \

1 \| 1

1 10* 10- 10- 10- 10° 10'

Reduced flaw size,Cf/a

Critical load to' cone fracture as fiinction of starting flaw si

Limiting cases of Eqns. (1) and' (2) shown as full, straight li

(Note, controlling equation is determined by which predicts th

larger value of P^.) Real behavior shown as curve. (Note

"tail" at large c° in real curve.)
150

100

t A

• C^ = 23 /jm

50

m • •

0 —*

150

0 1 2 3

100

50

••

0

150

100

1

•

1

'e 0

•

1 2 3

engtti 50 • •

C° r 16yL(m

•

CO 0 1 —« « T .1 1

It

bend

( 150)

•

1 2 3

lur-poir 100

50 •

•

•

C°f - 12pm

• o

u.

150

1 1

1

) 1 2 3

100

50

* M

•

•

• ! 1 »_

1 ) 1 2 3

Indenter load, P/kN

Strength degradation as function of indentation load, for giv

WC sphere, r_ = l.SSmm, on soda-lime glass surfaces containing

different abrasion flaw sizes (indicated) . Oil environment.

Crosshead speed for indentation tests 0.5mm min , for bend t

. SOrnm min ^
150,

. • 1 100 • r= 6.34 mm

50

* • • • •

• n 1. .- 1 1

^1 0 1 2 3 4 5 150

100 r = 3.17 mm

50 •

E2ste

3 4

1*= 1.68 mm

rs 0.40 mm 12 3 4 Indenter load, P/kN

J

3 4 5

rsO.yg mm

Strength degradation as function of indentation load, for sodalime glass surfaces containing given abrasion flaws c° ss lO'fjm,

using WC spheres o£ different radii (indicated) . Oil environnent.

Crosshead speed for indentation tests O.Smm min , for bend tests

50mm min
Al. Plane-crack representation of Hertzian cone configuration (side

view) . One seeks the crack increment at (£, 9) , with respect to

origin at tip of representative crack, which optimizes energ\'

release conditions.
Cone-crack Inclination angle, Ct

A2. Angular terms for cone-crack extension in tensile field.
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USCOMM-NBS-DC
NBS-114A (REV. 7-73)

U .S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBl.K ATION OR RI-;PORT NO. NBSIR 75-664

2. Gov't Accession

No.

3. Recipient's Accession No.

4. Tl 1 Lh AND SUBTITLh

Strength Degradation of Brittle Surfaces:

Blunt Indenters

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHOR(S)

B. R. Lawn; S. M. Wiederhorn; H. H. Johnson

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

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Department of the Navy

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7/74 - 6/30/75

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ONR

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16. ABSTRACT (A 200-word or less (actual summary of most si^ificant information. If document includes a significant

bibliography or literature survey, mention it here.) Indentation fracture mechanics is used to develop a theoretical basis for

pre-determining the strength properties of brittle surfaces in prospective

contact situations. Indenters are classified as "blunt" or "sharp," of which

only the first is considered in the present work. The classical Hertzian

cone crack conveniently models the fracture damage incurred by the surface

in this class of indentation event. Significant degradation is predicted to occur at a critical contact load; however, with increasing load beyond this

critical level the degradation rate becomes relatively slight. Bend tests on abraded glass slabs confirm the essential features of the theoretical predictions.

The role of controlling variables in the degradation process, notably

starting flaw size and indenter radius, is systematically investigated. An

indication is also given as to optimisation of material parameters. The

analysis leads to some novel suggestions concerning surface preparation

procedures that might be followed in order to minimize strength losses.

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; cracks; degradation; fracture; Hertzian;

indentation; strength

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