NBSIR 75-730

Equilibrium Penny-Like Crack in

Indentation Fracture

B. R. Lawn and E. R. Fuller, Jr.

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

September 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

NBSIR 75-730

EQUILIBRIUM PENNY-LIKE CRACK IN

INDENTATION FRACTURE

B. R. Lawn* and E. R. Fuller, Jr.

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

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

Kensington, N.S.W. 2033, Australia

September 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

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Acting Director

ABSTRACT

A study is made of the mechanics of two basic

types of indentation fracture, cone cracks ("blunt"

indenters) and median cracks ("sharp" indenters) . The

common feature which forms the central theme in this

work is that both crack types, in their well-developed

stages of growth, may be regarded as essentially "pennylike". On this basis a universal similarity relation

is derived for equilibrium crack dimension as a function

of indentation load. Experimental measurements confirm

the general form of this relation. A more detailed

fracture mechanics analysis is then given, to account

for additional, contact variables evident in the data.

Notwithstanding certain analytical limitations, the study

serves as a useful basis for investigating a wide range

of contact-related problems, both fundamental and applied,

in brittle solids.

1 . INTRODUCTION

Considerable interest has recently been shown in

the crack patterns produced during the indentation of brittle

surfaces. Apart from providing a novel means for measuring

important fracture parameters (e.g. fracture surface energies,

crack velocities) , the individual indentation fracture event

serves as a convenient "basic microscopic unit" in the

ultimate description of a wide range of ceramics engineering

properties (degradation, abrasion, wear, erosion, etc.).*

By far the greatest attention has been directed to the

relatively well-defined crack configuration generated by a2,3 spherical indenter, the so-called Hertzian cons fracture.

The sphere typifies "blunt" indenters, in which the contact

prior to fracture remains predominantly elastic. By contrast,

the more complex fracture patterns produced by "sharp"

4 indenters (e.g. cone, pyramid), where limited irreversible

flow about the indenter point occurs as a necessary

precursor to crack growth, have gone largely unstudied. Yet

of the two extremes in indenter geometry, it is the sharp

indenter which emerges as more pertinent in real contact

situations where severity of surface damage is a prime

5,6 concern.

In this study we seek to establish a wider descriptive

basis for the cracking that occurs in the general indentation

of brittle solids, with particular emphasis on sharp indenters,

For this purpose it becomes convenient to identify various

phases in the evolution of the indentation fracture event,

from crack initiation in the contact field to full-scale
propagation in the advanced loading (and even unloading)

stages. While somewhat arbitrary, this approach usefully

highlights the similarities and differences in the blunt

and sharp indenter patterns, and permits an adequate

description of certain facets of a given fracture situation

in cases where the overall behaviour appears hopelessly

complex. It is found that the greatest differences occur in

the earliest stages of indentation, the very mode of crack

initiation depending critically on the geometrical distribution

of contact stresses. Conversely, the similarities become more

apparent in the well-developed stages, in which the far field

of the applied loading now controlling crack growth is

insensitive to details in the contact region. Ultimately, all

well-developed indentation cracks tend to expand on an everincreasing, near-circular front at advanced loading, thus

assuming a penny-like configuration. It is this last aspect

which forms the focal point in the present work.

2. GENERAL FEATURES OF INDENTATION FRACTURE PATTERNS

Fig. 1 illustrates the essential features of crack

geometry for both blunt and sharp indenters, along with the

relevant fracture mechanics parameters. In each case the

indenter sets up a contact stress field, the tensile

component of which provides the driving force for the ensuing

fracture. If the applied load P were to be effectively

concentrated at a point in the specimen surface (i.e. a+0) , the intensity of the stresses would vary according to a simple inverse-square relation (the so-called Boussinesq

1 4 field) . However, in reality the stress level cuts off
at some finite limit within the contact region: with blunt

indenters the cutoff is r.erely a manifestation of a redistribution in load over a nonzero, elastic contact area,

whereas with sharp indenters it is associated more closely

with the inability of the solid to sustain stresses greater

than some "yield" value'*. One may accordingly view the real

distribution of stresses along the prospective crack path in

terms of a hypothetical superposition of Boussinesq stresses

* and localised, near-contact "closure" stresses.

In this context we investigate the evolution of the two

situations depicted in Fig. 1, as follows:

2 .

1

Blunt Indenters

(i) Crack nucleat-Lon . Outside the elastic contact circle

(whose radius is determined by the load, indenter size, and

elastic constants) the stresses in a shallow "skin" layer,

2 3 8 where surface flaws pre-exist, are highly tensile. ' ' As

the stress intensity builds up with increasing load, one or more of the flaws nucleates a crack.

(ii) Crack formation. The "dominant" flaw runs around the

contact circle to form a shallow surface ring crack. Owing

to the restraining action of the "closure" stresses, the newly

formed ring does not, in general, extend spontaneously

downward: the system must first overcome an energy barrier.

Accordingly, the ring crack may be driven downward, either

stably by purely mechanical forces (i.e. by increasing the

2 indenter load) or subcritically by combined mechanical and

* The procedure is strongly analogous to that used in fracture

theory to represent the complex nonlinear stress field at the

7 tip of a propagating crack.
4 .

chemical forces (e.g. by introducing a reactive environment

9 into the system) , until some critical depth (typically ~ 0.1 a)

is attained.

(iii) Crack propagation . Once beyond the critical, formation

depth the surface ring propagates unstably into the full

Hertzian cone, at a depth ~ a. The crack is then said to be

"well-developed", in that details in the near-contact stress

distribution no longer greatly influence the fracture

mechanics. Increasing the driving force still further simply

causes the cone to continue its propagation, in a stable manner.

(iv) Crack unloading . Upon unloading the system the cone

10 crack tends to close (but rarely to heal)

.

J

" In certain

contact situations frictional tractions between indenter and

specimen may actually enhance growth of the cone system."^

2 . 2 Sharp Indenters

(i) Crack nucleation. In this case the contact is partially

plastic, necessarily so because of a singularity about the

4 sharp indenter point in an otherwise linear elastic field.

The plasticity somewhat relieves the tension locally about the

contact area (whose dimension is now determined by the load,

indenter shape, and material hardness) . Instead, the

greatest concentration of tensile stress occurs directly below

the indenter point, and the crack nuclei are created there by

the deformation process itself.

(ii) Crack formation. The deformation-induced crack

nuclei grow as near-pennies, wholly contained beneath the

contact zone, on median planes (planes containing the normal
5.

load axis) . As with the Hertzian surface ring fractures,

these so-called median cracks grow stably, but are susceptible to environmental effects. Depending on the

indenter geometry, this formation stage may continue to a depth z a or more. Several, mutually intersecting median

cracks may form during the course of indentation.

(iii) Crack propagation . At some stage in the growth the

median cracks begin to "break through" to the specimen surface;

the restraining, compressive "hoop" stresses outside the contact

4 area can no longer contain the expanding pennies. This

critical stage is not as well defined as the corresponding

cone development in the Hertzian test, and the different co- existing median cracks may "pop-in" successively as the

loading proceeds. The cracks then tend to the well-developed

configuration of more or less symmetrical, stably propagating

half-pennies centred on the contact point.

(iv) Crack unloading . Unloading the system causes the

median cracks to close. However, with sharp indenters there

are additional, residual-stress effects, attributable to

incompatibility between the plastic zone and surrounding elastic

12 material; the mechanical mismatch appears to set up a "reversed field" just prior to complete withdrawal of the

indenter. Thus the stresses which, on loading, acted to open

up the median cracks, tend now to compression, thereby

enhancing closure beneath the indenter. In the near-surface

region, on the other hand, the reversal is of opposite sign,

the development of a hoop tension actually opening up any

partially contained penny cracks to the specimen surface.
The net result is therefore always a crack configuration close

to the ideal half-penny shape, regardless of whether the fully

propagating stage was reached during loading or not. A

secondary manifestation of the residual stress field is the

initiation of an entirely different, laterally extending

system of cracks (not shown in Fig. lb); these lateral cracks

emanate from the deformation zone and grow in a saucer-shaped

configuration toward the specimen surface, corresponding to a 12 "chipping mode" of fracture.

_3- SIMILARITY RELATIONS FOR THE PENNY-LIKE CRACKS

In the interest of simplicity we proceed to an analysis

of the indentation fracture problem on the assumption that

crack growth is determined predominantly by point-load,

Boussinesq stresses. Such an assumption effectively restricts

the applicability of any fracture mechanics solution to the

fully-developed stages of crack propagation.- Preceding,

formation stages of crack growth may then be seen as precursor

stages, in which the near-contact field exerts an initial

perturbing restraint on the evolution of the fracture pattern.

With this simplification the way is open to an analysis

in terms of a scaling argument, in which mathematical formalism

13

may be conveniently circumvented. Roesler used this approach

to obtain a solution for the cone crack configuration, and in

the present section we merely generalise Roesler* s treatment.

The idea is to make use of the geometrical similarity of the

well-developed cracks, in conjunction with the fundamental

14 Griffith energy-balance condition for crack extension, to

derive a relationship between the size of the cracks and the

indenter load. In this way, noting that all well-<developed
indentation cracks are of the same geometrical for:?., i.e.

penny-like (in the sense that they extend on a near-circular

front) , we may obtain a general expression for equilibrium

growth

.

In accordance with the Griffith condition, one seeks

the configuration for which the rate of increase of total

surface energy just balances -he rate of decrease of total

mechanical energy as the crack expands. Since the area of

the crack surface must scale with the square of the

characteristic crack dimension, 2 say, the total surface

energy must be of the form

f

s

» re

2

, (1)

where f is the fracture surface energy. Again, we may note

that the intensity of the indentation stress field will be

determined by the point leaf divided by a characteristic area

2 (=?/j ) , that the strain energy density will be

given by the square of the stress divided by an elastic

2 <i modulus (<=? /o E, with E Young's modulus), and that the volume

of stressed material associated with the field of the crack

will scale with the cube of the crack dimension (=2

J

) , such

that the total mechanical energy may be written

U

n

= P

2 /Eo .

• (2)

The energy-balance requirement, d:'-,/de = -d: w/d:?, then

gives

D2. 3 P t /e = const, (3)

for equilibrium cracks.

In the point-load approximation used here all details

of the contact conditions are effectively "washed out".
However, we should expect the indenter size or shape to have

some influence in the fracture mechanics, for, as we have

seen, the very nature of the crack pattern (i.e. whether cone

or median) is determined to a large extent by the contact

geometry. Before taking up this issue in depth we shall first

examine pertinent experimental data in connection with the

verification of Eqn.3.

4_. EXPERIMENTAL MEASUREMENTS OF CONE AND MEDIAN CRACKS

Observations have been made of the manner in which the

size of both cone and median cracks depends on indenter load.

All tests were conducted on soda-lime float glass, 12.7 mm

thick, under conditions close to equilibrium (viz. under

conditions effectively excluding water vapour from the

cracks'^) . The progress of the cracks was followed either

photographically or by travelling microscope; inaccuracies in

the measuring techniques were found to be insignificant in

** comparison to the scatter in results from crack to crack.

A considerably greater proportion of the experimental effort

was devoted to the median cracks, these being relatively

undocumented in the literature.

4 .1 Cone Cracks

The cone crack tests were conducted within an environmental chamber, with the force on the indenter delivered

* *As seen in the data of Figs . 2 to 5 below. Together with the

data points, these figures include least-squares fits of Eqn.3

3/2 (corresponding to a minimisation of the error in P/c ) ; the

slopes of the lines thus fitted have a typical standard

deviation of 5%.
16

by a dead-weight loading machine. The indenter itself was a tungsten carbide sphere, bur with a polished flat of 1 ram rad

.

to maintain an invariant contact area (thus to ensure that the

surface trace of the cone crack not be enveloped by an expanding

contact circle) . A light surface abrasion of the glass test

slabs (e.g. in a slurry of No. 1000 SiC grit) prior to indentation

served to introduce an abundance of suitable "pre-existing flaws"

for crack initiation. The cracks were started by loading the

specimens slowly, in moist air; this allowed for the controlled

formation of well-defined surface rings, from which highly

regular cones could be propagated. Having thus developed a -4

crack, the chamber was evacuated (-10 Pa) , and the base raaius

R of the stably propagating ccr.e (representing the characteristic

dimension a of Sect. 3) recorded as a function of increasing load P.

Fig. 2 shows the results, for four cracks. The plot is seen

to be linear, within the scatter of the data. Eqn.3 thus

adequately represents the fracture mechanics for cone cracks.

4.2 Median Cracks

For the median crack tests a standard Instron universal

testing machine was used to deliver the indentation load.

Indenters included six tungsten carbide cones ranging in halfangle from v = 30° to 80° in 10° intervals, and a conventional

Vickers diamond pyramid in a heavy-duty mount. The glass

surfaces were indented in their as-received state. Initial

loading was slow, to permit the median cracks to be formed in a controlled manner. For the sake of reasonable speed in the data

accumulation, all subsequent crack propagation measurements were

made in normal laboratory test environment.

The first set of such measurements involved following

the depth 0 of the dominant median crack as a function of

steadily increasing load ?. In these tests the crosshair of
10.

a travelling microscope was continually translated to

predetermined positions along the path of the expanding crack,

and the load at the instant each crosshair position was

reached by the crack tip appropriately recorded. This procedure

conveniently provides the means for reducing the kinetic

contribution to crack growth (due to the inevitable presence

of water vapour) to a tolerably low level, through a simple

adjustment of the Instron crosshead speed. For instance, with

a crosshead speed of 0.5 mm min ^ in the present experiments

kinetic effects were estimated to give rise to increases of

*** considerably less than five percent in the crack lengths.

The data obtained are plotted in Figs. 3 and 4. As

with the cone crack data, these plots confirm the essential

form of Eqn.3, within the limits of experimental reproducibility.

However, certain additional features in the median crack results

warrant further attention here. First, as a result of a greater variability in the crack pattern, the scatter in data

becomes relatively pronounced at higher indentation loads.

This trend could be directly associated with discontinuities

in the downward growth of the leading median crack,

corresponding to the sudden pop-in and expansion of

intersecting neighbours. In general, the indentation process

was typified by an extended sequence of abrupt load drops,

these being most marked for the conical indenters of smaller

half-angle. A second complicating feature was the tendency

in the case of the most "blunt" conical indenter (i.e. = 80°)

for cone crack formation to precede median cracking. Here the

*** As adjudged from comparison with control tests in dry

nitrogen atmosphere, in which slow crack growth effects were

imperceptible

.
plastic component of the contact is relatively small, and

the effectiveness of pre-existing surface flaws as nucleation

centres for fracture accordingly matches that of the

deformation-induced flaws. The presence of the enveloping

cone crack noticeably restricted the development of the

contained median pennies . A second set of measurements, prompted by the observation

(Sect. 2. 2) that any median cracks still in the formation stage

tend to complete their development into the half-penny

configuration during unloading, was taken from the residual

crack patterns on the indented test surfaces. In these

experiments the Instron crosshead velocity (0.5 mm min was simply reversed upon the attainment of some prescribed peak

indentation load P

1 . The glass surfaces were then lightly

etched in dilute hydrofluoric acid until the cracks were

clearly visible, and the trace 2D 1 of the dominant median

crack appropriately recorded for each indentation. Plots of

the data thus obtained for conical indenters are given in

Fig. 5, and may be compared with their analogues of Fig. 3.

5_. MORE EXPLICIT FRACTURE MECHANICS ANALYSIS

While the data of the previous section verifies the

functional relationship between crack size and indenter load

given in Eqn.3, our fracture mechanics description remains

incomplete. The emergence of contact geometry as an important

factor in Figs. 2 to 5 emphasises a major inadequacy in our

simplistic treatment of Sect. 3. To account for this

additional factor a more detailed analysis is necessary. We

adopt the conventional fracture mechanics approach here,

seeking solutions for the crack-extension force G or stress-
intensity factor K, from which equilibrium equations may

once again be obtained via the Griffith condition, expressible

2 2 17 as G = K (1-v )/E (plane strain)

.

5 .

1

Cone Cracks

The case of well-developed cone cracks has already been

treated rigourously in the paper by Roesler,''"^ and we shall do no more than quote Roesler's solutions. The crackextension force, per unit width of crack front, is calculated

as

G = K^(v)P2/ER

3

(4)

p . where k is a dimensionless constant uniquely determined by

it Poisson's ratio v: the superscript p denotes a fully

propagating crack and the subscript R identifies the

characteristic crack dimension. In conjunction with the

Griffith condition, Eqn.4 gives the equilibrium expression

2 3 r> pVi? = 2TE/k*(v) . (5)

The dimensionless constant, although independent of the

geometry of the indenter (provided the contact remains blunt)

,

directly involves the cone crack angle a (Fig. la) according to

1

kJl(v) = /(v) [cos al . (6)

This result therefore represents only a minor elaboration on the similarity relation, Eqn.3. A numerical analysis by

Roesler for v=0.25 (ot=22°) gives k^=2 . 75xl0~ 3

, whereas from

n the slope of Fig. 2 we obtain for soda-lime glass, using

T=3.9 Jm"

2 and £=7.0xl0 10

Pa,

18

the value k

P

r

= (1 .19+0 .10) xl0~ 3

.
5 . 2 Median Cracks

For the median cracks we consider the centre-loaded halfpenny configuration of Fig. lb. We envisage a single welldeveloped crack driven primarily by the wedging component

of indentation force, P , normal to the median plane. In

the case of a smooth contact the wedging force is readily

resolved in terms of the characteristic half-angle ip of the

indenter,

P^ = P/2tani{; . (7a)

More generally, for a rough contact, Eqn.7a modifies slightly

to

P^

= P/2tan:'« , (7b)

with

' = \i) ± arctan u, (8)

u being the coefficient of sliding friction; the plus sign

in Eqn.8 refers to the loading half-cycle, the minus sign to the

unloading half-cycle. Contact friction may thus be regarded as either "blunting" (loading) or "sharpening" (unloading) the

indenter

.

The stress-intensity factor for the crack system under

19 consideration is obtainable from fracture mechanics handbooks,

K = 2P±

/(*D) 3/2

. (9)

(This equation omits a small correction factor associated with

free-surface effects.) Incorporating Eqns . 7b and 9 into the

Griffith condition, we have

P

2 /D 3 = 2TE/KP

D

(y> t y) (10)

in analogy to Eqn.5, with the dimensionless constant

.P

'D <P(v,^') = (l-v 2 )/- 3 tanV . (11)
The geometrical term ty' , unlike its counterpart a in the

cone crack system, varies independently of v. Using the

values of v, T and E already specified, we may evaluate

K^(if>) for soda-lime glass from the slopes of Figs. 3 and 5,

and thence investigate the validity of the present analysis

2 3 P 1/2 by plotting arctan {[(1-v )/ir k^] } as a function of t|s: According to Eqns. 11 and 8, such a plot should have a slope

of unity and an intercept on the ordinate dependent on y.

However, the results of the present experiments, Fig. 6, are

too uncertain to warrant an appropriate curve-fitting exercise

we recall in particular from Sect. 4. 2 the tendency for the

crack pattern to depart markedly from the ideal, single halfpenny configuration at extremes in half-angle Instead, we

simply plot the zero friction curve in Fig. 6 for comparison

with the data.

6. DISCUSSION

Our treatment of the indentation fracture problem

derives from the rationalisation that the well-developed

cracks are basically penny-like in form. This is tantamount

to assuming that the fracture mechanics are predetermined by

a point-loading, Boussinesq-type stress field, in which case we should recognise certain limitations of the approach. For

a start, as mentioned in Sect. 2, near-contact "closure"

stresses operate to retard crack growth in the "formation"

stage. The nature of these near-contact stresses determines,

among other things, the "critical load" at which growth

2 4 proceeds to the fully-developed stage ' Thus, in using

the present analysis to model ceramics engineering problems,

we are restricted to the more severe indentation damage
situations. A second limitation concerns the fact that the

Boussinesq picture appears to be inadequate for describing

the lateral cracks which form upon unloading sharp indenters

(Sect. 2. 2). Although its saucer-like geometry does preserve

the most essential feature of the penny configuration (namely

the outward expansion on a near-circular front) , the lateral

crack system evolves in an ill-defined residual stress field

12 whose very source is not well understood. This secondary

crack type warrants further attention, for it relates closely

1 20

to surface removal processes in brittle materials. '

At the same time, our circumvention of detailed

descriptions of the near-contact stresses in the loading halfcycle, and of the residual stresses in the unloading halfcycle, has not been entirely restrictive. We have still been

able to identify some parameters associated with the nature of

the contact, such as a in Eqn.6 for cone cracks and ifi and u in

Eqns.8 and 11 for median cracks, although the role of these

parameters could not be unequivocally established in the presen

experiments. Again, due recognition has been given to the

important function of residual stresses in expanding partially

contained median penny cracks to the specimen surface,

thereby permitting one to infer the relevant fracture mechanics

from examinations of crack traces on test surfaces after

21 indentation ; the potential application of this correlation

as a means for diagnosing the history of indentation damage

events in brittle materials, especially in opaque materials,

becomes obvious

.

Indentation fracture presents itself as a useful

addition to the mechanical testing repertoire of the materials

scientist. 1 Where the measurement of basic fracture mechanics

parameters is the objective the theory given here should serve
16.

as an adequate framework for analysing the data, provided the

indentation conditions are chosen such that a single, welldefined penny crack is produced: the cone crack system

appears to be best suited for this purpose. Although we have

considered only equilibrium cracks, an extension of the

analysis to include kinetic effects is not impracticable:

indeed, cone crack tests have actually been used to obtain

22 crack velocity data on glass in moist environments. Where

an analysis of more practical contact damage situations is

required one needs to bear in mind that the calculations are

likely to overestimate the crack dimensions. This constitutes

no real disadvantage in strength degradation studies, from

which we emerge with a sound basis for conservative

5,6 engineering design. Moreover, the distinction between blunt

and sharp indenters, representing extremes in contact severity,

provides convenient bounds for the description of real contact

situations.

ACKNOWLEDGEMENTS

The authors are indebted to S.M. Wiederhorn and

A.G. Evans for valuable discussions throughout this work.

Sponsorship by the Office of Naval Research under Contract No.

NR-032-535 is acknowledged.
REFERENCES

1. B.R. Lawn and T.R. Wilshaw, J . Mater. Sci. 10_ (1975)

2. F.C. Frank and B.R. Lawn, Proc. Roy. Soc. Lond. A299

(1967) 291.

3. T.R. Wilshaw, J. Phys. Dz Appl. Phys. 4_ (1971) 1567.

4. B.R. Lawn and M.V. Swain, J. Mater. Sci. 10_ (1975) 113.

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

J. Amer. Ceram. Soc.

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

J. Amer. Ceram. Soc.

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

(Cambridge University Press, Cambridge, 1975) Ch.4.

8. B.R. Lawn, J. Appl. Phys. 39 (1968) 4828.

9. A.G. Mikosza and B.R. Lawn, J. Appl. Phys. 4_2 (1971) 55

10. J.S. Williams, B.R. Lawn and M.V. Swain, Phys. Stat. So

(a) 2 (1970) 7.

11. K.L. Johnson, J.J. O'Connor and A.C. Woodward,

Proc. Roy. Soc. Lond. A334 (1973) 95.

12. B.R. Lawn, M.V. Swain and K. Phillips, J. Mater. Sci.

10, (1975)

13. F.C. Roesler, Proc. Phys. Soc. B69 (1956) 981.

14. A. A. Griffith, Phil. Trans. Roy. Soc. Lond. A221

(1920) 163.

15. S.M. Wiederhorn, J. Arier. Ceram. Soc. 50_ (1967) 407.

16. J.J.H. Beek and B.R. Lawn, J. Phys. E: Sci . Instrum.

5 (1972) 710.

17. Ref. [7],Ch. 3

18. S.M. Wiederhorn, J. Ar.er. Caran. Soc. 52 (1969) 99.
18.

19. P.C. Paris and G.C. Sih, .4ns J?. Soo. Test. Mater.,

Spec. Tech. Publ. 3_81 (1965) 30.

20. B.R. Lawn, Wear, 33. (1975) 369.

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22. M.V. Swain and B.R. Lawn, Internat. J. Fraet. 9_ (1973)

481.
19.

FIGURE CAPTIONS

1. Basic indentation fracture systems. (a) Cone crack

system, associated with "blunt" indenter: crack

nucleates from pre-existing surface flaw (small dashes)

outside contact (shaded) , forms into surface ring

(broken line) , and finally becomes critical and

propagates into fully developed cone. (b) Median crack

system, associated with "sharp" indenter: crack

nucleates from plastic contact zone (shaded) , forms into

contained penny (broken circle) and ultimately develops

into full half-penny.

2. Results from observations of fully-developed cone cracks

in soda-lime glass. Data points from vacuum tests, using

truncated spherical indenter. Each symbol represents

separate crack.

3. Results from observations of median half-penny cracks in

soda-lime glass. Data points from tests in air, with

crack dimension recorded directly as a function of

increasing load. Conical indenters, with half-angles

as indicated. Each symbol represents separate crack.

4. Results from observations of median half-penny cracks

in soda-lime glass. Data points from tests in air,

with crack dimension recorded directly as function of

increasing load. Vickers pyramid indenter. Each symbol

represents separate crack. Compare slope with those

of Fig. 3 (half-angle between opposing pyramid edges is

74°).
5. Results from observations of median half-penny cracks

in soda-lime glass. Data points from tests in air,

with crack dimension recorded from residual surface

trace as function of peak load. Conical indenters,

with half-angles as indicated.

6. Plot of median crack data to investigate the validity

of Eqn.ll.

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Characteristic crack parameter, D

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NBS-1 14A (REV. 7-7 3)

U.S. DEPT. OF COMM. BIBL.OGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 75-730

No.

3. Recipient \s Accession No.

4. TITUK AND SUBTITLE

EQUILIBRIUM PENNY-LIKE CRACKS IN INDENTATION FRACTURE

5. Publication Date June 197 5

6. Performing Organization ( ode

B.R. Lawn and E.R. Fuller, Jr.

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. J± JU43

11. Contract/Grant No.

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

Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

ONR

15. SUPPLEMENTARY NOTES

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

bibliography or literature survey, mention it here.) A study is made of the mechanics of two basic types of indentation fracture,

cone cracks ("blunt" indenters) and median cracks ("sharp" indenters) . The common feature which forms the central theme in this work is that both crack

types, in their well-developed stages of growth, may be regarded as essentially

"penny-like". On this basis a universal similarity relation is derived for

equilibrium crack dimension as a function of indentation load. Experimental

measurements confirm the general form of this relation. A more detailed

fracture mechanics analysis is then given, to account for additional, contact

variables evident in the data. Notwithstanding certain analytical limitations,

the study serves as a useful basis for investigating a wide range of contactrelated problems, both fundamental and applied, in brittle solids.

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

Contact fracture; degradation; Hertzian cracks; indentation fracture; median

vents; penny crack

18. AVAILABILITY [jg<Unl imited 19. SECURITY CLASS

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21. NO. OF PAGES

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26

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22. Price

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