NBSIR 75-659

A New Mode of Chipping Fracture

in Brittle Solids, and Its Application

in a Model for Wear Under Fixed

Abrasive Conditions

I. Mode of Chipping Fracture

II. Wear Model

B. R. Lawn

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
L^m, B. R., Swain, M, V., Phillips, K.,

On the mode of chipping fracture in brittle

solids, J. Mater Sci. Lett. 10, 1236-1239

(1975). ' / y

313.

Lavjn, B. R,, A model for the wear of brittle

solids under fixed abrasive conditions.

Wear Short Commun. 33, 369-372 (1975).

313
NBSIR 75-659

A NEW MODE OF CHIPPING FRACTURE

IN BRITTLE SOLIDS, AND ITS APPLICATION

IN A MODEL FOR WEAR UNDER FIXED

ABRASIVE CONDITIONS

I. MODE OF CHIPPING FRACTURE

II. WEAR MODEL

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

B. R. Lawn

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

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

the Mode of Chipping Fracture in Brittle Solids

Chipping processes in brittle solids, despite their unquestionable

relevance to a diversity of technologies from ceramics finishing

to geological engineering, are not well understood at the fundamental

level. Some recent studies of microfracture patterns beneath standard

J. , z. hardness indenters do, however, provide some insight into the problem ,

and it is our objective here to indicate how this insight may be

applied to construct a physical model of chipping fracture.

Essentially, the picture which emerges is that depicted in Fig. 1:

(i) Upon loading the indenter, a confined zone of irreversible

(plastic) deformation forms about any sharp points or corners (thereby

accounting for the residual hardness impression) , from which

"median vent" cracks first initiate and siobsequently propagate radially

outward along suitable planes of symmetry (e.g. as defined by the

diagonals of a pyramid indenter, or by preferred cleavage planes)

containing the contact axis; (ii) Upon unloading the indenter, the

median vents close up, but, just prior to complete removal, "lateral

vent" cracks initiate and extend laterally from the deformation zone

toward the specimen surface. Of the two types of cracking it is

clearly the second which relates more directly to brittle chipping.

Yet up till now a detailed fracture mechanics analysis has been

attempted only for the median vent system. This system is relatively

well defined, since the indentation stress field, which uniquely

determines the extent of crack growth^, can reasonably be represented

2 3 in terms of the classical Boussinesq field for normal point loading '

On the basis of the fundamental Griffith energy-balance condition

4

for fracture , it may readily be argued that brittle cracks will generally

1
tend to follow trajectories of the lesser principal stresses within

the indentation field, such that the path maintains near-orthogonality

to a component of major tension: the best studied illustration of

5 6 this principle is the Hertzian cone crack ' , which, in the absence

of any deformation-induced nucleation center, initiates from an

incipient surface flaw and flares downward into the specimen.

In the scheme of Fig. 2, in which the principal stresses are defined

such that ^ii'^°'22^'^33 ^P^^^^-"-^® values denoting tension) nearly everywhere, median vent geometry may be specified in terms of families

of Q^^ and trajectories, cone crack geometry in terms of families

of and O^^ trajectories.

The conditions under which the lateral vents form are, unfortunately,

less easily modelled. Since the lateral system operates only as the

indenter is withdrawn from the specimen surface it is evident that

the driving force for propagation must originate from some residual

stress field associated with the irreversible deformation zone.

This conclusion is substantiated by microscopic investigation

of the damage patterns as a function of indenter geometry (e.g. "sharp"

or "blunt") : in general, the extent of lateral venting is found to

increase markedly with expanding zone size. A graphic illustration

of the effect is obtained by loading a soda-lime glass plate with

a small ( 1mm diam.) spherical indenter: at comparatively low load

the contact is elastic, and the only fracture is that of cone cracking,

whereas at higher load some plasticity develops beneath the penetrating

sphere, and lateral venting becomes evident in the unloaded plate.

7 A related effect was reported by Culf j who observed otherwise

2
regular cone cracks in glass to deflect upward ("hat brim" effect)

upon sudden release of the indenter load. Culf also observed considerable residual stress birefringence in association with this phenomenon,

over distances large compared with the scale of the deformation zone

itself. That residual stresses exist about hardness impressions in

most brittle materials has been amply demonstrated by a number of

g strain-sensitive techniques . That these stresses can also be

moderately long-range in nature is seen most clearly in the distances

over which relaxation by plastic flow (e.g. dislocation loop punching)

9 occurs m annealing experiments . Neither the existence nor the

intensity of the residual elastic fields should come as any surprise,

for the stress levels achieved beneath the indenter in hardness tests

on highly brittle solids tend to be of the order of the intrinsic bond

strength of the structure'^^' and the relief of these high stresses

would ideally require the impressed region to restore completely to

its original unstrained state

.

These observations, coupled with a reexamination of the Boussinesq

field, provide us with a working model upon which to base an analysis of

lateral vent formation. We note that the lateral vents extend in all cases

on surfaces closely delineated by families of a^^ and a^^ trajectories

in Fig. 2 (although the paths are modified somewhat by the deformation

zone itself, and by free surfaces, including any preexisting median vents

or cone cracks) ; it is as if the applied load were actually reversed upon

indenter withdrawal, so that the O stress normal to the lateral vent

becomes the dominant component of tension in the filed. Of course, it is

3
physically meaningless to associate a reversed applied load with a surface

in the unloaded state, but an effectively similar net result may obtain if

the deformation zone were to act as a center of contraction with respect

to the surrounding elastic matrix. This effect is depicted schematically

in Fig. 3. The distribution of stresses at the zone boundary must

inevitably depend strongly on the nature of the irreversible deforma11 12 tion (which itself remains an issue of some controversy ' ) .

Nevertheless, one can proceed by making reasonable assumptions as to

this distribution (e.g. that the tractions are of constant magnitude,

and are directed such that the net force is zero) , and evaluate the

residual field in the matrix by taking expressions for the stresses

13 due to elemental point forces (e.g. Mindlin ) and integrating around

the boundary. One may then construct a stress trajectory pattern for

the field, in analogy to Fig. 2, and thereby trace out prospective

fracture paths from the deformation zone. Full details of such calculations will be discussed elsewhere; we simply report here that

the predicted paths do indeed curve toward the specimen surface in

essentially the manner shown in Fig. 1.

The scope of the present model extends well beyond the establishment of a suitable basis for evaluating an "index of brittleness"

14 in standard hardness testing . It provides physical insight into

a number of seemingly unrelated phenomena in brittle solids:

(i) Strength degradation . Surface damage introduced into a brittle

surface as a result of contact (either static or impact) with hard

particles constitutes a potential source of weakness . The mechanics of the

damage process may be conveniently simulated in a simple indentation

test"""^. 4
(ii) Glass cutting. A glass cutter's wheel is designed to produce a

continuous "trailing" median vent as a linear starting crack for subsequent plate fracture in flexure. However, lateral venting invariably

occurs in the wake of the moving "indenter", thereby damaging the edges

of the final cut. Clearly, the need here is for a m.eans of

suppressing the chipping mode.

(iii) Surface removal processes . Individual chipping events in the

machining, drilling, grinding, abrasion, erosion and wear of brittle

surfaces in general (e.g. ceramics, gemstones, rocks) are of the type

depicted in Fig. I"''. By summing over an appropriate distribution of such

microscopic events, it should be possible to describe macroscopic surface

removal parameters at a fxmdamental level

(iv) Geophysical impact phenomena . Meteorite-induced craters ranging

16 17 in scale from geological land masses to lunar fines bear a resemblance

to the damage pattern in Fig. 1 which can only be described as striking.

While thermal and stress-wave effects associated with the high-velocity

18 impacts are undoubtedly important factors in these cases , the possible

role of residual stresses about the central "deformation zone" in

determining crater morphology may warrant further attention.

B. R. Lawn

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

U. S. A.

M. V. Swain

Martin Marietta Laboratories

1450 South Rolling Road

Baltimore, Md. 21227

U. S. A.

K. Phillips

Division of Materials Science

University of Sussex

Falmer, Sussex BNl 9QT

England

5
References

1. Lawn, B. R. and Wilshaw, T. R. , J. Mater. Sci ., in the press.

2. Lawn, B. R. and Swain, M. V., J. Mater. Sci ., in the press.

3. Boussinesq, J., Application des Potentiels a 1 'Etude de I'Equilibre

et du Mouvement des Solides Elastiques , (Gauthier-Villars , Paris,

1885). Discussed in Timoshenko, S. P. and Goodier, J. N.,

Theory of Elasticity (McGraw-Hill, New York, 1970), pp. 398-402.

4. Griffith, A. A., Phil. Trans. Roy. Soc. Lond ., A211 , 163 (1920).

5. Hertz, H., J. Reine Angew. Math ., 92 , 156 (1881); Verhandlungen

des Vereins zur Beforderung des Gewerbe Fleisses , 61 , 449 (1882).

Reprinted in English, in Hertz's Miscellaneous Papers (Macmillian,

London, 1896), Chs. 5, 6.

6. Frank, F. C. and L%wn, B. R, , Proc . Roy . Soc . Lond . , A299, 291 (1967).

7. Culf, C. J., J. Soc. Glass. Tech ., 41, 157 (1957).

8 . Hockey , B . J . , in The Science of Hardness Testing and its Research

Applications , Symposium Proceedings, Eds. Westbrook, J. H. and

Conrad, H. (American Society for Metals, Metals Park, 1973), Ch. 3.

9. Wagatsume, R. , Sumino, K., Uchida, W. and Yamamoto, S. J . Appl .

Phys . , 42_, 222 (1971) .

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

11. Hill, M. J. and Rowcliffe, D. J., J. Mater. Sci ., in the press.

12. Ernsberger, F. M., Ann. Rev. Mat. Sci ., 2, 529 (1972).

13. Mindlin, R. D., Physics , ]_, 195 (1936).

14. Westbrook, J. H., in The Science of Hardness Testing and its Research

Applications , Symposium Proceedings, Eds. Westbrook, J. H. and

Conrad, H. (American Society for Metals, Metals Park, 1973)

,

' pp. 491-494.

6
15. Evans, A. G., J. Amer. Ceram. Soc , 56 , 405 (1973).

16. Nadai, A., Theory of Flow and Fracture of Solids (McGraw-Kill , New

York, 1963), pp. 247-249.

17. Carter, J. L. and MacGregor, I. D., Proc. Apollo 11 Lunar Sci .

Conf . , 1, 247 (1970)

.

18. Vedder, J. F. and Mandeville, J.-C, J . Geophys . Res . , 79 , 3247 (1974).

7
Figure Captions

1. Fracture geometry beneath sharp indenter. Central deformation zone

shown as dark region, median vent cracks as broken lines, lateral

vent cracks as heavy lines. (a) Section view schematic, (b) plan

view schematic, (c) surface view of fused silica indented with sharp,

irregular particle (scanning electron micrograph, field width 3mm)

.

2. Stress trajectories (curves whose tangent indicates direction of

principal stress) for Boussinesq field, showing half-surface view

(top) and section view (bottom). Cone cracks initiate from

incipient surface flaws and propagate everwhere orthogonally to O^^

(tensile outside contact area) , median vents initiate from central

defomation zone and propagate orthogonally to O^^ (tensile below

contact zone) , lateral vents initiate from deformation zone and

propagate nearly orthogonally to (compressive everywhere, but

tensile if applied load reversed)

.

3. Schematic representation of distribution of mismatch tractions at

boundary between central deformation zone and surrounding elastic

matrix, at indenter withdrawal.

8
(a)

I

i

I

Ii

I

I

I

I

I

DEFORMATION

ZONE

ELASTIC

MATRIX

II. A Model for the Wear of Brittle Solids

Under Fixed Abrasive Conditions

other than that the wear mechanism involves some microfracturing

,

and the wear rate is remarkably high, relatively little is known

about the abrasion of highly brittle solids '

; this despite intense

current interest in the machining and finishing of brittle surfaces

3 within the ceramics engineering industry . However, with the advent

of "indentation fracture mechanics" a new approach has become avail4

able for investigating a wide range of small-scale cracking phenomena .

The purpose of the present note is to use this approach to construct

an explicit model of the wear process in brittle solids, for the

simple case of a "fixed" abrasive medium ("two-body" process) in

which the grit particles are "ideally sharp."

A schematic representation of the wear mechanism is given in

Figure 1. Macroscopically , one measures the wear rate V = dV/dt (V =

volume removed) appropriate to a specified total load and velocity

_v_^ for the abrasive medi\im relative to the specimen. Microscopically,

attention focusses on the individual chip-removal mechanism,

characterized by an "indenter" load and velocity ^ (all "indenters"

traverse the specimen with the same velocity in the two-body configuration) . The idea is to start with a mechanical description of the

removal process for the i_ th indenter, and thence to sum over all

such _i events to predict the macroscopic behavior.

To this end we resort to observations in "model" brittle solids

(notably glass) of the fracture patterns beneath standard sharp

indenters (e.g. cones, pyramids ) ^ ' ^ , to build up the following

picture. We consider the sliding particle i_ to produce a "plastic"
2.

deformation track of width 2a^. Then, for geometrically similar

impressions, the mean indentation pressure at any instant of contact

7 may be identified with the material hardness ,

2

p. = P ./aira. « H, (1) -1 -11

where a is a factor deteirmined by indenter geometry. Upon unloading ,

residual stresses, associated with incompatibility between deformation

zone and surroxanding elastic matrix, initiate and propagate lateral,

chip-forming cracks (so-called "lateral vents"; other cracks form on

loading, but these extend straight downward, and play only a secondary

role in chipping) . In this view, the size of the prospective chip

is determined by the configuration of the hardness impression, so

the chip area may be written

2

where _n is a linear scaling factor. The volume of material removed

by the indenting particle in traversing through a distance in an

interval of time At is AV. = A.A£, whence, from (1) and (2), — —1 —1 —

V.= AV./At = A.A^/At = (nv /aTrH)P.. (3)

A straightforward summation operation now gives the macroscopic

wear rate;
3.

. n . N

V = Z V. = (nv/ctiTK) Z P, = nv P/ctiTH. (4)

1=1 x=l

This equation may be rearranged.

V/v P= n/aTTH, (5)

such that the left and right sides conveniently represent macroscopic

and microscopic parameters respectively. It would thus appear

possible to predetermine the abrasive wear rate of brittle ceramics

simply from quantities measured in standard hardness testing procedures.

Some data from soda-lime glass illustrate the principle. Taking

H « 1.0 X lO"*"^ Nm ^ ("dynamic" hardness)^, a ~ 1 '(conical particles),

-11 2 -1

_n W 1 , we predict ri/arrll « 3 x 10 m N as the wear rate . This

• -11 2 -1

compares with V/jv_^ « 1 x 10 m N measured under test conditions

in which chipping is pronounced (namely, spherical specimens on an

alxxmina grinding block pre-ground with 45um diamond paste, decyl alcohol

-1 9 environment, atP=iONv —

= lms ). / —

o

There are some interesting implications associated with the present

model

(i) The calculated wear rate is independent of the (apparent)

area of contact between work tool and specimen, and also of the number

and size of indenting particles. Thus, all arbitrariness and complication of a statistical analysis is avoided. Physically, this

*A term equivalent to that on the left of (5) , AV/P^A£, is often used

as an alternative expression of the macroscopic wear rate.
arises because of the essential "linearity" of the fixed-abrasive wear

mechanism: the chip volume is proportional to the load on the

indenting particle, so that the total voliame removed does not depend

on the way in which the total load is distributed.

(ii) The analysis tacitly assumes that the intensity of the

residual stress field about the deformation track is sufficiently high

to drive the chip-forming cracks to the surface. The indication from

indentation fracture mechanics studies^*^ is that the extent of microcracking relative to the size of the deformation zone diminishes with

decreasing load. Thus we might anticipate a brittle-to-ductile,

chipping-to-ploughing transition in wear mechanism at low abrasion

loads, small particle sizes, with an attendant fall in wear rate to

a value more typical of non-brittle solids'''. Again, it has been

assumed that geometrical similarity is preserved in the indentation

fracture process. In practice, initially sharp particles tend to

become "blunt" (either by fragmentation or by clogging with debris)

,

and intersections tend to occur between neighboring tracks,

as abrasion proceeds; these effects will further reduce the wear rate.

(iii) Most significantly, the wear rate under ideal chipping

conditions is uniquely determined by the material hardness ; by controlling the scale of the crack pattern behind the indenting particle,

the "plasticity" properties of the material assume a key role in the

abrasion process. However, hardness is a rate-dependent quantity which

can change markedly with the conditions of testing, e.g. environment,

8 11 load rate (sliding velocity) , etc. ' This bears strongly on the
correlations between a wide range of chemo-mechanical properties (e.g.

machining, drilling, grinding) and the hardness of brittle materials

12 reported by Westwood and co-workers . While the present model may

provide a sound basis for interpreting chemo-mechanical phenomena, it

needs to be emphasised that correlations of this type can be truly

meaningful only if the hardness values are measured under conditions

pertinent to the macroscopic situation.

Acknowledgements

The author is indebted to S. M. Wiederhorn and H. H. Johnson for

discussions on this work. The sponsorship by the Office of Naval

Research, under Contract No. NR-032-535, is acknowledged.
Figure Captions

1. Cross-sectional views of "fixed" abrasion process. (a) Macroscopic view: total load bears on specimen via abrasive grit

particles bonded to tool. (b) Microscopic view: _i th particle

experiences load and leaves in its wake a deformation track,

width 2a^, from which "lateral vents" propagate to form chip,

: section area A^. All particles translate across specimen surface

with velocity v .
References

1. E. Rabinowicz, Friction and Wear of Materials , Wiley, New York,

1965, Ch. 7.

2. M. M. Kruschov, Wear, 28 (1974) 69.

3. The Science of Ceramic Machining and Surface Finishing , Symposium

Proceedings, N. B. S. Special Publ. 348 , 1972.

4. B. R. Lawn and T. R. Wilshaw, J. Mater. Sci . (in press)

.

5. B. R. Lawn and M. V. Swain, J. Mater. Sci . (in press).

6. B. R. Lawn, M. V. Swain and K. Phillips, to be published.

7. D. Tabor, The Hardness of Metals , Clarendon, Oxford, 1951, Ch. 1.

8. S. P. Gunasekera and D. G. Holloway, Phys. and Chem. Glasses , 14 (1973) 45.

9. S. M. Wiederhorn and D. E. Roberts, Wear, 32 (1975) 51.

10. B. R. Lawn and E. R. Fuller, to be published.

11. R. E. Hanneman and J. H. Westbrook, Phil. Mag. 18 (1968) 73.

12. A. R. C. Westwood and N. H. Macmillan, The Science of Hardness

Testing and its_Research Applications , Symposium Proceedings,

American Society for Metals, 1973, Ch. 28.

A MODEL FOR THE WEAR OF BRITTLE SOLIDS

UNDER FIXED ABRASIVE CONDITIONS

B. R. Lawn*

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. V7. 2033, Australia

other than that the wear mechanism involves some microfracturing

,

and the wear rate is remarkably high, relatively little is known

1 2 about the abrasion of highly brittle solids '

; this despite intense

current interest in the machining and finishing of brittle surfaces

within the ceramics engineering industry"^. However, with the advent

of "indentation fracture mechanics" a new approach has become avail4

able for investigating a wide range of small-scale cracking phenomena

The purpose of the present note is to use this approach to construct

an explicit model of the wear process in brittle solids, for the

simple case of a "fixed" abrasive medium ("two-body" process) in

which the grit particles are "ideally sharp."

A schem.atic representation of the wear mechanism is given in

Figure 1. Macroscopically , one measures the wear rate V = dV/dt (V =

volume removed) appropriate to a specified total load P_ and velocity

for the abrasive medium relative to the specimen. Microscopically/

attention focusses on the individual chip-removal mechanism,

characterized by an "indenter" load P^^ and velocity (all "indenters

traverse the specimen with the same velocity in the two-body configuration) . The idea is to start with a mechanical description of the

removal process for the _i th indenter, and thence to sum over all

such i events to predict the macroscopic behavior.

To this end we resort to observations in "model" brittle solids

(notably glass) of the fracture patterns beneath standard sharp

indenters (e.g. cones, pyramids ) ^ ' ^ , to build up the following

picture. We consider the sliding particle to produce a "plastic"
2.

deformation track of width 2a^. Then, for geometrically similar

impressions, the mean indentation pressure at any instant of contact

7 may be identified with the material hardness ,

2

p. = P./a7ra. « H, (1) -1 -XI ~

where is a factor determined by indenter geometry. Upon unloading ,

residual stresses, associated with incompatibility between deformation

zone and surrounding elastic matrix, initiate and propagate lateral,

chip-forming cracks (so-called "lateral vents"; other cracks form on

loading, but these extend straight downward, and play only a secondary

role in f:hipping) . In this view, the size of the prospective chip

is determiiTyd, by the configuration of the hardness impression, so

the chip . are* may ' fc>e written

where jn is a linear scaling factor. The volume of material removed

by the indenting particle in traversing through a distance Ai in an

interval of time At^ is AV^ = h.j_^A' whence, from (1) and (2) ,

•V.= AV./At = A.A2/At = (nv /airH)?.. (3) —X -X - -X ~ - ---o ~ - ~x

A straightforward sxammation operation now gives the macroscopic

wear rate;
3.

. K . N

V = S V. = (nv /airK) E P. = nv P/ctrrH. (4) - . , -1 ---o - - . -1 --0- - 1=1 1=1

This equation may be rearranged.

y/v^P= T^/atiK, (5)

such that the left and right sides conveniently represent macroscopic

and microscopic parameters respectively. It would thus appear

possible to predetermine the abrasive wear rate of brittle ceramics

simply from quantities measured in standard hardness testing procedures.

Some data from soda-lime glass illustrate the principle. Taking

10 ~2 8 H « 1.0 X 10 Nm ("dynamic" hardness) , a w 1 "(conical particles),

-11 2 -1

rj_« 1, we predict n/ornH « 3 x 10 m N as the wear rate. This

• -11 2 -1

compares with V/_v^ w 1 x 10 m N measured under test conditions

in which chipping is pronounced (namely, spherical specimens on an

alumina grinding block pre-ground with 45yra diamond paste, decyl alcohol

-1 9 environment, atP=10NV —

= lms ). / —

o

There are some interesting implications associated with the present

model

(i) The calculated wear rate is independent of the (apparent)

area of contact between work tool and specimen, and also of the niomber

and size of indenting particles. Thus, all arbitrariness and complication of a statistical analysis is avoided. Physically, this

*A term equivalent to that on the left of (5) , AV/PAil, is often used

as an alternative expression of the macroscopic wear rate.
4.

arises because of the essential "linearity" of the fixed-abrasive wear

mechanism: the chip volume is proportional to the load on the

indenting particle, so that the total volume removed does not depend

on the way in which the total load is distributed.

: (ii) The analysis tacitly assumes that the intensity of the

residual stress field about the oieforraation track is sufficiently high

to drive the chip- forming cracks to the surface. The indication from

indentation fracture mechanics studies'''^ is that the extent of microcracking relative to the size of the deformation zone diminishes with

decreasing load. Thus we might anticipate a brittle-to-ductile,

chipping-to-ploughing transition in wear mechanism at lov/ abrasion

loads / small particle sizes, with an attendant fall in wear rate to

a value more typical of non-brittle solids'''. Again, it has been

assumed that geometrical similarity is preserved in the indentation

fracture process. In practice, initially sharp particles tend to

become "blunt" (either by fragmentation or by clogging with debris)

,

and intersections tend to occur between neighboring tracks,

as abrasion proceeds? these effects will further reduce the wear rate.

(iii) Most significantly, the wear rate under ideal chipping

conditions is uniquely determined by the material hardness ; by controlling the scale of the crack pattern behind the indenting particle,

the "plasticity" properties of the material assume a key role in the

abrasion process. Hov/ever, hardness is a rate-dependent quantity which

can change markedly with the conditions of testing, e.g. environment,

8 11 load rate (sliding velocity) , etc. ' This bears strongly on the
5.

correlations between a wide range of chemo-mechanical properties (e.g.

machining/ drilling, grinding) and the hardness of brittle materials

12 reported by VJestwood and co-workers . While the present model may

provide a sound basis for interpreting chemo-mechanical phenomena, it

needs to be emphasised that correlations of this type can be tarul

/

meaningful only if the hardness values are measured under conditions

pertinent to the macroscopic situation.

Acknowledgements

The author is indebted to S. M. Wiederhorn and H. H. Johnson for

discussions on this work. The sponsorship by the Office of Naval

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

1. E. Rabinowicz, Friction and Wear of Materials , Wiley, New York,

1965, Ch. 7.

2. M. M. Kruschov, Wear, 28 (1974) 69.

3. The Science of Ceramic Machining and Surface Finishing , Syiriposium

Proceedings, N. B, S. Special Publ. 348 , 1972.

4. B. R. Lawn and T. R. Wilshaw, J. Mater. Sci . (in press)

.

5. B. R. Lav/n and M. V. Swain, J. Mater. Sci . (in press).

6. B. R. Lawn, M. V. Swain and K. Phillips, to be published.

7. D. Tabor, The Hardness of Metals , Clarendon, Oxford, 1951, Ch. 1.

8. S. P. Gunasekera and D. G. Holloway, Phys. and Chem. Glasses , 14 (1973) 45.

9. S. M. Wiederhorn and D. E. Roberts, Wear, 32. (1975) 51.

10. B. R. Lawn and E. R. Fuller, to be published.

11. R. E. Hanneman and J. H. Westbrook, Phil. Mag. 18 (1968) 73.

12. A. R. C. Westwood and N. H. Macmillan, The Science of Hardness

Testing and its Research Applications , Symposium Proceedings,

American Society for Metals, 1973, Ch. 28.
Figure Captions

1- Cross-sectional views of "fixed" abrasion process. (a) Macroscopic view: total load ]P bears on specimen via abrasive grit

particles bonded to tool. (b) Microscopic view: _i th particle

experiences load P^^, and leaves in its wake a deformation track,

width 2a^/ from which "lateral vents" propagate to form chip,

section area A^. All particles translate across specimen surface

with velocity v . -o

On the Mode of Chipping Fracture in Brittle Solids

Chipping processes in brittle solids, despite their unquestionable

relevance to a diversity of technologies from ceramics finishing

to geological engineering, are not well understood at the fundamental

level. Some recent studies of microfracture patterns beneath standard

1, 2 hardness indenters do, however, shed some light on the problem

Essentially, the picture which emerges is that depicted in Fig. 1:

(i) Upon loading the indenter a confined zone of irreversible

(plastic) deformation forms about any sharp points or corners (thereby

accounting for the residual hardness impression) , from which

"median vent" cracks first initiate and subsequently propagate radially

outward along suitable planes of symmetry (e.g. as defined by the

diagonals of a pyramid indenter, or by preferred cleavage planes)

containing the contact axis; (ii) Upon unloading the indenter the

median vents close up, but, just prior to complete removal, "lateral

vent" cracks initiate and extend laterally from the deformation zone

toward the specimen surface. Of the two types of cracking it is

clearly the second which relates more directly to brittle chipping.

Yet up till now a detailed fracture mechanics analysis has been

attempted only for the median vent system. This system is relatively

well defined, since the indentation stress field, which uniquely

determines the extent of crack growth^, can reasonably be represented

2, 3 m terms of the classical Boussinesq field for normal point loading

On the basis of the fundamental Griffith energy-balance condition

4 for fracture it'may readily be argued that brittle cracks will generally

1
tend to follow trajectories of the lesser principal stresses within

the indentation field, such that the path maintains near-orthogonality

to a component of major tension: the best-studied illustration of

this principle is the Hertzian cone crack ^' ^

, which, in the absence

of any deformation-induced nucleation center, initiates from an

incipient surface flaw and flares downward into the specimen.

In the scheme of Fig. 2, in which the principal stresses are defined

such that ^ii^^22^^33 values denoting tension) nearly everywhere, median vent geometry may be specified in terms of families

of and O^^ trajectories, cone crack geometry in terms of families

of and O^^ trajectories.

The conditions under which the lateral vents form are, unfortunately,

less easily modelled. Since the lateral system operates only as the

indenter is withdrawn from the specimen surface it is evident that

the driving force for propagation must originate from some residual

stress field associated with the irreversible deformation zone.

This conclusion is substantiated by microscopic investigation

of the damage patterns as a function of indenter geometry (e.g. "sharp"

or "blunt") : in general, the extent of lateral venting is found to

increase markedly with expanding zone size. A graphic illustration

of the effect is obtained by loading a soda-lime glass plate with

a small ( 1mm diam.) spherical indenter: at comparatively low load

the contact is elastic, and the only fracture is that of cone cracking,

whereas at higher load some plasticity develops beneath the penetrating

sphere, and lateral venting begins to extablish itself^. A manifes7 tation of this behavior was reported by Culf , who observed otherwise
regular cone cracks in glass to deflect upward ("hat brim" effect)

upon sudden release of the indenter load. Culf also observed considerable residual stress birefringence in association with this phenomenon,

over distances large compared with the scale of the deformation zone

itself. That residual stresses exist about hardness impressions in

most brittle materials has been amply demonstrated by a niamber of

g Strain-sensitive techniques . That these stresses can also be

moderately long-range in nature is seen most clearly in the distances

over which relaxation by plastic flow (e.g. dislocation loop punching)

9

occurs m annealing experiments . Neither the existence nor the

intensity of the residual elastic fields should come as any surprise,

for the stress levels achieved beneath the indenter in hardness tests

on highly brittle solids tend to be of the order of the intrinsic bond

strength of the structure^^' , and the relief of these high stresses

would ideally require the impressed region to restore completely to

its original unstrained state.

These observations , coupled with a reexamination of the

Boussinesq field, provide us with the basis for an analysis of lateral

vent geometry. We note that the lateral vents extend in all cases

on surfaces closely delineated by families of and cr^^ trajectories

in Fig. 2 (although the paths are modified somewhat by the deformation

zone itself, and by free surfaces, including any preexisting median vents

or cone cracks) ; it is as if the applied load were actually reversed upon

indenter withdrawal, so that the a^^ stress normal to the lateral vent

becomes the dominant component of tension in the filed. Of course, it is

3
physically meaningless to associate a reversed applied load with a surface

in the unloaded state, but an effectively similar net result. may obtain if

the deformation zone were to act as a center of contraction with respect

to the surrounding elastic matrix. This is depicted schematically in

Fig. 3. The distribution of stresses at the zone boundary must

inevitably depend strongly on the nature of the irreversible deforma11 12

tion, which itself remains an issue of some controversy '

Nevertheless, one can proceed by making reasonable assumptions as to

this distribution (e.g. that the tractions are of constant magnitude,

and are directed such that the net force is zero) , and evaluate the

residual field in the matrix by taking expressions for the stresses

13 due to elemental point forces (e.g. Mindlin ) and integrating around

the boundary. One may then construct a stress trajectory pattern for

the field, in analogy to Fig. 2, and thereby trace out probable

fracture paths from the deformation zone. Full details of such calculations will be discussed elsewhere; we simply report here that

the predicted paths do indeed curve toward the specimen surface in

essentially the manner shown in Fig. 1.

The scope of the present model extends well beyond the establishment of a suitable basis for evaluating an "index of brittleness"

14

in standard harndess testing . It provides physical insight into

a number of seemingly unrelated phenomena in brittle solids

(i) Strength degradation . Surface damage introduced into a brittle

surface as a result of contact (either static or impact) with hard

particles constitutes a potential source of weakness. The mechanics of the

damage process may be conveniently simulated in a simple indentation

test^^. 4
(ii) Glass cutting . A glass cutter's wheel is designed to produce a

continuous "trailing" median vent as a linear starting crack for subsequent plate fracture in flexure. However, lateral venting invariably

occurs in the wake of the moving "indenter", thereby damaging the edges

of the final cut. Clearly, the objective here is to find a way of

suppressing the chipping mode.

(iii) Surface removal processes . Individual chipping events in the

machining, drilling, grinding, abrasion, erosion and wear of brittle

surfaces in general (e.g. ceramics, gemstones, rocks) are of the type

depicted in Fig. 1"^. By summing over an appropriate distribution of such

microscopic events it should be possible to describe macroscopic surface

removal parameters at a fundamental level

.

(iv) Geophysical impact phenomena . Meteorite-induced craters ranging

X 6 X 7 in scale from geological land masses to lunar fines bear a resemblance

to the damage pattern in Fig. 1 which can only be described as striking.

While thermal and stress-wave effects associated with the high-velocity

18 impacts are undoubtedly important factors in these cases , the possible

role of residual stresses about the central "deformation zone" in

determining crater morphology may warrant further attention.

B . R . Lawn

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

U. S. A.

M. V. Swain

Martin Marietta Laboratories

1450 South Rolling Road

Baltimore, Md. 21227

U. S. A.

K. Phillips

Division of Materials Science

University of Sussex

Falmer, Sussex BNl 9QT

England

5
References

1. Lawn, B. R. and Wilshaw, T. R. , J. Mater. Sci ., in the press.

2. Lawn, B. R. and Swain, M. V., J. Mater. Sci . , in the press.

3. Boussinesg, J., Application des Potentiels a 1 'Etude de I'Equilibre

et du Mouvement des Solides Elastiques , (Gauthier-Villars , Paris,

1885). Discussed in Timoshenko, S. P. and Goodier, J. N.,

Theory of Elasticity (McGraw-Hill, New York, 1970), pp. 398-402.

4. Griffith, A. A., Phil. Trans. Roy. Soc. Lond ., A211 , 163 (1920).

5. Hertz, H., J. Reine Angew. Math ., 92 , 156 (1881); Verhandlungen

des Vereins zur Beforderung des Gewerbe Fleisses , 61 , 449 (1882)

Reprinted in English, in Hertz's Miscellaneous Papers (Macmillian,

London, 1896), Chs . 5, 6.

6. Frank, F. C. and L^wn, B. P., Proc. Roy. Soc. Lond ., A299 , 291 (1967).

7. Culf, C. J., J. Soc. Glass. Tech . , 41 , 157 (1957).

8. Hockey, B. J., in The Science of Hardness Testing and its Research

Applications , Symposium Proceedings, Eds. Westbrook, J. H. and

Conrad, H. (American Society for Metals, Metals Park, 1973), Ch. 3.

9. Wagatsxame, R. , Sumino, K., Uchida, W. and Yamamoto, S. J. Appl .

Phys . , 42, 222 (1971)

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

11. Hill, M. J. and Rowcliffe, D. J., J. Mater. Sci ., in the press.

12. Ernsberger, F. M., Ann. Rev. Mat. Sci ., 2, 529 (1972).

13. Mindlin, R. D., Physics , 1_, 195 (1936).

14. Westbrook, J. H., in The Science of Hardness Testing and its Research

Applications , Symposium Proceedings, Eds. Westbrook, J. H. and

Conrad, H. (American Society for Metals, Metals Park, 1973),

pp. 491-494.

6
15. Evans, A. G,, J. Amer. Ceram. Soc , 56 , 405 (1973).

16. Nadai, A., Theory of Flow and Fracture of Solids (McGraw-Kill , New

York, 1963), pp. 247-249.

17. Carter, J. L. and MacGregor, I. D., Proc. Apollo 11 Lunar Sci .

Conf . , 1_, 247 (1970) .

18. Vedder, J. F. ^nd Mandeville, J.-C, J. Geophys . Res ., 79 , 3247 (1974).

7
Figure Captions

- ' 1. Fracture geometry beneath sharp indenter. Central deformation zone

shown as dark region, median vent cracks as broken lines, lateral

vent cracks as heavy lines. (a) Section view schematic, (b) plan

view schematic, (c) surface view of fused silica indented with

sharp, irregular particle (scanning electron micrograph)

2. Stress trajectories (curves whose tangent indicates direction of

principal stress) for Boussinesq field, showin-f half-surface view

(top) and section view (bottom) . Cone cracks initiate from

incipient surface flaws and propagate everwhere orthogonally to cr^^^

(tensile outside contact area) , median vents initiate from central

deformation zone and propagate orthogonally to O^^ (tensile below

contact zone) , lateral vents initiate from deformation zone and

propagate nearly orthogonally to C723 (compressive everywhere, but

tensile if applied load reversed)

3. Schematic representation of distribution of mismatch tractions at

boundary between central deformation zone and surrounding elastic

matrix, at indenter withdrawal.

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U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

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L PUBLICATION OR REPORT NO. NBSIR 75-659

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

A New Mode of Chipping Fracture in Brittle Solids, and Its

Application in a Model for Wear Under Fixed Abrasive

Conditions I. Mode of Chipping Fracture II. Wear Model"

5. Publication Date February 1975

6. Performing Organization Code

7. AUTHOR{S)

Brian R. Lawn

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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~j ^ \j ''t —f ^

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 Interim

7/1/74 - 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 description is given of the mode of chipping fracture observed in highly

brittle solids. It is pointed out that residual stresses about indentation

deformation centers play a vital role. The implications of this mode in a number of mechanical phenomena are discussed.

An explicit model for the wear of brittle surfaces under fixed abrasive

conditions is presented in terms of indentation fracture concepts. The predicted

wear rate for glass agrees with that observed experimentally to within an order

of magnitude. Some implications concerning the parameters which influence the

abrasion process, particularly the hardness, are 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)

Abrasion; brittle solids; brittle surfaces; chipping; fracture; hardness;

indentation; residual stress; stress analysis; wear rate

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