Microfracture About Scratches in Brittle Solids

$Page 1: Microfracture About Scratches in Brittle Solids$
Microfracture About Scratches in Brittle SolidsAuthor(s): M. V. SwainSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 366, No. 1727 (Jul. 26, 1979), pp. 575-597Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79821 .

Accessed: 08/05/2014 14:40

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 169.229.32.137 on Thu, 8 May 2014 14:40:42 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/79821?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Proc. R. Soc. Lond. A. 366, 575-597 (1979)

Printed in Great Britain

Microfracture about scratches in brittle solids

BY M. V. SWAINt

Physics and Chemistry of Solids, Cavendish Laboratory, Cambridge

(Communicated by D. Tabor, F.R.S. - Received 26 May 1978 -

Revised 29 November 1978)

[Plates 1-4]

Observations of the microcracking about scratches in a number of brittle solids including sapphire and a variety of glasses are presented. The scratching was carried out primarily with a Vickers pyramid indenter under various loads at one scratching velocity. It is observed that the nature of the cracking is very similar to that occurring about a quasi-static pointed indenter. Directly beneath the indenter a well-defined median crack is formed, behind the indenter small partial Hertzian cracks develop within the track, and subsurface lateral cracking initiates from the region of the plastically deformed zone. With a Vickers pyramid the cracking phenomena may be conveniently divided into three regions with increasing load. At low loads (P < 0.05 N) no cracking is observed, intermediate loads (0.1 N < P < 5 N) well-defined median and lateral cracks occur, at higher loads (P > 5 N) the plastically deformed track appears to shatter and the extent of lateral and median cracking is less than that occurring for the higher loads in the intermediate region. The present observations and those of others in the literature have been interpreted in terms of an approximate indentation fracture mechanics analysis, and there is reasonable agreement between theory and experiment. The paper concludes with a discussion of the implications of this work to abrasive wear rate and to the residual strength of scratched bodies.

INTRODUCTION

The shaping of brittle solids invariably involves grinding or abrading these solids with sharp and elastically harder grits. However, despite the increasing use of brittle materials, the mechanisms by which material is removed during machining are still poorly understood. Of particular interest is the nature of microcracking that occurs about the contact area between abrasive and specimen. The microcracking controls to a large extent the rate of material removal, the size of subsurface flaws and hence the residual strength of ground or abraded materials.

Previous studies of scratching of brittle solids have adopted two approaches; one is to determine the coefficient of friction and hence the force ratios likely to be encountered during machining (Broese van Groenou et al. 1975), whereas other

t Present address: Service Central de Recherche, 39 Quai Lucien Lefranc, 93 304 Auber- villiers Cedex. France.

[ 575 ]



576 M. V. Swain

studies have concentrated on the microfracturing and deformation phenomena about the scratch (Peter i964; Peter & Dick I967). However, there has recently been considerable advance in the subject of indentation fracture mechanics (Lawn & Swain 1975; Lawn & Wilshaw 1975; Lawn & Fuller 1975), and the extension of these concepts to scratching of brittle solids is now possible.

The following study is an extension of quasi-static pointed indentation studies (Lawn & Swain I975; Swain & Lawn 1976) and considers the case of a point contact subjected to both normal and horizontal tractions. The indentation stress field differs from that of an axially loaded point contact in that it is asymmetric and this asymmetry may reflect itself in the microcracking phenomena. Scratching experi- ments, primarily with a Vickers pyramid indenter, were carried out on a number of model brittle materials including a variety of glasses, single crystal and polycrystalline alumina. Observations of the track after scratching do indicate that microcracking occurs behind the indenter as predicted by the stress fields. The extent of the subsurface microcracking is predicted by a simple fracture mechanics treatment, and observations are in good agreement with theory. The magnitude of the residual stresses about a scratch are determined in a similar manner. The fracture mechanics treatment explains some observations by Holland & Turner (1937) that initially appear anomalous; it also accounts for the variation of crack depth below the scratch as a function of sliding velocity as observed by Peter (I964), and glaziers' experience that the residual strength of a scratched plate of glass is least when the glass is 'hot'. Finally, from the fracture mechanics treatment it is possible to derive expressions for the abrasive wear rate of brittle materials. The relationships derived for abrasive wear rate are similar to those of Evans (1976) who used a somewhat different approach to the problem. It is also possible to infer from this analysis whether the cracking responsible for material removal occurs ahead of the scratching point or as a result of the residual stress about the deformed zone behind the indenter.

STRESS FIELD BENEATH A POINTED INDENTER UNDER

COMBINED NORMAL AND HORIZONTAL FORCES

(a) Elastic stress field

In a previous paper (Lawn & Swain 1975) the stresses beneath an axial-symmetric pointed indenter were considered in detail. In the present case an elastic half space is subjected to a static normal as well as a horizontal point load (a somewhat similar analysis has been carried out by Freeman (I975) to simulate the stresses about an oscillating pendulum selerometer), the resultant of which is P (figure 1). The problem has been previously considered by Mindlin (1936) and is summarized in the appendix. For an isotropic material of Poisson's ratio v, the stress components in spherical polar coordinates (figure 1) are of the form

o~ij = (P/Itr2) [A1Fi(0)n + aFij(0)t]p, (1)



Microfracture about scratches in brittle solids 577

where Flj(Ob)f and ljj(5b)t are the normal and horizontal (transverse) components respectively, and A and 1a are constants denoting the relative ratios of these two components. For instance, in a sliding friction test A would be unity and It the coefficient of friction, whereas in a grinding experiment A and ,t would denote the vertical and horizontal forces respectively. From equation (1) it is evident that the stress field varies as the inverse square of the radial distance from the contact point.

y~~~ P/

r -

ZR

-r'

FIGURE 1. Coordinate system for point loading, P orientated at an angle to the surface.

The stress distribution suggested by equation (1) is valid only for an ideally sharp point which in theory will give rise to a stress infinity and produce inelastic deformation. However, for the case of a conical indenter the mean contact pressure is independent of load and is given by (Sneddon 195 1)

PM = Ea(i-/)(2) 2(1-V2)

where E is the Young modulus and Vf the half angle of the indenter. Provided Pm exceeds the hardness of the solid, permanent deformation will occur over most of the contact area. It is then possible, following a procedure outlined by Lawn & Swain (X975), to replace an ideal point contact by a model that takes into account indenter geometry and a non-zero contact area. This is achieved by considering the contact zone to have a diameter 2a, such that the mean contact pressure (or hardness) may be written Po = P/ana2, (3)



578 M. V. Swain

where a is a dimensionless constant determined by indenter geometry. For the case of a sliding pointed indenter creating plastic deformation the contact pressures given by equation (3) are likely to be too high because the rear half of the contact area will be partially released from load. This would suggest that the contact pressures may be too low by a factor of ca. 2. Substituting for P from equation (3) into equation (1) we obtain

ca(xa/r)2 AFij(q)n +tijF(q5)tIv = a (a/r)2 IRii(0) IV (4)

From the expressions given in equation (A 1) the principal stresses were determined. The significant features of the stress field are illustrated in figures 2-4 for P at 30? to the normal corresponding to a coefficient of friction of 0.58. Figure 2 is a plot of the stress trajectories of the two principal stresses oAl1 and cr33; these are curves whose tangents indicate the directions of the principal stresses at each point

/ /

'/' // '

/ 33

FIGURE 2. Side view of stress trajectories in stress field for inclined point contact. Plotted for v - 0.25 and y 300.

and are such that ol > C33 nearly everywhere in the field. In figure 3 are plotted contours of the three principal stresses. Finally, in figure 4a, the angular variation of the relative magnitudes of the three principal normal stresses are shown to better advantage. Plots of the maximum principal shear stress and hydrostatic compression are shown in figure 4 b.

The principal features of the stress field are that the Cr11 and or33 components are almost entirely tensile and compressive respectively, whereas the 0-22 'hoop' stress component is tensile in a region below the indenter but compressive near the surface.




The most significant difference between the axial point loading stress field and that of the present case is the asymmetry in the latter. The other important factor is the strong tension near the surface in the 0o,1 component behind the point contact, figure 3 a. In this sense the stress field is somewhat similar to the sliding Hertzian stress field when friction is present. The stress trajectories and principal tensile stresses for the sliding Hertzian stress field (Hamilton & Goodman i966; Lawn i967), under the same conditions of friction as in figures 2-4, are shown in figure 5. The actual elastic stress distribution beneath grits of grinding wheels and fixed abrasives in contact with a specimen probably lies somewhere between an ideally sharp contact, figures 2-4, and that of a blunt indenter, figure 5.

(a)

0.003 0.001

a -0.0-0.01 0.10. .1-.05

-0.00 0.05

-0.001 0.03

-0.001

0.01

(C)

-0.1

-0.05

-0.01 -0.03

FIGURE 3. Contours of principal normal stresses (a) ol; (b) p'22; and (c) 0o33 under point loading as defined in figure 1, with y = 30?. Plotted for v = 0.25 and in the plane containing the contact axis. Unit of stress is po and tioe contact diameter is shown. Note the slight asymmetry of the stress field.



580 M. V. Swain

(b)

(a) R(?O) (a)~ ~ ~ (O

1

-1~~~~

2W \\C 1%~~~~~~~-T 2'T Byr7

FIGURIE 4. Angular variation of principal stress components plotted in terms of dimensionless function Rij(o) (equation 4). (a) Principal normal stresses, O1 022 and a (b) maximum principal shear stress, cr13, and hydrostatic compression, p. All plotted for y =300 and v 0.25.

0.05 0

0.2/

0.5

S S A A

0.05 c.z.

0.01 / 005

FIGURE 5. Hfalf-surface view (top) and side view (bottom) of stress trajectories (broken lines) and contours of principal stress 0,ll beneath a sliding Hertzian contact with friction. Plotted for v _ 0.33 and t =0.5, SS is the surface, AA the diameter of contact, and c.z. is the compressive zone in which all three principal stresses are negative. The unit of stress is the mean indentation pressure po. Direction of motion is from left to right, Lawn

( I967) .

(b) Inelastic stress field

As mentioned above, plastic deformation invariably occurs about a sharp indenter on loading. The stresses in the immediate vicinity of the contact site are then somewhat different from those presented above. An adequate elastic-plastic analysis of this problem does not exist although, following a number of simplifying




assumptions, Marsh (i964) and Johnson (I973) have provided a means of approxi- mating this situation. Following Johnson the plastic zone about a hardness impression, which is similar to that in front of a moving indenter, may be likened to the expansion of the front half of a spherical cavity under internal pressure. The stresses outside the plastic zone, in spherical polar coordinates, are given by (Hill

1950) 2r -- Y(c0/r)3, (

#9 = SY(Co/r)3,5

where Y is the yield stress and c0 the plastic zone size. The actual situation just outside the plastic zone probably lies somewhere between the ideal elastic case and plastic case presented here. However, once a normal crack has nucleated subsurface its propagation will be controlled by the wedge opening force rather than by exact details of the stress field. This will be considered in more detail later.

(c) Unloading stress field

On unloading, the elastically deformed hinterland lying beyond the plastically deformed zone is constrained from returning to its original equilibrium position and so gives rise to a residual stress. An attempt to determine the residual stress upon unloading, following a procedure outlined by Hill (1950), has been made by K. L. Johnson (unpublished work; see, for example, Murray & Hannink I974).

However, as discussed by Swain & Hagan (I976), this leads to values of the radial and hoop stresses which exceed the yield stress of the material. Instead these authors proposed that the residual stress was simply some fraction, because of relaxation effects, of the plastic induced stresses. This approach gives little appreciation of the distribution of the stresses or the likely trajectories of any cracks driven by this source. For the indentation produced by a normally loaded indenter the resulting residual stress field is axially symmetric about the axis of contact. However, for a sliding indenter the residual stress is only symmetric in a plane perpendicular to the scratch direction. A model that is physically realistic and at the same time sustains the cavity analogy is that of a cylindrical cavity under pressure in an elastic half-space adjacent to the surface. Such a model might give some indication of the crack patterns during unloading. Strong support for this approach comes from photoelastic observations by Dalladay & Twyman (i92i) of a subsurface ' centre of stress' beneath the track of a scratch on glass from a diamond indenter. This problem has been considered by Jeffery (i92i), and the stress on the cavity wall in bipolar coordinates is given by

,f,8 = p(t + 2 tan2 S), (6)

where p is the pressure inside the cavity, d the distance from the centre of the cavity to the surface, r the radius of the cavity and q5 the angle of the ray as shown in

figure 6. The stress on the surface of the half space is given by

l,8/ = -4pr2(x2-d2+r2)/(X2+d2+ r2)2 (7)



582 M. V. Swain

From a more detailed examination of the stress field the tensile stress trajectories may be determined and these are shown in figure 7.

The stress field about the cavity model presented above is in good agreement with observations of the residual stresses about diamond scratches in glass made by Dalladay & Twyman (i92i). Holland & Turner (I937) also examined the residual stress field about scratches in glass and found that it remained virtually unchanged in form with load although the magnitude of the stress increased with load, as observed by photoelastic techniques. Dalladay & Twyman determined the principal

(a)

x 0

d

r

(b)

FIGURE 6 FIGuRE 7

FIGURE 6. Geometry (a), and coordinato system (b) for stresses about a hole under pressure in a plate adjacent to the surface.

FIGURE 7. Stress trajectories about a holo under pressure adjacont to the surface. Plotted for d = 1.2 r. Trajectories emanating from the cavity are tensile and those from the surface compressive.

stress trajectories from a photoelastic study of the strain field about scratches in glass; these are shown in figure 8. As pointed out by Preston (i921x) in the discussion of Dalladay & Twyman's paper, these figures are in slight error because the tensile trajectories should intersect the surface at right angles. However-, the agreemeilt between the pressurized subsurface cavity model and Dalladay & Twyman's analysis is very encouraging.




S: flaw (a) r WP /" 900 (b)

~~~~~~\ _

glassl Xls >4~~~~~~5

900 450 900

FIGURE 8. Principal tensile and compressive stress trajectories of the residual stress about a scratch in glass from photoelastic studies. (b) is for the case when no flaw was observed beneath the scratch. - , Isoclinic lines of stress; ---, direction of compression; -**----,

direction of tension. After Dalladay & Twyman (I 92 I) .

MICROFRACTURING ABOUT SCRATCHES

Previous work has established that microcracking accompanies the scratching of brittle solids with sharp points. Busch & Prins (I972) have photographed the cracking that occurs during the scratching of glass with sharp diamond points and their observations are shown schematically in figure 9. The main features are in essential agreement with recent observations of the microfracture about quasi- static pointed indenters (Lawn & Swain I975). When a pointed indenter is pressed against a surface, a 'median crack', or crack normal to the surface, nucleates from plastically induced incipient flaws formed beneath the contacting point. On con- tinued loading, a favourable disposed flaw expands into a subsurface penny shaped crack which then grows in a stable manner. On unloading, because of the irreversible nature of the deformation in the plastic zone, substantial tensional strains develop in the region of the elastic/plastic interface. The magnitude of the residual strain increases during unloading and eventually becomes sufficient to nucleate and develop a system of sideways spreading 'lateral cracks'. In the case of a sliding Vickers pyramid indenter, at initial contact two median cracks form along the indentation diagonals, but only the one aligned along the prospective track ulti- mately survives. This favourably disposed crack simply translates along with the indenter to produce a linear fissure, while the other crack remains immobile at the starting site. The lateral cracks form continuously in the wake of the indenter from the plastically deformed zone and propagate normally outward from the median plane. These observations are further confirmed by Gielisse et al. (I972)



584 M. V. Swain

(a) motion

l~~~~c.-C

.c.C (b?

. c~~~.C

FIGURE 9. Schematic view of the cracking accompanying the scratching of brittle solids, (a), side-on view; (b), end-on view. In front of the scratching point a median crack (m.e.) is observed below the surface, to the side and back of the point contact lateral cracks (l.c.) are observed to form.

who have monitored the surface view of the cracking about a single point diamond grinding wheel simulator. These workers noted that lateral chipping about the machined groove did not occur until some time after the diamond point had left the surface.

EXPERIMENTAL

Studies were carried out on a number of commercially available glasses and polycrystalline and monocrystalline alumina. All surfaces were highly polished. Scratching was carried out by using a microhardness tester with a small motor coupled to one of the micrometer manipulators. The indenter was a Vickers pyramid and the sliding direction was parallel to the pyramid diagonal. The load was varied from 0.25 to 5 N and the scratching velocity was 1 mm s- for the scanning electron microscope (s.e.m.) study and 125,jm s-1 for lateral crack study. Most tests were done in air and measurements of the lateral cracking were made approximately one hour after scratching. A few tests were carried out on a single point grinding simulator with a 1200 diamond cone mounted on a balanced disk of radius 9 cm, rotating at 12 000 rev/min and at a feed rate of 4.5 cm ski. The details of this apparatus have been described elsewhere (Gielisse & Stanislao I972). The glass samples were cross-sectioned for s.e.m. observation by propagating a slow crack perpendicular to the scratch. The other samples were sectioned by using standard metallographic techniques.



Proc. R. Soc. Lond. A, volume 366 Swain, plate 1

i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o

FIGURE 10. Surface and cross -sectional views with the scanning electron microscope of scratches made with a Vickers pyramid on soda-lime glass; velocity of indenter in all cases was 1 mm s51. Force on indenter was (a), 0. 25N; (b), 1.0 N; and (c), 4 N.

(Facing p. 5 84)




i.~~~~~~~~~~~~~~~~~4.

FIGURE 11. (a) Surface view of the microcracking about a scratch made with a Vickers pyramid on fully dense hot pressed alumina (grain size ca. 10/tm), force on indenter 0.5N; (b) higher magnification micrograph of the slip bands within the grains deformed beneath the sliding indenter.




FIGURE 12. Scanning electron micrograph of the microcracking about a scratch made with a Vickers pyramid on the basal plane of single crystal sapphire; force on indenter, 0.5 N. Note the presence of both median cracks and bow-like Hertzian cracks within the track.




0d

'4-)

i~~~~~~~~~~~~~~~~ti

-4 )

.0 QD

CO




OBSERVATIONS AND DISCUSSION

(a) Scanning electron microscope observations

Here we shall illustrate the general features of the scratching of brittle solids, and in the following sections we shall treat more specifically various aspects of the fracturing phenomena. Observations of the surface trace of a scratch formed by a Vickers pyramid drawn across the surface of soda-lime glass at 1 mm s- under various loads are shown in figure 10. Considering first the surface views of the scratch, at low loads (figure 10a) the scratch appears almost completely plastic with little evidence to suggest subsurface microcracking. Other workers have observed turnings about the scratch in the case of soda-lime glass (Gruver & Kirchner 1974; Aghan & Macpherson I 973 ). On increasing the load on the indenter, microcracking about the scratch becomes evident (figure 1Ob). Lateral cracks frequently intersect the surface, causing the removal of a considerable volume of material. At even higher loads (figure 10c), evidence for the plastic deformation associated with the scratch has disappeared and all that remains is shattered material about the scratch. As shown in figure 10c, the debris was removed by cleaning before examination in the s.e.m. Optical examination sometimes revealed the entire plastic zone about the scratch lying intact adjacent to the path of the indenter. The trace of the median crack is also very clear in figure 10c. It deviates to left and right in its path along the track. The reason for this deviation is not known at present. The other interesting feature is that the extent of lateral cracking is not as large as at lower loads, presumably because the driving force for this cracking system has been removed or greatly reduced.

Cross-sectional views of these scratches in soda-lime glass are also shown in figure 10. It is apparent that, even at the lowest loads studied, median cracks are always present, as are lateral cracks. In figure 10a it appears as though there is a well-defined plastic zone beneath the track of the indenter, and it is from this zone that the median and lateral cracks emanate. At higher loads the extent of the median crack is greater (figures lOb and 10c), and often it is difficult to distinguish between diverging median and lateral cracks.

Observations were also made on hot pressed fully dense polycrystalline alumina (ca. IO um grain size), and single crystal sapphire. Tests were only carried out under low loads (0.5N) on polycrystalline alumina. At this load there is little evidence of either lateral or median cracking; however, there is a considerable amount of grain boundary failure about the scratch (figure II a). At higher magnification, slip or twinning within the individual polycrystalline grains may be seen (figure II b), and failure occurs where these slip bands intersect a grain boundary. This observation is in agreement with transmission electron microscope studies of twin- grain boundary interaction by Hockey (I972). There is no evidence for the transmission of these slip bands from one grain to the next in figure II b. Gruver & Kirchner (1974) have monitored the depth of median cracks in polycrystalline alumina after scratching and this form of cracking is discussed in the next section.



586 M. V. Swain

Observations of the scratches in monocrystalline sapphire are somewhat similar to those in glass. All scratches were carried out on the basal plane and little evidence for anisotropy was observed. The most significant feature of the scratches, however, was at low loads (figure 12), where both median cracks and bow-like Hertzian cracks within the track may be seen. Again note how the median cracks deviate to left and right from the base of the track.

Cracking about the grooves machined by the single point grinding wheel simulator are shown in figure 13 for soda-lime glass and sapphire. The fracturing observed in both cases is somewhat similar to that beneath sliding point contact. It was found that the forces measured on the transducers attached to the diamond point varied significantly with the two materials for the same depth of cut, as would be expected because of their different hardness. Cross-sectional views of the machined grooves (figure 13) revealed the presence of well-defined median and lateral cracks aligned along favourable (1012) type cleavage planes of sapphire.

. ~~~~~~~~~~~~~v/frim s-1) T 2.5

60 1

T ./ T 102 S _

T A

40 - T

T T

0 0.2 0.4 indenter load/N

FIGURE 14. Median crack depth in soda-lime glass as a function of load beneath a Vickers pyramid at three different scratching velocities. After Peter (i964).

(b) Median cracking

Previous work by Peter (i964) has quantified the variation of median crack depth with load beneath scratches made with a Vickers pyramid in soda-lime glass (figure 14). Peter also observed that the median crack depth varied with scratching velocity under the same load (figure 15). Gruver & Kirchner (1974) measured the depth of median cracking beneath scratches with a sharp diamond on polycrystalline alumina. The extent of the median cracking may be interpreted with the aid of indentation fracture mechanics. After crack nucleation beneath the indenter the stress intensity factor is determined by the wedge opening forces in the surface perpendicular to the direction of travel of the indenter. However, in the present




60

4-; 40 - ;

20 S +, _~~~~~~~~~~030 N

40-

0.15 N

0 I . I . I I . 1 100 102 i04

scratching velocityl(gm s'1)

FIGURE 15. Extent of median cracking beneath a Vickers pyramid as a function of scratching velocity. After Peter (I964).

case, in addition to the normal load P, there is a tangential load ptP (where It is the coefficient of friction) opening up the crack, the resultant of which is P(1 + t2)1.

In the case of a well-developed median crack beneath a quasi-static pointed indenter, Lawn & Fuller (I975) have shown that the stress intensity factor at the crack-tip is given by

K ta f( )4' (8)

where Vf is the half angle of the indenter and P the normal load on the indenter. On substituting for P in equation (8) for the case of the sliding indenter we obtain

K- me P(1 +t2)I (9)

where me is a modifying factor that takes into account slight differences in crack shape from that considered by Lawn & Fuller (I975) and also allows for the increased indentation pressure as mentioned in ? 2. When the results in figure 14 and those of Gruver & Kirchner (I974) are plotted against ct (figure 16), they approximate to a straight line relation except perhaps at low loads. However, in order to obtain from equation (9) values of K that are in reasonable agreement with those of other workers a modifying factor of me I 4 must be used.

Note that below a critical load on the indenter (ca. 0.05 N, in figure 14) no median cracking is observed beneath the scratch. A similar phenomenon was observed about quasi-static Vickers indentations on soda-lime glass by Lawn et al. (1976a).

The transition from plastic to brittle behaviour was predicted at a load of ca. 10-2 N. The variation in the equilibrium stress intensity factor with scratching velocity

as shown in figure 16 is not surprising since the crack velocity in glasses is a very 20 Vol. 366. A.



588 M. V. Swain

sensitive function of the stress intensity factor; this is because of stress corrosion cracking assisted by water vapour. From the work of Wiederhorn & Bolz (I970) it is known that if the crack velocity in moist air is greater than ca. 10-4 mS-' the relevant equilibrium stress intensity factor Kic ( 7.5 x 1 05 N Ink for soda-lime glass), whereas below 10-7 m s'l the equilibrium value is - 2 to 1/3KI.. That is, in moist air at very low scratching velocities, the extent of cracking would be determined by a 'fatigue limit' or a 'crack arrest' limit of the glass. Similarly, if one were to scratch glass in a very inert environment having very low water vapour pressure (e.g. toluene, paraffin), the depth of median cracking should be almost independent of the scratching velocity and be determined by the Kic value.

/ 0.05 -0.5

K/(MN m-2) /0.67 5.2 0.42 0 X 0.31

U/ (FM s4 04 i0 102

2.5 0.03 0.

_ ~~~~A 00 0X

A o

0.01- 0.1

0 200 400 600 (crack depth, c) 2/jm2

FIGURE 16. Data from figure 14 and Gruver & Kirchner (1974), plotted according to equation 9 with me 4. The value of #f was 680 for the glass data and ca. 38? for the polyerystalline alumina. The values of the equilibrium stress intensity factor at the different scratching velocities are labelled on the curves. x , o, A, Glass; a, alumina.

(c) Regidual stress about scratches As mentioned in ?2, photoelastic studies clearly indicate that residual stress

exists about scratches in brittle solids. Further evidence for the presence of residual stress is the work of Holland & Turner (I937) who measured the effect of scratch velocity on the fracture strength of scratched glass beams. These authors found that at constant load the fracture strength, in contrast to the results shown in figure 16, decreased with increasing scratch velocity; their results are replotted in figure 17. The range of scratching velocities is such that the crack depth is probably independent of scratch velocity as shown by Peter (i964) (figure 14). All fracture




strength determinations by Holland & Turner were done at very fast loading rates in three point bending and hence the results are essentially independent of the environment. It is possible to determine the stress intensity factor Kb at fracture for Holland and Turner's results. From standard fracture mechanics handbooks (for example, Sih I973)

Kb = YO~ c', (10)

where Ad is the outer fibre bend stress, Y is a geometric factor ca. 1.12 i, o, 22.4 MPa, and c = 13 gm from Holland & Turner's measurements. The stress intensity value from equation (10) is ca. 0.47MNm-4 which is substantially less than the

K,.,. value of ca. 0.75 MN m- for soda-lime glass. Similar differences in the value of the measured stress intensity value versus the K1.,. value have been noted in

30

28

0

2 26 0

0

24 -

0 20 40 60 80 scratching velocity/(mm s-1)

FIGURE 17. The effect of scratching rate on the fracture stress of scratched glass. Forco on indenter was 5 N. After Holland & Turner (I937).

indentation induced flaws in glass and other brittle materials. When the residual stress about an indentation impression is either ground away or released by annealing, the stress intensity value and K1, value determined by using other techniques are in very good agreement. Thus in both cases the residual stress about the scratch or indentation gives rise to an additional stress intensity factor at the crack tip, that is:

Kr=Ki..c b, (11)

where the residual stress intensity factor Kr may be thought of as a line load at the mouth of the crack, in which case (Sih I973)

Kr = 4PIr/(c), (12)

where Pir is the equivalent line load due to the residual stress, and c is the crack depth. It is possible to convert the residual line load to a residual stress by knowing

20-2



590 M. V. Swain

the dimension over which it operates. This may be assumed to be equal to d, the width of the track; then

0r Pir/d. (13)

We now wish to normalize the residual stress intensity factor Kr to the critical stress intensity factor K,., for a crack of depth c. However, as the residual stress resulting from plastic deformation operates over the entire length of the scratch, the appropriate indenter geometry is that of a wedge of length equal to the scratch length. The critical stress intensity factor is then (Swain & Lawn 1976)

KI..= PI/(tan /i(c)t), (14)

where f is the wedge half angle, c the crack depth and P1 the line load (P/i). It is now possible to normalize the residual stress intensity factor Kr with respect to

K1.,. and obtain an explicit expression for 0r. From equations (1 2), (13) and (1 4) we have

Kr _ Pir tan ~/r 7tcli 4 tan ~f o-r d (5 KI.c n Pi a ( 1)

However, Pi may be related to the hardness of the material by

Pi Hd. (16)

Substituting for PI in equation (13) we have

Kr 4tan /r (1 K1 (tt) (17

When appropriate values are inserted for Kr/Ki.c ( 0.3), and tank i/- 2, the residual stress is 0r H//18. Previous estimates of the residual stress remaining about the permanent impressions after Knoop and Vickers pyramid indentations gave a value of 0r H/20 for a wide range of materials (Swain 1976).

This interesting result suggests that the residual stress is directly related to the hardness of the material. Previous work by Marsh (i964), and by Gunasekera & Holloway (I973), has shown that the hardness of glass is strongly dependent upon the indentation time. Figure 18 shows the hardness variation of soda-lime glass with indentation time according to these authors. The range of interest for com- parison with Holland & Turner's (I937) results is 10-1-2 x 10-2 s. Over this range the variation of the hardness is ca. 12 0 according to Gunasekera & Holloway, and ca. 30 0 according to Marsh, whereas the variation in the fracture stress of the scratched beams shown in figure 17 is Ca. 17 0.

Holland & Turner also noted that the residual strength of the scratched beams systematically increased with the time which elapsed after scratching as shown in figure 19. They also observed a significant reduction in the magnitude of the stress birefringent pattern about a scratch as a function of time. This observation confirms glaziers' experience that scratched glass is easier to break when the scratch is 'hot'. The results shown in figure 19 confirm the idea of residual stress about the scratch



tlcrofracture about scratches in brittle solids 591

10

0 8

0 ~~~~0 0 0 t ~~~~~o

0 tA 60 ~~? ~ ?

4

io-4 10-2 10?

indentation time/s

FIGURE 18. Vickers hardness numbers (V.h.n.) for natural surfaces of float glass as a function of duration of loading. o, Marsh; Ei, Gunasekera & Holloway.

1min 10min lh 5h 24h lOh 1000h

32 - SL

0

a 28 - o

e 24 0

20, 0

101 103 10V0 elapsed time/s

FIGURE 19. Fracture stress of scratched glass as a function of elapsed time after scratching. Rate of scratching was 8 cin s-1 and force on indenter was 5 N. After Holland & Turner (1I937) r

lowering the fracture stress. The difference between the instantaneous and long time fracture stress is ca. 30 0, in agreement with the Kr/Ki.C. ratio used above, if one assumes that the residual stress is zero after a sufficiently long time. It is also glaziers' experience that scratching glass in the presence of paraffin facilitates fracture and improves the edge quality. Gunasekera & Holloway (I 973) have shown that paraffin does increase the hardness of glass more than air. Additionally, in the presence of paraffin one might expect the extent of lateral cracking to be less



592 M. V. Swain

because of the very low water vapour pressure in this liquid, hence accounting for the better edge quality of the fractured glass. Peter (1964) also noted that after scratching glass at low loads, lateral cracks took some time to become visible, presumably because of slow crack growth assisted by stress corrosion. It is interesting to point out that Lawn et al. (1976b)1 in a recent study of strength degradation of glass resulting after quasi-static indentations with cones, failed to observe residual stress effects. This emphasized that the environment, magnitude of the load, and time interval between indenting or scratching and fracturing are important parameters when considering plastically induced residual stress.

T,

W 102

glass; A Ca~l~i glass; A, sapphire.

.~~~~~~~~~~~~~~~~~~~~~~~~~~~~1

2a

10-' 10' 101

loadIN FieuRE 20. The extent of lateral cracking, 2Q, about scratches mnade with a Vickors pyramnid

on three of the materials studied, error bars show the, range of lateral cracking. Also shown is the width of the track, 2a, formed by the indenter translating at 125ram ssf. 0, Lead glass; A CaAlSi glass; k , sapphire.

(d) Lateral cracking Observations of the extent of lateral cracking about scratches made with a

Vickers pyramid are shown in figure 20. The variation of lateral cracking with load is in reasonable agreement with the results of Broese van Groenon et al. (1975) on a variety of ceramic materials and Adewoye (1976) on single crystal silicon carbide.

In order to be able to interpret these observations the residual stress field about the scratch must be known. As shown in ? 2 (c), a reasonable approximation is to



M] crofracture about scratches in brittle solids 593

relate the residual stress to the pressure inside a cylindrical cavity. The stresses outside the plastic zone, in cylindrical coordinates, are given by (Hill i950)

o=-i1fH(co/r)2 r > c (18)

0' fH(c0/r)2J where f is a constant -Ll (as shown in the previous section), H is the indentation or scratch hardness, and co is the elastic/plastic boundary assumed equal to a, half

T I X

10 | | t 1 1 1 1 - 0.5 I I

10 -~~ 5 0\

0 no

005 0.10.10 A

0.05-

11 I ii lii I ~~~~~~~~~~~~~~~I I I - I

0.05 0.1 1.0 5 10

KlHal c,/a

FIGuRE 21 FIGuRE 22

FIGURE 21. Data in figure 20 replotted according to equation (2 1). The solid line is the predicted variation from equation (21), whereas the broken line has a slope of - 2. m, CaAl glass; o, CaAlSi glass; A, lead glass; *, quartz; o, sapphire; v, soda-lime glass.

FIGURE 22. Data from Evans & Wilshaw (1976) for lateral cracking about quasi-static indentations. The solid line has a slope of - R in agreement with equation (23). m, Sapphire; A,

spinel; A, silicon nitride; 0, zinc sulphide.

the diameter of the scratch. The tensile hoop stress of? is responsible for the lateral cracking. The stress intensity factor for a crack of length c in a varying stress field is approximately given by (Sih 1973)

(c -ff C ol, dr(19 K = m'e (?-JC( '(1 9)

where on is the prior distribution of normal tensile stresses along the crack path and me is a modifying factor to account for edge effects, here we assume me = 1.0. Substituting o- for Orn and integrating we obtain

K - ffl a(c2- a )ci, (20)



594 M. V. Swain

where 2 a is the diameter of the scratch, or, in normalized nomenclature following Evans & Wilshaw (I976),

K/(fHal) =-2(a/c)' [1 -(a/c)]. (21)

The results in figure 20 are replotted in figure 21 according to equation (21) and a reasonable fit is observed. Typical values obtained for f are in the range L-- I

again in reasonable agreement with the previous section. Following a similar procedure, the extent of lateral cracking about a static

axi-symmetric indentation may be predicted. The appropriate stress fields are given in equation (5) and the corresponding three dimensional stress intensity factor is (Sih I973)

K f rOJ n dr (22)

On substituting o-r for 0n from equation (5) and integrating, we obtain, in normalized nomenclature,

Kl(fHal) = 2,0(a/c)l I |- (a/c)2 11. (23)

The results of observations by Evans & Wilshaw (1976) for the extent of lateral cracking about axi-symmetric indentations are shown in figure 22. The agreement is surprisingly good.

It is now possible to predict the volumetric loss of material for two body abrasion from the two dimensional treatment presented above in equation (21). Following Lawn (I975) the volume of material removed in the case of a fixed abrasive is, to good approximation, given by

V = 2c11d, (24)

where cl is the extent of lateral cracking, d the mean depth of lateral cracking and 1 the distance of travel. Again following Lawn and from the observations in figure 1 0, d is approximately equal to a, the radius of the track width. Substituting for cl and d in equation (21) we obtain

V/oc H'P3/K2K (25)

On the other hand if we assume that lateral cracking proceeds ahead of the scratching indenter then the appropriate relation for cl is given by the three dimensional analysis, equation (23). In this case, upon substituting for cl, it may be shown that

V/lcc P`K-1H-1 (26)

which is precisely the relation derived by Evans (I976) from a semi-empirical approach for the volumetric wear rate about a scratch. It must be emphasized that neither equation (25) nor equation (26) takes into account lateral crack interaction between adjacent scratching points.

Recent observations by Broese van Groenou et al. (I 975) are very interesting with respect to the two formulae derived above for abrasive volumetric wear rate. These authors noted that the width of the lateral cracking about a scratch in some




materials is proportional to the load, whereas for other porous ceramics the width was proportional to the square root of the load. From equation (21) the two dimensional analysis predicts that the lateral crack width should be proportional to P, whereas the three dimensional analysis, equation (23), predicts lateral crack width proportional to Pi. These observations suggest that voids and pores may play an important role in nucleating cracks ahead and about the moving indenter.

CONCLUSIONS

Observations of the microfracturing about scratches in brittle solids have established that it is very similar to that occurring beneath quasi-static pointed indentations. The differences in behaviour may readily be accounted for by the asymmetric nature of the indentation stress field because of tangential tractions. There appear to be three distinct regions of crack formation with increasing load on a scratching Vickers pyramid. At low loads (< 0.05N) no visible surface or subsurface cracking occurs about the scratch. The remaining track is smooth and fully plastic; the process of material removal is plastically controlled. The accumu- lation of a large number of such plastically controlled contact events may result in a mechanical 'polishing' operation which is characterized by a relatively low material removal rate and a smooth mirror-like finish. For intermediate loads (0. t-5 N) a fully plastic track and well-developed median and lateral cracks occur about the scratch. Often at the lower loads (0.1-1 N) the lateral cracks do not intersect with the surface. At higher loads ( < 5 N) median cracks and poorly developed lateral cracks are observed, as well as considerable crushing of the region about the scratch. In the latter case material is removed by microcracking about the contact sites and corresponds to typical 'abrasion' operations on brittle solids. It is characterized by rapid material removal and invariably leaves a rough surface.

An approximate fracture mechanics analysis for the cracking about quasi-static pointed indentation tests was found to predict the extent of cracking beneath the scratch. This enabled an interpretation of observations by Peter (i964) as to the variation of median crack depth with scratching velocity. A simple extension of the above, and incorporation of a previous analysis of the residual stresses about indentations in brittle solids, permitted the residual stress about a scratch to be determined. The magnitude of the residual stress was estimated as ca. #A-H. The concept of residual stress about a scratch was found to explain other observations by Holland & Turner (1937), namely the variation of the residual strength of scratched bodies as a function of scratching velocity and also as a function of time elapsed after scratching.

The application of fracture mechanics also enabled the extent of lateral cracking about a scratch to be estimated. Again it is the residual stress about the deformed plastic zone which is the driving force for the growth of lateral cracks. The proposed variation of lateral cracking with load is different from that proposed by Evans (1976) because in our work a two dimensional rather than a three dimensional



596 M. V. Swain

analysis was used. However, if one assumes that lateral cracks occur ahead of the scratching point, a three dimensional analysis is appropriate, and an expression similar to that proposed by Evans is derived. For lateral crack growth ahead of the scratching point a material with microcracks or pores is needed. Broese van Groenou et at. (1975) have recently confirmed this implication; with porous materials they find a different relation between lateral crack width and load on the scratching point from that observed in homogeneous materials.

This work was started while the author was at Martin Marietta Laboratory (U.S.A.) and was partially supported by the National Science Foundation under grant G.H. 38455. Some of the s.e.m. micrographs were taken by M. H. Meyerhoff. The single point machining studies were carried out at the University of Rhode Island with the help of T. Rocket. The S.R.C. is thanked for financial support, and the Ministry of Defence for a grant to the laboratory. J. T. Hagan, J. E. Field and B. R. Lawn are thanked for their comments on the manuscript.

APPENDIX

Solutions for the stress field in an elastic half space under a normal point load were first derived by Boussinesq (i885). Mindlin (1936) generalized the original solutions to take into account an arbitrary point force located at or below the surface. In the curvilinear coordinates of figure 1 we have the following stress components:

o-rr = (P/li]2) [A{!(I - 2v) sec2 2 -3- COS 0 sin2 0}

+,u{- (I - 2v) sin 0 (1 + COS 0)-2 - 3 sin2 i}]

?00 = (P/rCR2) [A{ (I - 2v) (cos q- sec2 14)} +?,{sin 0(1 - (1 + cos 02}]

0-z = - (P/mB2) [A{32 cos3 0} +a{32 COS2 0 sin 4}],

=- (P/,tR2) [A{3- COS2 -a sin q} + a{3 cos 0 sin2 i}1,

='rO 00z 0, (A l)

where A and It are the magnitudes of the vertical and horizontal components respectively. The principal normal and shear stresses may be found in the usual manner from the above.

REFERENCES

Adewoye, 0. 0. I976 Ph.D. thesis, -University of Cambridge. Aghan, R. L. & Macpherson, R. 1973 J. Am. ceram. Soc. 56, 46. Boussinesq, J. I885 Application des potentials a l'etude de lNquilibre et du mouvement des

solides 6lastiques. Paris: Gauthier-Villars: See also Timoshenko, S. P. & Goodier, J. N. 1970 Theory of Elasticity, pp. 398-402. New York: McGraw-Hill.

Broese van Groenou, A., Maran, N. & Veldkamp, J. D. B. 1975 Philips Res. Rep. 30, 320. Busch, D. M. & Prins, J. F. 1972 in The science of ceramic machining and surface finishing

(ed. Schneider, S. J. and Rice, R. W.). Natn. Bur. Stand. spec. Publ. no. 348, p. 73. Dalladay, A. J. & Twyman, F. 192i Trans. opt. Soc. 23, 165. Evans, A. G. 1976 Int. Rockwell res. Rep., no. SC5023.3TR.




Evans, A. G. & Wilshaw, T. R. I976 Acta metall. 24, 939. Freeman, I. B. I975 Ph.D. thesis, University of Sussex. Gielisse, P. J. & Stanislao, J. I972 in The science of ceramic machining and surface finishing

(ed. Schneider, S. J. and Rice, R. W.). Natn. Bur. Stand. spec. Publ. no. 348, p. 5. Gielisse, P. J., Kim, T. J. & Choudry, A. I972 Final tech. Rep., U.S. Naval Air Systems

Command. Contract No. N00019-71-00244. Gruver, R. M. & Kirchner, H. P. I974 J. Am. ceram. Soc. 57, 220. Gunasekera, S. P. & Holloway, D. G. I973 Physics Chem. Glasses 14, 45. Hamilton, G. M. & Goodman, L. E. i966 J. appl. Mech. 33, 371. Hill, R. I950 The mathematical theory of plasticity. Oxford: Clarendon Press. Hockey, B. J. I972 Proc. Br. ceram. Soc. 20, 95. Holland, A. J. & Turner, W. E. S. 1937 J. Soc. Glass Technol. 21, 383. Jeffrey, G. B. i92I Phil. Trans. R. Soc. A 221, 265. Johnson, K. L. I970 J. Mech. Phys. Solids 18, 115. Lawn, B. R. i967 Proc. R. Soc. A 299, 307. Lawn, B. R. I975 Wear 33, 369. Lawn, B. R. & Fuller, E. R. I975 J. Mater. Sci. 10, 2016. Lawn, B. R., Fuller, E. R. & Wiederhorn, S. M. I976a J. Am. ceram Soc. 59, 193. Lawn, B. R., Jensen, T. & Arora, A. I976b J. Mater. Sci. 11, 573. Lawn, B. R. & Swain, M. V. I975 J. Mater. Sci. 10, 113. Lawn, B. R. & Wilshaw, T. R. I975 J. Mater. Sci. 10, 1049. Marsh, D. M. i964 Proc. R. Soc. A 279, 420. Mindlin, R. D. I936 Physics 7, 195. Murray, M. J. & Hannink, A. H. J. I974 Phil. Mag. 30, 213. Peter, K. i964 Glastech. Ber. 37, 333. Peter, K. & Dick, E. i967 Glastech. Ber. 40, 470. Preston, F. W. I92i Trans. Opt. Soc. 23, 168. Sih, G. C. I973 Handbook of Stress Intensity Factors. Bethlehem, Pa.: Lehigh University

Press. Sneddon, I. N. I95I Fourier Transforms. New York: McGraw-Hill. Swain, M. V. I976 J. Mater. Sci. 11, 2345. Swain, M. V. & Hagan, J. T. I976 J. Phys. D 9, 2201. Swain, M. V. & Lawn, B. R. I976 Int. J. Rock Mech. and mining Sci. 13, 311. Wiederhorn, S. M. & Bolz, L. H. I970 J. Am. ceram. Soc. 53, 543.



Documents

Microfracture About Scratches in Brittle Solids