Size Effects in Abrasion of Brittle Materials

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Size effects in abrasion of brittle materials

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1979 J. Phys. D: Appl. Phys. 12 195

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J. Phys. D: Appl. Phys. Vol. 12, 1979. Printed in Great Britain

Size effects in abrasion of brittle materialsK E Put tick, M A Shahid? and M M HosseiniDepartment of Physics University of Surrey Guildford Surrey GU2 5XH

Received 8 May 1978, in final form 15 September 1978

Abstract. Small indentations and scratches can be made in highly brittle materials suchas glass or silicon without any associated fracture. It is proposed that this effect whichis usually explained in terms of flaw statistics is in fact governed by a strain energycriterion which can be defined quantitatively if the tensile field which initiates fractureis known. Using a model of the field of residual stress around an indentation proposedby Swain and Hagan it is shown that the critical size of indentation for fracture to occurshould be about 12(Er/Y2), where E is Youngs modulus I fracture surface energyand Y the yield stress for plastic flow in uniaxial compression.The critical size parameter is evaluated for silicon. Observations are reported onthe process of indentation of single crystals under very light loads and it is concludedthat plastic flow begins not well beneath the indenter as required by the simple Hertziantheory of elastic penetration but at the indenter-specimen interface as a result of frictionbetween specimen and indenter. Using the value of Y calculated on this assumptiongood agreement is found between the theoretical critical dimension of an indentationand observed values.It is further suggested that the initiation of cracks by indentations or scratchesmarks the transition from polishing to grinding of brittle solids as a function of abrasionparticle size.

1. IntroductionIt is a well-attested fact that permanent impressions can be made without fracture evenin highly brittle materials provided th e load does not exceed a critical value. This isobserved in solids as diverse as inorganic glass (Peters and Knoop 1936, Taylor 1949,M ars h 1964) marble (Shaw 1954), organic glass (Puttick 1973) an d silicon (Puttick a ndShahid 1977). Th e sam e is true of scratches (van Gro enou et a1 1975, Shahid 1977). Sinceindentation and scratching model the basic processes of abrasion, the effect is of tech-nological importance quite apart from its implications for hardness testing.The effect is usually associated with flaw statistics: the smaller the impression, thelower th e probability th at it will engage an inherent surface flaw. Yet, as in the analogouscase of ring cracking du ring elastic indentation (Roesler 1956) there is no evidence of theincreased variability in the onset of indentation fracture to be expected on this basis.This is particularly ob vious in the case of scratches, where on e finds tha t below a criticalload n o surface fracture is initiated however long the scratch. We therefore propose tha tthe true criterion is the magnitude of the strain energy contained in the field of tensilet Present address : Physics Department Government Science College Lahore Pakistan.0022-3727/79/020195+ 1 1 01.00 979 The Tnstitute of Physics 195


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196 K E Puttick, M A Shahid and M M Hosseinistress surrou nding th e indentation, as suggested by Roesler (1956) for Hertzian indenta-tion of glass an d by P uttick (1978a,b) fo r the radial fracture initiated in o rganic glass byplastic-elastic indentation . This criterion implies th at fracture occurs at a characteristiclinear dimension of indentation.

2. Indentation fracture of highly brittle materialsIt has been convincingly dem onstrated by M ars h (1964).j. th at materials with a high valueof the ratio Y/E, where Y is the uniaxial yield stress for plastic flow and E Youngsmodulus, accommo date the deformation beneath a n indenter by a strain field in whichelastic as well as plastic compon ents are significant. An impo rtant consequence of thisconclusion is (Puttick 1973, Puttick et aZ1977) th at the tensile stress field is very differentfrom th at associated either with Hertzian (elastic) or with purely plastic indentation inparticular, surface tensile stresses predominate which are orthogonal to radii from theimpression centre, leading to the formation of radial cracks extending from the zone ofplastic deformation into the elastic strain field surrounding it.More recently, the work of Lawn and his colleagues (e.g. Lawn and Wilshaw 1975)has established that in inorganic glasses, and probably also in highly brittle crystals likesilicon, the removal of material by chipping around indentations and scratches occursduring unloading of the indenter. It is therefore the consequence of a residual tensilestress field due to the presence of a zone of permanent deformation constrained by asurrou nding field of elastic strain. Chipping, which is the type of frac ture responsible fo rmaterial removal during grinding, appears t o be the result of the intersection of two typesof crack: one no rmal an d on e roughly parallel to the free surface, the so-called radialcracks an d lateral vents. Th e residual stress fields responsible fo r propag ating each ofthese have been discussed by Swain and Hagan (1976).In this paper, we evaluate the critical size of plastic-elastic indentation to initiateradial fracture, according to the strain-energy criterion, in the field of residual tensilestress (in the surface aro un d an impression) proposed by Swain an d Hagan . Th e calcu-lated dimension is then compared with observation on indentation fracture in silicon.

3. Cra ck initiation in inhomogeneous stress fieldsIt has been pointed ou t by Puttick (1978a,b) in connection with the initiation of fractureduring loading of a ball indenter in polymethylmethacrylate, that the strain energyrelease rate G of a crack propagating in an inhomogeneous tensile stress field may bewritten

G= K a W ( F ( R / a ) } (1)where a is a characteristic length of the tensile stress field, W a measure of the strainenergy densjty per unit volume of the tensile stress field, and F a non-dimensionalfunction of crack length R relative t o a which is determ ined by the field geom etry. sa constant of the geometry of the crack. If we consider a series of geometrically similarf It should be noted that plastic-elastic models of indentation stress fields have a long history. Dugdale(1954) appears to have been the first to adapt the deep punching analysis of Bishop et a1 (1945) to indenta-tion particularly of steel by cones and wedges.


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Size effects irt abrasion of brittle materials 197stress fields of increasing size, fracture will be initiated at a dimension

ra,= ~.K W { F (Rdac)}where I is the fracture surface energy and RE he length of a Griffith flaw.

In plastic-elastic stress fields aro un d indenta tions, the scale of the tensile stress fieldis given by a fraction m of the yield stress Y in uniaxial compression. Th e correspondingscale of the strain energy field is then ( mY)Z/E,where E i s Young s modulus. No w the fullfield of Circumferential tensile stress which leads to radial cracking in m aterials such asinorganic glasses, semiconductors, etc. is apparently developed only during unloading.In the Sw ain-Hagan model the circumferential stress unde r load is assumed t o be thesum of two co mpo nents: one described by H ertz s equations, in which the stress is com -pressive; and a second component given by an expanded-hole model with a tensile cir-cumferential stress. During loading, these two com ponents opp ose each othe r; duringunloading the first subtracts from the second to give a total tension greater than thatattained during loading. We take this model as a guide in computing ac in (2 ) , hough aswill be a pp are nt we believe th at the result is not very sensitive to stress field detail.It is clear that for the case of an impression with radial symm etry, the field of surfacestress must fall off as inY) (c / r )nwhere c is the radius of the plastic-elastic boun dary an dY radial distance from the indentation centre. It seems reasonable to suppose tha t in thesurface conditions approximate to plane stress and therefore n -2 . Swain and Haganassume m=1/2/3, cEa,where a is the radius of the im pression itself, where (see appendix)

The critical size of indentation is therefore, taking K = 1 (the value appropriate to astraight crack front norm al to the free surface as compared with d2j7 rfo r a curved fron t),

(We have used the equa lity Gc=Kc2/E,where Kc is th e stress intensity, as we have assum edplane stress conditions. Swain and Hagan use G, =Kc2(1 9 ) / Ewhich, strictly speaking,is appropriate only for plane strain.)

4. Initiation of plastic flow and fracture at indentations in siliconW e now evaluate (4) fo r the specific case of silicon a nd a ttempt in particular t o establishan ap propriate value of Y, o which the calculation is particularly sensitive. One advant-age of the use of silicon in this context is the availability of dislocation-free crystals inwhich any region of plastic flow can be well defined. Accordingly we have observed bytransmission electron microscopy the deformed region created by indentation under verylight loads, in an attempt to eludicate the mechanism of yield initiation as well as toestimate th e pressure.The material used was dislocation-free, phosphorus-doped 6-10 C cm) grown bythe Czochralskj method and prepared by standard industrial procedure in the form ofslices about 300 pm thick with the polished surface misoriented by some 3 fro m (1 11)towards (110). Discs for microscopy were trepanned by an ultrasonic drill and thepolished surface was given a controlled pattern of damage in the form of alternate


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198 K E Puttick, M A Shahid and M M Hosseiniscratches and indentations with a diamo nd pyramid microhardness indenter. Load sranged from 10 to 0.3 g (- 10-2 to 3 x 10-3 N). The discs were then trepanned by astandard technique for viewing by transmission electron microscopy.A typical picture is seen in figure l? of an indentation made under 2 g load. Theindentation surface is covered with a series of irregular d ark patches which, though theyare difficult to resolve, alter in contrast as the reflection vector is changed in such a wayas to indicate th at they are seen by diffraction c ontrast. (Because of the high straingradient in the deformed region, it proved impossible to photograph the entire area in atwo-beam condition.) Here an d there loops of dislocation may indeed be resolved. Noradial cracks could be detected at this loa d; the first clearly defined fracture was associ-ated with indentation u nder 20 g load.No evidence was found in any of the micrographs or their associated diffractionpatterns either for twinning or phase transform ations, an d it seems reasonable to concludethat the indentation deformation is accommodated by an array of dislocations, of adensity in excess of 1016 m-2. This array appears, however, not in the interior of thecrystal, where yield should begin according to the classical Hertzian theory of elasticcontact, but at th e specimen-indenter interface. We therefore require some modificationof the conventional model of the onset of f low in these circumstances before we canestimate the value of Y.The most obvious reason why dislocation nucleation should appear at the specimensurface, rather th an at a depth of ab ou t half a n indentation radius below it, is the effectof th e difference in elastic con stants between indenter and specimen. This difference givesrise to a relative tangential motion which is opposed by friction, setting up a tangentialsurface traction. Th e effect has been considered by Joh nson et aZ(l973) who distinguishtwo extreme cases: tha t in which friction is great enough to prevent slip, and the situationin which slip is allowed over the whole interface. Where no slip occurs, the distributionof tangential stress is governed by the parameter

where GI, G2 are shear moduli and V I , VY Poissons ratios for th e specimen and indenterrespectively. Th e tangential stress rises from the centre of the indentation t o a maximumat the edge of the circle of contact. Comp lete slip, on the other ha nd, leads to a maximumtraction of order ppo , where p o is the mean normal pressure and ,U the coefficient offriction, at th e centre of the contact area. Johns on et aZ (1973) remark that the realsituation in most cases is likely to lie between these two extremes, with slip occurring inan annulus of the area of contact, and minimum stresses of the order of ppo occurring atthe edge of the annulus.The value of K for diamond on silicon is 0.5. The coefficient of friction does notappear to have been measured, but a reasonable estimate of this is given by the value fo rdiam ond on diam ond, which is 0.1. Th e relative tangential surface stresses duringindentation are therefore probab ly no t very different for those computed by Johnson et alfor steel on glass ( K=0*4 ,p=O.l)-that is, of the order of 0.1P O either for com plete slipor no slip. We conclude tha t the dislocation array at the interface has been nucleated byshear stresses of this order.Th e pressure required to form the indentations in the present work is abo ut 103 kgmm-2 (9.8 x 109 N m-2); for example, the value for the indentation in figure 1 ist All figures are in the plate section.


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Size ef ec ts in abrasion of brittle materials 1991.5 x l o3 kg mm2 (1,274x 1010 N m-2). As with most microhardness work, there is somevariation with load.) The value of the yield stress in shear T is therefore 1.27 x lO9Nm-2, Th e value of the shear mod ulus G (i.e. the stiffness constan t c44) is 7.96 x 1010 Nm-2, so th at th e shear stress given by the analysis of Joh ns on et a1 is abo ut 0-02G. Thismay be compared with other experimental measurements of the stress required fornucleation of dislocation loops in a perfect lattice, for example the determination byJones and Mitchell (1958) of the punching of dislocation loops in a perfect lattice bytherm al stresses rou nd inclusions in silver bromide, which yielded values of G/33.The value of the shear stress required to initiate plastic flow was also inferred fromobservations by transmission electron microscopy of scratches made by the diamondpyramid indenter under light loads. Once again the deformed a rea exhibited patches ofdiffraction contrast which, in the absence of twinning or transformation spots in thediffraction pattern, m ay reasonably be ascribed to a dense array of dislocations. Figure2, of such a scratch made on a near-(I 11) silicon surface in a [TI01direction under a loadof 5 g, also shows lines parallel to [ O i l ]of dark contrast on a light background, whichappear t o be individual dislocation lines. Th e pressure beneath the indenter was1.08 x 1010 N m-2; if the dislocations have been nucleated by tangential surface forces dueto the motion of the indenter, then the surface shear stress T is again (assuming p=O.I

l o 9 N m-2 an d the yield stress in uniaxial com pression is 2 / 3 7 ~.9 x lo9 N m-2,Co rrob oratio n of this value in order of magnitude is obtained fro m stan dard micro-hardness tests. Th e pyramid ind enter was loaded with 30 g (2.94 x 10-1 N) and th e sizeof the impression measured: this gave a pressure of 2 x I O 9 N m-2. The theoreticalexpression derived by Johnson (1970) (also used by Swain and Hagan) isp = z l + l n E / Y t a n /3+4(1-2v)

3 6(1- Vwhere /3 = 20 O for pyramid indenters.cribed above we find p = 2.6 x 109 N m-2.For the mean value of Y=2.1 x l o 9 N m-2 furnished by the two experim ents des-From the above values we have Y /E = 1.02 x 10-2 and Y2/E=2*1x IO N m-2.We next require some estimate of the value of F(Rf/a,) in equation (3) . Since the cracktip extends (figure 3) to a distance from the centre of the impression about equal to animpression diagonal, which we may t ak e to be t he effective diameter of the inden tation,we may put P I = R/a=2. Now the equation G = 2 r has in general two roo ts, o ne corres-ponding to dG /dpl< 0, the other to dG/d pl> 0, i.e. respectively to unstable propag ationfrom a flaw with its tip at Rp and to the stable crack situation actually observed. Th evalues of fo r these two crack lengths are likewise equal:

Hence we find &/a= 1.5. We therefore estimate

The value of I next requires consideration. The fracture surface energy of silicon hasbeen determined for cleavage on {111) to be 2.5 J m-2 (St John 1975). This would giveac = 1.5 pm. However, numerous experiments on indentation fracture on near (1 11)silicon surfaces show that in these circumstances fracture never follows cleavage planes


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200 K E Puttick, M A Shahid and M M Hosseinifor example, radial cracks d o no t f orm along traces of (1 11) in general, an d chip surfacesrarely show crystallographic facets (Shahid 1977). The tru e value of I must thereforeexceed the value for regular cleavage, and it seems reasonable to assume that a morerealistic magnitude would be 10 J m 2. The latter figure gives ac=6 pm.We therefore expect a, t o be in the range 1-10 pm. Experimentally, cracks were firstdetected at indentations produ ced by a diam ond pyramid microhardness indenter under aload of 25 g (0.245 N), with a mean linear size of inden tation of 7 pm , a s illustrated byfigure 3. This indenter shape differs fro m tha t of the theoretical model, so tha t the localtensile stress is probably higher an d the result is probably an und erestimate of a, for anindenter with circular symmetry. Nevertheless, it is interesting to note th at cracks ar eindecd initiated at a well defined size of indentation which agrees in order of magnitudewith the prediction of the model.

5. DiscussionThis apparent agreement may of course, in view of the necessary approximations anduncertainties in the d ata , be fortuitous, an d we emphasise th at the analysis is at this stageintended mainly to illustrate the underlying principle. However, accepting for themoment the reasonable consistency of calculation and experiment, we next recall thata criterion similar to (2) should apply in principle to any inhomogeneous stress field,and speculate that in particular the initiation of cracks by scratches is governed by suchan energy scaling law. Th e tensile stress field, either direct or residual, ar ou nd a plastic-elastic scratch is not known, but it seems likely that it bears to the indentation field arelation similar to th at between the tensile stresses developed by sliding and by indentationby a loaded ball under purely elastic conditions: that is, in the sliding case an extracomponent of tensile stress associated with the frictional forces opposing movementappears in the wake of the indenter. In practice we find th at the critical width of scratcheson near (1 11) silicon surfaces is of the same order as th at of indentations made with thesame indenter. Figure 4 shows a series of scratches made with a m icrohardness pyramidindenter in [llz] under loads of 15, 18 and 20 g (0.15, 0.18 and 0.2 N), and it is cleartha t at the lowest load no cracks whatever are visible; cracking begins at 18-20 g whenthe scratch width is about 7 pm.We therefore conclude that both indentations and scratches initiate fracture inaccordance with the strain energy criterion (2). Now these processes represent the basicmechanisms of abrasion, the first operating in the free particle and the second in thefixed particle situ ation , when th e abrading particles a re respectively free to roll over th esurface or are bound to a backing of paper or cloth. Th e removal of material duringgrinding of a brittle solid is essentially due t o chipping (Lawn an d Wilshaw 1975), an dchipping involves the form ation of lateral vents and radial cracks. The transition frompurely plastic-elastic indentation or scratching to chipping may therefore be identifiedwith the onset of radial cracking in the field of residual tensile stress created duringunloading when the linear size of the permanent impression attains a critical value.There is thus a qualitative difference between the processes of indentation below andabove a critical size, a statement which we may also apply t o th e processes of polishingan d grinding of brittle m aterials.If this is so we require some mechanism distinct from chipping by which materialmay be removed d uring polishing of such materials. Th e clue to this problem may liein the state of the material immediately below the surface of the impression or scratch


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Size effects in abrasion of brittle materials 20 1groove, observations on which have been briefly reported by Puttick and Shahid (1977).This material possesses a granular structure on a scale of 1 pm o r less, which is probablyassociated with the very high density of dislocations generated by plastic flow underhydrostatic compression, and it is conceivable that polishing removes this highly frag-mented layer. We suggest tha t this regime of abrasion deserves further study.

6 ConclusionsThe initiation of cracks by the residual tensile stress field around an indentation orscratch is governed, not by a flaw spacing criterion, but by the magnitude of the strainenergy in the sur round ing material. If the strain energy field preserves geom etricalsimilarity as it expands, fracture will occur at a critical size of indentation a which isproportional t o the ratio I/ F(Rf /a)where I s the fracture surface energy per unit area ,W a measure of the strain energy density per unit volume, and F(Rf /a) is a non-dimensional function of the Griffith flaw size Rf elative to a.For a plastic-elastic indentation WK Y2/E,where s the yield stress in uniaxia lcompression and E Youngs modulus. F or highly brittle materials inden tation fractureoccurs in the field of residual stress cre3ted durin g unloading of the indenter. Ado ptingthe model of the residual stress field in the surface proposed by Swain and Ha gan (1976),we find 6 E rac= Y2[ YRf /a>lwhere 2 may be calculated by standard fracture mechanics.

ac has been evaluated for silicon. Y is estimated from observations by transmissionelectron microscopy of indentations made by a diamond under very light loads; itappe ars th at dislocations are first nucleated by frictional forces nt th e indenter-specimeninterface. Using the resulting value together with the expression for t derived from thestress field proposed by Swain and Hagan, ac is found to lie in the range 1-10 pm ngood agreement with observed magnitudes of the critical size of indentations for fractureof abou t 5 Fm. It seems reasonable t o suppose tha t a similar criterion governs thetransition from polishing to grinding as the particle size is increased in abrasion.

AcknowledgmentsPart of the experimental work was carried out under the terms of a grant from theSRC. M A S was supported by a scholarship from the Government of Paskistan, andM M H by the Government of Iran.

Appendix. The strain energy release rat eTh e Swain-Hagan stress field for complete unloa ding of an indentation with circularsymmetry is

where C O , the radius of the plastic-elastic boun dary, is abo ut equa l to the radius of the


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202 K E Puttick, M A Slialzid and M M Hosseiniimpression a. We first calculate the stress intensity factor. Fo r consistency with aprevious calculation of this factor for a loaded indentation (Puttick 1978a) we ado pt aslightly different approx imation to th at of Swain and Hagan , an d idealise the radial crackas spreading from opposite ends of the diameter of the indentation with the tips at aradial distance R fro m its centre. In th e absence of detailed knowledge of its shape,however, we postulate that the crack front is straight and normal to the surface.This situation we now regard as an appro ximation to the problem of a crack of length2R in an infinite plate with th e crack faces loaded by the above stress field over the rangea< R


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J Phys. D : Appl . Phys., V o l . 12, 1979-K E Pi i r r i ck , M A Shahid citid M M Hosseitii (see pp 195-202)

Figure I Transniission electron micrograph o f an indentation made by the t ip o adiamond indenter under 2 g load in a near I I I ) surface of sil icon, showing dense a rrayof dislocations at the surface. Magnification x 500. Reflection 022.

Figure 2. Transmission clcctron m icrograph d a r k field) of I w a t c h in [TI01 made by ad iamond indenter under 5 g load in a near I I I ) silicon surfac e. Magnification x 3 3 OOOReflection 220.

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J Phys. D: Appl. Phys., Vol . 12, 1979-K E P / r r / i c k , M A S l i d i i d i u i M M Hoswiiri see pp 195-202

Figure 3. Indentations made by Vickers pyramid microhard ness indenter in near I 1 Isilicon surface. Magnification x 1680 (a) 20 g load; ( h ) 25 g load; c ) 30 g load.


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J . Phys. D : Appl. Phys. , Vol. 12, 1979--K E Pi i t t i , M A Slicrliicl rti l M M Hosseiiri see pp 195 -202)

Figure 4. Scratchcs made by Vickcrs pyramid niicrohardncss intlcntcr in [ I 121on ncarI I ) silicon surface: 1 ) 0.15 N load: h ) .18 N ; ( c )0.20 N

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Size Effects in Abrasion of Brittle Materials

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