The Twyman effect,1,2 originally described by Twy-man in 1905 and named for him by Preston, refers tothe formation of a residual compressive state of stresson a ground-glass surface. In Twyman’s words1:“If a very thin piece of glass, grey on both sides, be
polished so that it can be examined through its thick-ness and polarised light be applied, stress can dis-tinctly be seen near each surface; if now one side bepolished strain disappears from that surface, and theglass blows up ~sic!; on polishing the remaining greyside, the glass becomes parallel again, and strainentirely disappears.”Twyman noticed2 that the word blows was a mis-
All authors are with the Center for Optics Manufacturing, Uni-versity of Rochester, Rochester, New York 14627. J. C. Lambro-poulos, S. Xu, and T. Fang are also with the Department ofMechanical Engineering, Materials Science Program.Received 3 December 1995; revisedmanuscript received 11 April
5704 APPLIED OPTICS y Vol. 35, No. 28 y 1 October 1996
print for bows. Greying of a glass surface refers togrinding that surface in loose abrasive ~lapping! con-ditions.Dalladay3 studied the formation of the surface
stress by grinding the surface of an originally pol-ished hard crown thin glass plate ~with a thickness of;3.25mm and eventually ground to;3mm!. Whenone side of the strip was ground with abrasives ofvarious sizes, the plate appeared stressed as if it werebeing subjected to pure bending combined with ex-tension; i.e., the neutral axis was straight and nearerthe polished than the grey surface; that part of theplate adjacent to the grey surface was in tension, theother in compression. Dalladay3 concluded that thisstress system was the result of a uniform compres-sion in the grey surface itself, which, although in-tense, was so closely confined to the gray surface thatit could not be directly observed by polarized light.The abrasives used by Dalladay, in the indicated or-der, were carborundum ~SiC, 400 and 150 mm!, nat-ural corundum ~Al2O3, 110 mm!, and Naxos emery~impure corundum, 15–50 mm!. The compressiveforce ~see Fig. 1!was found bymeasuring the bending
moment required to counteract the bending inducedby grinding.Preston4 studied the formation of residual com-
pressive surface stresses during grinding and con-cluded that the ground surface consists of deep flawsextending far below the surface irregularities.Preston proposed that the formation of the surfacecompressive stress was due to the surface cracks be-ing slightly open at their mouths and held open byminute fragments of glass or other materials.Preston observed pits on the ground surface extend-ing to a depth of three wavelengths, estimated thesurface cracks to extend to a depth four times as greatbelow the surface, and noticed that a few moments ofetching in a very dilute solution of HF acid wasenough to remove the effect almost completely.Belowwe briefly describe three equivalentmethods
of measuring the Twyman effect. The methods areby Ratajczyk,5,6 Nikolova,7,8 and Rupp.9 All threemethods are equivalent and are based on measure-ment of the bending sag ~or radius of curvature! of athin plate, one side of which is in a compressive stateof stress from the Twyman effect.Ratajczyk5,6 studied the surface compressive stress
in optical glasses BK7 and SF3 by measuring withFizeau interferometry the number n of interferencerings ~Zahl der Interferenzstreifen! on the polishedside of a thin plate, the other side of which wasground with SiC abrasives ~3.5–150 mm in size!.The plate thickness was in the 0.75–9-mm range.Ratajczyk wrote5
n 5 ZSDt D2
, (1a)
where D is the diameter and t is the thickness of theglass plate. The dimensionless constant Z increasedmonotonically with abrasive size L ~typically from0.006 at L 5 3 mm to 0.013 for L 5 150 mm for BK7!and can be converted to the surface grinding force P0
Fig. 1. Dependence of surface compressive force P0 ~Twyman ef-fect! on abrasive size for grinding borosilicate crown glass. Thedata from the literature are for loose abrasive grinding ~lapping,fixed nominal pressure!. Podzimek’s data15–17 are for a cast ironbacking plate. Our current work is for Blanchard grinding andfor deterministic microgrinding ~at a fixed infeed rate!.
~Newtons per meter! through
P0 54E
3~1 2 n!SZ l
2D , (1b)
where E is the Young’s modulus, n is the Poissonratio, and l is the wavelength of light used ~Na light,l 5 0.59 mm!. The dependence of the compressiveforce on abrasive size for BK7 is included in Fig. 1.Ratajczyk’s results showed that the magnitude of
the compressive force strongly depended on the glasstype.5 For the softer, more brittle flint glass SF3 thecoefficient Z was two to three times higher than thatfor the harder, tougher borosilicate crown BK7.Ratajczyk also observed that for pressures as great as60 kPa and velocities as high as 6 mys the number ofrings was constant, indicating that the compressiveforce depends on the material ground and the abra-sive size but is independent of applied pressure andrelative speed. Ratajczyk observed that the groundsurfaces exhibited a significant amount of crackingand indicated that the Twyman forces were related tofracture processes during lapping.6 Figure 1 showsthat the data of Ratajczyk5 are in good agreementwith those of Dalladay3 corresponding to carborun-dum and corundum abrasives. However, the use offine Naxos emery by Dalladay produced lower surfaceforces than SiC by Ratajczyk, possibly because of thelower hardness of emery compared with carborun-dum.Nikolova7,8 performed similar experiments in
glasses and crystals and expressed the Twyman effectby correlating the radiusR of curvature ~measured byinterferometry! to the plate thickness t through coef-ficient A:
R 5 At2. (2a)
The compressive surface force can be found from con-stant A through
P0 5E
6~1 2 n!A(2b)
so that a greater value for A leads to a lower surfaceforce. Typical values for optical glasses are in therange A 5 40–70 mm21.7Nikolova, using much thinner plates ~t 5 0.1–0.5
mm! and abrasives ranging in size L from 3 to 40 mm,did not observe a dependence of the surface stress P0on the abrasive size,8 as shown in Fig. 1. This resultis in contrast to all other measurements of the Twy-man effect, which show that the surface force de-creases with decreasing abrasive size. Nikolova7explained the independence of P0 from L by the dis-ruption of the surface tension in the ground surface.The unbalanced surface tension on the polished sidethus causes the plate to bend. Note, however, thatthe surface tension thus measured is quite large:For BK7, P0 5 294 Nym 5 294 Jym2, whereas thetypical surface energy for brittle materials is lessthan a few Jym2.Still a third way of describing the Twyman effect
1 October 1996 y Vol. 35, No. 28 y APPLIED OPTICS 5705
has been described by Rupp,9 with the bending sagDh of the polished side of a thin plate, whose otherside is ground, correlated with the plate dimensions~diameter D, thickness t! as
Dh 5 CSDt D2
, P0 54E
3~1 2 n!C. (3)
The constant C ~nanometers!, which strongly de-pends on the abrasive size, is often referred to as theTwyman constant.The formation of surface stresses during grinding
has also been observed in semiconductors10–12 as wellas in ductile-mode grinding conditions.13,14 Goliniand Jacobs13,14 used the term loose abrasive micro-grinding to describe lapping with abrasives of lessthan 3 mm. The data by Golini and Jacobs are an-alyzed further in Section 4 because they reveal themagnitude of the surface compressive stress inductile-mode grinding conditions.Podzimek15–17 has conducted to date the most ex-
tensive study of the Twyman effect in optical glassesin lapping conditions involving material removal byfracture. He used SiC abrasives ~grades 800, 500,400, 320, and 240; size ranges, 15–38, 22–48, 25–55,34–67, 44–82 mm; average sizes, 22, 30, 35, 46, and59 mm! on 12 optical glasses, fused silica, quartz, Ge,and ZnSe. Backing plates were glass or cast iron,and the slurry liquid was water or decanol. Pod-zimek, who measured the apparent coefficient of fric-tion during and the surface roughness followinglapping, found that the surface stresses were inde-pendent of the lapping pressure and the relativespeed. The data for BK7 with a cast iron backingplate are shown in Fig. 1. These data converge atsmaller abrasive sizes to the measurements of Rataj-czyk.5Podzimek16 found that etching the ground surface
diminished the surface compressive force without re-moval of material. To establish the rate at whichthe surface force diminishes with the distance intothe glass he used ion milling to remove layers ofmaterial without altering the surface structure ~be-cause ion milling did not affect the surface rough-ness!. Podzimek showed that the rate of decrease ofthe surface compressive force is15–17
P~j! 5 P0 exp~2jyds! 5 s0ds exp~2jyds!, (4)
where j is the depth of removed material, ds is theequivalent depth of the compressive layer, and
s0 5 P0yds (5)
is the maximum value of the compressive surfacestress. Note that, although ds is the equivalentdepth of the compressive layer, the actual region ofcompression extends beyond j 5 ds because of theexponential decay in the compressive stress. Be-cause of their extent, the data of Podzimek16 are an-alyzed further in Section 3.To date the bulk of the measurements of the sur-
face compressive forces has been done for lapping
5706 APPLIED OPTICS y Vol. 35, No. 28 y 1 October 1996
conditions, in fixed nominal pressure. We describebelow measurements of surface forces resulting fromdeterministic microgrinding conditions ~at fixed in-feed rate!. The Center for Optics Manufacturing atthe University of Rochester has made significant con-tributions in the fabrication of precision optical com-ponents through deterministic microgrinding withrigid, computer-controlled machining centers andhigh-speed tool spindles. The manufacturing of con-vex and concave spherical surfaces with radii of 5 mmto `, i.e., planar, and work diameters from 10 to 150mm, has been achieved.18–20 Aspherical surfaceshave also been manufactured. Specular surfaces,resulting after less than 5 min of deterministic mi-crogrinding, have a typical rms microroughness ofless than 20 nm, 1 mm or less of subsurface damage,and a surface figure of better than 1y2 wave frompeak to valley ~p–v!.20 Typical infeed rates are 6–10mmymin with 2–4-mm bound abrasive diamond tools.An overview of the mechanics and materials issues indeterministic microgrinding has been presented byLambropoulos et al.21The bound abrasive tools used in this research con-
sist of single crystal or polycrystal diamonds embed-ded in a metal-bond ring, typically 50 mm indiameter. The size of the diamonds used in differenttools varies from ;100 mm down to 2–4 mm. Typi-cally, several tools are used with diamonds in thisrange. Each tool is used to remove the damagedlayer resulting from the previous tool and to reducethe surface microroughness further. Tool rotationrates for deterministic microgrinding of opticalglasses range from 5000 to 30,000 rpm ~15,000 rpmbeing typical, leading to surface speeds of approxi-mately 40 mys!, whereas the work rotation rates arefrom 50 to 300 rpm. Aqueous coolants are used tofacilitate the mechanical and chemical action of thebound abrasive tools in deterministic microgrinding.In what follows, we summarize the measurement
of the surface forces ~the Twyman effect! in determin-istic microgrinding conditions ~Section 2!, and we usethe data of Podzimek15–17 in Section 3 to show thatthe magnitude P0 of the surface compressive force,the depth ds of the compressive layer, the depth ofpenetration of an abrasive grain into the glass sur-face, and the resulting surface roughness can be cor-related with the cracking and the deformationaspects of the ground glasses. We then proceed inSection 4 to analyze the data of Golini and Jacobs13,14on loose abrasive microgrinding ~lapping with abra-sives of less than 3 mm in size! to estimate the surfacecompressive stress in ductile-mode grinding and tocorrelate the surface stress with the glass mechanicalproperties.
2. Deterministic Microgrinding
A. Glass Properties
The glass used was a borosilicate crown ~BK7 in theSchott notation!. The glass properties from theSchott catalog were22E5 81 GPa, n 5 0.21, r 5 2.51 gcm23, Knoop hardnessHk 5 5.2 GPa at 200 gf. The
fracture toughness was determined by microindenta-tion23,24 as Kc 5 0.8260.05 MPa =m when the mi-croindentation model of Evans25 was used. Themicroindentation measurement is in good agreementwith bulk measurements26–28 and other microinden-tation measurements.29–32
B. Deterministic Microgrinding
A 50.66-mm-diameter blank of glass BK7 was pre-pared with one side polished planar ~approximately1y2 wave p–v! and the second side to be used as thetest ~variable! surface. The starting blank thicknesswas 1.75 mm, and as the experiment progressed itdecreased to 1.66 mm as the second side was groundin various conditions. The aspect ratio ~approxi-mately 30:1! was sufficiently high to allow measur-able bending from grinding-induced compressivestress. Bending was measured on the polished pla-nar surface with Fizeau interferometry ~with a ZygoMark IV!. The ground surface is in compression andbends convex. The surface that is measured, how-ever, is the polished surface ~opposite surface! thattherefore becomes concave.We performed the first cut with a Blanchard
grinder, using a 300-grit wheel. The sag changecaused by the part bending was 1.38 mm p–v afterBlanchard grinding. The part was then etched withHF acid to relieve all machining-induced surfacestresses. The sag changed by 1.38 mm p–v, return-ing the part to its nominal stress-free state.We then proceeded with the deterministic micro-
grinding experiments where the infeed rate was keptconstant for a given bound abrasive size. Theground side was machined with a 220-grit ~70-mm!wheel on the Opticammachining center, reducing thethickness to 1.70 mm. The machining stress bentthe part by 7.05 mm ~p–v sag!. Next the part wasmachined with a 10–20-mm wheel, the bending wasreduced to 0.92 mm, and the thickness to 1.67 mm.Finally, the part was machined with a 2–4-mmwheel~at an infeed rate of 10 mmymin!, the bending sagdecreased again, to 0.83 mm, and the thickness de-creased to 1.66 mm. At the end of this cut the partwas etched with HF acid, after which the bending sagdecreased to 0.11 mm p–v.The variation in the surface compressive force with
abrasive size is shown in Fig. 1, which also shows thecollected data for lapping conditions. For abrasivesof less than ;40 mm in size, deterministic micro-grinding produces lower surface compressive stressesthan lapping. Still, for larger bound abrasives, de-terministic microgrinding produces surface forcesthat are higher than those in lapping. This is at-tributed to the infeed rate being constant in deter-ministic microgrinding, whereas lapping occurs at aconstant nominal pressure. Thus, for low infeedrates the equivalent pressure between the tool andthe glass surface is expected to be low, increasingwith the increasing infeed rate. As in lapping thepressure is constant; at some infeed rate the pressurepresent in microgrinding may exceed that in lapping,
thus leading to deeper fractures and higher grindingforces.
3. Lapping Data
Because of the great variety in conditions and mate-rials used by Podzimek,15–17 we proceed to analyzesome of that data on loose abrasive grinding.In Fig. 2 we show the correlation of the surface
compressive force P0 with the size of the indentationcrack c measured after indentation with a 2-N forcefor various materials. The data for Ge are from Mc-Colm.33 The correlation of depth ds of the compres-sive layer with the abrasive size L and theindentation crack size c is shown in Fig. 3. Thecorrelations in Figs. 2 and 3 show that the Twymansurface residual force and its extent below the surfaceare clearly dictated by the response of the glass tosurface-indenting forces, as originally visualized byPreston,4 Ratajczyk,6 and Phillips et al.34 This cor-relation agrees with the observation, reported byPreston4 among others, that etching, which opensand blunts the surface cracks, relieves the residualcompressive stress almost entirely.14In Fig. 4 we show the dependence of the stress
depth ds and the maximum surface compressive forces0 on the abrasive size L in brittle material-removalconditions in H2O for optical glasses BK7 and SF6.
Fig. 2. Dependence of the surface compressive force P0 on inden-tation crack size. P0 owing to lapping ~SiCyH2O slurry, 48-mmmaximum abrasive particle size! was measured by Podzimek15–17through interferometry.
Fig. 3. Dependence of the depth ds of the compressive layer onabrasive size L and indentation crack size; ds in lapping was mea-sured by Podzimek15–17 through ion milling of lapped surfaces andinterferometry.
1 October 1996 y Vol. 35, No. 28 y APPLIED OPTICS 5707
The surface stress s0 was calculated from P0 from Eq.~5!. Smaller abrasives L lead to shallower compres-sive depths ds and higher compressive stresses s0,although, as Fig. 1 indicates, the compressive force P05 s0 ds generally decreases with decreasing abrasivesize. We also observe from Fig. 4 that, becausestress depth ds is linearly related to the abrasive sizeL, the exponential decay of surface stress with depthin Eq. ~4! established by Podzimek16 equivalentlycould be cast as P~j! ; exp~2ajyL!.It is interesting to observe that flint glass SF6
~Schott notation with E 5 55 GPa, n 5 0.244, r 55.18 g cm23,Hk 5 3.1 GPa at 200 gf,22 and a fracturetoughness of Kc 5 0.5460.04 MPa=m as measuredby microindentation32 or bulk methods,35 has a lowersurface stress s0 but deeper stress depth ds than theborosilicate crown, BK7, which is both harder andtougher. Still, as Fig. 2 shows, the compressive forceP0 is higher in SF6 than BK7. This conclusionagrees with the results of Ratajczyk5 who observedthat flint glass SF3 had a significantly higher com-pressive force P0 than the borosilicate crown, BK7.In addition to the features of the Twyman effect
discussed above, we have considered the depth ofpenetration di of an abrasive of size L into the surfaceof the glass ~see Fig. 5!. The model by Phillips et
Fig. 4. Dependence of the maximum surface compressive stresss0 and of the depth ds of the compressive layer on abrasive size Lfor the borosilicate crown glass BK7 and the flint glass SF6. Thebacking plate used by Podzimek15–17 was made of glass. The lin-ear dependence of the stress depth ds on the abrasive size L wasnoted by Podzimek.16
Fig. 5. Lapping model by Phillips et al.34 involving the tumblingmotion of an abrasive grain. Angle ua can be determined from themeasurement of the nominal coefficient of friction ~the ratio of dragforce to normal force!.
5708 APPLIED OPTICS y Vol. 35, No. 28 y 1 October 1996
al.34 was used to extract the depth of penetration difrom the measurement of the coefficient of frictionduring lapping by Podzimek.16 In this model thetumbling angle ua is found from the coefficient offriction:
m 5 2ln ~cos ua!
ua, di 5
L2
~1 2 cos ua! . (6)
Figures 6 and 7 show the dependence of the depth ofpenetration di on the hardness of the glass for thecase of a glass-backing plate. The linear depen-dence of di on 1y=Hk indicates that the penetrationof the abrasive grain into the glass surface is similarto the indentation of a brittle surface by a normalindenting force of constant magnitude.The depth di of penetration also can be correlated
to the surface microroughness in lapping. Figure 8shows the correlation of the measured surface rough-ness, resulting from lapping with water slurry andSiC abrasives of maximum size, 48 mm, with thedepth of penetration di as extracted from the coeffi-cient of friction @see Eqs. ~6!#. Figures 6 and 8 leadto the correlation of Fig. 9, which shows that thesurface roughness in lapping is determined by theplastic properties of the glass.Buijs and Korpel-Van Houten26 have argued that
the surface microroughness should be identified withthe depth b of the plastic zone caused by an indenting
Fig. 6. Dependence of penetration depth di of abrasive grain inwater slurry into the glass surface on the hardness of the glass.The penetration depth di was measured by Podzimek.16
Fig. 7. Dependence of penetration depth di of abrasive grain indecanol slurry into a glass surface on the hardness of the glass.The penetration depth di was measured by Podzimek.16
particle. They used the correlation
b 5 diSEHDm
(7)
between the size b of the plastic zone created under-neath an indenter and the depth di of the penetrationof the indenter, where m is in the range of 1y3–1y2~Ref. 36! for an indenter of Vickers shape ~blunt in-denter, with indenter angle 2c 5 136°!. For the caseof a constant indenting force, di scales with 1y=Hk,so that the plastic zone size b should then scale with~=E!yHk ~for m 5 1y2!. The correlation of rough-ness resulting from lapping with ~=E!yHk, as sug-gested by Buijs andKorpel-VanHouten,26 is shown inFig. 10. This correlation has a correlation coefficientR 5 0.87 and is thus weaker than the correlation ofsurface roughness with 1y=Hk shown in Fig. 9.For the glasses examined by Podzimek, EyH
ranges from 12 to 20, so that the identification ofroughness with the plastic zone size b of Eq. ~7!wouldpredict a roughness of 2.3–4.5 times higher than thedepth of penetration di. Figure 8 shows that this isnot the case. We conclude that the surface rough-ness and depth of penetration measurements by Pod-zimek16 are consistent with the scaling of the surfaceroughness with 1y=Hk, as shown in Fig. 9.We also have performed a numerical simulation of
microindentation, using the finite-element package
Fig. 8. Correlation of the depth di of penetration of an abrasivegrain ~extracted from measurement of the apparent coefficient offriction! and the surface roughness measured by Podzimek16through profilometry.
Fig. 9. Correlation of the surface roughness measured by Pod-zimek16 through profilometry with the Knoop hardness of the glasssurface. The straight-line correlation coefficient is 0.91.
ABAQUS.37 The numerical simulation examined theaxisymmetric indentation by a rigid indenter of anelastic, perfectly plastic solid of Young’s modulus E,Poisson ratio n, and uniaxial yield stress sY. Thesematerial properties were selected to be close to thosefor a typical optical glass ~E 5 70 GPa, n 5 0.25, sY5 3 GPa!. No cracking was allowed to occur duringthe indentation simulations.The indenter was described by the indenter angle
2c and the radius of curvature Rtip of the indenter tip~taken to be 100 nm in the numerical simulations!.Angle 2c describes the overall shape of an abrasiveparticle, whereas Rtip describes the contact of a sharpabrasive point or dull ~worn! abrasive flat, both ofwhich occur in optics manufacturing and signifi-cantly affect the rate of material removal, as dis-cussed by Rupp.38 Thus, whereas angle 2c will notsignificantly evolve during manufacturing, Rtip itselfmay evolve due to dulling and creation of a flat.Even if Rtip does not evolve during optics manufac-turing, it may be used to describe more angular abra-sives versus more rounded abrasives.In the simulation of indentation the indenter nor-
mal load was increased from P 5 0 to Pmax and thenreduced to zero, while the penetration of the indenterinto the glass surface increased from 0 to Dmax andthen to Dres, respectively. The maximum load Pmaxwas high enough for the contact of the indenter withthe surface to encompass the indenter tip and a por-tion of the conical indenter sides. Because of thehigh glass hardness, we observed a significantspringback, in that the residual depth Dres was sig-nificantly lower than themaximumpenetration Dmax.Figures 11~A! and 11~B! show the shape of the
plastic zone under the indenter and the residualstresses s11 for the cases of indentation with an acuteabrasive particle ~2c 5 60°! and an obtuse abrasiveparticle ~2c 5 136°!. The plastic zone size is shownunder the maximum load Pmax, which was the samefor both simulations ~0.3 mN!. The stress s11 ~thecomponent parallel to the indented surface! is shownafter the load has been reduced to 0. Figures 11~A!and 11~B! show that acute ~low 2c! and obtuse ~high2c! abrasives are expected to produce significantly
Fig. 10. Correlation of the surface roughness measured by Pod-zimek16 through profilometry with the Knoop hardness andYoung’s modulus of the glass surface. The correlation coefficientof the straight line is 0.87.
1 October 1996 y Vol. 35, No. 28 y APPLIED OPTICS 5709
different indentation shapes and plastic zones wheninteracting with a brittle surface.We observe that, for the same maximum normal
force, the maximum penetration Dmax is considerablyhigher for the acute abrasive ~106 versus 52 nm!, thatthe acute abrasive produces less relative springback~12DresyDmax 5 0.19! than the obtuse abrasive~12DresyDmax 5 0.37!, and that the acute abrasiveproduces a significant pileup along the periphery ofthe contact region between the abrasive and the glasssurface. Such a pileup, resulting from the fact thatshear flow in glass occurs under constant volume, canbe used to define more sharply the boundary of theresidual indentation. For the same normal force theacute abrasive also produces a higher surface rough-ness for several reasons: The acute abrasive pene-trates significantly more deeply into the glasssurface, and the pileup near the periphery of thecontact region increases the apparent residual depthof penetration, which thus consists of the actual re-sidual penetration Dres and the height of the pileupregion @;20% of Dres from Fig. 11~A!#.For the acute abrasive the depth of the plastic zone
under the indenter is about twice the maximum pen-
Fig. 11. Finite-element simulation of the axisymmetric indenta-tion of an elastic, perfectly plastic solid by a rigid indenter ~char-acterized by indenter angle 2c and the radius of curvature Rtip ofthe indenter tip!. The load is increased from 0 to Pmax and thendecreased to zero. The depth of penetration changes from zero tothemaximumpenetrationDmax and then to the residual depthDres.The figures show only one-half of the axisymmetric patterns. Thehorizontal directions ~1, 3! are parallel to the free surface. Inden-tation occurs along the vertical direction, increasing along direc-tion 2. The plastic zone size is shown under the maximum loadPmax, whereas the residual stresses are shown after the load hasbeen applied and removed: ~A! acute indenter ~2c 5 60°!; ~B!obtuse indenter ~2c 5 136°!.
5710 APPLIED OPTICS y Vol. 35, No. 28 y 1 October 1996
etration, whereas the ratio of plastic zone depth b tomaximum penetration Dmax is ;3.5 for the obtuseindenter. On the other hand, the estimate ~EyH!m
of the ratio byDmax would be in the range of 3.3–3.7~for m 5 1y2, and depending on how the hardness isextracted from the shape of the residual indentation!and thus in fair agreement with the numerical com-putation for the obtuse indenter.The correlation of surface roughness caused by
loose abrasive lapping suggested from the lappingdata analysis and numerical computation discussedabove, namely, that surface roughness in lappingscales with 1y=Hk is in contrast with the correlationof surface roughness with glass mechanical proper-ties in deterministic microgrinding conditions ~fixedinfeed!, where the surface-roughness scales with theductility index J 5 ~KcyHk!2,39 a concept in reverseanalogy to the brittleness index.40 Thus the fracturetoughness of the glass does not enter the formation ofsurface roughness in three-body abrasion ~lapping ata fixed nominal pressure!, whereas it does in deter-ministic microgrinding ~two-body abrasion at a fixedinfeed rate!.
4. Loose Abrasive Microgrinding
The term loose abrasive microgrinding has been usedby Golini and Jacobs13,14 to describe loose abrasivegrinding with abrasive particles less than 3 mm insize. Depending on the chemistry of the slurry used@both brass and ceramic backing plates were used inlapping Corning ULE ~C7971! and Schott Zerodur#,Golini and Jacobs13,14 observed that the material-removal rate changed from brittle to ductile. Suchchanges also affected the magnitude of the compres-sive surface forces induced. In their words,14 “. . . atremendous amount of surface stress is introduced inloose abrasive ductile mode grinding. This stresswas observed when the Twyman effect in ULE platesincreased by a factor of 4 in the transition from thebrittle to the ductile mode.” It must be pointed outthat Golini and Jacobs used the term stress to de-scribe the surface compressive force P0. Similar ex-periments onULE had also been conducted by Rupp.9The experiments of Golini and Jacobs14 allow an
estimate of the actual surface compressive stress s0.Golini and Jacobs,14 who described the Twyman effectin terms of the constant C of Eqs. ~3!, observed thatpolishing a 0.022-mm layer from ULE ~previouslyground with 1–2-mm diamond abrasives and a brasstool! eliminated the surface compressive stress.Their measured constant C can be converted to thesurface force P0 through Eqs. ~3!, and the surfacecompressive stress can be estimated through Eq. ~5!with ds 5 0.022 mm. The resulting surface stress iss0 5 3.5 GPa.Rupp9 conducted a similar experiment inULE ~E5
68 GPa, n 5 0.17, Hk 5 4.6 GPa at 200 gf !. Aftergrinding with 1-mm diamond slurry, Rupp measuredthe constant C fromwhich the compressive force P0 isfound @see Eqs. ~3!#. By polishing the ground sur-face, he estimated the depth of the compressive layerto be ;0.15 mm, which leads to a compressive stress
of s0 5 3.1 GPa. Such a value for the surface com-pressive stress is in fair agreement with the valueresulting from the measurements of Golini and Ja-cobs14 discussed above and is considerably higher,almost by an order of magnitude, than the compres-sive stresses extracted from the Podzimek lappingdata and shown in Fig. 4.In Fig. 12 we show the dependence of the surface
compressive stress on the abrasive size for fused sil-ica, crystalline quartz, and Corning ULE ~92.5%SiO2, 7.5% TiO2!, which is quite similar to fused sil-ica.41 The large abrasive data for fused silica andquartz are based on the measurements of Pod-zimek.16 For ULE we averaged the values of thecompressive stress based on the measurements ofRupp9 and Golini and Jacobs14 discussed above. Weobserve that the correlations for fused silica andquartz are parallel, describing a power-law depen-dence s0 ; L23y4 of the surface compressive stress onabrasive size.This figure also shows the uniaxial yield stress sY
for these materials, as calculated from the measuredVickers hardness at a large load ~typically 2 N!, theYoung’s modulus, and the Poisson ratio from the ex-panding cavity model of Hill.42 The application ofthis model to many optical materials has been dis-cussed by Lambropoulos et al.,39 where Vickers datafor fused silica can be found. For quartz we used theVickers indentation hardness of 12.2 GPa reported byKurkjian et al.43 For ULE we converted the Knoophardness of 4.6 GPa ~at 200 gf ! to a Vickers hardnessbased on the correlation presented by Lambropouloset al.39In Fig. 12 we show that, in the limits of small
abrasive size, the surface compressive stress s0 de-scribing the Twyman effect in these glasses and crys-tals is very close to the uniaxial yield stress sY forthese materials. This observation is not surprisingin that small abrasives induce ductile-mode grinding,which in turn produces extensive plastic flow on the
Fig. 12. Dependence on abrasive size of the maximum surfacecompressive stress s0. The data for fused silica and quartz are fora glass backing plate and SiCyH2O slurry from Podzimek.16 Thedata for CorningULE are fromGolini and Jacobs14 and Rupp,9 whoused diamond slurry. The fitted line for s0 in quartz decreaseswith an abrasive size such as L20.7560.10. For fused silica s0decreases as L20.7460.02 does. The plot also shows the uniaxialyield stress, as calculated from the expanding cavity model ofHill.42
glass surface. Such plastic flow requires stressescomparable with the yield stress of the deformingmaterial.Similar conclusions, namely, that in the case of
small abrasives the surface stress reaches the uniax-ial yield stress, can be drawn for glass ceramics suchas Zerodur. Shore et al.44 measured a surface forceof 160 Nym extending to a depth of ;50 nm, thusgiving a surface compressive stress of 3.2 GPa, whichis practically identical to the yield stress of 3.1 GPafor Zerodur ~Young’s modulus, 91 GPa, Poisson ratio,0.24; Vickers hardness, 7.960.2 GPa at a load of 200gf !.The argument that plastic flow on a glass surface
induces residual compressive stresses comparablewith the yield stress is further extended in Figs. 11,showing the numerically computed residual stressess11 after the indenting load Pmax has been applied onand then removed from a glass surface through in-dentation by an acute indenter @2c 5 60°, Fig. 11~A!#or an obtuse indenter @2c 5 136°, Fig. 11~B!#. Theresidual stresses s11 are normalized with respect tothe yield stress of 3 GPa.Figures 11 show that the near-surface residual
stresses are compressive, as required for the Twymaneffect, and their magnitude is commensurate with thematerial’s yield stress. It is also seen that there is asignificant gradient in the near-surface stresses withdistance into the indented solid: The near-surfaceresidual stresses are large and compressive, whereasthe subsurface stresses are considerably smaller andtensile.The subsurface residual stresses directly beneath
the indenter are tensile and of lower magnitude thanthe near-surface stresses ~s11ysY 5 0.08 for both theacute and the obtuse indenters!. Such tensile resid-ual stresses, which are seen to occur near the elastic–plastic boundary beneath the indenter, areresponsible for the formation of median–radialcracks, known to occur on unloading in many brittlematerials.23,24,33,36,45,46In spite of the similar magnitudes of the subsurface
tensile residual stresses, the detailed distribution ofthe near-surface residual compressive stresses de-pends on the shape of the indenting abrasive particle.An acute particle @Fig. 11~A!# produces residual com-pressive stresses that exceed the uniaxial yield stressby as much as 60% and that show a high gradient inshifting from compression to tension below the sur-face. For the acute abrasive grain the maximum inthe residual compressive stress occurs at some dis-tance underneath the indented surface. The obtuseabrasive produces lower residual compressivestresses ~approximately equal to the yield stress!,which decay more slowly to the subsurface tensilevalue, and the maximum residual compressive stressnow occurs at the indented surface.We observe that the depth of the layer of the re-
sidual compressive stresses is approximately equal tothe depth of the elastic–plastic boundary. For ex-ample, in Fig. 11~A! the surface roughness is approx-imately equal to the residual penetration Dres,
1 October 1996 y Vol. 35, No. 28 y APPLIED OPTICS 5711
whereas the compressive depth would be twice thatmuch. The presence of the subsurface residual ten-sile region would further diminish the extent of anequivalent compressive layer. Thus, depending onthe abrasive shape, the compressive depth is ex-pected to be higher than the surface roughness Ra.This is indeed in qualitative agreement with the ex-perimental observations: Figure 4 shows that forthe 48-mm SiC abrasives the equivalent compressivestress depth dsmeasured by Podzimek in BK7 is;1.3mm ~Ref. 16! @implying that the actual compressivelayer must be higher, because ds is the length scaledescribing the exponential decay of P~j!, see Eq. ~4!#,whereas the surface roughness is 0.8 mm from Fig. 8.For SF6 the effective compressive depth ds is 1.8 mm,whereas the roughness is 1.15 mm.The strong dependence on the abrasive shape of the
magnitude and gradient in the surface and subsur-face residual compressive stresses may be the reasonfor the variance, as seen in Fig. 1, in the measuredsurface force P0 or constant C reported by variousresearchers, presumably using nominally identicalabrasive sizes that may still exhibit a great variabil-ity in shape.
5. Conclusions
We have examined the Twyman effect as manifestedin three optics manufacturing operations: deter-ministic microgrinding ~bound abrasive ring tools un-der a fixed infeed rate, leading to two-body abrasion!,loose abrasive grinding ~or lapping, with abrasives ofrelatively larger size under fixed nominal pressure,leading to three-body abrasion!, and loose abrasivemicrogrinding ~lapping with abrasives smaller than 3mm in size!. These manufacturing operations leadto significant residual surface compressive stresses,i.e., the Twyman effect.We have shown that in deterministic microgrind-
ing of BK7 ~under typical infeed rates! with boundabrasive diamonds smaller than 40 mm in size, theresulting surface compressive force ~Twyman effect!is smaller than the force in lapping experiments~fixed nominal pressure! where the compressive forceincreases with increasing abrasive size. The corre-lation is reversed at larger abrasive sizes, probablybecause deterministic microgrinding occurs at a fixedinfeed. The Twyman effect is determined by the re-sponse of glass surfaces to applied forces, describingthe interaction of an abrasive grain with the glasssurface in brittle material-removal conditions.The Twyman effect is described by the magnitude
P0 of the surface compressive force ~Newtons permeter! induced by grinding, the depth ds of the com-pressive layer ~micrometers!, and the maximum com-pressive stress s0 ~Newtons per square meter!induced by grinding. The interaction of abrasivegrains with a glass surface is described by depth di ofpenetration of the grain into the surface and the re-sulting surface roughness.The surface compressive force P0 scales with inden-
tation crack size, and the depth ds of the compressivelayer scales with abrasive size and indentation crack
5712 APPLIED OPTICS y Vol. 35, No. 28 y 1 October 1996
size in loose abrasive grinding. These correlationsclearly show that the Twyman effect is due to theextensive cracking of a brittle surface during lapping.The surface compressive stress increases whereas thedepth of the compressive layer diminishes with di-minishing abrasive size.The depth di of the penetration of an abrasive grain
into the glass surface scales with 1y=Hk, indicatingthat the engagement of an abrasive particle with thesurface is similar to the indentation of the surfacewith a sharp indenter.The surface roughness in lapping correlates well
with the depth di of the abrasive grain into the glasssurface or equivalently with 1y=Hk. This correla-tion shows that the near-surface glass plastic prop-erties primarily determine the features describingthe surface geometry in loose abrasive grinding ~lap-ping!.For loose abrasive microgrinding ~abrasives less
than 3 mm in size! of ULE, the surface compressivestress increases to a value near the uniaxial yieldstress of the glass. Similarly, when larger abrasivelapping data for fused silica and quartz are extrapo-lated to abrasives ;1 mm in size, the surface com-pressive stress attains the uniaxial yield stress of thematerial, thus indicating the presence of surface re-sidual stresses approximately equal to the uniaxialyield stress. For the case of ductile-mode grindingthe surface compressive stresses are essentiallybound by the glass yield stress. Similar initial con-clusions have been drawn for Zerodur.A finite-element model was used to study the axi-
symmetric indentation of glass surfaces by acute andobtuse rigid indenters, described by the tip radius ofthe curvature and the angular shape of the conicalindenter. Acute indenters lead to pileup around theperiphery of the contact area, and residual surfacecompressive stresses in excess of the yield stress,with the maximum compression occurring at a sub-surface depth. Obtuse indenters lead to maximumcompressive stresses, approximately equal to theyield stress, occurring at the glass surface. In bothcases there is a significant gradient in the compres-sive stress distribution, which depends on the in-denter shape. The dependence of the surfacecompressive stress distribution on abrasive shapemay explain the variability in the measured-surfaceTwyman forces when abrasives of nominally thesame size are used. Such finite-element simulationsprovide a useful tool for modeling the interaction ofan abrasive grain and a glass surface.
We acknowledge financial support by the Na-tional Science Foundation under Award MSS-8857096 and many insightful discussions with andsources made available by Stephen D. Jacobs fromthe Institute of Optics and the Laboratory for LaserEnergetics at the University of Rochester. We alsoacknowledge insightful and constructive commentsby the reviewers.
References1. F. Twyman, “Polishing of glass surfaces,” in Proceedings of the
2. F. Twyman, Prism and Lens Making ~Hilger &Watts, London,1952!, Chap. 9, p. 318.
3. A. J. Dalladay, “Some measurements of the stresses producedat the surfaces of glass by grinding with loose abrasives,”Trans. Opt. Soc. London 23, 170–173 ~1922!.
4. F. W. Preston, “The structure of abraded glass surfaces,”Trans. Opt. Soc. 23, 141–164 ~1921–1922!.
5. F. Ratajczyk, “Die Abhangigkeit des Twymaneffekts von denScheifbedingungen des optischen Glases,” Feingeratetechnik15, 445–449 ~1966!.
6. F. Ratajczyk, “Die Abhangigkeit des Twymaneffekts von derDicke der abpolierten Schicht der geschliffenen Glasober-flache,” Feingeratetechnik 16, 254–256 ~1967!.
7. E. G. Nikolova, “Review: On the Twyman effect and some ofits applications,” J. Mater. Sci. 20, 1–8 ~1985!.
8. E. G. Nikolova, Bulg. Univ. Ann. Tech. Phys. 18, 21–28 ~1980!~in Bulgarian!.
9. W. J. Rupp, “Twyman effect for ULE,” in Optical Fabricationand Testing Workshop ~Optical Society of America, Washing-ton, D.C., 1987!, pp. 25–30.
10. B. G. Cohen and M. W. Frocht, “X-ray measurement of elasticstrain and annealing in semiconductors,” Solid-State Electron.13, 105–112 ~1970!.
11. V. A. Perevoshchikov and V. D. Skupov, “Accumulation ofdefects in silicon wafers from operation to operation duringmechanical finishing process,” Sov. J. Opt. Technol. 56, 302–305 ~1989!.
12. I. Blech and D. Dang, “Silicon wafer deformation after back-side grinding,” Solid State Technol. 37 ~8!, 74–76 ~1994!.
13. D. Golini and S. D. Jacobs, “Transition between brittle andductile mode in loose abrasive grinding,” in L. R. Baker, P. B.Reid, and G. M. Sanger, eds., Proc. SPIE 1333, 80–91 ~1990!.
14. D. Golini and S. D. Jacobs, “Physics of loose abrasive micro-grinding,” Appl. Opt. 30, 2761–2777 ~1991!.
15. O. Podzimek, “Residual stress and deformation energy underground surfaces of brittle solids,” Ann. CIRP 35, 397–400~1986!.
16. O. Podzimek, “Residual stress and deformation energy underground surfaces of brittle solids,” Technical Report WB-85-16~Twente University of Technology, Enschede, The Nether-lands, 1986!.
17. O. Podzimek, “Deformation energy under optical surfaces,” inHigh Power Lasers: Sources, Laser-Material Interactions,High Excitations, and Fast Dynamics, E. W. Kreutz, A. Quen-zer, and D. Schuoecker, eds., Proc SPIE 801, 221–225 ~1987!.
18. D. Golini and W. Czajkowski, “Microgrinding makes ultra-smooth optics fast,” Laser Focus World 28~7!, 146–150 ~1992!.
19. H. H. Pollicove, D. Golini, and J. Ruckman, “Computer aidedoptics manufacturing,” Optics Photon. News 15–19 ~June1994!.
20. J. Liedes, “Opticam SM update,” in Current Developments inOptical Design and Optical Engineering II, R. E. Fischer andW. J. Smith, eds., Proc. SPIE 1752, 153–157 ~1992!.
21. J. C. Lambropoulos, P. D. Funkenbusch, D. J. Quesnel, S. M.Gracewski, and R. F. Gans, “Mechanics andmaterials issues inoptics manufacturing,” in Proceedings of the Ninth AnnualMeeting of the American Society for Precision Engineering~ASPE, Raleigh, N.C., 1992!, pp. 370–373.
23. R. F. Cook and G. M. Pharr, “Direct observation and analysisof indentation cracking in glasses and ceramics,” J. Am. Ce-ram. Soc. 73, 787–817 ~1990!.
24. M. Sakai and R. C. Bradt, “Fracture toughness testing of brit-tle materials,” Int. Mater. Rev. 38, 53–78 ~1993!.
25. A. G. Evans, “Fracture toughness: the role of indentationtechniques,” in Fracture Mechanics Applied to Brittle Materi-als, S. W. Freiman, ed., ASTM STP 678 ~American Society forTesting and Materials, Philadelphia, Pa., 1979!, pp. 112–135.
26. M. Buijs and K. Korpel-Van Houten, “A model for lapping ofglass,” J. Mater. Sci. 28, 3014–3020 ~1993!.
27. S. M. Wiederhorn, H. Johnson, A. M. Diness, and A. H. Heuer,“Fracture of glass in vacuum,” J. Am. Ceram. Soc. 57, 337–341~1974!.
28. S.Wiederhorn andD. E. Roberts, “Fracturemechanics study ofskylab windows,” report 10892 ~NASA Manned SpacecraftCenter, Structures and Mechanics Division, PR1-168-022,T-5330A, National Bureau of Standards, U.S. Department ofCommerce, Washington, D.C., 1972!.
29. T. S. Izumitani, “Lapping hardness of optical glass,” HoyaTechnical Report HGW-O-7E ~Hoya Glass Works, Tokyo, 20Feb. 1971!.
30. T. S. Izumitani and I. Suzuki, “Indentation hardness and lap-ping hardness of optical glass,” Glass Technol. 14, 35–41~1973!.
31. T. S. Izumitani, Optical Glass ~American Institute of Physics,New York, 1986!, Chap. 4.
32. M. Cumbo, “Chemo-mechanical interactions in optical polish-ing,” Ph.D. dissertation ~The Institute of Optics, University ofRochester, Rochester, N.Y., 1993!.
33. I. J. McColm, Ceramic Hardness ~Plenum, New York, 1990!.34. K. Phillips, G. M. Crimes, and T. R. Wilshaw, “On the mech-
anism of material removal by free abrasive grinding of glassand fused silica,” Wear 41, 327–350 ~1977!.
35. I. M. Androsov, S. N. Dub, and V. P. Maslov, “Unique featuresassociated with using the indentation method for determiningthe crack resistance of brittle materials,” Sov. J. Opt. Technol.56, 691–693 ~1989!.
36. S. S. Chiang, D. B. Marshall, and A. G. Evans, “The responseof solids to elasticyplastic indentation, I. Stresses and resid-ual stresses,” J. Appl. Phys. 53, 298–311 ~1982!.
37. Hibbitt, Karlsson and Sorensen, Inc., ABAQUS, Version 5.4~Providence, R.I., 1994!.
38. W. J. Rupp, “Mechanism of the diamond lapping process,”Appl. Opt. 13, 1264–1269 ~1974!.
39. J. C. Lambropoulos, T. Fang, P. D. Funkenbusch, S. D. Jacobs,M. J. Cumbo, andD. Golini, “Surfacemicroroughness of opticalglasses under deterministic microgrinding,” Appl. Opt. 35,4448–4462 ~1996!.
40. B. R. Lawn, T. Jensen, and A. Arora, “Brittleness as an inden-tation size effect,” J. Mater. Sci. 11, 573–575 ~1975!.
41. H. H. Karow, Fabrication Methods for Precision Optics ~Wiley,New York, 1993!, Chap. 1, pp. 35–50.
42. R. Hill, The Mathematical Theory of Plasticity ~Oxford U.Press, New York, 1950!, Chap. 5, pp. 97–105.
43. C. R. Kurkjian, G. W. Kammlott, and M. M. Chaudhri, “In-dentation behavior of soda-lime silica glass, fused silica, andsingle-crystal quartz at liquid nitrogen temperature,” J. Am.Ceram. Soc. 78, 737–744 ~1995!.
44. P. Shore, P. McKeown, S. Impey, and D. Stephenson, “Surfaceand near surface conditions of “ductile” mode ground Zerodur,”in International Progress in Precision Engineering, in Proceed-ings of the Eighth International Precision Engineering Semi-nar ~Elsevier, New York, 1995!, pp. 365–368.
45. B. R. Lawn, A. G. Evans, and D. B. Marshall, “Elasticyplasticindentation damage in ceramics: the medianyradial cracksystem,” J. Am. Ceram. Soc. 63, 574–581 ~1980!.
46. S. S. Chiang, D. B. Marshall, and A. G. Evans, “The responseof solids to elasticyplastic indentation, II. Fracture initia-tion,” J. Appl. Phys. 53, 312–317 ~1982!.
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