Nature of the Deformation and Flow of Metals at and Near the Interface During Abrasion, and its Relation to the Friction BY D. S. LIN AND H. WILMAN Applied Physics and Chemistry of Surfaces Laboratory, Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, London, S. W.7. Received 12th June, 1972 The direction of flow of various metals (Pb, Al, Au, Ag, 62.1 At. % Au-Ag alloy, Cu, Ni, 25.7 At. % Ni-Cu, Mo, W, U, Ti, Mg, and Zn) past the faces of a ploughing Vickers diamond pyramid of dihedral angle 136" is shown in ploughing (a) parallel to the side of the initial square indentation, (b) parallel to the diagonal, (c) in an intermediate asymmetrical direction. In all these cases, with this obtuse indenter, the metal flow at the interface is close to the direction lying in a plane through the ploughing direction, normal to the surface of the specimen.Approximate theoretical expressions are derived for the friction coefficient p for such conditions. The form of the flow past an indenter is also indicated by observations on the ploughing of macro- scopic metal pyramids and cones in Plasticene built from superposed layers of different colours. Observations are also made on the widths of grooves ploughed by the diamond on various metals, compared with the width of the static indentation. These are considered in relation to the theoretical expressions and the factors which may affect the groove development. To elucidate the frictional behaviour of metals sliding against multi-asperity surfaces, as in typical abrasion, it is desirable to investigate the action of single indenters of known shape.The experiments below explore the flow of many metals at and near the interface with a ploughing Vickers diamond pyramid (cf. also Buttery and Archard for steels). only about 10 % of the total groove volume ploughed out is removed as wear (similarly for Sic papers 5 ; cf. also Stroud and Wilman 6), thus the friction coefficient p is more characteristic of the shape and orientation of the non-wear-producing asperities. Goddard and Wilmar~,~ in a theory of abrasive friction and wear, derived expressions for the p of not too obtuse cones and pyramids with axis vertical, assuming that flow of metal round the indenter is practically in a horizontal plane. However, for the other extreme of obtuse indenters, such as a Vickers diamond square pyramid of dihedral angle 136" between opposite faces, the experiments below show the flow past the front face(s) is in or near a vertical plane parallel to the sliding direction, and expressions for p for this case are derived.Emery particles appear to be mainly relatively o b t ~ s e , ~ * ~ * ~ * ~ thus a ploughing Vickers diamond gives a useful and relevant comparison. preferred " smooth-cut " steel files as a standard abrasive surface for quantitative studies ; these are coarse enough to avoid clogging, and on them metals give p values of a similar order to those on coarse emery papers, and thus seem to involve many obtuse asperities. p for metals in relation to hardness and structure were broadly similar to the p on emery papers, and remarkably similar to the adhesion-coefficient data of Sikorski.lo Studies were also made on the p, wear rate, hardness and surface 46 In abrasion of metals on bonded abrasives such as emery papers 3 9 To avoid clogging and abrasive-pick-up effects, Alison and WilmanD .S . LIN AND H . WILMAN 47 structure in abrasion of Zn-Cu alloys,l Ni-Cu 1 g and Au-Cu alloys,l and the age-hardening alloy A 1 4 wt. % Cu,14* l 5 and on the relation of p to the state of work-hardening of metals.16 EXPERIMENTAL The purity and sources of the metals and alloys used are given in table 1. Prior to measurement of the static microhardness and the groove widths, the specimen surfaces were first fully abraded unidirectionally to work-harden their surface region to the maximum extent, as in normal abrasion conditions.This was done by abrasion on carefully cleaned and dried “ smooth-cut ” steel files,g for about 400 cm under hand pressure N 1 kg load. They were then smoothed by about 1000 cm abrasion in the same direction on emery papers from grade 0 (35 pm particle dim.) to 0000 (5 pm). In both processes the surfaces were well wetted with propyl alcohol to minimize heating and oxidation and also pick-up of emery particles. The microhardness tests and experiments on groove formation in all cases followed immediately the above initial surface preparation. The Vickers diamond-indentation microhardness H of the surface region was measured using a load in grams equal to +H, TABLE 1 .-GROOVE WIDTHS (mm) AND HORIZONTAL PROJECTIONS OF CONTACT AREAS (mm2), AT x 610 MAGNIFICATION, FROM A VICKERS DIAMOND PYRAMID UNDER LOAD +H GRAMS, PLOUGHING AND INDENTING IN FULLY ABRADED METAL AND ALLOY SURFACES* metal/alloy W f structure b.c.c.H 509 bi bm (D)25.1 (S)16.3 (D)24.3 (S)16.6 (D)23.9 (S)16.7 (D)25.2 (S) 1 7.8 (D)26.3 (S)18.3 (D)27.3 (S)18.3 (D)26.5 (S)18.5 (D)27.3 (S)18.8 (D)18.4 (S) 1 5.1 b Astat. slp from eqn (3) 1.61 1.36 1.07 1.74 2.00 1.91 1.65 1.74 1.16 164 164 181 173 182 190 198 203 196 (D)20.3 (S)13.1 (D)20.6 (S) 13.8 (D)20.7 (S)13.6 (D)21.6 (S) 1 3.8 (D)21.7 (S) 1 3.6 (D)21.7 (S) 13.8 (D)22.1 (S)14.0 (D)23.1 (S)13.9 (D)22.1 (S)15.5 (D)24.8 (S)15.8 (D)24.1 (S)15.4 (D)25.0 (S)15.4 (D)24.8 (S)16.4 (D)27.2 (S)17.4 (D)27.4 (S) 1 7.3 (D)25.9 (S)17.4 (D)27.4 (S)18.0 (D)21 .O (S)16.5 (i)lll 164 ($237 (i)100 154 ($219 (i)103 148.5 (ii)227 (i)124 165 (ii)260 (i)159 193 (ii)3 1 1 (i)176 204 ($326 (i)138 190 (ii)290 (i)141 200 (ii)302 (i)109 115 (ii)247 U 2 b.c.c.390 MO Ni b.c.c. f.c.c. f.c.c. f.c.c. f.c.c. f.c.c. 267 182 132 cu5 Ag Au A1 112 101 36 5 Pb f.c.c. 62.1 At % Au/Ag lo f.c.c. 125 (D)20.9 (S) 1 3.7 (D)25.8 (S)16.7 (D)26.0 (S)15.6 (i)102 178 (ii)224 177 1.14 25.7 At % Ni/Cu l1 f.c.c. c.p.h. c.p.h. c.p.h. 166 233 52 46 (D)21.1 (S)13.9 (D)22.1 (S)13.4 (D)19.9 (S)13.6 (D)19.3 (S)13.4 (D)27.4 (S)16.7 (D)24.4 (S)15.4 (D)22.2 (S)15.6 (D)19.7 (S) 1 5.1 (D)28.1 (S)16.2 (D)29.0 (S)20.4 (D)22.6 (S) 1 5.1 (D)21.1 (S)15.0 (i)108 185 (ii)240 (i)232 368 (ii)442 (i) 84 139 (ii) 195 (i) 95 124 ($203 178 161 182 181 1.42 3 .O 0.38 0.55 Ti l2 Mg l3 Zn l4 [References to Table I overleaf48 FLOW OF METALS AT INTERFACES DURING ABRASION where His in kg/mm2 and using where possible values found by Alison and W i l r n a ~ ~ , ~ thus indentations of the same size were made in the different metals. The load used for the formation of the grooves by the diamond was then taken as, in grams, half the H value just measured.RESULTS 1. MICROSCOPIC OBSERVATIONS OF THE GROOVE MORPHOLOGY I N METALS AND ALLOYS THE INITIAL BROADENING OF THE GROOVE 1 . 1 Generally there is a very slight narrowing just ahead of the initial indentation, cf. fig. 1. On starting to slide, the diamond has to over-ride the initial adjacent piled-up metal (note slightly curving sides of the initial indentation) and becomes slightly lifted, and also there is less displaced metal at the corners of the initial indentation.The indenter loses contact at the back face(s) and the depth of indentation increases until the reaction of the metal on the remaining faces again balances the load. After moving about twice the diameter of the static indentation, a dynamic equilibrium is reached in which the pile-up of metal at the front face is displaced continuously round to the sides of the indenter, giving the well-known ridge of piled-up metal at the sides of the groove, and a constant maximum groove width. 1.2 THE TRANSVERSE VERTICAL CROSS-SECTION OF THE GROOVES By oblique illumination at 45", the groove profile in transverse section can be seen Fig. 1, 5(a) from the displacements of the shadow lines cast by fine graticule lines.and 6(a) show considerable pile-up of metal at the lateral groove edges. References to Table I ] * Purity 99.99 % except 99.98 for Pb, 99.95 for Mg and somewhat less for Au. H is the Vickers microhardness in kg/mm2 of the fully abraded metal surface measured at a load fi 3H (kg/mm2) in g. All subsequent grooves ploughed using a load = +H (kg/mm2) in g, bi, bm, bf are the the groove widths x 610 in inm at the initial indentation, halfway along the groove and at the front end of the groove respectively. (D), (S) denote ploughing direction of the diamond indenter along the square diagonal and parallel to the square side of the indeutation respectively. As is the projected horizontal contact area x 6102 in mm2 when the diamond indenter is ploughing in a direction parallel to the square side of the indentation.(i) A: denotes the projected contact area of the front face of the indentations including the contact with the pile-up in front of the diamond indenter. (ii) A: as (i) plus the projected contact area of the side faces. AD is the projected horizontal contact area x 6102 in mm2 when the diamond indenter is ploughing in a direction along a diagonal of the square indentation. Asat. is the projected horizontal contact area x 6102 in mm2 of a static square indentation made in the vicinity of the grooves at the same load and during the same period of time. l From Murex Ltd., - rod, From Metallography Division, A.E.R.E., Hanvell-commercial purity. From Murex Ltd.-rod. From The International Nickel Co.(Mond) Ltd.-cast from highest-purity nickel pellets in vacuo. From the British Aluminium Co. Ltd.-" super-purity " A1 rod. From Hopkin & Williams Ltd.-Analar. lo From Johnson, Matthey & Co Ltd.- From the International Nickel Co. l2 From Metallography Division, A.E.R.E., Harwell-commercial purity Ti. l3 From Magnesium Elektron Co. Ltd. l4 From Hopkin & Williams Ltd.-Analar, re-cast. 5-7 From Johnson, Matthey & Co. Ltd.-Spec pure.(b) FIG. 1.-Micrographs showing grooves made by Vickers diamond pyramid ploughing ]I to the square side of the indentation, and I/ to the initial fine surfacing grooves, on (a) Mo (x 425), H = 267 kg/ mm2; (b) Au (x S O ) , H = 101 kg/mm2. In (a) the vertical lines are oblique shadows of fine graticule lines.( a ) (b) FIG. 3.-(a) Front end of groove ploughed /I to square side, 1 to the initial surfacing giooves, in Mo (x 1450); (6) as (a) but 11 to initial surfacing grooves (x 1150)-note several initial surfacing fine grooves passing down the front face, approx. I1 to groove, in this plan view. (b) FIG. 4.-Micrographs showing profile of front end pile-up of (a) a groove in Ag ( x 500) ploughed 11 to square side ; (b) two grooves in 25.7 At. % Ni-Cu alloy ( x 215). [To face page 48(4 (6) FIG. 5.-Micrographs of front end of grooves in Mo, ploughed I/ to the square diagonal, (a) x 820, (b) x 610; note " twin " pile-ups ahead, and several initial fine surfacing grooves continuing down the front face at -10" to groove direction in this plan view. (6) (c) FIG.6.-Micrographs of grooves ploughed in Au in a direction at 25" to square diagonal, and /I to initial fine surfacing grooves : (a) x 590, (6) x 700, (c) x 680. Note asymmetric pile-up ahead, " chatter " marks in the groove, and oblique shadow lines showing groove profile in (a). (a) (6) FIG. 7.-Micrographs of front end of grooves ploughed /I to square side in (a) 25.7 At. % Ni-Cu ( x 1100), (b) 62.1 At. % Au-Ag ( x 1100). Note irregular shape of front face indicating strong adhesion to diamond.(a) (6) FIG. 8.-Micrographs of (a) front end of groove ploughed j / to the square diagonal, in Ti ( x 600). Note the two " chips " prqjecting from the surface ahead of the groove, (6) Ti adhering strongly to the diamond. FIG. 9.-Micrograph of front part of a groove ploughed 11 to the square side on U.Note cracks and fractures in the groove surface. FIG. lO(6) FIG. 10 (a) FIG. lO.-(a) Plan view ( x 1 .l) and (6) longitudinal median vertical section ( x 1.7) through the groove of (a), which was ploughed by a 90" square pyramid of brass, I1 to the square side, in a stratified Plasticene block. Note front pile-up and lateral " flaking " at edges of groove in (a) ; and initial forward and upward, then downward flow in (6).FIG. 11 (a) FIG. 11 (c) FIG. 11.-As fig. 10 but ploughed ll to square diagonal ; (a) plan ( x 1.0)-note forward flow in the twin pile-ups ahead of the front of the groove; (6) transverse vertical section ( x 1.7) through tip of indenter at front end of groove ; (c) longitudinal median vertical section ( x 1.7) showing profile of pile-up, and subsurface flow directions.FIG. 1 1 (b) FIG. 12 (a) FIG. 13.-Plan view of the front end o f a groove ploughed in stratified Plasticene by a cone. FIG. 12 (6) FIG. 12.-(a) Oblique view ( x 1.7) on to surface of a Plasticene block, showing chip formation by an acute- angled square pyramid indenter, ploughing /I to the square side. (6) plan view of a groove made by an acute-angled square pyramid plough- ing in Plasticene \ / to the square di- agonal ( x 0.2). Note twin " chips ", cf. fig. 8(a).FIG. 8.-Stereoscan micrograph showing a FIG. 9.-Stereoscan micrograph of the fracture sheath of nylon surrounding a pulled out fibre. surface of a 0.15 Vf surface treated carbon fibre showing little pull out of fibres. FIG. 10.-Stereoscan micrograph of fracture surfaces of a 0.15 Vf carbon fibre/nylon com- posite showing the effect of matrix morphology on fracture (note fibre, lower left).FIG. 11 .-Stereoscan micrograph of the fracture surface of a 0.1 5 Vf uncoated carbon fibre/nylon composite showing high degree of pull out.D. S . LIN AND H . WILMAN 49 The volume of metal displaced from below the original specimen surface level is seen to be approximately matched by the volume of metal left behind at the groove edges, as is also shown by the groove profiles recorded by the Rank-Taylor-Hobson " Talysurf " apparatus, as in fig. 2. FIG. 2.-" Talysurf " trace across the groove of fig. l(a). 1.3 THE LONGITUDINAL VERTICAL CROSS-SECTION OF THE FRONT END OF THE GROOVES Fig. 4 shows typical profiles of the pile-up of metal at the front end of the grooves, in Ag and in 25.7 At.% Ni-Cu alloy, with the indenter sliding parallel to the square side. Fig. 3(a) and (b) show the appearance of the front pile-up, in plan view, and indicate the tendency for some concentric wrinkling to occur. Fig. 5(a) and (b) show the groove form for the diamond sliding along the square diagonal direction-note less pile-up at the front edge than towards the sides; and fig. 6 shows the groove form for an indenter at an oblique angle of sliding, with the square diagonal at 25" to the direction of sliding, giving asymmetric pile-up. 1.4 OBSERVATIONS O N THE FLOW OF METAL DURING PLOUGHING The initial fine surfacing abrasion grooves give a useful indication of the direction of flow of the displaced surface metal during ploughing.A continuation of the surface abrasion grooves on to the front contact face is clearly visible in micrographs such as fig. 3(a), 3(b), 5(a) and 5(b). In fig. 3(a), the lines are curves concentric with the upper edge of the front pile-up; in fig. 3(b) the lines pass down the front face parallel to the sliding direction; and in fig. 5(a) and 5(b) the lines on the front faces are at about 10" diverging outwards from the sliding direction, then parallel to it at the side faces of the groove. 2. MACROSCOPIC OBSERVATIONS ON GROOVE FORMATION I N STRATIFIED PLASTICENE To examine the sub-surface flow, which is not revealed by the microscopic exam- ination of grooves on metals, experiments were made on stratified Plasticene blocks about 3 in.thick, the coloured strata being about 1/16 in. thick. Plasticene has a stress-strain relation resembling that of metals and has been used to investigate the flow in various types of def0rmation.l' Brass indenters were used, machined to a fine finish and polished smooth on emery to grade 4/0 followed by metal polish. The square pyramidal indenters were50 FLOW OF METALS AT INTERFACES D U R I N G ABRASION of dihedral angle 40" and 90" between opposite pyramid faces, and the cone had a semiapical angle of 30". These were clamped at a fixed level in a milling machine, and the Plasticene slabs, held between steel blocks on a steel base, were raised until the indenter penetrated about 5/16 in. and then traversed horizontally so as to plough a groove at about 3 in./s.Fig. lO(a) shows a plan view of a groove ploughed by the 90" square pyramid sliding parallel to the side of the initial square indentation (cf. fig. 1 and 3). Note the similarity between the " flaking " effect at the lateral sides of the groove with that in the grooves made by the diamond in metals like gold, fig. l(b). Fig. 10(b) shows a median longitudinal section of the groove of fig. lO(a). It is seen that some of the surface and sub-surface material is first pushed up and forward by the front indenter face before flowing down past the indenter and along the groove sides. Fig. 1 l(a), (b) and (c) similarly show a plan, a view of the front end (and transverse section through the apex) from along the groove, and a median longitudinal section, of a groove ploughed by the 90" square pyramid in a direction along a diagonal of the square indentation.Note in this case the lower pile-up at the front edge than towards the sides of the two front faces, as is seen in fig. 5(a), (b), and fig. 1 l(c). Fig. 12(a) is with the acute-angled square pyramid (dihedral angle 40") sliding along the square side, and fig. 12(b) along the square diagonal direction. Chip formation is seen (cf. Avient, Goddard and Wilman and Sedriks and Mulhearn 18), with two symmetrical chips in the case of sliding along the square diagonal (fig. 12(b)). Fig. 13 illustrates the flow pattern of the sub-surface material for ploughing by the cone of semiapical angle 30" (plan view). Note the practically horizontal flow round this acute cone.3. GROOVE MEASUREMENTS I N P U R E METALS A N D ALLOYS Table 1 shows the mean side of the square indentations measured at the beginning of the grooves made with the diamond pyramid sliding parallel to a square side. The initial indentations at the beginning of the groove agree satisfactorily with static indentations, such as those made during the microhardness measurements. The Vickers hardness is given by H = 1854 W/d2 (kg/mm2), where W is the load applied, in g, and d is the diagonal of the square indentation in pm. Thus when W = +H (Hin kg/mm2), in g, d = 0.030 45 mm. The diagonal length on the projected image, at the standard magnification used, x 610, should thus be 18.6 mm for all the metals, and the side of the square indentation 13.15 mm. It is seen from table 1 that apart from Pb, which is known to recrystallize at room temperature,lg the initial groove breadths bi(S) for all the other metals and the two alloys, agree to within 10 %.The slight discrepancies observed, especially in the case of the softer metals, may well be due to the initial vibrations on turning the micrometer screw to start the ploughing. For the diagonal ploughing direction, however, the initial groove breadths bi(D) are all high, ranging from 19.3 to 23.1 mm, instead of the calculated 18.6 mm in the image. This may be associated with the more uneven sliding in this direction; for example note the " chatter " marks in fig. 5 and 6. The median groove breadth b, and front-end breadth 6, show good agreement for all the metals and alloys except for a few metals such as Ti and Pb, where b, is notice- ably less than b,, probably associated with strong adhesion of metal to the indenter faces during ploughing (cf.fig. 8(b)), causing a large drag or tension on the metal behind the advancing indenter (cf. also fig. 10(a) at the edges of the groove). Under the heading of As in table 1, for the horizontal projected area of the frontD . S . LIN AND H . WILMAN 51 contact, when sliding is parallel to the square side, (i) A: is the area for the front face only, (ii) A: the area including also the two side faces, which is the expected operative area against which the metal flow pressure acts. It is seen that there is an appreciably wide spread of values of A,(i) and (ii) for the metals investigated.This spread is noticeably less for the projected contact area AD when the ploughing is along the square diagonal. It is also seen from table 1 that except for the softer metals the measured horizontal projected static-indentation areas A,,,,. are all in reasonable agreement with the calculated value of (13.15)2 = 173 mm2. All the projected horizontal areas were estimated by tracing on transparent mm-squared paper the curved periphery of the front contact area and that of the side faces, or the square indentation, as seen on the microscope projection screen. DISCUSSION 1. THE CONTACT AREAS AT THE FRONT OF THE GROOVES DURING PLOUGHING OF THE VICKERS DIAMOND When an abrasive asperity or an indenter such as a Vickers diamond pyramid is under load W against a metal, the area of the static indentation is given by W/p, where p is the maximum flow pressure of the metal.20* 21 When the loaded indenter is caused to plough by applying a sufficient force F to it parallel to the metal surface, there is loss of contact at the back face of the indenter, and the remaining faces must support the load, thus the indentation becomes deeper, and the groove width increases, as is observed, e.g., in fig.1. The estimates in table 1 of the horizontal projected areas of contact AD made by the indenter when sliding parallel to the square diagonal agree quite well with the pro- jected area Astat. of the static indentation, except for Pb and the hexagonal metals Zn, Mg, Ti. A$ is mostly considerably larger than Astat., by a factor too large to be explained by the presence of a few furrows passing down the interface, making the real area of contact less than the nominal apparent contact area (in any case this would also apply for AD which agrees well).Also, if the dynamic flow pressure p' during ploughing differs from the static flow pressure p , it would be expected to be larger thanp,20 and this wouldreduce the observed A? relative to Astat,, which is in the sense opposite to the observations in table 1. &T- PLAN P L A N FIG. 14. In the case of the relatively obtuse Vickers pyramid, as seen above from micro- graphs such as fig. 3, 5 and 6, the flow of metal down the front face of the indenter is practically in a vertical plane parallel to the sliding direction when this is along the square side, and not far from the sliding direction (only about 10" in the horizontal projection, i.e., plan view as in fig.5) when this is along the square diagonal, or at intermediate azimuths. For such flow, the vertical component (downwards in this52 FLOW OF METALS AT INTERFACES DURING ABRASION case) of the adhesive shear force (s/unit area) of the metal along the front face (see fig. 14) must be added to the load, and the total then balanced by the product of p and the horizontal projected area, in a way similar to that applied by Sedriks and Mulhearn l 8 in the case of a ploughing machine tool. We may consider the following directions of ploughing. (i) PLOUGHING PARALLEL TO THE SQUARE SIDE W = p(A, +2A2) sin 8’-sA, cos 8’ = pA;-sA; cot 8’ A; = w / { p - ~ ( A ; / A ; ) cot el) = ( w / p ) / { 1 - ( s / p ) ( ~ ; / A ; ) cot el] sip = (A;-- A,,,,.)/A; cot 81, (1) ( 2 ) (3) thus, where A , = area of contact on the front face, A2 = area of contact at each side face, A: = horizontal projected area of the whole area of contact = ( A , + 2A2) sin Of, A; = horizontal projection of Al, = A l sin Of, and 8’ is the angle between the pyra- mid axis and the bisecting line of a face.Thus, A{ involves not only (s/p) and 8’ but also the ratio (&/A;) which has a mean value of 0.47 from the data in table 1. Resolving the forces horizontally, and neglecting any slight deviation of the flow from a vertical plane, the coefficient of friction p is given as the resultant horizontal frictional force I: divided by W (given by eqn (1)) :- F = 2sA2 +sA, sin 8’ +PA1 cos 8’ (4) ( 5 ) = s(2A2 + A , ) +sAl(sin 8’ - 1) +PA1 cos 8’ Thus, p involves the ratio (s/p) and Of, and also the ratio &/A{.(ii) PLOUGHING PARALLEL TO THE SQUARE DIAGONAL W = 2pA, sin 8’ -2sA1 cos 8 = pAD-sAD cos O/sin 8’ A D = (W/P)/{l - (s/P)(cos 8/sin of)), (7) thus, (8) where 8 is the angle between the pyramid axis and an edge of the square pyramid, thus tan 8 = ,/?tan 8’. In this case, AD does not involve any ratio of front contact area to side contact area. Also, (9) which again only involves (s/p) and 8‘. p = (COS 8‘ cos 45” + (s/p) sin e]/(sin 8‘ - (s/p) cos el, (iii) PLOUGHING I N AN INTERMEDIATE DIRECTION AT ANGLE 4 TO THE SQUARE SIDE If the areas of contact on the two front faces are A , and A2, W = p(A1 +A2) sin 8’-s(A, cos <, +A2 cos c2) F = s(A, sin cl +A2 sin c2)+p sin 8’(A1 cos 4 + A 2 sin (6) = pAI-s(Al cos 5 1 +A2 cos 5 2 )D .S . LIN AND H . WILMAN 53 where c, and c, are given by tan rl = tan B'/cos 4 ; tan C2 = tan B'/sin 4, and A , = ( A , +A,) sin 8' = horizontal projected area of contact. In the above equations it might appear that (s/p)-O.4 to 0.6, since the friction coefficient of a smooth diamond surface on various metals at room temperature in air was found to be of this order.22* 23 However, if we use the data in table 1 for A:, A; and Astat. in eqn (3), we obtain values for s / p given in the last column of table 1, the mean being about 1.7 for the (fully-work-hardened) cubic metals, if we exclude Pb and U as being extremes of softness and brittleness, respectively.This high s / p may possibly indicate that shear occurs in the metal rather than at the true interface. Fig. 15 and 16 shows the p values as a function of 8, to be expected from eqn (6) and (9), for ploughing square pyramids, for various s / p values and taking &/A," = 0.47 as the average from the data of table 1. Since these equations only apply for sufficiently obtuse square pyramids, the curves calculated from the expressions of Goddard and Wilman for acute square pyramids for which practically horizontal flow round the indenter occurs (if there is no " chip " formation), are also shown in fig. 15 and 16. In the intermediate region of 8, transitional values can be expected as 8 FIG. 15.-The relation between p and the angle 8 between the square pyramid axis and an edge, for various s / p values and taking &/A; = 0.47 (cf.table 1) ; ploughing I[ to square side. The curves through points El are calculated from eqn (6) assuming flow in a vertical plane round the obtuse indenter ; the other curves are calculated from Goddard and Wilman's ' expression assuming hori- zontal flow for acute pyramids. Curves . . . . are the suggested transition.54 FLOW OF METALS AT INTERFACES DURING ABRASION indicated by the dotted curves, corresponding to more or less inclined flow. Although we have not as yet estimated the p experimentally, it appears that if s/p- 1.7 a value of -3-4 can be expected for the ploughing Vickers diamond.* 0 6 5 4 P 3 2 I I0 20 30 do ;o do Yo' 8'0 goo 0 . 5 There are, nevertheless, uncertainties such as whether the effective flow pressure may be less for the piled-up metal just ahead of the front face, than for deeper metal more surrounded by the mass of the specimen.If much temperature rise of the metal at and near the indenter surface occurs during ploughing, this could also result in a lower p for the metal there than that under static conditions, at room temperature. 2. GROOVE MORPHOLOGY I N RELATION TO HARDNESS AND CRYSTAL STRUCTURE Fig. 8(a) shows the front contact face of a groove ploughed in Ti. This is a hexagonal metal of quite high hardness, about 233 kg/mm2, so clean-cut grooves of the type seen in metals like Mo and Cu would be expected. In fact, the grooves show a considerable amount of adhesion effects, and fig. 8(6) also shows the strong adhesion of Ti to the diamond surface.Lead, an extremely soft rnetal, surprisingly did not show similar strong adhesion (cf. Sedriks and Mulhearn 18). * see Discussion, p. 60: p 2 0.45; contact area < apparent area.D . S . LIN AND H . WILMAN 55 The alloys investigated also showed some interesting features, particularly the uneven nature of the contact at the front face as seen in fig. 7(a) and (b). Metals harder than about 250 kg/mm2 show some cracks and fractures at the side of the grooves, as in uranium, fig. 9, such metals being increasingly brittle in nature. The above experimental work was carried out under a contract between the National Engineering Laboratory (Ministry of Technology) and the Imperial College of Science and Technology, London, SW.7. D. S. Lin, Ph.0. Thesis (University of London, 1967). T. C. Buttery and J. F. Archard, Proc. Inst. Mech. 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