Chapter Slice–Push Ratio Oblique Cutting and Curved Blades, Scissors, Guillotining and Drilling Contents 5.1  Introduction 5.2  Floppy Materials 5.3  Offcut Formed in Shear by Oblique Tool 5.4  Guillotining Edges 5.5  Drills, Augers and Pencil Sharpeners 111 113 119 123 134 5.1  Introduction In the kitchen or at the dinner table, cutting may be performed simply by ‘pressing down’ with a knife It is common experience, though, that even with the sharpest knives, cutting seems to be easier when sideways motion as well as vertical motion is incorporated in the cutting action By easier, we mean that the vertical force is reduced Even when just ‘pushing down’, without sideways action, angling the cutting blade to the direction of cut is beneficial in some way, for example by giving a better surface finish Why is there this difference in cutting forces for angled blades, and for blades having sideways slicing motion as well as the normal pushing motion? In the case of a loaf of bread, it might be thought that this is to with the serrated teeth found on many breadknives, but the phenomenon is just as evident with plain smooth edges on blades The effect seems disproportionate in that the pressing force is reduced quite markedly by even the smallest sideways motion of the cutting blade The greater the sliding velocity relative to the pressing velocity the greater the reduction in the pressing force Captives whose wrists are tied by rope find it necessary to rub their bindings back and forth as well as press hard against the best available edge to cut through their bindings In orthogonal cutting (Chapter 3) the cutting edge is always at right angles across the workpiece When a straight blade is angled to the direction of motion of the workpiece, it is called oblique cutting All the different types of chip described in Chapter are found in oblique cutting The inclination of the cutting edge need not be constant: it changes as the straight blades of scissors are closed, and in devices with curved blades such as the scythe the inclination continuously changes Metal-cutting tools often have two cutting edges, both of which are angled to the direction of cutting, and in round-nosed tools the inclination continuously varies (Chapter 6) We shall discover that the slice–push ratio  given by (blade displacement or velocity parallel to the cutting edge/blade displacement or velocity perpendicular to the cutting edge) is important in making cutting seem easier, and that greater  gives easier cutting As shown in Figure 5-1, a slice–push ratio is obtained when (a) an orthogonal blade is driven sideways as well as down; (b) driven straight down but at an angle, since the cutting feed velocity has components along and across the inclined blade; and (c) when an angled tool fed into the workpiece with feed f has its own independent motion parallel to the Copyright © 2009 Elsevier Ltd All rights reserved 111 112 The Science and Engineering of Cutting ν h f i A B f h i C Figure 5-1  ‘Slice–push’  produced by various blade motions and orientations: (A) an orthogonal blade with displacements (velocities) both into the workpiece (v) and across the workpiece (h); (B) an oblique blade, inclined at angle i to the crossways direction of the workpiece, moving into the workpiece with feed displacement (velocity) f; the blade itself has no motion along its edge; (C) as in (B) but now where the blade has velocity h along its edge as well cutting edge Thin samples fed into a rotating disc cutter along the centreline are an example of case (a), where  is given by (wheel peripheral speed/workpiece feed speed) Feeding a sample either above or below the centreline into a stationary wheel is case (b); when the cutting disc rotates as well, it is case (c) where different  are given since the cutting edge of the disc is both inclined to the feed direction and has its own velocity The behaviour in case (c) depends on whether the cut is taken above or below the centreline as, in the one case, the edge speed augments the geometrical effect of the inclined blade, and in the other it subtracts from the geometrically induced  Why the cutting force is reduced when there is slice–push can be explained using energy arguments It is not surprising that if energy is put ‘sideways’ into the system, less energy and hence a smaller force will be required in the vertical direction But it is not as simple as that because, as we shall show, a non-linear coupling occurs between the two forces which causes the vertical force to drop markedly as soon as the slightest horizontal motion is introduced Reduction of forces by slice–push produces better surfaces whatever the material In the laboratory, the best sorts of junctions with fibre-optic cables and scintillators are obtained when a slicing cut is made with a warm razor blade; there are similar proprietary devices Lower tractions across a cut surface reduce the tendency for components in the microstructure to separate (fat from meat in bacon slicing) Cutting with slice–push is the only way that some materials 113 Slice–Push Ratio  can be cut easily For example, attempts to cut highly deformable soft foams by pressing down alone are rarely successful, but slicing with an inclined blade readily cuts such a material Thus it is difficult to cut foam with nail clippers, but cutting is achieved with scissors (Bonser, 2005) If, in addition, the foam can be prestressed in bending across where a cut is to be made, cutting is even easier Slice–push is the reason why one’s tongue is sometimes cut when licking envelopes The reduction in forces when cutting thin sheets with slice–push stops prows and buckles forming ahead of the blade which stop or interfere with the process Cutting with a steeply inclined blunt blade may be possible when, at smaller angles, cutting fails Spades and shovels on the continent of Europe often have pointed blades, along which there will be slice–push, in contrast to the square-ended tools found in the UK They also have long handles and are operated differently 5.2  Floppy Materials 5.2.1  Frictionless thin blade In Figure 5-1(A), a thin knife (negligible wedge angle) cuts a block of material of width w The knife blade is long enough always to overhang the workpiece (or it is a ‘band blade’, like a band saw but having no teeth) The blade is orthogonal to the workpiece and, additionally, it moves across as well as down; it is thus case (a) of the Introduction Forces V (normal to the cutting edge) and H (parallel) have associated displacements v and h, respectively The incremental work done is [Vdv  Hdh] This provides the fracture work required for the increment of new cut area, which is given by Rwdv, assuming frictionless conditions and that the growth of cut keeps steady with the movement of the blade Thus Vdv  Hdh  Rwdv (5-1) The resultant force is given by [V2  H2]1/2 and the resultant displacement is [(dv)2  (dh)2]1/2 When there is no permanent distortion of the offcut, and when the wedge angle of the blade is small, these increments are coincident, so that we may also write [V  H ]1/ [(dv)2  (dh)2 ]1/  Rwdv (5-2) The slice–push ratio  is given by (dh/dv), whence solution of these two simultaneous equations gives [H/Rw ]  ξ/[1  ξ2 ] (5-3a) [V/Rw ]  1/[1  ξ2 ] (5-3b) H  ξV (5-3c) and i.e The resultant force is given by (V2  H2), so the non-dimensional resultant force (FRes/Rw) is (FRes /Rw)  (1/[1  ξ2 ])1/ (5-4) 114 The Science and Engineering of Cutting The variation of normalized H/Rw and V/rw with  is shown in Figure 5-2 For   0, H/Rw  0 and V/Rw  1 For  → 1, H increases to a peak at   1 (when H/Rw  V/ Rw  0.5) and then diminishes as  increases V diminishes for all  The smallest normalized forces occur for largest , i.e the sideways speed has to be as great as possible to reduce cutting forces so long as R is constant (strain rate effects may very well affect R) The effect of friction, curves for which are also shown in Figure 5-2, suggests that there is no point in increasing  indefinitely The common experience of V diminishing quickly as soon as some sideways motion is introduced is immediately apparent from Figure 5-2 The effect is noticeable because it is disproportionate: the non-linear coupling between V and H is because the vertical blade displacement and the area of new cut both depend upon V Since a knife failing to penetrate with only a vertical force will be almost at rest, the slightest horizontal motion will cause   and hence much reduce V, as found practically When a workpiece approaches a stationary blade whose normal is inclined at an angle i to the direction its motion, a slice–push effect exists because the approach feed velocity f has components fsini parallel to the edge of the blade and fcosi perpendicular to the edge (Figure 5-1B); in orthogonal cutting f  v A familiar example is planing wood with the plane angled to the length of the workpiece (although wood is not really floppy) It follows that   sini/cosi  tani so that greatest  is obtained with the steepest inclination Note that the effective width weff of the sample becomes (w/cosi) for use in Eq (5-3/4) to give H and V whose directions are along and across the inclined blade (not along and across the direction of f) The sign of the inclination angle i is immaterial for magnitudes of forces, the only difference being the direction of H The forces in the direction of f and across are given by resolution, i.e.: Falong f  Vcosi  H sini  V(cosi  ξsini) Facross f  Hcosi  V sini  V(ξcosi  sini) 4.5 (5-5b) θ = 6°, μ = 0.3 V/Rw and H/Rw (5-5a) θ = 12°, μ = 0.3 3.5 Frictionless 2.5 1.5 0.5 0 ξ Figure 5-2  Reduction in normalized force V/Rw at increased slice–push ratio  and initial increase in H/Rw up to   1 followed by decrease, when cutting floppy materials Frictionless case is the same for all included angles of blade but, with friction, predictions depend on both  and  Examples shown for   6° and   12°, both with   0.3 115 Slice–Push Ratio  using Eq (5-3) When   tani, Falong f  V/cosi  Rw eff /[1  ξ2 ]cosi  Rw Facross f  (5-5c) (5-5d) Thus, in frictionless cutting with an inclined blade, the force required to cut in the direction of tool or workpiece motion is simply Falong f  Rw, with zero sideways force, as expected since the only work is separation work 5.2.2  Cutting with friction Still with case (a) of the Introduction, the orthogonal cutting edge has displacements dv and dh as above, but the blade now has an angle (including clearance) of  The resultant displacement of the offcut over a flank of the blade has two components: dh parallel to the cutting edge and (dv/cos) along the line of greatest slope of the rake face of the blade (Figure 5-3) This gives a resultant displacement dr on the rake face of magnitude dr   [(dh)2  (dv/cosθ)2 ]  (dv/cosθ)  [(ξcosθ)2  1] (5-6) since   dh/dv; dr acts at an angle q  tan1[dh/(dv/cos)]  tan1[cos] with respect to the line of greatest slope The resultant friction force between offcut and rake face of the tool is assumed to act in the same direction as the resultant displacement Hence the incremental friction work for Coulomb friction on one flank of the blade is Ndr and is given by µNdr  µ[V/(sinθ  µcosθ)cosθ] [(ξcosθ)2  1]dv (5-7) Blade dν Cos θ dh dh dr θ dν Figure 5-3  Motion of slice over blade has two components: (i) dv/cos along the rake face of the tool having included angle ; and (ii) dh along the cutting edge 116 The Science and Engineering of Cutting substituting for N in terms of V from Appendix The expression in Eq (5-7) may equivalently be obtained by summing the work done by the component Ncosq of N along the line of greatest slope of the wedge times (dv/cos), plus the component Nsinq of N parallel to the cutting edge times dh Equating external and internal work increments for an orthogonal cut with a sidewaysmoving blade gives V/Rw  1/{1  ξ2  [(2)µ  ((ξcosθ)2  1)/cosθ(sinθ  µcosθ)]} (5-8a) and H/Rw  ξ(V/Rw) (5-8b) The bracketed ‘2’ with  is to be used when both sides of a blade are in contact with the work­piece Figure 5-2 includes curves for V/Rw and H/Rw that include friction according to Eq (5-8) When a stationary blade is inclined at angle i to the crossways-dimension of the workpiece,   tani, and the forces V across and H along the edge are given by Eqs (5-3a,b) noting that w is replaced by the inclined length of contact weff  w/cosi The feeding force Falong f and the crossways force Facross f are obtained using Eqs (5-5a,b) to give Falong f /Rw eff  1/cosi {1  ξ2  [(2)µ  ((ξcosθ)2  1)/cosθ(sinθ  cosθ)]} (5-9a) and Facross f  (5-9b) since   tani and, in this case, the blade is stationary There may be optimum inclination angles i for least cutting force owing to the competition at large  between smaller forces on the one hand, but larger frictional contact length on the other It will depend on  and  A similar effect is found when cutting materials with an initially slack wire (Chapter 12) 5.2.3  Inclined separately propelled blade: the disc slicer Cutting on a delicatessen slicer involves workpieces of bacon, salami and so on which are relatively thick compared with the diameter of the cutting disc Here we consider laminae fed into a rotating disc cutter, where  is approximately constant across the thickness Cutting of thick workpieces that cover considerable parts of the blade is considered in Chapter 12 Consider a sheet fed, below the centreline, into a cutting disc of radius  (Figure 5-4A) Point P is located at angle i measured from the centreline of the wheel where positive i is anticlockwise The disc rotates with angular velocity , where positive  is anticlockwise The feed rate of material into the wheel is f from left to right We bring the workpiece to rest by adding a velocity (f) which means that in addition to rotating in a clockwise sense, the disc now has a forward speed f from right to left, which has a tangential component fsini in an anticlockwise direction, and a radial component given by fcosi The local velocities (displacements) at P normal to the cutting disc dvdisc, and parallel to the edge of the disc dhdisc, are thus dv disc  fcosi (5-10a) 117 Slice–Push Ratio  ω ρ i f f P i fsini fcosi ρω A 0.3 0.25 Facross f 0.2 0.15 0.1 0.05 –180° –90° i Falong f 0 i +90° +180° –0.05 –0.1 –0.15 B Figure 5-4  Thin sheet fed into a disc cutter below the centreline with speed f (A) Geometry of device where zero for i is along the centreline and positive i is anticlockwise; disc has radius  and rotates anticlockwise with angular velocity ; (B) variation of feeding force and vertical force with position i of cutting for /f  5,   0.1 and   6° Negative values for feeding force mean that the workpiece has been ‘grabbed’ by the cutting disc and dhdisc  (ρω  fsini) (5-10b) where positive dhdisc has the sense of  The slice–push ratio at P in an anticlockwise sense is ξdisc  dhdisc /dv disc  (ρω  fsini)/fcosi  (ρω/fcosi)  tani (5-11) 118 The Science and Engineering of Cutting For cutting above the centreline, i is negative and tani changes sign If (/f)  sini, the contribution of tool obliquity to the push/slice effect will not be noticeable except at large i (cutting at the top and bottom of the disc) The forces Falongf in the feed direction and Facrossf perpendicular to the feed table are given by Eqs (5-5a,b) using V and H from Eqs (5-3a,b) that includes friction, i.e Falongf  V[cosi  ξdiscsini ] (5-12a) and Facross f  V[ξdisc cosi  sini ] (5-12a) The variation of Falongf and Facrossf with position above (i negative) and below (i positive) the centreline is shown in Figure 5-4(B) for /f  5,   0.1 and   6° The negative values of Falongf indicate that the workpiece has been ‘grabbed’ and requires no positive force to push it through the disc cutter This is familiar to anyone who has used a hand grinding wheel or circular saw Calculations show that, unsurprisingly, overall cutting forces increase with greater friction and with smaller , but the pattern of disproportionate decrease in V as  increases is retained Atkins et al (2004) performed experiments with a disc cutting cheese and salami and demonstrated the effect of slice–push in reducing cutting forces as the speed of the disc was altered at constant feed In that paper, the friction force was modelled not by the Coulomb relation, but rather as a frictional stress  that was some fraction m ( 1) of the workpiece shear yield stress, i.e   mk, acting over some finite contact area between offcut and blade (this approach is often employed in metal cutting; Appendix 1) It may be shown that for an orthogonally orientated blade V/Rw  [1  S  (1  ξ2 )]/(1  ξ2 ) (5-13) and H  V, in which S  (2)mLk/R with L the contact length along the rake face The bracketed (2) relates to whether one or two faces of a blade contact the workpiece There are similar expressions employing S for Falong f and Facross f when the blade is both inclined and independently moving Whichever way friction is modelled, calculations show that there is probably no benefit in increasing disc indefinitely, by increasing the speed, owing to increased work against friction, and experiments confirm this Instead of determining Falongf and Facrossf using H and V as intermediate values, there are other ways of obtaining the feed and across-feed forces directly, employing the effective wedge angle eff of the disc (not the line of greatest slope in the cutting bevel, rather the slope along which the offcut passes for which taneff  cosi tan) and ieff (where tan ieff  disc), but these alternative lines of attack are, perhaps, confusing Similar alternatives occur in modelling the formation of ductile chips during oblique cutting (Section 5.3) 5.2.4  Pizza cutter: disc harrows A similar analysis may be performed for the pizza cutter disc that rolls along the base of the pizza It may be shown that at the base pizza is infinite as the motion is instantaneously all slice and no push so, in theory, requires no force (rather like an extremely thin sheet cut at the top or bottom of a delicatessen slicer) While it is possible to define a mean slice–push 119 Slice–Push Ratio  ratio pizza*, it is unbounded The force Fpizza in the direction of cutting with a frictionless rolling disc is simply Rh where h is the thickness of the pizza With friction, the procedures in Section 5.2.2 may be employed In practice, there will be additional friction as the bottom of the cutter rolls up and emerges behind the wheel Pizza wheels are used to cut cork in Sardinia (Negri, 2008) Discs are used in some designs of harrow for improving the tilth of seed beds (Chapter 14) They must act rather like pizza cutters, but in a complicated way, as the plane of the disc is often angled to direction of tractor motion, and the disc itself may be dished, in order to improve disturbance of the soil Godwin et al (1987) show that haulage forces arise from two components, namely a passive reaction on concave faces and scrubbing action on convex faces The associated forces were estimated using pressure-dependent soil yielding mechanics As explained in Chapter 14, this is equivalent to plasticity-and-friction-only analyses of cutting but, as also explained in Chapter 14, toughness work in soils may be swamped by other components of work done such as lifting the soil 5.2.5  Reciprocating blades Reciprocating blades have slice–push but, unlike blades moving continuously in the same direction,  varies at different positions in the stroke There is zero slice–push at the ends of the stroke where the blade is instantaneously at rest The maximum  will be at mid-stroke Owing to the continuing changes in , force plots from a high-speed reciprocating blade are very spiky If the reciprocating motion is approximated by h  hosint,   (ho/f)sint, where f is the feed displacement in orthogonal cutting, Eqs (5-3a,b) and (5-4) give for frictionless cutting [H recip /Rw]=(ho /f)sinωt/[1+(ho /f)2sin2ω t] (5-14a) and [Vrecip /Rw]  1/[1  (ho /f)2sin2ωt] (5-14b) and (FRes /Rw)  (1/[1  (ho /f)2sin2ωt])1/ (5-14c) The variation of the forces in one cycle is shown in Figure 5-5, for (ho/f)  10 Forces are always low in the middle of the stroke, but high at the ends, the more so when h  v Benefits of large  are evident only in the centre of the stroke The analysis is easily modified to demonstrate the effects of friction Similar considerations apply to hedge cutters, hair trimmers, sheep-shearing comb cutters and electric carving knives (see Chapter 10) 5.3  Offcut Formed in Shear by Oblique Tool When a chip is formed in shear in orthogonal cutting of ductile materials, it has the same width as the uncut chip thickness but a different thickness It also has curvature caused by the non-uniform width of practical primary shear zones, and also by secondary shear, which together with bending forms the chip into a spiral When a straight-edged tool is angled to the direction of feed and used to cut a ductile material, there are two main differences from orthogonal cutting: (i) the offcut has not only a different thickness, but also a different width 120 The Science and Engineering of Cutting 1.2 0.8 Fres/Rw 0.6 V/Rw 0.4 H/Rw 0.2 0 0.5 1.5 2.5 3.5 – 0.2 Reciprocating displacement of knife (arbitrary units) Figure 5-5  Variation of V/Rw, H/Rw and Fres/Rw for frictionless reciprocating cutting Slice–push  varies during the stroke of the blade It is a maximum in the centre but zero at the ends of the stroke Benefit of  lost except at central portion of stroke (caused by primary shear over a longer angled contact length between tool and workpiece); and (ii) more complicated curvature that bends the chip into a permanent helix with the axis of rotation approximately parallel to the cutting edge The greater the inclination angle i to the direction of feed, the wider the chip and the tighter the curl In a simple single shear plane model of oblique cutting, the shear plane connecting the cutting edge to the free surface is skewed at the obliquity angle i to the feed direction (Figure 5-6) There are three velocity components: the workpiece approach velocity VW, the shear velocity VS in the shear plane, and the chip velocity VC in the plane of the tool rake face In orthogonal machining, all three velocities and the hodograph lie in the plane of cutting, the direction of shear is along the line of steepest slope in the primary shear plane, and the direction of chip flow is along the line of steepest slope of the rake face of the tool In oblique cutting, both the primary shear direction in the shear plane and the chip flow direction across the tool are no longer in the directions of steepest slope VS is now at an angle S (the shear flow angle) to the normal to the cutting edge in the shear plane; the shearing action at angle S results in the final cocked direction of the chip over the rake face of the tool, which is defined by the angle C (the chip flow angle) to the normal to the cutting edge in the rake face Since all three velocities VW, VS and VC form a hodograph in one plane they are related by geometry (e.g Amarego & Brown, 1969, p 80) It was pointed out earlier that when cutting thin sheets not along the centreline of a disc cutter, it was possible to calculations in terms of the effective blade included angle eff rather than the usual included angle given by the line of greatest slope When shear planes are formed in oblique machining, there is again a number of alternative definitions of tool rake angle and of shear plane angle (for a discussion see Shaw, 1984; Amarego & Brown, 1969) That usually employed in analyses of ductile materials is the rake angle n, given by the rake 126 The Science and Engineering of Cutting Force F Displacement δ θ 2t h j Scissors load (N) y 0 A B 10 20 Scissors displacement (mm) 30 Figure 5-8  (A) Geometry of scissors and cut material; (B) comparison between experimental results of Pereira et al (1997) for cutting palmar skin with scissors and predictions of theory The upper experimental curve is the total cutting scissors force; the lower is the force to close the scissors from the same 0 and indicates friction The thick line is the prediction of the theory using R  2.4 kJ/m2; the experimental value estimated from the work area between the two force plots is some 2.4 ( 0.2)  kJ/m2 The thick line in Figure 5-8 is the prediction of theory for R  2.4 kJ/m2 Pereira et al (1997), from the work area bounded by the two experimental force plots in Figure 5-8(B), gave R  2.4 ( 0.2) kJ/m2 along the skin creases and 2.6 ( 0.4) kJ/m2 across The motion of some element in contact with the material along the blade is perpendicular to a line joining the element and the pivot The slice–push ratio  is given by the ratio of velocities along and across the edge of the blade and may be shown to vary along the blade The biggest variation in  will be at the beginning of a cut when  is smallest, but even so the range is not marked The variation is quite regular and it may be shown that use of the  value at the mid-point of the blade is adequate in calculations Large  promotes low cutting forces at the beginning of a cut, when the mean  is greatest, and the forces at the handles of scissors are low, particularly for thin slices Later in the stroke, however,  decreases, which increases the cutting forces In addition, the effective lever arm of the cutting force decreases, so that the handle forces increase even more, as shown in Figure 5-8(B) Lucas and Pereira (1991) used both scissors and the guillotine to cut newsprint (sheet thickness 70 m) tested singly and in layers, applying the graphical method to the force– displacement results (compensating for friction and the set of the blade) to obtain toughness Toughnesses by both methods are comparable but they show that values depend on the number of layers Lucas and Pereira attribute the different R to fracture mode mixity; there is Slice–Push Ratio  127 probably also an effect from the (blade clearance/thickness) ratio changing as the number of layers change The effect of bluntness on scissor toughness, discussed in Chapter 9, has been studied by Arcona and Dow (1996) and Meehan (1999) Curved-bladed scissors, secateurs, pruning shears and specialist scissors such as pinking shears (with zig-zag notched edges to prevent fraying of cloth), hairdresser thinning scissors, nail scissors and so on, can all, in principle, be analysed in the way given here Some scissors have spring-loaded blades whose natural position is open That may be produced by a separate spring between the handles, but in old-fashioned sheep shears and cloth sampling shears, clever design enabled the two blades to be manufactured from one piece that crossed over in a spring loop behind the gripping positions When permanent deformation of the offcut occurs, as with tinsnips or devices for clipping coins (in the USA, two bits  one quarter), the analysis of the next section is required 5.4.3  Ductile materials The cut edges of guillotined plates of ductile material are similar whether cropped with an orthogonal blade or with an inclined blade (Figure 3-24A), where a smooth tool indentation region lies above a rough separated region The critical depths cr at the transition in cropping, guillotining (and in punching, Chapter 8) are very similar, despite questions about shear in different modes, and mode mixity The critical depth depends on blade sharpness and clearance; similarly, an estimate of mixed-mode toughness is given by R  kcr (Section 3.8) when the tool is sharp In guillotining, deformation occurs only over the small region around that part of the guillotine blade currently cutting The action in guillotining is a steady-state composite of the sequential actions which occur progressively as increasing travel of the blade in orthogonal shearing, but with additional features On the one hand, (i) the offcut must bend to conform to the inclination of the blade (bending about EB in Figure 5-9A), which is not found in orthogonal cropping On the other hand, (ii) the offcut in orthogonal cropping bends about the cut face (about an axis perpendicular to that in (i) above) and the rotation increases with blade travel (Figure 3-22C) Because blade contact in guillotining occurs over a range of indentation depths from first contact to down to f (the blade travel at which separation is complete), this rotation gets progressively greater through the contact zone The shape BFB’ of the (hidden) crack profile alongside the blade in the deformation zone is very important in determining both sorts of offcut rotation Since the type (ii) rotations start at B and get progressively greater until F is reached, the offcut comes away as if it were twisted (Figure 5-9B), although the deformation is really the result of plastic bending of cantilever elements of diminishing built-in thickness, not torsion Twisting in ductile plates is most marked at offcut overhang widths less than the plate thickness in size Narrow offcuts of paper deform into open helices in a similar manner, where the permanent deformation is a result of irreversible stretching of fibres that permanently slide over one another (paper bent or torn to a tight radius gives permanent curling) Helix formation occurs in most materials that can experience permanent deformation In guillotines whose cutting angle does not change during the cut (disc cutters), the pitch of the helix depends on the width of offcut and thickness of material; the thinner the offcut the tighter the helix 128 The Science and Engineering of Cutting α A F Fʹ Aʹ E Bʹ Tapered burnished land Uncracked B F C Cʹ Eʹ D Dʹ A Cracked B Figure 5-9  (A) Sketch of plate deformation mode during guillotining Note triangular zone BFB of uncut material, and interfacial crack profile, beneath inclined blade Sideways bending rotations (corresponding with those in Figure 3-22C) vary along varying uncut cross-section BFB, and the offcut comes away twisted (B) Progressive fan-like rotation of elements experiencing sideways bending in the guillotining zone Burnished land not uniform as in Figure 3-22(C) but now tapers out to its full width When the cutting angle changes during the stroke (lever guillotines) the pitch will change along the cut and, with scissors, the range of pitch depends on what ‘gape’ (angle of opening) the scissors had at the beginning of the cut Whatever the sheet material, the helical deformation is less marked as the offcut becomes wider A consequence of the progressive twisting with permanently deformed offcuts is that the burnished land seen on ductile metals, which has a uniform width in orthogonal bar cropping, now tapers out to the full developed width at which the offcut parts company with the blade (Figure 5-9B) A work rate analysis for guillotining includes components relating to: (a) shear and fracture in the plane of the cut face (b) bending to the inclination of the blade (c) differential sideways bending and shear (offcut ‘twist’) (d) friction Experiments on ductile metal plates suggest that these components are uncoupled An approx­ imate analysis incorporating all work components was derived (Atkins, 1987a) but, except at small offcut width w, the forces and work associated with component (c) are comparatively small compared with the total forces measured experimentally Also, the friction component (d) is small compared with other components (dry and lubricated cuts requiring virtually the same load) Consequently, a simplified algebraic expression is adequate to interpret the experimental results Figure 5-10 shows how the steady-state guillotining force over the whole contact length is predicted from the non-steady orthogonal cropping force element by element The guillotining force Fguillotine at w  t is given by Atkins (1990) Fguillotine /t  (kψ )w  (R*/tanα) (5-17) 129 Orthogonal force/Unit width Slice–Push Ratio  δcr δf δ Tool travel Blade α δ δcr x dx δf Figure 5-10  Schematic of how the non-steady force vs blade stroke for orthogonal cropping is used to predict the steady-state force component of guillotining in the cut plane of intense shear Blade movement to the left Top surface of sheet is at top of diagram where t is plate thickness, R* is the effective fracture toughness in the plane of intense shear (see below),  is blade inclination, k is the shear yield stress resisting bending under the blade, and w is width of offcut;  ( [(l  n)/8n]sin 2, with n the workhardening index in   0n) is a function connected with where the bent offcut becomes tangential to the guillotine blade We see that a linear relation between Ftotal/t and w is predicted, the slope of which is k and the ordinate intercept is (R*/tan) R* is defined as the mean total work per area performed in the cut face from   0 to   f It thus sums the indentation plastic work up to   cr and the subsequent plasticity and fracture between   cr and   f Insofar as the cut face cannot be produced with given tooling without some combination of flow and fracture in the cut plane, R* is the effective fracture toughness that may be analysed separately from the accompanying plastic rotations and friction in guillotining R* is not the same as the ‘true’ specific essential work of fracture (the fracture toughness) R, which is the work of fracture alone devoid of the remote flow component The use of R* is similar to the effective toughness that applies when tearing ductile sheets (Mai & Cotterell, 1984) A burr is left on the torn edge and separation by tearing is impossible without The specific work of burr formation is added to the fracture toughness to give the effective toughness for tearing Figure 5-11(a,b) shows representative plots of guillotining force against offcut overhang width w for 6 mm plates of copper cut with blades of angles 10° and 25° At w  t, Fguillotine varies linearly with w according to Eq (5-17), the different lines (all with the same slope) corresponding 130 25 Clearance 0.3 mm 20 –3 m /k y r y R* 15 20 25° blade mm thick copper 83 VPN 15 –3 m T Clearance 0.9 mm 10 Intercepts 12–15-4 kN ∴R* = 352-453 kJ/m2 Slope = 160 kN/m ∴k = 133 MPa ∴(R*/ky) = (2.6–3.4)10–3 m ry o he Guillotine force (kN) /k y = r y R* Theo 10 2.6 x –3 m ry eo /k y R* 5x The Science and Engineering of Cutting Guillotine force (kN) Theo x 10 = 3.4 Increasing clearance 10° blade mm thick copper 83 VPN –3 m /k y R* 4x Th 10 Slope = 375 kN/m ∴k = 125 MPa Intercept = 6.5 kN ∴R* = 505 kJ/m2 (R*/ky) = x 10–3 m 0 A 12 18 Offcut overhang width w (mm) 24 B 12 18 24 Offcut overhang width w (mm) Figure 5-11  Experimental results for guillotine force vs offcut overhang width at various clearances for 6 mm thick copper plate of 83 VPN following   440 0.27 MPa Dashed lines give theoretical predictions Linear approximation to theory at w  t shows how R* and k may be determined from Eq (5-17) (A) 10° sharp guillotine blade; (B) 25° sharp blade 131 Slice–Push Ratio  to different clearances between blade and baseplate Increasing clearance at constant offcut width required less force, giving smaller back-extrapolated intercepts Quoted on the graphs are the derived k from the slopes and R* from the ordinate intercepts At small w, Fguillotine increases rapidly owing to the increasing importance of work of twisting of the offcut Also shown in the figures as dashed lines are the theoretical plots using the full relationship for Fguillotine (Atkins, 1987) for the appropriate ranges of R*/k The overall agreement between theory and experiment is satisfactory: that was the case also for other materials (mild and stainless steel, and brass) not shown here The value of k for copper determined from Figure 5-11 (about 130 MPa) is that to be expected, at the level of bending strain under the blade, both from the   0n relation for copper and, simply, from the hardnesses H using the approximate relation H  (56)k in consistent units That is, for copper’s 83 VPN we expect k  830/6  140 MPa In other experiments on 6 mm brass plate guillotined with both 10° and 25° blades, k  220 MPa; its hardness was 110 VPN, and we should expect k  1100/6  200 MPa The clearance-dependent R* values of 350–500 kJ/m2 for copper (and 500–650 kJ/m2 for brass) are what would be expected on the basis of independent orthogonal cropping of the same material, as discussed in Chapter R* decreases at increasing clearance owing to a changing mode of fracture from nearly all shear to a combination of shear and tension, with increasing bending across the clearance gap The true mixed-mode specific essential work of fracture (the fracture toughness) R is given approximately by R  kcr  (VPN/6) cr in consistent units For 83 VPN with cr  2 mm in 6 mm thick plate we expect R  (830/ 6)2  277 kJ/m2 The larger values for R* reflect the inclusion of remote plastic work, which has nothing to with the process of fracture Experimental results for cr from guillotining thin sheets with sharp workshop shears with negligible clearance between blade and anvil are given in Table 5-2 Guillotining of thin sheets is related to the slitting of sheets, and its finite element method (FEM) modelling, discussed in Section 5.4.4 Guillotined cuts taken on plates tapering in thickness give linear traces of Fguillotine vs blade travel , increasing uphill and decreasing downhill (Figure 5-12) In both cases, (Fguillotine/t) is (Fguillotine/) are constant since t is proportional to the horizontal length of cut or equivalently to the vertical blade travel  This suggests a way of obtaining lots of results from one experiment and is a useful experimental trick for orthogonal cutting too The idea has also been employed in milling by Sinn et al (2005) (see Section 4.7.3) Inspection of the edges of cut tapered plates shows that cr increases with increasing plate thickness (and vice versa) Table 5-2  Experimental results from guillotining thin sheets Material Low-carbon steel Thickness (mm) Hardness (kg/mm2) cr (mm) R  (H/6)cr (kJ/m2) 200 0.17 57 200 0.40 133 158 0.30 79 158 1.00 263 Brass 110 0.95 171 Stainless steel 2.4 175 0.40 114 Soft low-carbon steel Source: Atkins (1988b) 132 The Science and Engineering of Cutting Tapered copper 10° blade w = 18 mm Guillotine force (kN) 15 25° blade w = 18 mm 10 10 20 30 40 50 Blade travel (mm) 60 B 0 A 10 20 30 Blade travel (mm) Figure 5-12  Guillotine force vs blade travel for cutting tapered plates with increasing thickness: (A) 10° blade; (B) 25° blade Theory predicts a linear variation Discontinuity in slope for the 10° blade concerns a change in the relative sharpness between blade and current plate thickness; early in the stroke there is much shear before separation but always remains the same proportion of the thickness, i.e cr  t Were cr a fixed size, it would suggest that plates whose thickness was smaller than cr could not be cut But changing cr has strange implications for the toughness in shear: if hardness is fixed, and cr alters, then R ( kcr) must alter with thickness It might be expected that R ought to have one value characteristic of the material and its thermomechanical state, as in ‘normal’ fracture mechanics testing (The well-known variation of RI with plate thickness caused by different plane stress/plane strain constraint is something different; see Atkins & Mai, 1985.) It is believed that changing cr and hence changing R comes about because the plastic deformation zone (shear band) is set up through the whole thickness from the outset of cutting In the usual type of fracture mechanics testpiece, the crack tip zone is limited in its forward extent and does not reach the back face of a specimen until late in the test Metallographic examination of guillotined tapered sections reveals that h, the width of the shear band, changes with thicknesses in the same way as cr, i.e h  t (Atkins, 1988) The thickness of primary shear bands in orthogonal cutting is also found to be proportional to the length of the slip band (Stevenson & Oxley, 1970–71; Childs et al., 2000) Thus when both cr and h vary directly in proportion to t, cr  (cr/h) remains constant for all t An alternative, equivalent, interpretation of R varying directly with plate thickness is that R/h is constant since h varies in proportion with t (R/h) represents a critical plastic work/volume for cracking for particular blade sharpness (There is an assumption of uniform deformation in the shear band which will not be quite correct as the local strains near the cutting edge where the fracture process zone forms will be greater than average.) Slice–Push Ratio  133 The variation of R with t has implications for punching holes and for the scaling of energies in plug formation in armour penetration, and perhaps for scaling teeth, and is discussed further in Chapters and 13 5.4.4  Slitting and shredding sheets A type of office paper guillotine consists of an undriven sharp cutting wheel fitted to a block that slides on a long bar parallel with the direction of cut Sheets are cut by pushing the block holding the wheel along the bar to cut off paper hanging over the edge of the baseplate On production lines, thin materials of all sorts (floppy to ductile) are slit into different widths by pulling the sheet through a similar sort of wheeled device There can be multiple cutting wheels to produce many strips from a wide roll Paper shredders operate in a similar fashion The problems of slivers and burr on the edges of trimmed sheets of metal, which were discussed in Chapter 3, can occur in slitting Ma et al (2006) and Lu et al (2006a) apply Li’s ideas of slitting at an angle to eliminate this difficulty, and it is found that the arrangement is relatively insensitive to clearance and gap between the rotary knives The process has been simulated with FEM (Ghosh et al., 2005) in which a variety of separation criteria were tried Gurson–le Rousselier porous plasticity models were not successful at replicating the experimental results for the orientation of the shear plane, height of burr and so on What worked best was a criterion of critical equivalent plastic strain (affected by hydrostatic stress in the separation zone) The work ought ultimately to be able to predict the critical depth cr , at which separation begins in a ductile sheet, for different combinations of clearance, tool sharpness and material properties 5.4.5  The can opener A canister is any sort of small container with a removable lid for storing things such as tea or coffee (from the Latin canistrum for wicker basket) The familiar ‘can’ or ‘tin’ in which food or drink remains fresh for a long time is completely sealed and has to be opened in order to consume the contents Sealing was originally done by soldering, later by wrapped-around joints Steel sheet for canning was plated with tin against corrosion, and helped soldering Advice on the opening of Victorian hand-soldered big cans (that could weigh up to 7 lbs empty) was to use a hammer and chisel Many modern cans have ring-pull devices by which the lid, or a portion of the lid in the case of beverage containers, is removed Steel corned-beef cans once had a flap on which a key could be wound to open them The torque required for this elastoplastic fracture mechanics operation is given in Atkins and Mai (1985); the solution for ring-pull cans following prescribed crack paths is similar The reason why the parallel tear sometimes runs to a point before completion around the circumference of the can is discussed in Chapter 15; other types of sealed tin not have in-built release devices and a can opener has to be used to reach the contents There are various designs but all involve first making a hole followed by propagation of the slit around the rim of the can A basic type of can opener has a thin sharp point and cutting edge that is used to stab a hole in the lid by hitting with the palm of the hand, after which a series of discrete leverings cuts around the lid to give a wavy edge to the removed lid In another design, the knife-indenter is attached to one of a pair of hinged handles and the initial piercing is made by squeezing the handles At the same time the device latches on to the underside of the 134 The Science and Engineering of Cutting rim by means of a toothed wheel Winding a handle attached to the wheel drives the cutting edge around the rim progressively detaches the lid A similar device works in a horizontal plane to remove the lid by cutting around the top of the cylindrical wall of the can rather than cutting the lid just inside the rim The mechanics of cutting around the rim of a can made of ductile sheet is similar to that of guillotining where the length of cut is the circumference of the lid However, there is more constraint across the lid than for the overhanging free edge of simple guillotining A simplified analysis might go as follows The deformation of the lid around the initial pierce consists of plastic bending under the inclined knife (i) around the circumference (as in guillotining) and (ii) in the radial direction of the lid Both plastic bend radii are determined by the sloping geometry of the knife, and may be represented by a single effective radius of curvature  Observation of can opening suggests that the radial distance over which plastic bending takes place is approximately equal to  A rotation d of the toothed wheel gives a circumferential movement (D/2)d, where D is the diameter of the lid, and hence an incremental fracture work requirement of Rt(D/2)d, where t is the thickness of the lid The incremental volume of material plastically bent is [(D  )t/2]d The mean bending strain is (t/4), so for a yield strength y, the incremental bending work is y(t/4)[(D  )t/2]d These two components of internal work are provided by rotation of the toothed wheel Hence for torque T Tdθ  Rt(D/ 2)dθ  σy (t/8)[(D  ρ)t]dθ i.e T  Rt(D/ 2)  σy (t/8)[(D  ρ)t] or (T/RtD)  0.5  (σy t/8R) [(1  (ρ/D)] (5-18) The torque is a steady value This treatment ignores friction and indentation of the toothed wheel into the underside of the rim, and is unable to predict the value of  However, it may be possible to couple the fracture work and the plastic bending work via the critical crack opening displacement as described in Section 8.6.1 and thus obtain a more comprehensive answer 5.5  Drills, Augers and Pencil Sharpeners Shards from flint knapping in the form of parallelepipeds were once used for drills They were inserted into a cleft in stick, and glued in place with tree resin heated up with charcoal, and then bound tightly with thread made from nettle stems (Sim, 2008) The string of a bow was wrapped a few times round the stick and, by reciprocating motion, rotated the drill clockwise and anticlockwise in succession Glass may be drilled using a copper rod with emery powder, and when holes are required through gemstones (e.g beads), a small rotating rod or tube with a diamond tip is used to drill through the stone, sometimes aided by a slurry of silicon carbide and coolant The process is ‘rotating scratching’ and depends on the factors described in Chapter 6, including point geometry, and the toughness and hardness of the material A wimble is a marbleworker’s brace for drilling; the word is also used for a device in mining for extracting spoil from the drilled hole A proposal by Jagger (1897) to determine the hardness of minerals used drilling A diamond ‘point’ (of undefined geometry) rotated on an orientated mineral section under uniform rate and uniform weight The number of rotations to penetrate to a given depth was Slice–Push Ratio  135 found to vary with the resistance of the mineral to abrasion by diamond What was being measured appears to have been some composite behaviour of the toughness and strength The simplest type of drill having a controlled geometry is the spade drill, as used in a braceand-bit woodworking The drill, of full radius rdrill, consists of a spike or screw thread at the tip which indents or screws into the workpiece and keeps the hole central, with flat pieces extending sideways, the bottom edges of which cut the base of the hole and the corners of which cut the sides Every element of the bit cuts as if it were a zero rake angle tool (the edges of the spade are angled underneath to provide clearance for the normal direction of drilling, i.e clockwise from above) The length of an elemental chip produced by one cutting edge of the spade depends on how far it is from the centre, being given by r, so is zero at the centre and rdrill at the outside The tangential speed of cutting, and the steepness of the helical path followed by an element of the cutting edge, also vary with radius Since the elemental chips are all joined up, the short portions of the chip from near the centre of the drill are stretched (and may fracture), while those from the outside are compressed and may buckle The whole chip therefore must curl in space Experiments where a spade drill is used on a block of butter demonstrate that the offcut rises in a circular fan shape to cover the face of the drill When drilling wood with a spade bit, the torque will fluctuate owing to grain orientation and anisotropic mechanical properties In a single rotation, a drill will encounter a variety of orientations and the effect of this is revealed in the surface quality of a drilled hole The growth direction of the tree from which the workpiece has been cut may be identified from the appearance of the surface of a drilled hole (particularly large-diameter holes): a smooth surface is produced where the cutting tool has turned in the same direction as that in which the tree grew, and a rough torn-out surface results on the opposite side of the same hole Drilling energy alters as the drill passes through the early and late wood in the growth rings Such effects are experienced in other woodworking operations such as turning on the lathe As pointed out by Effner (1992), however, the effect is noticeable only for a hand-held tool such as a brace and bit An auger is an ancient drill made by twisting a narrow strip of steel into a helix One end is sharpened like a chisel: there is a point in the middle to mark the centre of the hole and spur knives to cut cleanly round the outside of the hole in advance of the main blade The spur cutters act like the coulter on a plough (Chapter 14) Material removed from a hole made by an auger is lifted out along the helix to the surface Unlike augers, spade bits not have spur cutters on the outsides, resulting in rough holes, and removal of debris from a deep hole made with a spade drill can be difficult The holes for wooden pegs or trenails, which held together wooden-framed buildings and ships, would have been made by augers Unlike a spiral (properly called helical) staircase, there is no central core to stiffen the tool in the simplest sorts of auger Also, simple augers with a single helix cut on only one side of the axis so that the action is unbalanced With two or more helices balance is restored and the auger runs centrally The present-day parallel-sided twist drill was a development of multiplestart augers: debris is discharged along channels (flutes) formed out of the solid body of the twist drill The point of a simple twist drill has two main cutting edges, each of which has a rake angle and relief angle (with indexable drills the cutting edges are formed of separate carbide inserts) Were these edges orientated radially, they would meet at a point, but such a point would be likely to break off in use In practical drills the cutting edges are arranged as shown in Figure 5-13, and have a secondary cutting edge (the straight chisel edge web) at the centre The secondary edge, while solving the problem of tip strength, creates other problems The rake angle of the secondary edge varies from perhaps 60° at the centre to say 10° at its ends (Black, 1961), and the cutting speeds similarly vary from very low near the centre to faster at the outsides The centre of a drill appears to pierce the workpiece by a different 136 The Science and Engineering of Cutting Main cutting edges Web thickness Figure 5-13  Tip geometry of flute drill action from the actual cutting edges (Shaw, 1984) Built-up edges are often observed in drilling operations with steels and other metals (e.g Ventakash & Xue, 1996) All these observations indicate that drilling is overall inefficient Again, because the ends of the secondary edge of twist drills are never perfectly parallel to surface of the workpiece, the drill tends to wander and so produce non-round holes that are also not straight This problem is absent in a trepanning saw, but not in a hole saw with its own centre drill, nor in flycutters A pencil sharpener is an inside-out drill that cuts a taper on the end of a pencil to reveal the pencil ‘lead’ In the days of 78 rpm records, gramophone needles were made from steel or from bamboo (fibre needles) and were sharpened with a pencil-sharpener-like device (All needles tended to wear the grooves, as discussed in Section 6.4.) The angle of the taper of a pencil sharpener corresponds to the point angle on a drill The nearest device to the hollow drill equivalent of the pencil sharpener is perhaps a type of gimlet As with drills and augers, the action of a pencil sharpener relies on both axial and rotary motion with associated thrust/ feed force and torque The combined work done in one revolution by the axial force Faxial and the torque T performs (i) toughness work in separating the shaving; (ii) work against friction; and (iii) deformation in the shaving Experiments show that the wood shaving is thin, floppy and broken up, so for simplicity we neglect work component (iii) Figure 5-14 shows a pencil sharpener with a taper angle p The radius of the pencil is r and the feed per revolution of the pencil into the device is f The thickness t of the shaving is t  fsinp (from equating the volume of pencil r2f consumed in one revolution to the conical surface area r2/sinp times the thickness t) In a pencil sharpener, the same depth of cut is a bigger percentage of the circumference at the conical tip than further back The feed f has components fsinp perpendicular to the cutting edge and fcosp parallel to it There is therefore slice–push of magnitude   cotp Insofar as the motion can be considered similar to the oblique cutting in Section 5.2, an element of cutting edge of length ds  dr/sinp experiences a force dV perpendicular to the blade (circumferentially to the device) and dH along the blade The incremental component dH may be resolved into an axial component dHcosp and a radial component dHsinp For simplicity, consider frictionless cutting where the deformation in the shavings is negligible We have dV  Rds/(1  ξ2 )  Rds/sinp  Rdr/sin2 p (5-19a) and dH  Rξds/(1  ξ2 )  Rcospdr/sin3p (5-19b) 137 Slice–Push Ratio  Faxial, f Cutting edge T, ω V H p Figure 5-14  Geometry of a simple pencil sharpener having a single blade aligned along the surface of a cone of slope angle p The pencil is fed into the device with force Faxial and feed f, and rotated with angular velocity  by torque T where R is the toughness of the wood of the pencil The axial component dHcosp is dFaxial, where Faxial is the thrust force feeding the pencil into the device The radial component Hsinp is reacted by the walls of the conical cavity It follows that Faxial  Rrcos2 p/sin3p (5-20a) ignoring the difference between wood and graphite at the tip The element of torque dT  rdV, whence T ∫ Rrdr/sin p  Rr / 2sin2 p (5-20b) These results have been obtained without writing out the work equation because the use of  gives a connexion between Faxial and T The expression for Faxial suggests a minimum value at p  51° (very high) in this special case Friction may be included by employing the equations of Section 5.2.2 that will modify an optimum blade angle When drills cut ductile materials, the drillings are skewed across the cutting edges of the drill and permanently curled into helices, because of obliquity The slice–push interpretation of pencil sharpening may be applied to the drilling of ductile solids in terms of the obliquity angle i using   tani and   cotp, using the relations for cutting forces at the oblique edge given in Section 5.3 This would be expected to predict the sorts of Faxial and torque relations depicted in Figure 5-15 where thrust and torque vary almost linearly with feed for drilling polyethylene and acrylic resin with twist drills (Kobayashi, 1967); similar results are found when drilling cancellous bone (Shuaib & Hillery, 1995) Flutes of drills may vary in number from one to three and must be free-flowing so as not to become clogged Drills intended for cutting metals have ‘slower’ flute helix angles, i.e are less steep than the flutes of woodworking drills since, other things being equal, wood is easier to remove from the cutting face Typical flute angles are about 36° (quick helix), 27° (normal) 138 The Science and Engineering of Cutting 0.8 Torque (Nm) 0.6 Rotational drill speed N (rpm) 2000 0.4 1000 Thrust 4000 2000 0.2 1000 Torque 4000 0.05 0.1 0.2 Feeding distance per revolution f (mm/rev) Figure 5-15  Change in torque and thrust with feeding distance per drill revolution when polyethylene having a density of 0.95 g/cm3 is drilled Drill diameter was 8.1 mm, helix angle was 27°, and point angle was 120° (after Kobayashi, 1967) and 14° (slow) Since the cutting edges of drills remain in contact with the workpiece until the job is completed and are down at the bottom of the partially drilled hole, cooling may be difficult and flutes will clog As with the clearing of saw gullets (Chapter 7), periodical removal of a drill out the hole helps to unclog flutes The temperature rise may be significant if the feed rate is too high The need to have adequate channels along which swarf can escape and not jam, yet to retain sufficient torsional strength in a tool, causes problems in operations such as reaming (enlarging previously bored holes), tapping (cutting threads) and broaching (cutting slots), where cutting forces may be high This means that some of these operations have to be performed relatively slowly To ensure a ‘clean’ hole when making a hole right through a workpiece, the design of the point angle of drills is important With wood, in particular, there is the danger of leaving a torn edge on the undersurface of the workpiece when the drilling thrust force is high: a total included tip angle of not more than 60° minimizes this effect With hand-driven devices, when withdrawing the drill, it is customary to keep the rotation the same as when making the hole The type of woodworking drill having a flat end, a centre point and spur cutters on the periphery was originally intended to cut dowel holes in the furniture industry but is now widely used for all sorts of jobs To force blunt drills to cut requires a high thrust force with its associated problems of uncontrolled breakout when drilling a through hole: instead of all the displaced material being removed as chips, some may appear in a lip around top edge of the hole as a result of having to press so much – it is like a hardness impression with a rotating indenter Miller et al (2006) discuss so-called friction drilling in which a rotating conical disc simultaneously pierces and flanges sheet material Owing to the action of drills by which the whole cross-section of the hole is deformed, the material drilled out of a hole is waste, unlike punching holes in materials where either the hole or the disc or both may be used later There is some waste when using hole saws, but Slice–Push Ratio  139 the cut-out discs can be used for other purposes A form of hollow drill is the cork borer that removes a plug whole (Chapter 8), like an apple corer The sharp bottom edge of these devices separates material by splitting/cleavage, not by shear plane formation through the chip Cork borers have slice–push given by   /f, where  is the tube radius and  the angular velocity It is possible to drill apples, but rather silly as they are easily punched through by hand Geological core samples are made with hollow drills Hollow (cannulated) drills are used in surgery (Chapter 11) but for a slightly different reason, where they slide on prefixed stainless steel guide rods that direct the cutting edges to the right place Some materials, such as cellular foams, present special problems for drilling With metal foams, to make blind holes into which studs or screws can be inserted, flow drilling is employed where a polygonal ‘drill’, rotating at over 200 rpm, is pressed into the face of a sheet or plate The cell material ‘plastifies’ and becomes easily formable, so much so that material from the face flows into the hole Depths of holes are typically about three to four times the thickness of the face material (Seeliger, 2001) 5.5.1  Civil engineering drills Fence posts can simply be banged into the ground or a hole dug with a post-hole digger This device is a steel rod, turned by hand, at the bottom of which is fixed a flat disc having two cutting edges and flaps that permit spoil to pass up through the disc; when the device is lifted up out of the hole, the flaps close and the debris comes away It is not always possible to lift out spoil in this way and special narrow spades to reach down into deep holes form part of the kit Augers are used to make holes for telephone or electricity poles They not need the circumferential cutting spurs found in wood augers because the helix just scoops out loose earth like an Archimedean screw Civil engineers use large augers for taking foundation samples, preparation of some types of piling and in tunnelling In large-scale drill-like machines, a cutting head rotates about a central axis while penetrating parallel to that axis Cutting edges can variously be blades, discs, abrasive grains, studded or toothed rollers In order to cut a hole, material has to be removed across the whole radius Instead of a continuous cutting edge as in a drill, large machines employ a series of smaller tools arranged in a staggered pattern Overlapping knives give smooth and unbroken cutting like the wide cuts of a cylinder lawnmower As the cutting head rotates and penetrates at constant speed, the trajectory of every tool point is a helix, the angle of which becomes steeper at locations closer to the axis of rotation; along the axis, it is theoretically infinite (Mellor, 1975–81) Since tool linear speed varies with radius, a finite-width tool located close to the axis of rotation has different speeds at its two ends that travel along helical paths of quite different inclination Furthermore, since the depth of cut of every individual tool should strictly be taken with respect to its helical trajectory, it follows that the depth of cut varies with radius, and that tool penetration normal to the work face also varies with radius The behaviour of a cutter near the centre of the hole bears little relation to orthogonal cutting on which modelling is usually based In practical equipment, a central core of uncut material is sometimes left and allowed to break off periodically or some sort of ‘pilot bit’ like the point of a spear is employed As modelled earlier in this chapter, and with cutting devices for earth that are on chains (Chapter 14), the resultant force in civil engineering drills is usually resolved into components that are normal and parallel to the work surface, even though the head thrust down the axis of drilling FT is not strictly parallel to the axis of rotation since the tool cuts a helical path FT is provided either by deadweight and friction, or by reacting the machine against the surfaces 140 The Science and Engineering of Cutting of the hole When deep drilling in rock, the weight of the drill string can exceed the desired head thrust, so the drill string has to be held in tension at the upper end For devices such as rock augers (where tools follow shallow helical paths), the ratio of axial velocity to the cutting velocity is about 1:100 (rather like a pencil sharpener) so that the head thrust is given by the sum of the axial components of force on every cutter The broken-up debris/spoil of some materials occupies a greater volume than the solid from which it was cut; for example, loose crushed stone has about 45 per cent voids Thus in deep drilling with augers, the helical flight channels may have to accommodate a greater volume than that removed by the cutter In reports published by the US Army Cold Regions Research and Engineering Laboratory (CRELL), Mellor (1975–81) reviewed the relationship between head thrust and cutting head diameter in actual large machines, to see whether there was consistency in design Since installed power often drives the systems for clearing and hoisting spoil, however, it was difficult to get a clear picture between, for example, the relative cutting power and hole size In oil-well drilling, the spoil is usually removed by circulation of special fluids (muds), with up-hole velocities between 25 and 60 m/min In such conditions rotary power is consumed largely in shearing the drilling fluid and by hole-wall friction, rather than for actual cutting