Slice Push Ratio Oblique Cutting And Curved Blades, Scissors, Guillotining And Drilling

We shall discover that the slice–push ratio  given by blade displacement or velocity parallel to the cutting edge/blade displacement or velocity perpendic-ular to the cutting edge is im

111Copyright © 2009 Elsevier Ltd All rights reserved.

Slice–Push Ratio

Oblique Cutting and Curved Blades, Scissors,

Guillotining and Drilling

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 push-ing motion? In the case of a loaf of bread, it might be thought that this is to do 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 slid-ing 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 4 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 con-

tinuously 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 perpendic-ular 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

cutting edge Thin samples fed into a rotating disc cutter along the centreline are an ple of case (a), where  is given by (wheel peripheral speed/workpiece feed speed) Feeding

exam-a sexam-ample either exam-above or below the centreline into exam-a stexam-ationexam-ary wheel is cexam-ase (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 lab-oratory, 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 trac-tions 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

h f

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

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, ting 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

cut-a steeply inclined blunt blcut-ade mcut-ay be possible when, cut-at smcut-aller cut-angles, cutting fcut-ails Spcut-ades 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 incre-mental 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

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

[V2H ] [(dv)2 1 2 / 2(dh) ]2 1 2 / Rwdv (5-2)The slice–push ratio  is given by (dh/dv), whence solution of these two simultaneous equa-tions gives

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 ized 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

normal-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 dis-placement 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  

0 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

dif-ference 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ξsin )i (5-5a)

Facross f HcosiV sini V( cosξ isin )i (5-5b)

θ = 6°, μ = 0.3

θ = 12°, μ = 0.3 Frictionless

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

using Eq (5-3) When   tani,

Falong f V/cosi Rw /[eff 1ξ2]cosi Rw (5-5c)

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/cos ) ]θ2 (dv/cos ) [( cos )θ  ξ θ2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µcos )cos ] [( cos )θ θ ξ θ21]dv (5-7)

Blade dν

dν

θ

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

substituting for N in terms of V from Appendix 1 The expression in Eq (5-7) may lently be obtained by summing the work done by the component Ncosq of N along the line

equiva-of greatest slope equiva-of the wedge times (dv/cos), plus the component Nsinq equiva-of N parallel to the cutting edge times dh

Equating external and internal work increments for an orthogonal cut with a moving blade gives

sideways-V/Rw 1 1/{ ξ2[(2)µ(( cos )ξ θ21)/cos (sinθ θµcos )]}θ (5-8a)and

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/Rweff 1/cos {i 1ξ2[( )2µ(( cos )ξ θ21)/cos (sinθ θcoos )]}θ (5-9a)and

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 ini-tially 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

anti-clockwise The disc rotates with angular velocity , where positive  is antianti-clockwise 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 wise direction, and a radial component given by fcosi The local velocities (displacements) at P

anticlock-normal to the cutting disc dvdisc, and parallel to the edge of the disc dhdisc, are thus

f f

i

i

fcosi fsini

P

ρω

ρ ω

A

0 0 0.05

–0.05 –0.1 –0.15

0.1 0.15 0.2 0.25 0.3

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

where positive dhdisc has the sense of  The slice–push ratio at P in an anticlockwise sense is

ξdisc dhdisc/dvdisc (ρωfsin )/fcosi i( /fcos )ρω i tani (5-11)

For cutting above the centreline, i is negative and tani changes sign If (/f)  sini, the tribution of tool obliquity to the push/slice effect will not be noticeable except at large i (cutting

con-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ξdiscsin ]i (5-12a)

and

Facross f V[ξdisccosisin ]i (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 cir-cular 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ξ 2) (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 model-ling 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

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 trac-tor 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 pas-sive 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 tion,  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

direc-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

[Hrecip/Rw]=(h /f)sin t/[1+(h /f) sin t]o ω o 2 2ω (5-14a)

and

[Vrecip/Rw]1 1/[ (h /f) sin t]o 2 2ω (5-14b)

and

(F /Rw)Res ( /[1 1(h /f) sin t])o 2 2ω 1 2 / (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 mers, sheep-shearing comb cutters and electric carving knives (see Chapter 10)

trim-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

(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

cut-ting 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 do calculations in terms of the effective blade included angle effrather 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

H/Rw V/Rw

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

angle measured in the plane normal to the cutting edge It is the angle of greatest slope and is the same as the tool rake angle used in orthogonal cutting; n is variously called the ‘normal’,

‘oblique’ or ‘primary’ rake angle Similarly, the angle of greatest slope of the shear plane or

‘normal shear plane angle’ fn is usually employed to define the inclination of the primary shear plane in oblique cutting

The resultant force Fres has components FC parallel with the velocity approach vector VW,

FT perpendicular to the finished work surface, and FR perpendicular to the other two FC is the ‘power’ force, FT is the ‘thrust’ force and FR is the ‘radial’ (sideways) force These are the forces usually measured by a dynamometer They are related by force resolution (Amarego &

Brown, 1969; Shaw, 1984) It is not clear that the force vectors are co-linear with their tive displacement vectors (Shaw et al., 1952) In the plane of the finished surface, the compo- nents of the resultant force and resultant displacement are coincident; it is what happens out

respec-of that plane that is uncertain In what follows we shall assume for simplicity that the force and velocity components are co-linear; it turns out to be an acceptable approximation when comparing theory and experiment Another approximation often employed is ‘Stabler’s rule’, which says C  i The reader interested in the detail of oblique cutting of ductile materials

should consult original papers and standard texts, in particular those by Amarego and Brown (1969) and Oxley (1989)

By equating the external and internal work rates that include toughness as well as plasticity and friction, an expression is obtained for the power force FC (Atkins, 2006):

FC ( /Q1 shear oblique) [(kwγoblique)tRw] (5-15a)

F /Rw=(1/QC shear oblique) [(γoblique/Z)+1] (5-15d)

in which Z  (R/kt) is the non-dimensional term incorporating the toughness/strength ratio and uncut chip thickness; (1/Z) represents the non-dimensionalized uncut chip thick-ness Relation (5-15) has the same form as Eq (3-32), but with different expressions for the shear strain  and the friction factor Q Without the fracture term, i.e when R  0,

FC  (kwoblique)t/Qshear oblique is the plasticity-and-friction-only solution for oblique cutting first given by Amarego (1967) Experiments show that fn is essentially independent of the angle of obliquity, other things being equal, and is almost constant at sufficiently large t (sufficiently small Z) as in orthogonal cutting (Atkins, 2006) It follows that for a given workpiece mate-rial, f in orthogonal cutting and fn oblique cutting are the same It is also found that S does not alter too much with obliquity, whence from Eq (5-15c), oblique is approximately constant For a given tool rake angle and friction, quasi-linear plots of FC and FT vs uncut chip thickness

should therefore be approximately independent of i Equation (5-15a) says that there will be a

positive force intercept in plots of cutting force vs uncut chip thickness, and that it is a measure

of the material toughness R As in traditional analyses, the shear yield strength k is obtained from the slope of the plots The power required for cutting over this same range of obliquity

is also approximately constant The corresponding quasi-linear plots of the sideways force FR

vs uncut chip thickness do depend on i and, for given tool rake angle, increase as i increases

mainly because Qoblique decreases Experimental data from Brown and Amarego (1964)

con-firm these dependencies If, instead of varying i at constant n, n is varied at constant i, plots

of FP and FQ vs uncut chip thickness now depend on n, but this time experimental results in Brown and Amarego (1964) show that FR is apparently independent of n At small t (large Z), fn is predicted to become smaller, so oblique becomes greater; but Qshear oblique increases at smaller t, and the net result is that FC vs t plots droop downwards near the origin and have an intercept of Rw since Qoblique  1 at zero t when f  0, exactly as for orthogonal cutting.For given material (R/k) ratio, tool rake angle, friction and uncut chip thickness, the primary shear plane angle fn may be predicted by minimizing the total work done or, equivalently, by minimizing Eq (5-15a) Experimental data for oblique cutting give reasonable agreement with predictions (Atkins, 2006)

The specific cutting pressure (‘unit power’) given by FC/wt becomes

F /wtC ( /Q1 oblique)[γoblique kR/t](k/Qoblique)[γobliqueZ]] (5-16)

As with orthogonal cutting, (FC/wt) in oblique cutting is expected to rise disproportionately at small t owing to the inverse-dependent final term on the right hand side of the relation (Section 3.6.7) Experiments confirm that the specific cutting power does indeed rise to large values

at small t But when data are analysed in terms of the plasticity-and-friction-only theory, the shear yield stress k (given by FCQoblique/wtoblique) must also rise to extremely large values.

5.3.1 Napier’s rotary cutting tool

The analysis given above concerns a straight-edged tool that is angled to the direction of motion

of the workpiece, in which slice–push results simply as a consequence of i (Figure 5-1B) There

is no reason why a tool for cutting ductile materials should not have independent motion lel to its cutting edge and produce enhanced  Napier invented such a tool in Victorian times

paral-It consisted of a small chunky disc that was driven and could be rotated independently of the motions of the rest of the cutting machinery Thus, on a lathe, it acted like a round-nosed tool that cut in the usual way with set depth of cut, feed and speed, but additionally revolved

Shaw et al (1954) analysed the behaviour of rotary cutting tools using an equilibrium approach that, as explained in Chapter 3 and Appendix 1, is acceptable when toughness is omitted To employ equilibrium for the internal work components requires that (FC/w  R)

be employed in place of (FC/w) The reduction in forces was not explained in terms of ,

rather in terms of i, but they are related In Shaw and colleagues’ work with a driven rotary

tool, the greatest  was 2.5; given the much greater  possible with disc cutters, it would be interesting to know the performance of rotary tools at greater  The ability of the rotary tool

to become self-propelled is related to what happens with the disc cutter when the feed force goes negative (Figure 5-4B), the workpiece being grabbed by the wheel

In conventional cutting, the tool tip is in continuous contact with the workpiece, and is perpetually hot at commercial cutting speeds even with coolants and this can lead to tool failure Some respite occurs with tools taking interrupted cuts: for example, it is possible to use high-speed steel (HSS) tools in interrupted milling if the tool is adequately cooled in the idle phase In the Napier rotary tool, parts of the rotating cutting edge continuously move out of the hot cutting zone to cool before recutting and should therefore experience reduced wear and longer tool life However, the device has the extra complication over ordinary tool-ing of requiring a very stiff holder that allows the tool to rotate and/or to be driven Since modern tool materials have long lives and can withstand heavy cutting, the benefit of in-built cooling of the tool may no longer be important Even so, given that slice–push reduces cut-ting forces, and that surface damage is less with high , there should be applications of the device for difficult-to-cut materials Chang et al (1995) remark that because the nose radius

of rotary tools is much larger than conventional tools, feed rate has much less influence on the machined surface roughness There is similarity in action between ball end-mills and the rotary tool (Section 4.1)

5.4 Guillotining edges

Instead of cutting the whole length at one fell swoop, it makes sense to incline the blade in cropping and perform the cut progressively Although the total work required to cut an edge may be comparable in the two cases, forces in guillotining are lower since a longer stroke is involved to cut the same edge area Equipment can be lighter and potential damage to the cut edge reduced In contrast to unsteady blanking and orthogonal cropping, there is a steady-state region for much of the stroke in guillotining

Some guillotines have a cutter in the form of an undriven cutting disc; in others the blade may be straight and move through the workpiece always having the same orientation, as when

cutting paper with an inclined razor blade and, of course, in guillotines used for execution (Chapter 11) In the case of a disc cutter, the forces may be derived using the analysis given in Section 5.2.3; in the second case, from the expressions in Section 5.2.2 Alternatively, the guil-lotine may have a long straight cutting edge pivoted at one end which is levered down through the workpiece In this case the angle changes continuously, although some pivoted blades are curved with the intention, it seems, of maintaining the same angle at the point of cutting Curved blades are also employed in hand tools like secateurs and scythes Whether there are optimum shapes of curved cutting edge for least cutting forces is explored in Chapter 10.Whether the offcut is permanently deformed or not depends on the R/k ratio for the mate-rial The gearing equivalence (that the work given by the force on the handle of the blade times its stroke remains constant) is lost when cuts are formed by ductile shear owing to the different planes in which offcuts curl It is also lost when friction is significant In orthogonal cropping the offcut is comparatively undeformed, except at the sheared edges, but guillotined offcuts of ductile materials are permanently bent owing to the inclined tool.

5.4.1 Floppy materials

An office paper guillotine mounted in the frame of a testing machine was used by Atkins and Mai (1979) to determine the fracture toughness of thin sheets of materials A graphical approach was employed, as displayed in Figure 5-7, where the forces for cutting are shown together with forces on second cuts, after the material has been parted, in order to establish fric-tion A third cut would establish the forces required to scrape the ‘set’ blade over the baseplate

of the device; a set blade is curved and crosses the baseplate to give clean cuts See Table 5.1

It is common experience that narrow offcuts of paper form into permanently deformed open helices, but that wider cuts remain flat for all practical purposes Irreversibilities in paper come about from fibres that permanently slide over one another, as happens in simple tearing of paper where permanent curling may be produced Analysis of the formation of helices is given in Section 5.4.3

In high-quality bookbinding, the edges are not guillotined but rather ploughed A plough

press in bookbinding is a vice in which the book is held while edges are orthogonally planed

The blade of a book plough has a convex half-round shape and a large rake angle (it is nearly parallel to the cut surface).

5.4.2 Scissors

Figure 5-8(A)shows the geometry of a typical pair of scissors The half-thickness t of the rial being cut, the angular opening  of the scissors during cutting, and the half-separation 

mate-of the handles, are related (Atkins & Xu, 2005) Figure 5-8(B) shows the experimental results

of Perieira et al (1997) in which samples of palmar skin were cut by scissors The mens were some 1.4 mm thick The upper of the two experimental force–displacement plots gives the cutting load, and the lower the forces to close the scissors after the cut has been made The frictional contribution to the total force is about 25 per cent at all displacements

speci-table 5-1 Fracture toughness values determined from guillotine experiments.

Rubber reinforced with