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

Chapter 5

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 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 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 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 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


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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


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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 2  H 2 ]1/ 2 [(dv)2  (dh)2 ]1/ 2  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/ 2



(5-4)


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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  
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 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

4
V/Rw and H/Rw

(5-5a)

θ = 12°, μ = 0.3

3.5

Frictionless

3
2.5
2
1.5
1
0.5
0
0

1

2

3

4

5

6

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.


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Slice–Push Ratio 
using Eq. (5-3). When   tani,


Falong f  V/cosi  Rw eff /[1  ξ2 ]cosi  Rw



Facross f  0

(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

Cos θ
dh

dh
dr

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.


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substituting for N in terms of V from Appendix 1. 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  0

(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)


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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
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)


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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


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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/ 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 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


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1.2
1
0.8

Fres/Rw

0.6

V/Rw

0.4
H/Rw
0.2
0

0

0.5

1

1.5

2

2.5

3

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 do 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


121

Slice–Push Ratio 

Shear plane

Tool rake face
Vchip

Vshear
Zc

α

Zs
i
Vwork

Figure 5-6  In orthogonal cutting, the shear velocity VS is along the line of steepest slope in the
primary shear plane and the chip velocity VC is along the line of steepest slope of the rake face of
the tool. In oblique cutting, VS is at an angle S (the shear flow angle) to the normal to the cutting
edge in the shear plane, and the chip flows over the rake face of the tool at angle C (the chip flow
angle) to the normal to the cutting edge.

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 respective displacement vectors (Shaw et al., 1952). In the plane of the finished surface, the components of the resultant force and resultant displacement are coincident; it is what happens out
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  (1/Qshear oblique ) [(kwγ oblique )t  Rw]



(5-15a)


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The Science and Engineering of Cutting

where


Qshear oblique  1  (F/FC )[sinφn /cos(φn  α n )]

(5-15b)



with F the friction force along the rake face, and


γ oblique  [cotφn  tan(φn  α n )]/cosηS

(5-15c)



(Note: there is a misprint on p. 87 of Amarego and Brown (1969) in their Eq. (4-37) for
oblique, where ‘cos’ is written in the numerator instead of ‘cot’.)
Alternatively,


FC /Rw=(1/Qshear oblique ) [(γ oblique /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 thickness. 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 material, 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) confirm 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


FC /wt  (1/Qoblique )[γ oblique k  R/t]  (k/Qoblique )[γ oblique  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


Slice–Push Ratio 

123

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 parallel to its cutting edge and produce enhanced . Napier invented such a tool in Victorian times.
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 tooling 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 cutting 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 steadystate 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


124

The Science and Engineering of Cutting

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 guillotine 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 material. 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 friction. 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.
250

Load F (N)

200
B

A

150

100

C
D

50

0

2

4

6

8
10
12
14
Deflection δ (mm)

16

18

20

Figure 5-7  Load–deflexion traces for guillotining a single layer of manila folder, 0.26 mm thick.


125

Slice–Push Ratio 

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 material being cut, the angular opening  of the scissors during cutting, and the half-separation 
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 specimens 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.
Table 5-1  Fracture toughness values determined from guillotine experiments.
Material

t (mm)

R (kJ m--2)

Remarks

Interleaving paper

0.06

16.93

1 layer

0.24

10.32

4 layers

Copier paper

0.36

12.70

4 layers

Cardboard paper

0.48

13.60

Drawing paper

0.27

18.50

0°  angle of cut with fibre direction

26.00

45°

29.00

90°

0.26

15.16

1 layer

0.52

13.63

2 layers

Waxed paper

0.18

15.10

2 layers

Rubber reinforced with
cotton fabric

1.05

4.35

Aluminium foil

0.105

10.68

8 layers

Shim brass

0.05

53.18

1 layer,  rolled direction

0.10

33.38

2 layers,  rolled direction

0.05

63.93

1 layer,  rolled direction

0.10

42.86

2 layers,  rolled direction

0.12

41.45

 rolled direction

0.11

41.39

 rolled direction

0.11

18.47

 rolled direction

0.11

25.95

 rolled direction

0.22

21.41

2 layers,  rolled direction

0.22

22.59

2 layers,  rolled direction

0.16

42.06

 rolled direction

0.16

47.15

 rolled direction

Manilla folder

Copper foil

Shim steel


126

The Science and Engineering of Cutting
4

Force F
Displacement δ

3

θ
2t
h

j

Scissors load (N)

y

2

1

0
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



E



Tapered burnished
land

Uncracked

B
F

C



D


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
6 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

5

ry

o
he

Guillotine force (kN)

/k y =

r y R*

Theo

10
2.6 x

–3 m

0

ry
eo

/k y

R*

1
5x

The Science and Engineering of Cutting

Guillotine force (kN)

Theo

x 10
= 3.4

Increasing clearance

10° blade
6 mm thick copper
83 VPN

–3 m

0

/k y

R*

1
4x

Th

10

Slope = 375 kN/m
∴k = 125 MPa
Intercept = 6.5 kN
∴R* = 505 kJ/m2

5

(R*/ky) = 4 x 10–3 m
0
0
A

6

12

18

Offcut overhang width w (mm)

0

24
B

0

6

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 3. 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 do 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)

2

200

0.17

57

5

200

0.40

133

1

158

0.30

79

3

158

1.00

263

Brass

6

110

0.95

171

Stainless steel

2.4

175

0.40

114

Soft low-carbon steel

Source: Atkins (1988b).


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The Science and Engineering of Cutting
Tapered copper
10° blade
w = 18 mm

Guillotine force (kN)

15

25° blade
w = 18 mm

10

0
5

10

20 30 40 50
Blade travel (mm)

60

B

0
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.)


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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 8 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 do 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


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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 do 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


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