Slice–push, formation of grooves and the scale effect in cutting

Abstract

Three separate aspects of cutting are investigated which complement other papers on the mechanics of separation processes presented at this interdisciplinary Theo Murphy meeting. They apply in all types of cutting whether blades are sharp or blunt, and whether the material being cut is ‘hard, stiff and strong’ or ‘soft, compliant and weak’. The first topic discusses why it is easier to cut when there is motion along (parallel to) the blade as well motion across (perpendicular to) the cutting edge, and the analysis is applied to optimization of blade geometries to produce minimum cutting forces and hence minimum damage to cut surfaces. The second topic concerns cutting with more than one edge with particular application to the formation of grooves in surfaces by hard pointed tools. The mechanics are investigated and applied to the topic of abrasive wear by hard particles. Traditional analyses say that abrasive wear resistance increases monotonically with the hardness of the workpiece, but we show that the fracture toughness of the surface material is also important, and that behaviour is determined by the toughness-to-hardness ratio rather than hardness alone. Scaling forms the third subject. As cutting is a branch of elasto-plastic fracture mechanics, cube-square energy scaling applies in which the important length scale is (ER/k²), where E is Young's modulus, R is the fracture toughness and k is the shear yield strength. Whether, in cutting, material is removed as ductile ribbons, as semi-ductile discontinuous chips, or by brittle ‘knocking lumps out’ is shown to depend on the depth of cut relative to this characteristic length parameter. Scaling in biology is called allometry and its relationship with engineering scaling is discussed. Some speculative predictions are made in relation to the action of teeth on food.

1. Slice–push in cutting

It is common experience that, however sharp a knife, cutting is easier when the cutting blade has motion both parallel to the cutting edge and perpendicular to the cutting edge, i.e. both ‘slice’ and ‘push’. By ‘easier’ we mean that the force required when simply pushing a knife down into a workpiece is reduced when the blade is simultaneously moved along the surface of the workpiece. ‘Paper cuts’ (cutting the tongue on licking the flap of an envelope) are caused by this phenomenon. The effect does not depend on there being teeth on a blade, such as on a breadknife; a loaf may be cut perfectly well with a sharp plain blade (table 1).

Table 1.

Nomenclature.

Facross f	force perpendicular to a sheet fed into a rotating disc
Falong f	force required to feed a sheet into a rotating disc
FC	cutting force parallel to the surface during groove formation
FT	cutting force perpendicular to the surface during groove formation
f	feed rate of sheet into disc cutter
H	‘horizontal’ force parallel to cutting blade
h	‘horizontal’ displacement parallel to cutting blade
Fres	resultant force acting on cutting blade
KC	critical stress intensity factor
k	shear yield strength
R	fracture toughness of the workpiece
r	radius of delicatessen disc slicer
t	depth of groove or depth of cut
V	‘vertical’ force perpendicular to cutting blade
VPN	Vickers indentation hardness
v	‘vertical’ displacement of blade
W	dead-weight load
w	width of cut
α	rake angle of tool measured from the normal to the cut surface
β	angle of friction between tool and workpiece (tanβ = μ)
δ	semi-angle of the point of the grooving tool
θ	included angle at tip of wedge-like blade
φ	angle of the primary shear plane
μ	coefficient of effective friction between tool and offcut
λ	attack angle of grooving tool. λ = (90 + α)°; meaning will be clear from the context
λ	magnification factor in scaling; meaning will be clear from the context
ω	angular velocity of delicatessen slicer disc
ξ	the slice–push ratio given by the displacement or velocity along the cutting edge divided by the displacement or velocity across the cutting edge

Work is required to form a new cut surface, and work is provided by forces applied to the blade moving their points of application (along and across the blade). If work is supplied ‘sideways’, less work, and hence a lower force, will be required ‘vertically’ to produce the same cut area. Were the effect to be linear, provision of 30%, say, of the total work by sideways action would mean that only 70% of the work would have to be supplied vertically and that, over the same vertical displacement, the vertical force would be reduced to 70% of the value required when there is no slice action. However, experiments show that the effect is disproportionate in that even a small slice action results in a large reduction in normal force, and that the greater the slice velocity the greater the disproportionate reduction in normal force. There is a nonlinear coupling between the forces.

Calling the vertical force V, the horizontal force H and the resultant force F_res, it may be shown that for the case of a thin sharp blade having zero friction cutting floppy materials that are incapable of storing elastic energy and that do not dissipate energy irreversibly, we have [1]

1.1

1.2

1.3

1.4

where ξ is the slice–push ratio given by the displacement or velocity along the cutting edge divided by the displacement or velocity across the cutting edge; R is the fracture toughness of the workpiece and w the width of cut. Figure 1 shows V and H (both normalized by Rw) plotted against ξ for various conditions. Equations (1.2) and (1.3) are the two curves marked ‘frictionless’. We see that (V/Rw) decreases continuously from unity as ξ increases. (H/Rw) initially rises to a peak of 0.5 when ξ = 1 and then decreases continuously. Also included in figure 1 are plots for (V/Rw) and (H/Rw) in the presence of friction (μ) and when the blade is wedge-like with finite included angle θ [2]. The overall levels of cutting forces are higher, but the general effects are similar. Forces are smaller the greater the slice–push ratio, but the rate of decrease in force diminishes at increased ξ so that the ‘law of diminishing returns’ applies when operating cutting devices at extremely large ξ. Experiments [1] confirm these general trends.

Figure 1.

How the normalized vertical (V/Rw) and horizontal (H/Rw) forces change as the slice–push ratio ξ is increased.

It should be noted that the above analysis presumes that the resultant force vector is parallel to the resultant displacement vector. It is not obvious why this should be the case, but experiments show that in more complicated cases when the blade has a large finite included angle, and in the presence of friction, and with permanent deformation of the offcut, the assumption is reasonable (see ch. 5 of [2]).

Combined motion may be achieved in various ways, by simple angling of the blade with only vertical motion, and by sliding the blade as well as angling it when moving vertically (figure 2). Tools such as wood planes and scrapers are often angled in use. Without motion along the cutting edge, ξ = tan i for an angled blade where i is the angle of inclination [2]. For a 60° inclination ξ = √3 so, practically, it is difficult to achieve large ξ simply by angling: the blade has to be driven. This is shown by the action of a rotating delicatessen slicer, which also illustrates the difference between ‘uphill’ and ‘downhill’ slice–push. In figure 3a, a disc cutter of radius r, rotating at angular velocity ω, has a tangential velocity of rω. A thin frictionless sheet is fed into the disc at speed f that has components fcosi towards the centre of the disc and fsini tangential to the disc, where the angle i indicates the level of entry of the sheet relative to the horizontal centre of the cutter. For entry below the centreline (downhill, as shown in figure 3a), the tangential component of f and the tangential velocity of the disc are in the same direction and sum together to give an enhanced slice component. For entry above the centreline (uphill), they are in opposition and the effective slice is reduced. Figure 3b shows how the force (F_{along f}) required to feed the sheet into the rotating disc, and the force (F_{across f}) perpendicular to the sheet vary with angle i that describes the level of entry [1]. Of note is that while the perpendicular force is always positive and pushes the sheet downwards, (F_{along f}) becomes negative at large-enough angle below the horizontal centreline. This means that the sheet no longer requires a force to push it against the cutting disc, rather that the disc grabs the sheet and pulls it into the disc. This sort of thing is well known with grinding wheels and circular saws, and can be dangerous.

Figure 2.

Combined motion can be achieved by various combinations of blade orientation and motion.

Figure 3.

(a) Geometry for a thin sheet cut by a rotating delicatessen disc. (b) Variation of horizontal (F_{along f}) and vertical (F_{across f}) forces for cutting versus position of entry into the rotating disc defined by angle i.

Napier [2] invented a tool for cutting ductile metals that consisted of a small chunky disc that could be rotated independently of other motions of the machinery: on a lathe it acted like a round-nosed tool that cut in the conventional way but with independent motion parallel to the round cutting edge to give enhanced ξ. The device became self-propelled when the feeding force became negative (cf. the negative forces in figure 2).

Knife sharpness is sometimes assessed by cutting a vertical sheet of paper with an inclined blade. A downstroke requires a smaller force than an upstroke for the reasons given above, so the blade appears to be sharper. Since slice–push applies as much to blunt blades as to sharp, angling a blade to cause slice–push and thus reduce forces may compensate for bluntness. Rotary lawnmowers cut quite well with badly damaged blades owing to the high rotational speed giving high ξ; it is the tangential edge, rather than the radial edge, of the blade that does most of the cutting. Workpieces such as blades of grass and cereal crops are constrained only within the ground but otherwise can wave about. It turns out that even for the sharpest blades, a minimum speed is required to cut, owing to inertia of the stalk, see ch. 10 of [2].

Note that when slice–push is achieved by reciprocating a blade, as in carving meat, the slice–push effect is the greatest at mid-stroke but is zero at the ends of the stroke, since the blade is then instantaneously at rest. Consequently, although the average vertical forces are reduced in carving, the benefit is not as great as when the blade continues to move in the same direction (see fig. 5–5 in [2]).

Figure 5.

The two modes of deformation when forming a groove with a pointed tool.

When slack wires are used to cut (as in the grocer's cheese cutter) the slice–push ratio ξ determines the curved shape taken up by the wire [3]. The motion of the hand at the end of the wire when operating such a device is one that results in minimum work done (M. Charalambides 2014, personal communication). Experiments show that all cut surfaces have better quality with fewer blemishes, and less sub-surface damage, when produced at lower forces, i.e. at high ξ. A taut wire with motion along its length can have large ξ, and when used to cut juicy fruit there is far less loss of liquid through sub-surface cell damage than with conventional methods of cutting.

Instead of being circular, cutting blades can have variable curvature, and thus variable ξ along the edge. This leads us to ask what is the best geometry for blades of a scythe or sabre, what might be the optimum helical angle on a cylinder (US: ‘reel’) lawnmower, and what are the curved shapes taken up by cords and flexible wires during cutting in devices such as strimmers [4]. When only a portion of a rotating blade having variable curvature is in contact with the workpiece, the instantaneous force and torque required to cut is obtained from integrating along the current contact length. This is then repeated for successive arcs of contact, from which the variation of force and torque with time throughout whole cut is given [4]. Similar analyses should be possible for the shapes of animal beaks, claws, teeth, etc.

2. Formation of grooves: abrasive wear

Cutting often involves tools that cut with more than one straight edge. A simple example is the cutting of a ledge in a ductile solid (figure 4), where both the side and bottom of the tool cut new surfaces. This is how a ‘shaping machine’ functions. In the case of a tool with a round-nose between straight flanks, material is removed to leave a curved surface between the vertical and horizontal parts of the ledge.

Figure 4.

Cutting with two straight edges to form a ledge.

The formation of grooves concerns tools that cut with more than one edge (figure 5). It is found that material is removed from the groove by one of two mechanisms. When the tool ‘leans back’ (has a positive rake angle α measured from the normal to the surface) material is cut out of the groove and forms a chip. In ductile materials, the chip will be a continuous ribbon (figure 5a). (Instead of rake angle, the so-called ‘attack angle’ λ = (90 + α)° measured from the horizontal is often employed to indicate the inclination of the tool [5]). When the tool ‘leans forward’ sufficiently (has a large-enough negative rake angle, i.e. small attack angle), a different mode of deformation occurs. Instead of cutting, material is displaced from the groove by plastic flow into piled-up ridges alongside the groove by passage through a prow of material ahead of the tool, in a sort of breast-stroke swimming action (figure 5b). The same phenomenon occurs in two-dimensional cutting of ductile solids when the rake angle is very negative: instead of cutting, a wave of material is pushed ahead of the blade, rather like the method of thinning the walls of beverage cans by so-called ‘ironing’ [6]. In practical workshop machining, continuous/discontinuous chips are formed since the rake angles employed are positive or only slightly negative (the latter to put ceramic and similar hard brittle tools under compression when cutting).

Analysis of the forces required by the two modes of deformation in ductile materials was performed for facet-first sliding of pyramids [7]. For ridging, an upper bound analysis based on Childs [8] was used. For cutting, Amarego's ‘plasticity and friction’ analysis for the machining of a groove by a pointed tool [9] was used, with the addition of a toughness term for material separation at the tool tip (discussed further below). When plotted against attack angle, two sets of force curves are shown—one rising from left to right and the other from right to left—that intersect at λ_trans (figure 6). The mode of deformation that occurs will be that requiring the lower force (lower work over the same displacement). We see that displacing material up into ridges alongside the groove is favoured at small λ, and cutting at large λ. Unsurprisingly, friction increases the forces required for both modes, but a greater influence when cutting is the non-dimensional parameter Z = R/kt, where R is fracture toughness, k is the shear yield stress and t the depth of the groove. High values of Z (tough, soft materials with shallow grooves) delay cutting to larger λ; low values of Z (less ductile, harder materials with deeper grooves) result in cutting at smaller λ. For many metals and polymers, experiments show that 45° < λ_trans < 85°. The nominal angle between the three faces and the axis of a Berkovich indenter is 65.3°, so it suggests that facet-first nano-scratching experiments on some materials may result in ridges alongside grooves but in other materials may give cutting. Similar transitions between deformation modes occur for other-shaped tools when forming grooves (for pyramids slid edge-first, cutting prevails at lower attack angles). Differences in mode are important for engraving, linocuts and so on (see §6.9 of [2]). Similar mechanics apply to the formation of ‘ice gouges’, metres in size, in arctic shorelines [10]. When balls are used to form grooves, the attack angle is not constant and depends upon the depth of groove: shallow grooves give ridges as the contact angle is almost tangential to the surface; deeper grooves eventually cut material away, the greatest attack being λ = 90° when the ball is indented up to its waist. Determination of the transition attack angle when scratching a surface permits estimates to be made of the fracture toughness [7]. In archaeometallurgy, specimens are often too small for conventional mechanical property tests. Nevertheless, details of microstructure and microhardness can be obtained from small samples, but not fracture toughness. The transition attack angle method suggests a way forward: the performance of arms and armour cannot be assessed without knowledge of toughness (e.g. [11]).

Figure 6.

Variation of normalized forces with attack angle for the two modes of deformation when forming grooves by facet-first sliding of a pointed tool.

Mechanisms of abrasive wear and erosion concern the scratching of grooves on surfaces by hard particles. In erosion, particles are loose and ‘free-flying’ in a fluid stream but with abrasive papers, although the grits have many different shapes, orientations, and a wide range of attack angles, they are nevertheless fixed to the backing paper. In theory, there is no wear when material is piled up into ridges alongside the groove by grits since there is no separation of material. Only when a groove is formed by cutting does wear occur. While ridges can be removed by subsequent passage of particles across the initial groove, the efficiency of material removal by abrasive papers is not very high since most of the attached abrasive particles have low attack angles.

Early theories of wear [12] argue that the depth t and width w during sliding when forming a groove under load W, is the same as that produced in a static hardness indentation using as indenter the same tool that forms the groove under the same load. Using Vickers hardness VPN, for example, we have VPN ∝ W/t² [13] and it follows that wear rate ∝ t² ∝ W/VPN, and wear resistance ∝ (1/t²) ∝ VPN/W.

Kruschschov & Babichev [14] performed experiments in which materials of different hardness were worn against abrasive papers. Figure 7 shows their results between relative wear resistance and the hardness of the worn metals. The wear resistance of ‘pure metals’ increases in direct proportion to hardness. However, there are departures from a simple linear relation for metals that have been strengthened by thermomechanical treatments. It follows that hardness alone cannot be the determinant of abrasive wear resistance and that there must be an additional material property that is involved. Following the analysis of groove formation above, it seems that the missing factor must be Z = R/kt, that includes the fracture toughness R, as well the hardness (hardness is proportional to k, [13]).

Figure 7.

Relative wear resistance as a function of indentation hardness for metals rubbed against abrasive papers.

During groove formation under dead-weight load, the force FT acting on the tool perpendicular to the surface is the dead-weight W. FT is related to the cutting force FC through friction by FT = FC tan (β – α) where α is the rake angle of the tool and β is the friction angle given by tan⁻¹μ, where μ is the coefficient of effective friction on the faces of the tool (‘effective’ since there will be a mixture of stuck and sliding regions on the rake face [15]). Using Amarego's analysis [9] for cutting a groove of depth t with a pointed tool, but modified to include separation work [7], we have

2.1

where Q is the friction correction factor given by the expression [1 − {sinβ sinφ/cos(β − α) cos(φ − α)}] with φ the angle of the primary shear plane and δ the semi-angle of the point of the tool. This is the equation of the curves for FC on the right-hand side of figure 6. It can be concluded from equation (2.1) that the depth t of the groove during sliding is not the same as the depth of a static indentation under the same load, even for R = 0.

When equation (2.1) is applied to abrasive papers, some representative values have to be assumed for α, β, δ, φ and λ because there is a range of different grit geometries having different attack angles and different orientations to the direction of sliding. The equation can be solved for t to determine the influence of (R/k) on wear rate and wear resistance as the hardness changes. Alternatively, by dividing throughout by t², equation (2.1) may be expressed in terms of Z = (R/kt), after which it may be shown that

2.2

where c₁ and c₂ are constants whose magnitudes depend on the representative values assumed for α, β, δ, φ and λ for the particular grade of abrasive paper. Since hardness is proportional to k [13], equation (2.2) says that in wear resistance versus hardness plots of the sort shown in figure 7, there is really a series of straight lines emanating from the origin with slopes dependent on (c₁ + c₂Z) as shown schematically in figure 8. The lowest slope is for Z = 0 (i.e. R = 0) which would result from a ‘plasticity and friction only’ analysis for FC [9]. Materials having the same hardness, but increased fracture toughness, lie on steeper lines.

Figure 8.

Variation of wear resistance with hardness at different constant Z = (R/kt).

That many ‘pure metals’ in the Krushschev–Babichev plot fall on to a single line implies that they all have roughly the same Z; those thermomechanically treated harder metals on the right lie on lines of smaller Z, i.e. smaller R/k ratios at the same t, meaning less ductile solids. We note that if toughness were unimportant in abrasive wear, the Kruschschov and Babichev data would fall on the single line for Z = 0 and we should not expect any results to the right of the line. Experiments are underway at Imperial College to investigate the relevance of R/k in abrasive wear (M. Masen 2015, personal communication).

Work is also underway to study the wear of teeth in terms of this sort of analysis, as discussed by Lucas at this meeting [16].

3. Scaling

In engineering, scaling concerns the prediction of forces, stresses, strains, etc., in a large prototype from measurement of the same parameters in a small model. Wind tunnels, ship towing tanks and geomechanical centrifuges are employed to scale from model to full-size behaviour through the constancy between model and prototype of characteristic non-dimensional numbers associated with the names of Mach, Nusselt, Reynolds, Froude, Atkins (for ice-covered towing tanks), etc., as explained by Palmer [17] (see also Barenblatt [18]). Usually geometrically similar models of the prototype are used, where the scaling factors λ for ‘height’, ‘width’, ‘thickness’ and ‘crack length’ are identical, but sometimes non-proportional scaling is employed, as when it is possible to test in the laboratory a section of a pipeline but impossible to duplicate its length [19].

The term ‘scale effect’ or ‘size effect’ is used in materials science when behaviour with size does not follow expected/preconceived laws, such as properties changing owing to microstructural effects (cf. nano properties). In cutting, it was observed a long time ago that the so-called ‘specific cutting pressure’ or ‘unit power’, given by FC/wt, increases for no apparent reason at small depths of cut. In the metal-cutting literature, this seemingly anomalous result was called a ‘scale’ or ‘size effect’ [20] (a term still used unfortunately) but it is nothing of the sort since the modern analyses of orthogonal cutting that include fracture toughness R show that FC/wt depends on R/t, so that it is inevitable that FC/wt increases disproportionately at small t [2]. In grooving, unit power is given by FC/t², and equation (2.1) divided by t² predicts that unit power will increase disproportionately at small t.

Scaling works fine when the modes of deformation in model and prototype are identical, but if they are not, extrapolation from small to large gives erroneous predictions. Application of scaling to the mechanics of fracture causes problems because the energy stored or dissipated in a cracked body depends upon its volume (λ³), but the energy required for cracking depends only on the area of the cracked surface (λ²). In fracture, it means that large structures made from material that behaves in a ductile fashion in laboratory-sized test-pieces, behave in a less ductile—and even brittle—fashion in the large [21]. For example, in crashworthiness studies, it is found that large-size energy-absorbing components made of ductile materials absorb less energy than would be expected on the basis of the scaled-up energy absorbed by a smaller model made of material having the same mechanical properties [22]. The explanation for the failure of World War II Liberty ships is usually described in terms of inadequate Charpy impact energy values, but that only serves to exacerbate the consequences of elasto-plastic fracture scaling.

Such ‘cube-square’ scaling is found elsewhere: it determines the size of droplets in condensation—their weight depends on volume, but surface tension (an area term) holds them together. Again, in 1838 PS Great Western arrived in New York from Bristol with coal to spare, as Brunel appreciated that the resistance to the ship through the water depended on the wetted area of the hull, but the storage capacity for coal depended on the volume of the hull, so a large-enough ship will have sufficient fuel. In biology, cube-square scaling determines the size of the largest prehistoric animals: the body weight of a dinosaur depends on its volume, but the stress in its legs depends on their cross-sectional area. Again, the size of the largest bird that can fly depends upon the ability to lift off and the resistance through the air set against the volume of muscle that can be incorporated within its body [23].

Since the mechanics of cutting forms a branch of elasto-plastic fracture mechanics, we expect to see the consequences of cube-square scaling in cutting. Figure 9 shows some stills from a video of orthogonal cutting of PMMA at progressively deeper depths of cut [24]. At smallest depths, material is removed in the form of continuous curly ribbons; at slightly deeper depths, the continuity of the chip begins to be lost through fractures that emanate from the cutting edge and move into the chip. At progressively deeper depths, the cracks propagate more deeply into the chip, and eventually pass to the free surface giving discontinuous chips. At yet deeper depths, material is removed by a brittle process of ‘knocking lumps out’ where the crack path extends below the intended depth of cut. Figure 10 shows schematically the associated cutting force versus tool travel diagrams for these different depths of cut. Continuous ribbons have steady cutting forces; loss of chip continuity is associated with oscillating forces, the steady amplitudes of which increase the more the chip is broken up. Finally, the lower oscillating force drops to zero when the tool loses contact with the workpiece owing to a brittle crack having run far ahead of the cutting edge; the cutting force does not rise again until contact is regained between tool and workpiece. Prediction of ‘continuous ribbon’ forces at the smallest depth of cut may be made using rigid-plastic fracture mechanics in which elasticity is neglected as the plastic strains are so much greater than elastic; brittle chipping at the greatest depths is analysed using elastic fracture mechanics; at depths in between, elasto-plastic fracture mechanics is required. Experiments that display these sorts of cutting force trace on different materials may be found in [2].

Figure 9.

Stills taken from videos of orthogonal cutting of PMMA showing the gradual transition from ductile to brittle behaviour at progressively deeper depths of cut.

Figure 10.

Schematic scaling diagram showing how cutting forces vary with production of different types of chip at increasing depths of cut.

The different types of chip come about because of cube-square scaling and are determined by the magnitude of the depth of cut t compared with the material length parameter given by (ER/k²)—which is the same as (K_C/k)², where K_C is the so-called critical stress intensity factor. A simplified version of the length parameter, given by (R/k) that is appropriate for rigid-plastic analyses where elasticity is neglected, was introduced in §2 on grooving. (We note that as groove depth is increased, all the different types of chip discussed here can occur in scratching and other types of cutting.)

Roughly speaking, when t < (ER/k²), ductile cutting with continuous ribbons will occur, but as t is increased, broken-up chips appear; eventually pieces of the workpiece are knocked out by the tool and recover to their undeformed condition. Glass has (ER/k²) ≈ 50 µm so is normally brittle when cut, but Puttick et al. [25] showed that continuous curly chips of spectrosyl glass were removed at micrometre depths of cut. This is also the reason for grinding glass and why powders cannot be comminuted below sizes dependent on their (ER/k²) values. Mild steel has (ER/k²) ≈ 0.25 m and is normally ductile when machined, but would split at metre depths of cut. PMMA happens to have (ER/k²) ≈ 1 mm, so that the whole spectrum of chip types can be demonstrated easily in the laboratory.

The fact that a given material can show all types of behaviour from brittle to ductile simply by altering the depth of cut, demonstrates that classification of materials on the basis of the response of laboratory-size test-pieces is misleading. Note that transitions between different types of chip can be produced not only by changing t but also by the effects of rate, temperature and environment on E, R and k, so as to alter the magnitude of (ER/k²) [26].

In biology, scaling is called allometry (from measurement of growth). Isometric scaling in biology is the same as geometrically similar scaling in engineering. However, we note that there may be differences in application, since in biology one does not always get wide ranges of size of identical materials as in engineering; in the latter case, sheets of steel cover the range from the thickness of a tin can to the plates of an ocean liner. Cube-square scaling in relation to the size of the biggest animals and birds has already been mentioned above.

In the particular case of teeth acting as cutting tools to separate items of food into smaller pieces, we should like to know what cutting force or stress will separate food; and what force or stress is required to break a tooth.

The answers to both questions will depend on elasto-plastic fracture mechanics and the mechanical properties of food and teeth. When we apply these ideas to scaling, further questions have to be posed, such as:

• — Does the masticatory system (or parts of it such as teeth or jaw) scale in proportion to body weight?

• — Does the rate of chewing vary with size of animal?

• — What size of foodstuffs is ingested by animals of different size?

• — Do the properties of different food change with size?

• — Are foods that have a limited size range eaten by different size animals?

• — Does an animal of fixed size eat a range of sizes of different foods?

• — Does the amount ingested depend upon the gape of the animal?

The following is speculative but indicates what might be the effects of scaling in relation to ingestion. The force generated across the interface between food and tooth depends on muscle action, in particular on the cross-sectional area of muscle. Hence in two geometrically similar scaled bodies, the bite force should vary as λ², where λ is the scaling (magnification) factor. But the contact area between geometrically similar teeth also scales elastically as λ², so the contact stress given by force/area has the same magnitude across the interface whether the teeth are small or large.

Tooth enamel is inherently brittle in bulk and the stresses and strains at fracture are controlled by elastic fracture mechanics (e.g. [27,28]) (but note the production of very thin ductile shavings off the surface of brittle enamel—exactly as in PMMA in figure 9 at sufficiently small depths of cut when t < (ER/k²)—illustrated by Watson et al. [29]). The microstructural features that act as starter cracks in teeth are the lamellae, variable in depth, but well spaced between the 7 µm diameter prism-like microstructure of enamel. Conversely, these lamellae, called ‘tufts’ in the inner third of the enamel, may also enable crack stopping from external tooth stresses: they are closely related to the more elastic dentine that supports the enamel via the scalloped enamel–dentine junction. Cutting mechanisms for more elastic dentine are described by Atmeh & Watson [30].

If the starter cracks remain the same size whatever the absolute size of tooth, there will be no scaling factor for ‘crack length’ (i.e. λ_a = 1). But there will be scaling factors for ‘height’ λ_h, ‘width’ λ_W and ‘thickness’ λ_B, to reflect the different size of the bigger tooth. This means that we have non-proportional scaling. In the special case of λ_h = λ_W = λ_B = λ (i.e. geometrically similar scaling for the outside proportions of a tooth but with no scaling on initial crack defect), it may be shown [19] that:

3.1

3.2

3.3

so that the stress to fracture a large tooth is exactly the same as for a small tooth. The same conclusion is reached for brittle foodstuffs that have size-independent starter flaws.

If the food is ductile, it is more difficult to perform calculations. However, when separation is produced by a shearing action, it may be shown that the work required to effect separation of the item of food into two pieces scales as λ³ and also that F_large/F_small = λ² [2]. It follows, from ‘work = force × displacement’, that the ratio of average tooth displacements is

3.4

i.e. a proportionately larger shear displacement in larger mouths when larger pieces of the same ductile food are ingested. The force for shearing scales as λ², exactly the same as the force acting on the tooth, so that the stress on the tooth should be size-invariant even though the tooth is moving through a bigger distance in the larger mouth to effect separation by ductile shearing.

4. Conclusion

Increasing values of the slice–push ratio ξ during cutting reduce cutting forces. While, in theory, merely inclining a blade can give large values of ξ, the contact length with the workpiece becomes very long, so large values of ξ are better attained when the cutting edge is separately driven as in a delicatessen slicer. When slice–push is achieved by reciprocal motion of a blade, the effect is not as good as when the motion is continuous, as ξ = 0 at the ends of the stroke. Slice–push determines the shape taken up by initially slack wires when cutting and informs the best shapes of blade for tools such as scythes and lawnmowers. The reduction in forces achieved by large ξ reduces blemishes on cut surfaces and reduces sub-surface damage. In this way, juice loss from damaged sub-surface cells is much reduced when cutting fruit with a high-speed wire.

Two modes are identified by which grooves are formed in ductile materials—one, at small attack angles, where the displaced material is moved into ridges alongside the groove by plastic flow and the other, at large attack angles, where material is cut away. Analysis of the latter involves fracture toughness, as well as plastic flow and friction, since material is separated from the surface. When applied to abrasive wear of ductile materials it shows that events are controlled by the non-dimensional parameter Z = R/kt. This demonstrates that it is really workpiece toughness-to-strength ratio R/k that is important in abrasive wear, not merely hardness. The critical angle of attack between ridging and cutting is a way of estimating fracture toughness and this is important in archaeometallurgy when large samples are usually unobtainable. The different modes of groove formation are also important for prints obtained from engraved plates.

The common meaning of scaling in engineering is discussed, together with allometry (scaling in biology). Cube-square scaling applies in cutting since it is a branch of elasto-plastic fracture mechanics, and this sets scaling in cutting apart from conventional engineering scaling. The type of chip that is formed in all types of cutting is controlled by the magnitude of the non-dimensional parameter (ER/k²t). Large values result in ductile continuous chips (at very small t in a given material, say); small values (at very large t in a given material) result in brittle ‘knocking lumps out’. In between is a spectrum of discontinuous chips. The fact that the same material can display different behaviours simply by altering the depth of cut demonstrates that classification of materials on the basis of the response of laboratory-size test-pieces to determine mechanical properties is misleading. Some speculative consequences of scaling when applied to the forces between teeth and foodstuffs are made.

Competing interests

I declare I have no competing interests.

Funding

I received no funding for this study.