## Abstract

Flow and fracture of some soft solids may be described by the ‘solid’ mechanical properties of elastic modulus, yield stress and fracture toughness, all being dependent on rate, temperature and environment. Other soft solids behave more like very viscous materials. When cutting soft solids, friction is often high between the blade and the material, and cutting is made easier when performed with a thin wire. The wire may be held taut in a frame like a fretsaw, but cutting is often done using an initially slack wire pulled into the solid by hand or machine. For both types of material behaviours, we investigate the curved shape taken by a loaded wire, elements along which cut into the material both radially and tangentially.

For soft materials displaying solid properties, the treatment is based on the analysis of bi-directional cutting by Atkins *et al*. (Atkins *et al*. 2004 *J. Mater. Sci.* **39**, 2761–2766), in which it was shown that the ratio *ξ* of tangential to radial displacements strongly influences the cutting forces. The shapes of wires of various lengths arranged as bowstrings, and the loads in the wires, are assessed against experiments on cheddar cheese. The resultant force takes a minimum value for a particular length of the wire, owing to the competition between lower cutting forces, but higher friction at large *ξ* and vice versa.

Passage of a wire through very viscous materials is flow at very low Reynolds number. To determine the path swept out, we make use of the property of all slender bodies of revolution in highly viscous flow, namely, that the drag exerted across the body is approximately twice as large as along. Comparison is made with the experiments on weighted threads falling under gravity in glycerine.

Regelation is another example of passage of a wire through a solid. The mechanism is completely different but, in the context of the present paper, we provide in appendix A the solution for the typical hours-long school demonstration where, unlike most reported studies, non-uniform temperature fields develop in the block of ice. Comparison is made with experiment.

## 1. Introduction

It is a common experience that, even with a sharp knife, mere pressing down vertically on a material does not achieve much of a cut but, as soon as some horizontal slicing motion is introduced, cutting seems to become much easier. We have explained this using an energy analysis, in terms of the ‘slice–push’ parameter *ξ* given by the ratio of the relative velocity along the edge of the cutting blade to that across it. Cutting occurs by a ‘splitting’ mode of separation in the direction of the deepening cut, as it occurs with plain knives or tools having small wedge angles, where offcut deformation is by bending (Williams 1998). There is no waste, unlike in sawing. The effect depends neither on the existence of teeth along the cutting edge nor on the prestressing of the surface of the material. These ideas have been applied to food cutting machinery to predict the behaviour of the existing blading and to design new blade profiles that maximize *ξ* (Atkins *et al.* 2004; Atkins & Xu 2005; Atkins in press).

Some material–cutting blade combinations display high friction, and cutting is often performed with a fine plain wire in order to reduce the contact area and cutting forces. The separation of blocks of cheese into smaller pieces is a well-known example; in porcelain factories, extruded clay is cut similarly. Single crystals and semi-conductor wafers are cut by a plain wire in a slurry of abrasive material, or by a wire itself coated with abrasive particles. A large-scale example of cutting by a diamond-coated reciprocating cable concerned the removal of a sunken ship in the English Channel, which was a danger to navigation (Anon 2004). These processes are different and are similar to sawing where waste is produced.

Soft solids and foodstuffs can be strangely behaved materials. On the one hand, their flow and fracture behaviour at large deformations can often be well described by the usual solid mechanical properties of elastic modulus, yield stress and fracture toughness, all these properties being rate, temperature and environment dependent (e.g. Luytens 1988; Kamyab *et al.* 1998; Atkins *et al.* 2002). In such cases, the external work done by a wire when cutting provides the necessary component works of (i) fracture, (ii) friction, and (iii) any permanent deformation. On the other hand, some very soft foodstuffs behave, perhaps, more like very viscous materials (e.g. Scott Blair & Coppen 1939; Scott Blair *et al.* 1947; Wilkinson 1960). Here, the process of ‘cutting’ might be thought of as viscous flow past the wire, with the material not recombining behind the wire (some rehealing may occur of course, particularly with ‘sticky’ materials like dough). In what way might the path of a wire passing through a very viscous medium (flow at very low Reynolds number) differ from that for a medium having solid mechanical properties?

Another form of passage of wire through a medium is the well-known case of regelation, where a weighted wire passes through a block of ice under gravity, leaving the block intact. Pressure melting of ice below the wire, followed by refreezing above the wire, is the basic cause of regelation, and the physics has been investigated by many workers (for references, see Drake & Shreve 1973). Surprisingly, we have found no reference to the shape taken by the wire in the usual sort of school demonstration performed at room temperature where the block of ice warms up during the few hours of the experiment. In appendix A, we have provided the solution to this problem too.

We have performed experiments in all three cases against which to assess the different solutions for the same experimental set-up, *viz,* a weighted wire passing symmetrically through the solid or liquid. In the case of passage through solids, loops of wires of different lengths are attached to the ends of a weighted bar wider than the block of solid material. This gives bowstring-like shapes with no contact between the wire and the sides of the block. In the case of passage through a very viscous liquid, weights are attached to strings that are launched from initially horizontal positions on the surface of the liquid. For a detailed list of all nomenclature in this article see table 1.

## 2. Equilibrium of a flexible wire

The equations of equilibrium of a wire having negligible bending resistance, as in a catenary (Prescott 1923, ch. VIII), are common to all three problems. In these problems, the different and varying loading across and along the wire produces different paths swept out.

An element Q_{1}Q_{2} of length d*s* at point P (figure 1), has a local angle of inclination *ψ*. The tension in the wire is *T* at P, (*T*−d*T*/2) at Q_{1} located at (*ψ*−d*ψ*/2) and (*T*+d*T*/2) at Q_{2} located at (*ψ*+d*ψ*/2). The radial and tangential loadings on Q_{1}Q_{2} at P are d*F*_{R} and d*F*_{T}, respectively. Equilibrium of forces gives

along the tangent at P:

(2.1)

radially at P:(2.2)from which(2.3)and(2.4)

Dividing, we obtain(2.5)Equation (2.3) may be integrated to give the variation of *T* with *ψ*, for particular d*F*_{T} and d*F*_{R}, in terms of the unknown tension in the wire *T*_{0} at the apex of the curve. This result may be used in equation (2.4) to give the shape of the flexible wire in terms of *T*_{0} employing tan *ψ*=d*y*/d*x*. The magnitude of *T*_{0} is obtained from the length of the wire.

For the simple catenary, of a uniform chain of mass *m* per unit length, equations (2.1) and (2.2) become and , giving . Integration yields *y*=(*T*_{0}/*mg*)sec *ψ*+const. The constant may be made zero by choosing to place the *x*-axis at a distance (*T*_{0}/*mg*) below the apex of the curve, whence .

## 3. Analysis

### (a) Cutting soft solids with a wire

When there is negligible elastic and plastic deformations away from the cut surface, the incremental work required for cutting has two components, *viz,* work of surface separation and work of friction. The behaviour of an element of wire Q_{1}Q_{2} in figure 1 is then locally the same as that of an element of a stationary knife at an inclination *ψ* to the motion of the workpiece.

In the case of frictionless cutting (figure 2), a straight sharp blade cutting a block of width *w*, and moving with incremental displacement d*v* perpendicular to the blade and d*h* parallel to the edge of the blade, requires forces (Atkins *et al.* 2004)(3.1a)and(3.1b)where *R* is the specific work of separation (fracture toughness) and *ξ* is the ratio of parallel and normal displacements or speeds given by *ξ*=d*h*/d*v*, and is called the slice–push ratio. Since d*v* coincides with the increase in extent of cut surface, a nonlinear coupling occurs between *F*_{H} and *F*_{V} in equations (3.1*a*) and (3.1*b*), so that although the total work is independent of *ξ*, there is a disproportionate reduction in *F*_{V} as soon as simultaneous slicing occurs.

Consider an initially slack wire anchored at points A and B (figure 3) against which the block of material to be cut is driven at constant velocity. From the resolution of the velocity of the block along and across the wire,(3.2)From equations (3.1*a*) and (3.1*b*), the components of radial and tangential force, d*F*_{R} and d*F*_{T}, acting on an element of wire of length d*s* are(3.3)and(3.4)Equation (2.3) becomesi.e.(3.5)where *T*=*T*_{0} at *ψ*=0, the apex of the profile taken by the two symmetrical halves of the wire. Using equation (3.5) in (2.4), we obtain(3.6)Whence, on integration,(3.7)(the constant of integration is zero, since *ψ*=0 at *x*=0). Replacing tan *ψ* by d*y*/d*x*, we obtain(3.8)where the constant of integration is again zero, since *y*=0 at *x*=0. The shape of a bowstring wire during frictionless cutting is therefore a parabola. The unknown *T*_{0} depends on the length of the wire. We have(3.9)since d*y*/d*x*=*Rx*/*T*_{0} from equation (3.8). Integration gives(3.10)for the length of the wire within the block from −*x* to +*x*. Equation (3.10) gives, in terms of *T*_{0}, the *x*-coordinates where the wire exits the block at a given prescribed length *s*. Alternatively, for a given width of block 2*w*, the length of wire within the block is given in terms of *T*_{0}. When the total length of the wire is longer than that actually cutting the block, the lengths AC and DB outside the block will be straight and inclined at *ψ*_{exit}. *T*_{0} is obtained from the resolution of forces in the horizontal (*x*) and vertical (*y*) directions on an element of wire at any point P having local inclination *ψ* and local wire tension *T*, and from equilibrium of the whole section of wire from the apex at O to point P. It may be shown that increments of horizontal force d*X* along the wire are zero, so that *X*=*T*_{0} and also that(3.11)using equation (3.7). Hence, the vertical component of the frictionless cutting force at any point along the wire is given by the fracture toughness multiplied by the span of horizontal distance to that point from the apex of the parabola. This simple relationship is reminiscent of the property of the catenary that the tension at any point in the chain is equal to the weight of a portion of the chain that would extend in a vertical line from that point to the *x*-axis (Inglis 1951).

From equation (3.6), the radius of curvature along the wire is given by(3.12)For bi-directional cutting by a knife in the presence of friction, Atkins *et al*. (2004) showed that the normal cutting force *F*_{V} and parallel force *F*_{H} are given by(3.1c)and(3.1d)where is derived from Coulomb-friction modelling of cutting soft solids by a taut wire (Kamyab *et al.* 1998), in which contact forces were obtained by assuming that a thin flow zone exists around the bottom half of the wire, which presses radially on the wire with the yield stress *σ*_{y} and produces circumferential friction stresses *μσ*_{y}, where *μ* is the coefficient of friction. Integration around the lower half of the wire gives 2*rσ*_{y}[1+*μ*] for the frictional force per unit length of wire of radius *r*. (In the case of cutting with sharp flat blades having much greater contact with the soft solid than wires, Coulomb friction may still be employed (Williams 1998) or friction can be modelled in terms of a stress *τ*_{f} acting over a contact length on each face of the blade (Atkins *et al.* 2004)). The Williams–Kamyab model gives a value for friction even when *μ*=0, the residual friction force/length of 2*rσ*_{y} representing the formation of the thin yielded boundary layer around the wire, in which the micromechanisms of material surface separation come into play. (In passing, we note that this raises questions about the meaning of ‘sharpness’ of a cutting edge. Tool sharpness in cutting plays a similar role to the starter crack-tip radius in fracture toughness specimens, i.e. its size relative to the natural crack-tip opening at which a crack will propagate, the so-called critical crack-tip opening displacement (CTOD). In fact, sharpness of a cutting edge is more important, in the sense that in cutting there is no starter crack to help separation of surfaces during the initial indentation phase. It is a moot point whether the concept of sharpness is absolute (the tip radius) or relative to the microstructure and method of cutting.)

Equations (3.1*a*)–(3.1*d*) predict that, whatever the magnitude of the friction, *F*_{H} and *F*_{V} are lower the higher the *ξ*, but it was shown that friction limits the reduction in forces at highest *ξ*, so that there is no point in increasing *ξ* indefinitely (Atkins *et al*. 2004). Nevertheless, the theory explains why, in a bacon slicer, the wheel tangential speed is that much greater than the speed at which food is fed to the wheel.

Solution of equations (2.3)–(2.5) for the shape of the wire in the presence of friction follows the same procedure as before. Equation (3.5) remains the same, but equation (3.6) becomes(3.13)Integration gives(3.14)to replace equation (3.7). Equation (3.7) is recovered from equation (3.14) when *M*=0, using L'Hôpital's rule. Equation (3.13) gives(3.15)so(3.16)Again, for *M*=0, equation (3.8) is recovered.

The length of the wire is(3.17)with tan *ψ* given by equation (3.15). There is no closed-form solution. The radius of curvature in the presence of friction is obtained from equation (3.13) and is(3.18)For *M*=0, equation (3.12) is recovered from equation (3.18).

Equation (3.16) may be used to establish the appropriate (*T*_{0}/*Rw*) to produce various exit heights *y* (and hence exit angles *ψ*_{exit} from equation (3.14)) for a range of values of *M*, by making *x* the half-width of the block. The shapes taken up by wires of different lengths producing different exit angles may then be predicted by equation (3.16) for various *M*. It is convenient to plot these in non-dimensional form as *y*/*w* versus *x*/*w*, and figure 4*a* shows that for a given exit height (*y*/*w*), the parabolic shape for frictionless wire cutting is made flatter as *M* increases, reducing *ψ* along the wire, and giving different wire exit angles from the block. Discrimination between different M values is more easily made with longer wires producing larger (*y*/*w*).

The tension along the wire, and in particular the tension *T*_{exit} on exit from the block, is given by equation (3.5). It is convenient to plot the forces non-dimensionally as (*T*/*Rw*). Figure 4*b* shows how the predicted (*T*_{exit}/*Rw*) varies with exit angle for various values of *M*. For all *M*, the forces in the wire are high at small *ψ*_{exit} because the wire is ‘tight’ (as in a fretsaw frame). At greater *ψ*_{exit}, the forces fall but then increase again, passing through minima that are more marked the greater the *M*. The vertical and horizontal components of the non-dimensionalized exit force in the wire, given by (*T*_{exit}/*Rw*)sin *ψ*_{exit} and (*T*_{exit}/*Rw*)cos *ψ*_{exit}, also vary with *ψ*_{exit}. At small *ψ*_{exit}, the horizontal forces are very high but they decrease continuously as *ψ*_{exit} increases; this is vice versa for the vertical component of force in the wire.

Minimas in wire force arise owing to the competition along the wire between (i) the local reduction in force caused by the increased slice–push ratio (*ξ*) on elements of wire having high inclinations *ψ* (equation (3.2)), and (ii) the increase in local frictional force caused by the longer incremental contact lengths d*s* (=d*x*/cos *ψ*) between wire and material at higher wire inclinations. For frictionless cutting by wire, there is *no* minimum in wire tension; it decreases continuously as *ψ*_{exit} increases with longer wires and, as expected, the force component resolved in the direction of cutting is always equal to *Rw* as in simple microtoming. For finite *M*, the minimum in force is at 62° for *M*=0.1, 60° for *M*=0.2, 57° for *M*=0.6, 53° for *M*=1 and 48° for *M*=3 and greater. The absolute values of the non-dimensional loads increase, of course, with *M*. For example, the minimum values of (*T*_{exit}/*Rw*) are 1.4 (*M*=0.1), 1.6 (0.2), 2.4 (0.6), 3.1 (1), 5 (2), 6.8 (3), 10.3 (5) and 19.2 (10).

### (b) Passage of a wire through a very viscous fluid

It is well known that for very viscous flows around submerged bodies where viscous forces are much more influential than inertia forces (flow at low Reynolds number, *Re*), the drag force *D* is directly proportional to the flow velocity *V*, the absolute viscosity *μ* and some characteristic dimension *L*, and *D* is given by(3.19)where *C*_{D} is a dimensionless drag coefficient. A property of all slender bodies of revolution in highly viscous flow is that the drag exerted on the body is approximately twice as large when it is moving normal to its length as when it is moving parallel to its length (Batchelor 1967). Hence, a rod falling broadside on in a very viscous fluid takes twice the time to descend a given distance as the same rod falling with its axis vertical. Taylor (1966) demonstrated that an inclined rod falls obliquely and drifts to one side. The flight path of a rod inclined at 45° may be analysed by saying that the velocity along its axis is twice that across it giving a flight path of 18.4° to the vertical. This is in spite of the fact that the net drag force on the body is vertical, being equal and opposite to the gravitational pull. It may be shown that the greatest inclination of the flight path to the vertical is 19° when the rod is 35° to the horizontal, as indicated in figure 5.

Consider a flexible thread AB (figure 5) whose ends are constrained to move downwards at constant velocity *V*, with the distance AB remaining constant. An element of thread of length d*s*, inclined at angle *ψ*, will experience a velocity component *V* sin *ψ* along its axis and *V* cos *ψ* across. If we assume that buoyancy effects can be neglected and that the drag coefficient for flow across the wire is twice the drag coefficient along the wire, then equations (2.3)–(2.5) become(3.20)Integrating with *T*=*T*_{0} at *ψ*=0 yields(3.21)(3.22)since from equation (2.4), with *C*_{D} being the drag coefficient along the axis, and d*x*=d*s* cos *ψ*. It follows therefore that(3.23)and since d*y*=tan *ψ* d*x*,(3.24)The local radius of curvature *ρ* will be(3.25)Equation (3.24) can be integrated with when *ψ*=0, giving(3.26)Likewise, equations (3.23) and (3.24) yield(3.27)

(3.28)Solutions for equations (3.26)–(3.28) are given in table 2.

## 4. Experiments and discussion of results

### (a) Soft solids

Wire cutting experiments were performed on blocks of cheddar cheese, having uniform width, which were stored in a refrigerator wrapped in clingfilm in order to minimize drying out. The properties of cheeses in particular depend very much on moisture content to the extent that, unless great care is taken, different samples behave as if they are different materials (M. Charalambides 2005, personal communication). Experiments were conducted at temperatures of 20 and 15 °C on cheeses A and B, respectively. These cheeses were supposedly identical, which will be discussed later.

Every block of cheese was equilibrated at the test temperature before being attached to the underside of the crosshead of a tensile testing machine. The testing machine then lifted each block against the initially slack threads (0.5, 1.5 and 2.0 mm diameters) of various lengths, the ends of which were anchored at two points a fixed distance apart on the lower platen of the machine (figure 3). The anchor points were attached to two load cells that gave the tension along the wire, the testing machine itself giving a check on the vertical load.

After a transient stage during which the wire cut into the sides of the block, a steady state was reached; the wire curvature and exit angle from the block depending upon the length of the wire. The machine was stopped under load, and the block moved perpendicular to the plane of the wire, the wire now cutting through the sample to reveal the path taken (figure 6*a*).

Figure 6*b*,*c* shows the experimental variation of the tension in the wire on exit from the cheese blocks plotted against the exit angle *ψ*_{exit}. Figure 6*b* shows that there is a minimum in force, at an angle between about 50° and 60°, for the experiments on cheese A at 20 °C using a 0.5 mm diameter wire. For the experiments on cheese B at 15 °C with three wire sizes, figure 6*c* shows that the forces decrease as *ψ*_{exit} increases but then do not rise as much as expected beyond about 60° as predicted by theory (figure 4). It was noted that the behaviour of cheese B was much more variable than that of cheese A. Given that the same test rig and instrumentation was employed, this implies variability in properties within the samples, and this was confirmed when independent determinations of the yield strength and fracture toughness were made.

The detailed shapes taken up by the wire, and the tension along it, will depend upon the parameter *M*=2*rσ*_{y}[1+*μ*]/*R* and therefore involves the radius of the wire, the coefficient of friction and the mechanical properties of the two cheeses. For a given type of soft solid at the same rate and temperature, it is expected that *σ*_{y} and *R* will be constant. It is also probable that the coefficient of friction *μ* between wire and soft solid will be the same for all wire sizes. Hence, for cheese B, the different loads required by wires of different diameters will be mainly down to the different wire size.

We performed independent determinations of *μ*, *σ*_{y} and *R* for the two cheeses at various temperatures, and at rates comparable with those in the wire cutting experiments, in order to calculate the expected *M*. For both cheeses, the friction coefficient was variable and quite high (0.9<*μ*<1.2) but comparable values of 0.7–1.0 have been reported by Kamyab *et al.* (1998) for similar cheese. Uniaxial compression tests on a series of cylinders 20 mm diameter by 20 mm high were employed to determine *σ*_{y}, using polytetrafluoroethylene (PTFE) films to reduce platen friction. The as-received long bars of cheese from which specimens were cut are anisotropic as they are sectioned from across the diameters of whole ‘rounds’ of cheese. Consequently, care was taken to orientate testpieces in the direction in which the wire was compressed and cut through the specimens. The load–displacement curves remained quite steep after initial yielding so that yield stresses, defined as the departure from linearity, were noticeably lower than those based on a 0.2% offset definition. For cheese A, consistent results gave *σ*_{y}≈10 kPa (deviation from linearity) but approximately 22 kPa (offset) (Atkins *et al.* 2002). For cheese B, there was more variability in strength where *σ*_{y}≈39±10 kPa (52±10 kPa offset). Although it is expected for cheddar cheese that the yield strength will be greater at 15 °C than at 20 °C, the suspicion that the two cheeses were different was confirmed by determinations of the yield strength of cheese B at the temperature (20 °C) of the tests in which cheese A was cut. It was discovered that *σ*_{y}≈20±7 kPa (29±10 kPa offset) in contrast to the consistently reproducible 10 kPa (22 kPa offset) for cheese A at 20 °C.

Notched three-point bend testpieces, nominally 20 mm by 20 mm with a 60 mm span, using 10 mm diameter rollers, were employed to determine the fracture toughness *R* using standard energy methods (e.g. Atkins & Mai 1985/88). These beams were cut from larger blocks of cheese, so that cracking took place in the same direction as during wire cutting. For cheese A, *R*≈15 J m^{−2} at 20 °C (Atkins *et al.* 2002) and was reproducible. For cheese B at 15 °C, there was variability in toughness similar to the variability in yield strength described earlier. It was found that *R*≈22±5 J m^{−2}.

Estimates for *M*, given by the above independent values of *μ*, *σ*_{y} and *R*, are as follows: for cheese A at 20 °C with a 0.5 mm string, *M*≈0.5×10^{−3}×10×10^{3}[1+1]/15=0.7 using *σ*_{y} given by the deviation from linearity, or *M*≈1.4 using the offset yield stress. Theory in §3*a* predicts that for 0.7∼*M*∼1.4, there should be a shallow minimum in *T*_{exit} at about *ψ*_{exit}=55°. Figure 6*b* shows that a minimum experimental force is displayed between 50° and 60° and has magnitude of about 1.5 N. The specimens were 84 mm in width, so with *R*=15 J m^{−2}, (*T*_{exit}/*Rw*)=2.4; theory predicts that (*T*_{exit}/*Rw*) should be approximately 2. The line shown in figure 6*b* for the 0.5 mm diameter wire data at 20 °C is for *M*=1.

The non-homogeneity of cheese B means that there will be a spread of values for *M*. For the 0.5 mm diameter wire, *M*≈0.5×10^{−3}×2×(49 or 29)×10^{3}/(17 or 27)=1∼*M*∼3 (deviation from linearity) or 1.5∼*M*∼3.6 (offset); for the 1.5 mm diameter wire, 3∼*M*∼9 (or 4.5∼*M*∼11 offset); and for the 2 mm diameter wire, 4∼*M*∼12 (or 6∼*M*∼14 offset). The average value of *R* (22 J m^{−2}) was employed to construct the predicted plots of *T*_{exit} versus *ψ*_{exit} at different *M* in figure 6*c*. In all three cases, for *ψ*_{exit}<60°, the experimental data are aligned approximately with the predictions for a constant *M* (about 1.5 for the 0.5 mm wire, 3.5 for the 1.5 mm wire and 5 for the 2 mm wire); these are within the ranges predicted above using the yield strength based on the deviation from linearity. At higher *ψ*_{exit}, however, the results for cheese B do not rise as much as demanded by theory. It is not clear that the observed lack of uniformity in properties should cause this pattern of behaviour.

### (b) Viscous fluids

Experiments on glycerine were first tried with threads having weights attached to the ends, launched from the free surface. However, the free-fall behaviour is modified by the sideways pull of the thread, so that the weights approach one another and no steady state is achieved.

Two approaches were tried to overcome this problem. Since an inclined rod on its own will drift to the left in downward motion (figure 5), attaching equal rods at the ends of the string ought to keep the ends apart. An equilibrium configuration ought to be reached in which the ends of the string are tangential to the rods, since there would then be no moment on the rods, and the whole rod–string–rod assembly should descend at constant velocity, with AB at a fixed distance. This attractive solution had to be abandoned, however, as consistent results were difficult to achieve. It was suspected that the axial component of the flow along the rods interfered with the flow around the string.

Success was achieved finally by the simple expedient of attaching six equal fisherman's lead weights spaced 60 mm apart (one set of experiments with six weights each of 0.2 g and another each with 0.3 g) along a cotton thread of 0.25 mm diameter. The weights were launched from the surface in a straight line. The middle three ‘arch’ distances (figure 7) were proved to be very constant and were used for measurement.

Solutions for the theoretical shape of the thread given by equations (3.26)–(3.28) are given in table 2 for values of *ψ* from 10 to 57.5°. In the two cases (*ψ*=46 and 57.5°), the values are compared with a simple catenary of the same half-length *s* suspended between two points at a distance *x* on either side of the *y*-axis. The values given by equations (3.26)–(3.28) are within 2.5% of those obtained using standard catenary equations. The experimental shape shown in figure 7, where the measured *ψ* is 57.5°, was similarly found to be indistinguishable from a catenary hung up against an enlarged image of the descending thread. In this illustration, the value of the non-dimensional central sag, *h*/*x*=2(15.5/47.5 mm)=0.65, is very close to the one computed for 57.5° given in table 2.

## 5. Conclusions

The authors have investigated both analytically and experimentally the paths swept out by loaded wires cutting through solids, passing through very viscous liquids and (in appendix A) regelating through an ice block having temperature gradients, all of which result in different catenary-like shapes summarized in table 3, where the cases are listed for a given *ψ* in descending order of radius of curvature.

In the first and last cases in the list, friction along the wire is assumed to be minimal and so the tension *T* is constant at all points and equal to *T*_{0}. For all other cases in the table, *T*_{0} is the tension at the centre span.

## Acknowledgements

The impetus for the investigation of the profile of a wire during the cutting of foodstuffs came from a DEFRA (erstwhile MAFF) LINK programme. Valuable conversations were had with the industrial and academic members of the teams involved, particularly with George Jeronimidis and Jo Lakeland. The continuing interest of Christina Goodacre of DEFRA is acknowledged. The interest in the wire profiles during very viscous flow, and during regelation, arose simply out of curiosity. John Frew and Trevor Orpin at Reading are thanked very much for their help on the experimental side. Keith Moore kindly provided our first reference.

## Footnotes

- Received December 14, 2005.
- Accepted May 18, 2006.

- © 2006 The Royal Society