Fundamentals of cutting

Abstract

The process of cutting is analysed in fracture mechanics terms with a view to quantifying the various parameters involved. The model used is that of orthogonal cutting with a wedge removing a layer of material or chip. The behaviour of the chip is governed by its thickness and for large radii of curvature the chip is elastic and smooth cutting occurs. For smaller thicknesses, there is a transition, first to plastic bending and then to plastic shear for small thicknesses and smooth chips are formed. The governing parameters are tool geometry, which is principally the wedge angle, and the material properties of elastic modulus, yield stress and fracture toughness. Friction can also be important. It is demonstrated that the cutting process may be quantified via these parameters, which could be useful in the study of cutting in biology.

1. Introduction

Cutting may be defined as a process in which new surfaces are created in a solid by the use of a tool. Such tools are usually sharp though, in reality, blunt tools can cut and the sharpness can be an important factor in the process. The creation of new surfaces in a solid has important connotations in mechanics because it implies a change in continuity such that there will be an energy change. This notion was first explored by Griffith [1] and he introduced the parameter energy release rate (G after Griffith), which is the change of stored (elastic) energy per unit area change (for a list of the nomenclature used throughout this article, see table 1). Griffith was concerned with fracture in cracked, loaded solids and not cutting but the basic physics is the same, as noted by Lake & Yeoh [2]. This energy change is necessary to drive the fracture and provide the energy per unit area, G_c, required to create the new surface area and is a material property, the fracture toughness. The fracture criterion is thus

1.1

and is the basis of the discipline of fracture mechanics. Cutting can be regarded as a fracture mechanics process in which the input energy is provided, at least in part, by a cutting tool.

Table 1.

Nomenclature.

G	energy release rate
Gc	energy per unit area at fracture (fracture toughness)
Gd	plastic work energy dissipation per unit area
h	cut depth
hc	chip thickness
b	specimen width
Fc	cutting force
Ft	force normal to cutting direction
S	shear force
N	normal force
dUext	external work
dUc	fracture energy
dUf	friction energy
dUd	plastic dissipation energy
dUs	shear energy
R	radius of curvature
Rp	radius of curvature at first yield
θ	tool angle
μ	friction coefficient
M	bending moment
E	Young's modulus
σY	tensile yield stress
	plastic zone length
eY	tensile yield strain
ϕ	shear plane angle

This energy-based criterion, equation (1.1), is a necessary, but not sufficient, condition for fracture. A local critical stress must also be achieved in order for fracture to occur. For sharp cracks in brittle solids, there is a high stress concentration and this condition is automatically satisfied as noted in [1]. This gives rise to linear elastic fracture mechanics, which embodies linear elastic behaviour and a single criterion, G_c, for fracture. For some biological materials, such as bone, this works well but for soft tissue large deformations result in crack blunting [3] and the second criterion, a critical stress, is necessary. A similar effect may occur with blunt tools though it should be noted that for soft solids a sharp tool may create a fracture without blunting [2], which can be advantageous in biological materials. In addition, it should be noted that large deformation around a tool may result in the tool contacting the fracture point. This is referred to here as ‘crack tip touching’ and can result in energy going into the fracture directly from the force on the tool and replaces, or adds to, G as the driving force.

Modelling of cutting has a long history and was mostly confined to the machining of metals. In the early analyses, a fracture energy term was included in the analysis along with models of chip deformation and friction energy dissipation. In the 1950s [4], it was dropped because it was said that G_c was the true surface work of the material, which is very small and could be ignored. This is still the case in many machining modelling numerical codes but it was pointed out by Atkins [5] that the G_c is not small and included the local plastic work as well as the surface energy necessary to create the surfaces. This is the same as the important contribution to fracture mechanics made by Orowan [6] and Irwin [7], who extended the Griffith analysis done for glass, where G_c is indeed the surface work, to metals, where it is enhanced by local plasticity.

Figure 1 shows the model of cutting which will be used here to demonstrate the various mechanisms involved. It is a model of orthogonal cutting in which a surface layer, or chip, is removed by a wedge cutting with the edge normal to the cutting direction. Other geometries such as symmetrical blades or wires [8] are used but the basic scheme of the analyses is the same. In figure 1a, the chip is bent as the cutting tool moves forward. In all cases, a steady state is assumed in that the tool moves at a constant speed and the cut moves at the same speed. Cutting initiation is a more complex problem and will not be considered here. In general, the tool tip will not touch the fracture but can do for a finite opening and slope at the fracture point. For large radii of curvature in the chip, the deformation is elastic with no energy dissipation. For smaller radii, the chip yields, dissipating energy via plastic bending and giving rise to permanent deformation and chip curling. In figure 1b, a shear plane develops at the fracture point and the chip remains straight. Here crack tip touching always occurs. Which process occurs is determined by which has the lowest energy dissipation. In both cases, the chip slides along the upper surface of the tool, leading to the dissipation of energy via friction. To understand the cutting process, we must model these mechanisms and determine the contributions to fracture, plasticity and friction.

Figure 1.

Deformation processes in cutting. (a) Chip bending and (b) chip shearing.

2. Energy balance

Figure 2 shows the parameters involved in cutting and are the cut depth, h, the chip thickness, h_c, the specimen width, b, and the tool angle, θ. The forces generated by the tool moving at a constant speed are F_c in the cutting direction, i.e. the cutting force, and the transverse force F_t. A clearance angle is shown on the lower surface of the tool, and it is assumed here that no forces are generated by the tool tip rubbing on the cut surface. Also shown are the forces F_c and F_t acting at the contact point of the tool and the chip. These may be resolved into a shear force along the interface S and a normal force N.

Figure 2.

Parameters involved.

When the tool moves a distance dx so does the cut, and we may define the following energy changes:

• (1) external work dU_ext = F_c. dx,

• (2) fracture energy dU_c = G_c. bdx,

• (3) friction energy dU_f = S. dx (for h_c = h), and

• (4) plastic dissipation energy dU_d.

The energy balance is now

2.1

The speed of cutting is assumed here to be sufficiently low that no kinetic energy is involved, though this can be included. It is also assumed that the dissipated energy does not give temperature changes, i.e. the process is isothermal, so the material properties are for the ambient temperature. In most biological cases, the cutting speeds are sufficiently low for this to be a reasonable approximation. Changes for non-isothermal conditions can be included for high-speed cutting as in adiabatic conditions, but such conditions are unlikely in biological systems. Some special cases may now be considered.

2.1. Elastic deformation (dU_d = 0) with no friction (dU_f = 0)

This is the case for high values of the radius of curvature, R, as shown in figure 2, so that the bending is reversed when the chip passes the contact point and there is no energy dissipation. Indeed in these steady-state situations, there is never a change in the elastic energy but here we exclude any other dissipation including friction by putting S = 0 and hence dU_f = 0.

Equation (2.1) now becomes

2.2

This situation is employed in razor blade cutting [2] and using scissors to find the toughness of thin polymer films and is used for leaves [9]. Such materials do not usually yield and the test requires only the exclusion of friction. For very low modulus materials, the chip is a ‘floppy offcut’ as defined by Atkins [10], where the low stiffness, and hence low forces, gives crack tip touching and no friction. High stiffness is encountered in the splitting of wood with a wedge and where there is elastic energy which gives rise to an energy release rate and no touching, though friction is usually involved.

2.2. Elastic deformation (dU_d = 0) with friction

Equation (2.1) is now

and at the contact point

2.3

Thus,

and

2.4

If θ is known and F_c and F_t are measured then G_c can be found without defining the nature of the friction process. For S = 0, F_t = F_c(cos θ/sin θ) and equation (2.2) is retrieved.

Coulomb friction is often assumed and is given by

where μ is the friction coefficient, and from equation (2.3) we have

2.5

where μ = tanβ, and from equation (2.4)

2.6

Note that F_t/F_c = 1/μ at θ = 0 and −μ at = 90°, i.e. a change of sign, and F_t = 0 for tan θ_o = 1/μ. This has been suggested as a criterion to give smooth machined surfaces [11] and gives a minimum F_c/b, i.e. when d/dθ(F_c/bG_c) = 0, and from equation (2.6) we have

2.7

This condition for a minimum in F_c is shown in figure 3 for μ = 0.2, β = tan⁻¹(0.2) = 11.3°, θ_o = 79° and (F_c/bG_c)_min = 1.24.

Figure 3.

Variation of cutting force with blade angle for μ = 0.2.

The elastic bending of the chip may be described by simple bending theory in which the radius of curvature, R, is related to the bending moment, M, by

2.8

where E is Young's modulus and h is the chip thickness (h_c = h here). We may then plot the bending moment M/b versus 1/R as shown in figure 4, which also shows the linear relationship of equation (2.8). When the cut moves forward the moment goes from zero via the two loading lines shown to point A, where M = M_o and R = R_o. As the cut progresses, this point moves along the chip following the unloading line until it reaches the contact point, where M = 0 and 1/R = 0. The elastic energy imparted to the chip on loading is the area under the curve, half of which is recovered while the remaining half (shaded area in figure 4) is the energy release rate G = G_c,

2.9

Figure 4.

Bending moment–radius of curvature diagram for elastic bending.

2.3. Elastic–plastic bending

The stress distribution for elastic bending of the chip is shown in figure 5a and is linear with the maximum of

In figure 5b, this maximum stress reaches the tensile yield stress, σ_Y, at the outer surface and

Figure 5c shows the stress distribution for M > M₁, where the elastic–plastic interface is a distance y = c from the central neutral axis. The upper limit for M is shown in figure 5d when c = 0 and M/b = M_p/b = σ_Yh²/4. For M₁/b < M/b < M_p/b, the moment is given by

i.e.

At y = c, the strain is e_Y so that e_Y = c/R, and noting that e_Y = h/2R_p we have 2c/h = R/R_p and

2.10

i.e. M/b = M_p/b at R = 0 and M/b = (M_p/b)(2/3) = M₁/b at R = R_p.

Figure 5.

Stress states in elastic–plastic bending. (a) Elastic bending, (b) first yield, c = h/2, (c) elastic–plastic bending and (d) fully plastic bending, c = 0.

Figure 6 shows the bending moment radius of curvature relationship for this elastic–plastic case. The elastic line changes to equation (2.10) for 1/R ≥ 1/R_p and the loading is as shown in figure 4 and unloading is a line parallel to the elastic loading. The intercept on the 1/R axis, 1/¯R, is the residual curvature of the chip. This upper shaded area is G_c, given by

2.11

and the lower shaded area is the energy dissipated in plastic work, i.e.

2.12

and

2.13

Figure 6.

Bending moment–radius of curvature diagram for elastic–plastic bending.

It should be noted that, in equation (2.11), when R → 0, i.e. G_c is given by the elastic energy release rate when the elastic stress limit is σ_Y. Thus, the parameter represents the maximum G_c which can be achieved for a constant σ_Y. For there will be no cutting since the required G_c cannot be achieved. The limit for the onset of plastic bending is when R/R_p = 1, i.e. G_d = 0 and Equation (2.12) may be expressed as a function of by substituting R/R_p from equation (2.11) and if we ignore friction since it is a constant factor times F_c/b in this case we have

2.14

where This parameter is the ratio of h to a plastic zone length

2.4. Plastic shearing

A more efficient mode of plastic deformation is shown in figure 7, in which there is crack tip touching and the chip shears along a plane at an angle ϕ and there is no bending. Invoking the Tresca yield criterion, the shear stress on the plane is σ_Y/2. In this case the chip thickness, h_c, is greater than h because of plane strain conditions and for finite friction

2.15

For a tool movement of dx the movement along the tool–chip interface, dx_c, is

2.16

The shear displacement along the shear plane is

2.17

and hence the energy dissipation terms may be derived

and

Figure 7.

Plastic shearing.

From equation (2.1), we now have

i.e.

2.18

This solution forms the basis of a method for finding G_c from cutting [12] in which h is varied and h_c measured, hence giving ϕ. If F_c and F_t are measured then F_c/b + (F_t/b) tanϕ may be plotted versus (h/2)(tanϕ + 1/tanϕ) to give G_c from the intercept and σ_Y from the slope of the straight line.

ϕ can be determined approximately by finding the value to minimize the external work done [13], as in the friction case discussed previously. Differentiating equation (2.18) with respect to tanϕ and equating to zero we have

2.19

where ϕ_o is the value of ϕ at this minimum, and substituting in equation (2.18) we have

2.20

with

In this case G_c and σ_Y may be found by fitting F_c and F_t data for a set of h values without the need for measuring h_c. If Coulomb friction is assumed then, for this touching case, equation (2.5) becomes

2.21

This enables tan ϕ_o to be determined in terms of μ and θ using equations (2.16)–(2.18), and after some manipulation

where tan β = μ, and equation (2.19) becomes

2.22

For β = 0, i.e. μ = 0, this becomes

and may be compared with equation (2.17) for bending,

Figure 8 shows the three regimes of chip formation where F_c/bG_c is plotted versus x. For x > 3, the deformation is elastic bending so the chips are straight, though such cutting is prone to cracking and unstable behaviour. For 1 < x < 3, there is elastic–plastic bending giving curled chips and smoother cutting behaviour. For 0 < x < 1, there is no bending and the chips are straight with a linear relationship between F_c/bG_c and h. This regime is useful for finding G_c and usually gives smooth surfaces and stable forces. Equation (2.22) may be written as

where δ_c = G_c/σ_Y is the crack opening displacement and is a characteristic length for fracture.

Figure 8.

Diagram of chip formation; for shearing e_Y = 0.02 and μ = 0.

3. Conclusion

The cutting process can be modelled as removing a chip using a wedge-shaped tool. The major parameter of the tool is the wedge angle θ though sharpness, as defined by the tip radius, can have an effect but this is not considered here. The chip thickness is also a major factor in defining behaviour. The material properties which are relevant are Young's modulus E, the yield stress σ_Y and the fracture toughness G_c. Friction is also involved and can be described via the Coulomb friction coefficient μ. The parameter that drives the cutting is the cutting force F_c, which gives rise to a transverse force normal to F_c and is F_t.

Here F_c provides the external work for the cutting process which is dissipated as fracture G_c, friction and plastic work via either elastic–plastic bending or shear. The former gives rise to chip curling and the latter to chip thickening. For elastic materials such as rubber, there is no plastic deformation so only fracture and friction are involved. Thin sheets give low forces so the latter can often be avoided. Leaf cutting tests using either blades or scissors are of this type.

Where there is plastic deformation then the type of behaviour can be defined by a non-dimensional thickness given by

and when x > 3 the facture is always elastic as observed when cutting thick chips which give cracking rather than forming smooth chips. For x < 3, plastically deformed curled chips from plastic bending occur. At small values of x plastic shearing occurs, which gives smooth cutting and thickened chips without curling. In this case, there is a linear correlation between cutting force and chip thickness which can be employed to determine material properties via cutting. The behaviour is quite complex and friction can influence the transition from shear to plastic bending.

It does appear that mechanics modelling and ideas derived from fracture mechanics enable the parameters involved in cutting to be quantified. In teeth, for example, rectangular specimens may be made and then layers of varying thickness removed with the forces F_c and F_t measured. From these readings, G_a and σ_Y may be found which characterize both fracture and wear. Inhomogeneity may be accommodated by choosing the location where the cutting occurs. As previously mentioned, G_c may be found for many foods including leaves, fruit and meat and the interaction of teeth and food may be quantified to describe chewing efficiency and tooth wear. It is hoped that this will be useful in biological studies relating diet, teeth and evolution.

Competing interests

We declare we have no competing interests.

Funding

We received no funding for this study.