Fundamentals of cutting J. G. Williams1,2 and Y. Patel1
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Research Cite this article: Williams JG, Patel Y. 2016 Fundamentals of cutting. Interface Focus 6: 20150108. http://dx.doi.org/10.1098/rsfs.2015.0108
1 2
Department of Mechanical Engineering, Imperial College London, South Kensington, London SW7 2AZ, UK Department of Aero, Mechanical and Mechatronics Engineering, University of Sydney, Sydney, Australia 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.
One contribution of 14 to a theme issue ‘Cutting science in biology and engineering’. Subject Areas: biomedical engineering Keywords: fracture, cutting, fracture toughness, shear, elastic – plastic bending, elastic bending Author for correspondence: J. G. Williams e-mail: [email protected]
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, Gc, required to create the new surface area and is a material property, the fracture toughness. The fracture criterion is thus G Gc
ð1:1Þ
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. 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, Gc, 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 Gc was the true surface work of the material, which is very small and could be ignored.
& 2016 The Author(s) Published by the Royal Society. All rights reserved.
Table 1. Nomenclature.
Gd
plastic work energy dissipation per unit area
h hc
cut depth chip thickness
b Fc
specimen width cutting force
Ft
force normal to cutting direction
S N
shear force normal force
dUext dUc
external work fracture energy
dUf
friction energy
dUd dUs
plastic dissipation energy shear energy
R Rp
radius of curvature radius of curvature at first yield
u m M E sY r eY f
tool angle friction coefficient bending moment Young’s modulus tensile yield stress plastic zone length tensile yield strain shear plane angle
This is still the case in many machining modelling numerical codes but it was pointed out by Atkins [5] that the Gc 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 Gc 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
2. Energy balance Figure 2 shows the parameters involved in cutting and are the cut depth, h, the chip thickness, hc, the specimen width, b, and the tool angle, u. The forces generated by the tool moving at a constant speed are Fc in the cutting direction, i.e. the cutting force, and the transverse force Ft. 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 Fc and Ft 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. When the tool moves a distance dx so does the cut, and we may define the following energy changes: (1) (2) (3) (4)
external work dUext ¼ Fc. dx, fracture energy dUc ¼ Gc. bdx, friction energy dUf ¼ S. dx (for hc ¼ h), and plastic dissipation energy dUd.
The energy balance is now dUext ¼ dUc þ dUf þ dUd :
ð2:1Þ
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 (dUd ¼ 0) with no friction (dUf ¼ 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 ¼ 0 and hence dUf ¼ 0. Equation (2.1) now becomes Fc : dx ¼ bGc dx,
i:e:
Fc ¼ Gc : b
ð2:2Þ
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.
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energy release rate energy per unit area at fracture (fracture toughness)
2
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G Gc
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.
(a)
(b)
3
chip
friction force
friction force
cutting tool
shear plane
cutting force cutting force crack tip touching
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Figure 1. Deformation processes in cutting. (a) Chip bending and (b) chip shearing. 2.6 q
Ro R b
h
Mo
Fc
2.4
hc 2.2
0
2.0
Ft contact point
Fc 1.8 bGc
S
1.6 Fc 1.4
N Ft
1.2
Figure 2. Parameters involved.
1.0
Fc dx ¼ bGc dx þ Sdx, and at the contact point
and
)
N ¼ Fc sinu þ Ft cosu:
50
0
100
150
Figure 3. Variation of cutting force with blade angle for m ¼ 0.2.
Equation (2.1) is now
S ¼ Fc cosu Ft sinu
qo = 79° Ft = 0
qo
2.2. Elastic deformation (dUd ¼ 0) with friction
ð2:3Þ
Thus, Fc dx ¼ bGc dx þ (Fc cosu Ft sinu)dx
Note that Ft/Fc ¼ 1/m at u ¼ 0 and 2m at ¼ 908, i.e. a change of sign, and Ft ¼ 0 for tan uo ¼ 1/m. This has been suggested as a criterion to give smooth machined surfaces [11] and gives a minimum Fc/b, i.e. when d/du(Fc/bGc) ¼ 0, and from equation (2.6) we have Fc 1 1 ¼ ¼ and bGc min 1 cos uo 1 sin b ð2:7Þ uo þ b ¼ 90 : W
and Fc Ft (1 cos u) þ sin u ¼ Gc : b b
ð2:4Þ
If u is known and Fc and Ft are measured then Gc can be found without defining the nature of the friction process. For S ¼ 0, Ft ¼ Fc(cos u/sin u) and equation (2.2) is retrieved. Coulomb friction is often assumed and is given by S ¼ mN, where m is the friction coefficient, and from equation (2.3) we have cos u m sin u Fc Ft ¼ Fc ¼ , ð2:5Þ sin u þ m cos u tan(b þ u) where m ¼ tanb, and from equation (2.4) Fc 1 ¼ : bGc (1 cos u) þ sin u=tanðb þ uÞ
ð2:6Þ
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chip
This condition for a minimum in Fc is shown in figure 3 for m ¼ 0.2, b ¼ tan21(0.2) ¼ 11.38, uo ¼ 798 and (Fc/bGc)min ¼ 1.24. 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 M Eh3 1 ¼ , ð2:8Þ b 12 R where E is Young’s modulus and h is the chip thickness (hc ¼ 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 ¼ Mo and R ¼ Ro. As the cut progresses, this point moves along the chip following the unloading line until it reaches the contact point, where M ¼ 0 and 1/R ¼ 0. The
M b
M b Mp A
A
b
b
Mo
2 Mp 3 b
loading Gc
b
Eh3 12 1 R
1 Ro
Figure 4. Bending moment –radius of curvature diagram for elastic bending. s < sY
(b)
sY
h
O
c 1 Rp
1 – R
1 Ro
1 R
Figure 6. Bending moment – radius of curvature diagram for elastic – plastic bending. when c ¼ 0 and M/b ¼ Mp/b ¼ sYh 2/4. For M1/b , M/b , Mp/b, the moment is given by ! ðc ð h=2 M y2 ¼2 sY dy þ sY y dy , b c 0 c i.e. M Mp ¼ b b
(c)
(d )
sY
y q
sY
c h
Figure 5. Stress states in elastic – plastic bending. (a) Elastic bending, (b) first yield, c ¼ h/2, (c) elastic– plastic bending and (d) fully plastic bending, c ¼ 0. 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 ¼ Gc, 1 Mo 1 6 Mo 2 ¼ 3 : ð2:9Þ Gc ¼ 2 b Ro Eh b
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 6 M 1 12 M and ¼ 3 : s¼ 2 h b R Eh b In figure 5b, this maximum stress reaches the tensile yield stress, sY, at the outer surface and 6 M1 1 1 12 M1 2 sY sY ¼ 2 ¼ with ¼ ¼ : 3 h R Rp Eh h E b b Figure 5c shows the stress distribution for M . M1, where the elastic –plastic interface is a distance y ¼ c from the central neutral axis. The upper limit for M is shown in figure 5d
! 1 2c 2 1 : 3 h
At y ¼ c, the strain is eY so that eY ¼ c/R, and noting that eY ¼ h/2Rp we have 2c/h ¼ R/Rp and ! M Mp 1 R 2 ¼ , ð2:10Þ 1 b 3 Rp b i.e. M/b ¼ Mp/b at R ¼ 0 and M/b ¼ (Mp/b)(2/3) ¼ M1/b at R ¼ Rp . 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/Rp and the loading is as shown in figure 4 and unloading is a line parallel to the elas¯ is the residual tic loading. The intercept on the 1/R axis, 1/R, curvature of the chip. This upper shaded area is Gc, given by 2 sY h 2 R 1 1 , 1 , ð2:11Þ Gc ¼ 2E 3 Rp R Rp and the lower shaded area is the energy dissipated in plastic work, i.e. 2 Rp sY h dUd 1 R 4 , ð2:12Þ ¼ Gd þ 3 Rp 3 2E dx R and Rp ¼ R
Rp 1 R 2 3 : þ 2 Rp 2 R
ð2:13Þ
It should be noted that, in equation (2.11), Gc ¼ ðs2Y h=2EÞ when R ! 0, i.e. Gc is given by the elastic energy release rate when the elastic stress limit is sY. Thus, the parameter s2Y h=2EGc ¼ 1 represents the maximum Gc which can be achieved for a constant sY. For s2Y h=2EGc , 1, there will be no cutting since the required Gc cannot be achieved. The limit for the onset of plastic bending is when R/Rp ¼ 1, i.e. Gd ¼ 0 and s2Y h=2EGc ¼ 3: Equation (2.12) may be expressed as a function of s2Y h=2EGc by substituting R/Rp
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O
(a)
equation (2.10) Gd
Gc
unloading Eh3 12
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loading
Mo
4
5
q
hc
15
Fc
q = 90°
sY /2 h
Ft
f
elastic bending
10
5
q = 40°
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Fc bGc
q = 20°
equation (2.14)
Figure 7. Plastic shearing.
1
from equation (2.11) and if we ignore friction since it is a constant factor times Fc/b in this case we have Fc 1 xþ1 (3 x), ð2:14Þ ¼1þ 6 x1 bGc where x ¼ s2Y h=2EGc : This parameter is the ratio of h to a plastic zone length r ¼ 2EGc =s2Y :
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 f and there is no bending. Invoking the Tresca yield criterion, the shear stress on the plane is sY/2. In this case the chip thickness, hc, is greater than h because of plane strain conditions and for finite friction hc sin (f þ u) : ¼ sin f h
ð2:15Þ
For a tool movement of dx the movement along the tool–chip interface, dxc, is dxc hc sin (f þ u) : ¼ ¼ sin f dx h
ð2:16Þ
The shear displacement along the shear plane is dus sin u ¼ , dx sin (f þ u)
ð2:17Þ
and hence the energy dissipation terms may be derived dUf dxc sin f ¼S ¼ [(Fc bGc ) cos u Ft sin u] dx dx sin (f þ u) and dUd sY bh dus sY bh sin u ¼ ¼ : 2 sin f sin (f þ u) dx 2 sin f dx From equation (2.1), we now have Fc Fc Ft sin f Gc ¼ bGc cos u sin u b b b sin (f þ u) sY h sin u þ , 2 sin f sin (f þ u)
0
1
2 s 2Y h 2EGc
Figure 8. Diagram of chip formation; for shearing eY ¼ 0.02 and m ¼ 0. This solution forms the basis of a method for finding Gc from cutting [12] in which h is varied and hc measured, hence giving f. If Fc and Ft are measured then Fc/b þ (Ft/b) tanf may be plotted versus (h/2)(tanf þ 1/tanf ) to give Gc from the intercept and sY from the slope of the straight line. f 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 tanf and equating to zero we have Ft sY h 1 , ð2:19Þ 1 ¼ 2 b tan2 fo where fo is the value of f at this minimum, and substituting in equation (2.18) we have
Fc Ft sY h 1 : tan f þ þ tan f ¼ Gc þ 2 tan f b b
ð2:18Þ
ð2:20Þ
with 1= tan2 fo ¼ 1 ð2=sY hÞðFt =bÞ: In this case Gc and sY may be found by fitting Fc and Ft data for a set of h values without the need for measuring hc. If Coulomb friction is assumed then, for this touching case, equation (2.5) becomes Ft Fc cos u m sinu : ð2:21Þ ¼ Gc b b sin u þ m cos u This enables tan fo to be determined in terms of m and u using equations (2.16)–(2.18), and after some manipulation 1 bþu ¼ tan , tan fo 2 where tan b ¼ m, and equation (2.19) becomes Fc 2 bþu ¼ 1 þ tan x: eY bGc 2 For b ¼ 0, i.e. m ¼ 0, this becomes
i.e.
3
Fc sY h , ¼ Gc þ tan fo b
Fc 2 u ¼ 1 þ tan x, eY bGc 2
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plastic shearing elastic–plastic bending
ð2:22Þ
where dc ¼ Gc/sY is the crack opening displacement and is a characteristic length for fracture.
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 u 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 sY and the fracture toughness Gc. Friction is also involved and can be described via the Coulomb friction coefficient m. The parameter that drives the cutting is the cutting force Fc, which gives rise to a transverse force normal to Fc and is Ft. Here Fc provides the external work for the cutting process which is dissipated as fracture Gc, friction and plastic work via either elastic –plastic bending or shear. The former gives
x¼
s2Y h , 2EGc
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 Fc and Ft measured. From these readings, Ga and sY may be found which characterize both fracture and wear. Inhomogeneity may be accommodated by choosing the location where the cutting occurs. As previously mentioned, Gc 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.
References 1.
2.
3.
4. 5.
Griffith AA. 1921 The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221, 163–179. (doi:10.1098/rsta.1921.0006) Lake GJ, Yeoh OH. 1978 Measurement of rubber cutting resistance in the absence of friction. Int. J. Fracture 14, 509–526. (doi:10.1007/ BF01390472) Williams JG. 2015 Stress at a distance fracture criteria and crack self-blunting in rubber. Int. J. Non-Linear Mech. 68, 33 –36. (doi:10.1016/j. ijnonlinmec.2014.06.008) Finnie I. 1956 Review of the metal cutting theories of the past hundred years. Mech. Eng. 78, 715–721. Atkins AG. 2003 Modelling metal cutting using modern ductile fracture mechanics: quantitative
6.
7. 8.
9.
explanations for some long standing problems. Int. J. Mech. Sci. 45, 373 –396. (doi:10.1016/S00207403(03)00040-7) Orowan E. 1952 Fundamentals of brittle behaviour of metals. In Fatigue and fracture of metals (ed. WM Murray), pp. 139 –167. New York, NY: Wiley. Irwin GR. 1958 Fracture. Encyclopaedia of physics, vol. 6, pp. 551–590. Berlin, Germany: Springer. Kamyab I, Chakrabarti S, Williams JG. 1998 Cutting cheese with wire. J. Mat. Sci. 33, 2763–2770. (doi:10.1023/A:1017517332112) Lucas PW, Choong MF, Tan HTW, Turner IM, Berrick AJ. 1991 The fracture toughness of the leaf of the dicotyledon Calophyllum inophyllum L. (Guttiferae).
10.
11.
12.
13.
Phil. Trans. R. Soc. Lond. B 334, 95 –106. (doi:10. 1098/rstb.1991.0099) Atkins AG. 2009 The science of engineering of cutting, p. 37. Oxford, UK: Elsevier Butterworth-Heinemann. Kobayashi S, Thomsen EG. 1959 Some observations on the shearing process in metal cutting. Trans. ASME J. Eng. Industry 81, 251–261. Patel Y, Blackman BRK, Williams JG. 2009 Determining fracture toughness from cutting tests on polymers. Eng. Fract. Mech. 76, 2711–2730. (doi:10.1016/j.engfracmech.2009.07.019) Merchant ME. 1945 Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip. J. App. Phys. 16, 267–275. (doi:10.1063/1.1707586)
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Figure 8 shows the three regimes of chip formation where Fc/bGc 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 Fc/bGc and h. This regime is useful for finding Gc and usually gives smooth surfaces and stable forces. Equation (2.22) may be written as Fc bþu h h 2 ¼ 1 þ tan , , , eY bGc 2 dc dc
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
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and may be compared with equation (2.17) for bending, Fc 1 xþ1 (3 x): ¼1þ 6 x1 bGc
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Research Cite this article: Williams JG, Patel Y. 2016 Fundamentals of cutting. Interface Focus 6: 20150108. http://dx.doi.org/10.1098/rsfs.2015.0108
1 2
Department of Mechanical Engineering, Imperial College London, South Kensington, London SW7 2AZ, UK Department of Aero, Mechanical and Mechatronics Engineering, University of Sydney, Sydney, Australia 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.
One contribution of 14 to a theme issue ‘Cutting science in biology and engineering’. Subject Areas: biomedical engineering Keywords: fracture, cutting, fracture toughness, shear, elastic – plastic bending, elastic bending Author for correspondence: J. G. Williams e-mail: [email protected]
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, Gc, required to create the new surface area and is a material property, the fracture toughness. The fracture criterion is thus G Gc
ð1:1Þ
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. 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, Gc, 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 Gc was the true surface work of the material, which is very small and could be ignored.
& 2016 The Author(s) Published by the Royal Society. All rights reserved.
Table 1. Nomenclature.
Gd
plastic work energy dissipation per unit area
h hc
cut depth chip thickness
b Fc
specimen width cutting force
Ft
force normal to cutting direction
S N
shear force normal force
dUext dUc
external work fracture energy
dUf
friction energy
dUd dUs
plastic dissipation energy shear energy
R Rp
radius of curvature radius of curvature at first yield
u m M E sY r eY f
tool angle friction coefficient bending moment Young’s modulus tensile yield stress plastic zone length tensile yield strain shear plane angle
This is still the case in many machining modelling numerical codes but it was pointed out by Atkins [5] that the Gc 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 Gc 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
2. Energy balance Figure 2 shows the parameters involved in cutting and are the cut depth, h, the chip thickness, hc, the specimen width, b, and the tool angle, u. The forces generated by the tool moving at a constant speed are Fc in the cutting direction, i.e. the cutting force, and the transverse force Ft. 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 Fc and Ft 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. When the tool moves a distance dx so does the cut, and we may define the following energy changes: (1) (2) (3) (4)
external work dUext ¼ Fc. dx, fracture energy dUc ¼ Gc. bdx, friction energy dUf ¼ S. dx (for hc ¼ h), and plastic dissipation energy dUd.
The energy balance is now dUext ¼ dUc þ dUf þ dUd :
ð2:1Þ
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 (dUd ¼ 0) with no friction (dUf ¼ 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 ¼ 0 and hence dUf ¼ 0. Equation (2.1) now becomes Fc : dx ¼ bGc dx,
i:e:
Fc ¼ Gc : b
ð2:2Þ
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.
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energy release rate energy per unit area at fracture (fracture toughness)
2
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G Gc
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.
(a)
(b)
3
chip
friction force
friction force
cutting tool
shear plane
cutting force cutting force crack tip touching
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Figure 1. Deformation processes in cutting. (a) Chip bending and (b) chip shearing. 2.6 q
Ro R b
h
Mo
Fc
2.4
hc 2.2
0
2.0
Ft contact point
Fc 1.8 bGc
S
1.6 Fc 1.4
N Ft
1.2
Figure 2. Parameters involved.
1.0
Fc dx ¼ bGc dx þ Sdx, and at the contact point
and
)
N ¼ Fc sinu þ Ft cosu:
50
0
100
150
Figure 3. Variation of cutting force with blade angle for m ¼ 0.2.
Equation (2.1) is now
S ¼ Fc cosu Ft sinu
qo = 79° Ft = 0
qo
2.2. Elastic deformation (dUd ¼ 0) with friction
ð2:3Þ
Thus, Fc dx ¼ bGc dx þ (Fc cosu Ft sinu)dx
Note that Ft/Fc ¼ 1/m at u ¼ 0 and 2m at ¼ 908, i.e. a change of sign, and Ft ¼ 0 for tan uo ¼ 1/m. This has been suggested as a criterion to give smooth machined surfaces [11] and gives a minimum Fc/b, i.e. when d/du(Fc/bGc) ¼ 0, and from equation (2.6) we have Fc 1 1 ¼ ¼ and bGc min 1 cos uo 1 sin b ð2:7Þ uo þ b ¼ 90 : W
and Fc Ft (1 cos u) þ sin u ¼ Gc : b b
ð2:4Þ
If u is known and Fc and Ft are measured then Gc can be found without defining the nature of the friction process. For S ¼ 0, Ft ¼ Fc(cos u/sin u) and equation (2.2) is retrieved. Coulomb friction is often assumed and is given by S ¼ mN, where m is the friction coefficient, and from equation (2.3) we have cos u m sin u Fc Ft ¼ Fc ¼ , ð2:5Þ sin u þ m cos u tan(b þ u) where m ¼ tanb, and from equation (2.4) Fc 1 ¼ : bGc (1 cos u) þ sin u=tanðb þ uÞ
ð2:6Þ
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chip
This condition for a minimum in Fc is shown in figure 3 for m ¼ 0.2, b ¼ tan21(0.2) ¼ 11.38, uo ¼ 798 and (Fc/bGc)min ¼ 1.24. 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 M Eh3 1 ¼ , ð2:8Þ b 12 R where E is Young’s modulus and h is the chip thickness (hc ¼ 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 ¼ Mo and R ¼ Ro. As the cut progresses, this point moves along the chip following the unloading line until it reaches the contact point, where M ¼ 0 and 1/R ¼ 0. The
M b
M b Mp A
A
b
b
Mo
2 Mp 3 b
loading Gc
b
Eh3 12 1 R
1 Ro
Figure 4. Bending moment –radius of curvature diagram for elastic bending. s < sY
(b)
sY
h
O
c 1 Rp
1 – R
1 Ro
1 R
Figure 6. Bending moment – radius of curvature diagram for elastic – plastic bending. when c ¼ 0 and M/b ¼ Mp/b ¼ sYh 2/4. For M1/b , M/b , Mp/b, the moment is given by ! ðc ð h=2 M y2 ¼2 sY dy þ sY y dy , b c 0 c i.e. M Mp ¼ b b
(c)
(d )
sY
y q
sY
c h
Figure 5. Stress states in elastic – plastic bending. (a) Elastic bending, (b) first yield, c ¼ h/2, (c) elastic– plastic bending and (d) fully plastic bending, c ¼ 0. 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 ¼ Gc, 1 Mo 1 6 Mo 2 ¼ 3 : ð2:9Þ Gc ¼ 2 b Ro Eh b
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 6 M 1 12 M and ¼ 3 : s¼ 2 h b R Eh b In figure 5b, this maximum stress reaches the tensile yield stress, sY, at the outer surface and 6 M1 1 1 12 M1 2 sY sY ¼ 2 ¼ with ¼ ¼ : 3 h R Rp Eh h E b b Figure 5c shows the stress distribution for M . M1, where the elastic –plastic interface is a distance y ¼ c from the central neutral axis. The upper limit for M is shown in figure 5d
! 1 2c 2 1 : 3 h
At y ¼ c, the strain is eY so that eY ¼ c/R, and noting that eY ¼ h/2Rp we have 2c/h ¼ R/Rp and ! M Mp 1 R 2 ¼ , ð2:10Þ 1 b 3 Rp b i.e. M/b ¼ Mp/b at R ¼ 0 and M/b ¼ (Mp/b)(2/3) ¼ M1/b at R ¼ Rp . 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/Rp and the loading is as shown in figure 4 and unloading is a line parallel to the elas¯ is the residual tic loading. The intercept on the 1/R axis, 1/R, curvature of the chip. This upper shaded area is Gc, given by 2 sY h 2 R 1 1 , 1 , ð2:11Þ Gc ¼ 2E 3 Rp R Rp and the lower shaded area is the energy dissipated in plastic work, i.e. 2 Rp sY h dUd 1 R 4 , ð2:12Þ ¼ Gd þ 3 Rp 3 2E dx R and Rp ¼ R
Rp 1 R 2 3 : þ 2 Rp 2 R
ð2:13Þ
It should be noted that, in equation (2.11), Gc ¼ ðs2Y h=2EÞ when R ! 0, i.e. Gc is given by the elastic energy release rate when the elastic stress limit is sY. Thus, the parameter s2Y h=2EGc ¼ 1 represents the maximum Gc which can be achieved for a constant sY. For s2Y h=2EGc , 1, there will be no cutting since the required Gc cannot be achieved. The limit for the onset of plastic bending is when R/Rp ¼ 1, i.e. Gd ¼ 0 and s2Y h=2EGc ¼ 3: Equation (2.12) may be expressed as a function of s2Y h=2EGc by substituting R/Rp
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O
(a)
equation (2.10) Gd
Gc
unloading Eh3 12
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loading
Mo
4
5
q
hc
15
Fc
q = 90°
sY /2 h
Ft
f
elastic bending
10
5
q = 40°
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Fc bGc
q = 20°
equation (2.14)
Figure 7. Plastic shearing.
1
from equation (2.11) and if we ignore friction since it is a constant factor times Fc/b in this case we have Fc 1 xþ1 (3 x), ð2:14Þ ¼1þ 6 x1 bGc where x ¼ s2Y h=2EGc : This parameter is the ratio of h to a plastic zone length r ¼ 2EGc =s2Y :
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 f and there is no bending. Invoking the Tresca yield criterion, the shear stress on the plane is sY/2. In this case the chip thickness, hc, is greater than h because of plane strain conditions and for finite friction hc sin (f þ u) : ¼ sin f h
ð2:15Þ
For a tool movement of dx the movement along the tool–chip interface, dxc, is dxc hc sin (f þ u) : ¼ ¼ sin f dx h
ð2:16Þ
The shear displacement along the shear plane is dus sin u ¼ , dx sin (f þ u)
ð2:17Þ
and hence the energy dissipation terms may be derived dUf dxc sin f ¼S ¼ [(Fc bGc ) cos u Ft sin u] dx dx sin (f þ u) and dUd sY bh dus sY bh sin u ¼ ¼ : 2 sin f sin (f þ u) dx 2 sin f dx From equation (2.1), we now have Fc Fc Ft sin f Gc ¼ bGc cos u sin u b b b sin (f þ u) sY h sin u þ , 2 sin f sin (f þ u)
0
1
2 s 2Y h 2EGc
Figure 8. Diagram of chip formation; for shearing eY ¼ 0.02 and m ¼ 0. This solution forms the basis of a method for finding Gc from cutting [12] in which h is varied and hc measured, hence giving f. If Fc and Ft are measured then Fc/b þ (Ft/b) tanf may be plotted versus (h/2)(tanf þ 1/tanf ) to give Gc from the intercept and sY from the slope of the straight line. f 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 tanf and equating to zero we have Ft sY h 1 , ð2:19Þ 1 ¼ 2 b tan2 fo where fo is the value of f at this minimum, and substituting in equation (2.18) we have
Fc Ft sY h 1 : tan f þ þ tan f ¼ Gc þ 2 tan f b b
ð2:18Þ
ð2:20Þ
with 1= tan2 fo ¼ 1 ð2=sY hÞðFt =bÞ: In this case Gc and sY may be found by fitting Fc and Ft data for a set of h values without the need for measuring hc. If Coulomb friction is assumed then, for this touching case, equation (2.5) becomes Ft Fc cos u m sinu : ð2:21Þ ¼ Gc b b sin u þ m cos u This enables tan fo to be determined in terms of m and u using equations (2.16)–(2.18), and after some manipulation 1 bþu ¼ tan , tan fo 2 where tan b ¼ m, and equation (2.19) becomes Fc 2 bþu ¼ 1 þ tan x: eY bGc 2 For b ¼ 0, i.e. m ¼ 0, this becomes
i.e.
3
Fc sY h , ¼ Gc þ tan fo b
Fc 2 u ¼ 1 þ tan x, eY bGc 2
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plastic shearing elastic–plastic bending
ð2:22Þ
where dc ¼ Gc/sY is the crack opening displacement and is a characteristic length for fracture.
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 u 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 sY and the fracture toughness Gc. Friction is also involved and can be described via the Coulomb friction coefficient m. The parameter that drives the cutting is the cutting force Fc, which gives rise to a transverse force normal to Fc and is Ft. Here Fc provides the external work for the cutting process which is dissipated as fracture Gc, friction and plastic work via either elastic –plastic bending or shear. The former gives
x¼
s2Y h , 2EGc
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 Fc and Ft measured. From these readings, Ga and sY may be found which characterize both fracture and wear. Inhomogeneity may be accommodated by choosing the location where the cutting occurs. As previously mentioned, Gc 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.
References 1.
2.
3.
4. 5.
Griffith AA. 1921 The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221, 163–179. (doi:10.1098/rsta.1921.0006) Lake GJ, Yeoh OH. 1978 Measurement of rubber cutting resistance in the absence of friction. Int. J. Fracture 14, 509–526. (doi:10.1007/ BF01390472) Williams JG. 2015 Stress at a distance fracture criteria and crack self-blunting in rubber. Int. J. Non-Linear Mech. 68, 33 –36. (doi:10.1016/j. ijnonlinmec.2014.06.008) Finnie I. 1956 Review of the metal cutting theories of the past hundred years. Mech. Eng. 78, 715–721. Atkins AG. 2003 Modelling metal cutting using modern ductile fracture mechanics: quantitative
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explanations for some long standing problems. Int. J. Mech. Sci. 45, 373 –396. (doi:10.1016/S00207403(03)00040-7) Orowan E. 1952 Fundamentals of brittle behaviour of metals. In Fatigue and fracture of metals (ed. WM Murray), pp. 139 –167. New York, NY: Wiley. Irwin GR. 1958 Fracture. Encyclopaedia of physics, vol. 6, pp. 551–590. Berlin, Germany: Springer. Kamyab I, Chakrabarti S, Williams JG. 1998 Cutting cheese with wire. J. Mat. Sci. 33, 2763–2770. (doi:10.1023/A:1017517332112) Lucas PW, Choong MF, Tan HTW, Turner IM, Berrick AJ. 1991 The fracture toughness of the leaf of the dicotyledon Calophyllum inophyllum L. (Guttiferae).
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Phil. Trans. R. Soc. Lond. B 334, 95 –106. (doi:10. 1098/rstb.1991.0099) Atkins AG. 2009 The science of engineering of cutting, p. 37. Oxford, UK: Elsevier Butterworth-Heinemann. Kobayashi S, Thomsen EG. 1959 Some observations on the shearing process in metal cutting. Trans. ASME J. Eng. Industry 81, 251–261. Patel Y, Blackman BRK, Williams JG. 2009 Determining fracture toughness from cutting tests on polymers. Eng. Fract. Mech. 76, 2711–2730. (doi:10.1016/j.engfracmech.2009.07.019) Merchant ME. 1945 Mechanics of the metal cutting process. I. Orthogonal cutting and a type 2 chip. J. App. Phys. 16, 267–275. (doi:10.1063/1.1707586)
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Figure 8 shows the three regimes of chip formation where Fc/bGc 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 Fc/bGc and h. This regime is useful for finding Gc and usually gives smooth surfaces and stable forces. Equation (2.22) may be written as Fc bþu h h 2 ¼ 1 þ tan , , , eY bGc 2 dc dc
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
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and may be compared with equation (2.17) for bending, Fc 1 xþ1 (3 x): ¼1þ 6 x1 bGc
