Tool Sharpness as a Factor in Machining Tests to Determine Toughness
B. R. K. Blackman*, T. R. Hoult*, Y. Patel*, J. G. Williams*+.
* Mechanical Engineering Department, Imperial College London
+
University of Sydney
Abstract
An orthogonal cutting test has recently been proposed for determining Gc in polymers. This is
of particular value when applied to tough and/or ductile polymers, when the conditions of
LEFM are often violated. However, concerns exist about the effects of tool sharpness and the
contribution of ploughing to Gc. Here, tools with varying sharpness have been employed and
the results critically analysed. It is shown that cutting with sharp tools does give Gc, and that
when cutting with blunter tools the ploughing contribution can be rationalised by comparing
the tool tip radius with the height of the fracture process zone.
Keywords
Cutting, polymers, fracture toughness, ploughing, tool sharpness.
1
Nomenclature
Greek alphabet

tool rake angle

radius of curvature (sharpness) of the tool tip
c
height of the fracture process zone

shear plane angle
o
shear plane angle obtained from the intercepts (Figure 11)

angle around the tool tip in contact with workpiece (or half angle in wire cutting)

coefficient of friction between tool and workpiece
Y
yield stress of the workpiece material
English alphabet
b
width of cut
Fc
driving force on the tool in the cutting direction
Ft
transverse force on the tool generated by Fc.
Gc
fracture toughness
h
depth of cut
he
depth of elastic recovery (following ploughing)
hp
ploughed depth
2
1. Introduction
The inclusion of a fracture toughness term in the analysis of cutting and machining has a long
history. The machining of metals literature has generally not included the fracture term on
the grounds that it would be small compared to that for plastic work [e.g. 1]. It was also
excluded because cracks were not observed in ductile machining experiments. Atkins [2, 3]
has revisited the issue and pointed out that the fracture term is not necessarily small and
rederived the analysis including fracture. He shows that the toughness/strength ratio of a
material is a controlling factor in cutting behaviour. Lake et. al [4] had previously studied the
cutting of rubber and successfully treated it as a fracture problem.
This approach has been pursued by the present authors [5, 6] for the machining of polymers
and it has been proposed as a method for measuring the Fracture Toughness, Gc, for
polymers. It is of particular interest for polymers of high toughness and low yield stress
which are difficult to test conventionally because of crack blunting. The method has proved
successful [6] and the toughness values obtained are in good agreement with those for
conventional tests when the latter are possible. The method consists of machining layers of
varying thickness from a plate and measuring the cutting and transverse forces. These are
then analysed and, by extrapolation, the plastic work to shear the chip is separated from the
fracture component.
Such methods require care since the fracture term is usually
significantly less than that dissipated by the plastic shearing and/or bending of the chip [7].
The tool is assumed to be “sharp” in the analysis in that no local plastic work is done around
the tool tip. In the experiments the tools used are sharpened to give radii in the range 5 –
10m.
In a recent paper, Childs [8] challenges the notion that a fracture term needs to be included in
the analysis. Using a numerical code, a finite radius tool tip is included but there is no
fracture term per se. The programme computes the plastic dissipation around the tip of the
tool, which here we will term the “ploughing” contribution. Since new surfaces are created
the programme requires some form of separation process to run and this is achieved by
remeshing in the tool tip region. Some extrapolations from FEM simulation are explored and
it is concluded that what is measured is the “ploughing” term and that the energy changes
1
associated with remeshing are small. By implication the use of the method to determined
fracture toughness is in doubt. This paper addresses the issue by performing cutting tests
with blunt tools, i.e. with tip radii of up to 400 m, to measure the ploughing term. The
analysis presented for the removal of a surface layer with a blunt tool includes both fracture
toughness and the ploughing term such that it will be clearly evident if it is indeed ploughing,
rather than fracture, that is measured.
2. Preliminary Observations
Before considering the experiments it is useful to explore some of the existing data in the
literature to see if they would suggest that the ploughing term, rather than fracture, was
dominant. A recent paper [7] looked at a range of surface layer removal processes starting
from elastic cutting with tools of high rake angles () as shown in Figure. 1. The tip of the
tool is shown as blunt and takes no direct part in the fracture process as it does not come into
contact with the crack tip. A fracture process zone is shown with a tip opening of c. In this
case there is no contact between the workpiece and the tip of the tool and there is no plasticity
such that it is a classical elastic fracture problem. If there is no friction along the tool-chip
interface then [7]
Fc
 Gc
b
Equation 1
This analysis may be extended to include plastic bending of the chip (again in the absence of
contact between the tool and crack tip) and a fracture process results.
Plastic shearing in the chip generally occurs when smaller rake angles ( are employed and,
in the simplest solution, we have the situation shown in Figure 2a. This shows an infinitely
sharp tool ( = 0) touching the crack tip. It would be remarkable if this configuration does not
involve fracture whilst that discussed above does. Figure 2b depicts a fracture process zone
and the tip opening, c, where  < c and the contact is within the process zone. There is no
ploughing in Figure 2b but when  > c a ploughing contribution outside the zone is possible
as shown in Figure 2c. The larger tool radius requires the fracture or separation to occur at
2
some point around the radius at an angle  such that the cut depth h is reduced by hp, the
ploughing depth, i.e.
hp =  (1-cos)
Equation 2
The material in the layer hp is both plastically and visco-elastically deformed and recovers to
he as the tool passes over and the surface plastic flow leads to lateral deformation and the
F 
F 
formation of burrs. The forces per unit width due to ploughing,  c  and  t  may be
 b p
 b p
computed from equilibrium in the two orthogonal directions; we assume that the radial stress
is equal to the yield stress, Y, and for a friction coefficient of  we have (see Figure 3)
 Fc 
    Y  1  cos     sin  
 b p
 Ft 
    Y  sin    1  cos   
 b p
Equation 3
The angle  is determined by the fracture or separation point, i.e. the “stagnation” point in [8].
There may be some contribution from the recovering material on the clearance surface but it
is not included here since in the short test time the recovery is likely to be small.
It is of interest to note that this deformation mode also occurs in wire cutting tests on soft
solids [9] as is shown in Figure 4, where the analysis is used successfully. In this case  

2
so that by adding each half we have,
Fc
 2 1     Y   GC
b
F
and t  0 , from symmetry
b
Equation 4
Wires of varying diameters are used in the test to cut a soft material and the cutting force per
unit width is measured. This is then plotted against the radius of the wire,  and the resulting
linear relationship gives an intercept on the Fc/b axis of Gc. This is an example of a case
3
where  > c and, of course, no chip is produced and Ft = 0 because of symmetry.
Unfortunately it will not work for polymers or metals, because sufficiently strong wires are
not available.
If the notion is correct that the cutting test measures an apparent Gc value dominated by
ploughing, and as the tests are performed with sharp tools with the same tip radius, it would
lead naturally to the conclusion that the apparent Gc measured would be proportional to the
yield stress of the cut material. Table 1 shows values of the yield stress on the shear plane, Y,
and Gc from cutting tests taken from three references in the literature including some metals.
The Y values are generally higher than the quasi-static tensile values by a factor of about 2.5
because of work hardening in the shear zone and because of constraint [5]. The first two
rows in Table 1 show that the proportionality does not exist since the Gc values are the same
but the Y values vary by a factor of 7. The remaining polymer values further demonstrate
the lack of correlation and this can be quantified by assuming that  c 
Gc
Y
i.e. that the
constrained yield stress is an approximation to the cohesive stress. If Gc is proportional to
y. and also proportional to y.c, then if  is constant, c should be constant. However, c
varies by a factor of 45. In most cases,  < c, although in some low toughness materials c
approaches the lower limit of . The metals data have markedly different values of Y and Gc
but the c values vary significantly and are greater than the tool radii, so that ploughing would
be expected.
3. Machining Tests
3.1 Sharp Tools
The normal “sharp” tool is made by lapping the two tool faces which produces a tip radius in
the region of 5-10 m. Optical microscopy reveals a rather uneven surface (see Figure 5) so
this is not a smooth radius. Similar problems in defining sharpness in razor blades have been
reported [4]. The blunt tools used here were made using a CNC grinding machine and thus
did have smooth surfaces around the nose. Tool radii of 33, 41, 100, 200, 300 and 400 m
were produced to an accuracy of ±2%. The radii were measured using lead indentation and
surface profilometry. Figure 6 shows a micrograph of the 200 m tool.
4
Testing was performed with a rake angle =10° and with a width of cut b=6mm on two
polymers; polypropylene (PP) and high impact polystyrene (HIPS). Steady-state values of
F
Fc
and t were measured as a function of cut thickness, h, for the two polymers using a
b
b
sharp tool. Figure 7a and b shows the results for PP and HIPS respectively. After each cut
the chip thickness hc was measured and the shear plane angle  was determined from [6].
tan  
cos 
hc
 sin 
h
Equation 5
Gc was determined by what is referred to in [6] as Method 2*, i.e.
h
1 
 Fc Ft

  Gc
  tan     Y  tan  
2
tan  
 b b

Equation 6
F F

This analysis method requires that the values of  c  t tan   be plotted against
 b b

h
1 
h   tan  
 . The result is linear and the regression gives Y as the slope and Gc as
2
tan  
the intercept. The results for the two polymers are shown in Figure 8 and both materials show
good linearity (R2 = 0.998). The values of Y and Gc measured were:
PP
Y = 79.0 ± 0.6 MPa Gc = 3.14 ± 0.07 kJm-2
HIPS
Y = 121 ± 1.0 MPa Gc = 0.57 ± 0.11 kJm-2
The yield stress values, deduced from the slope, have a standard error < 1% while the Gc
values, deduced from the intercept, have standard errors of about 2% for PP and 20% for
HIPS. A large number of small thickness cuts, i.e. h < 50 m, are required to define the
rather low Gc value for HIPS. As mentioned previously the Y values are elevated above the
tensile values due to work hardening and constraint.
5
This method is prefered since it does not involve any assumptions such as Merchant’s meothd to find . It does
however, involve an extra measurement, hc, with attendant errors. It also avoids any detailed consideration of
the friction effects.
*
Single edge notched bend (SENB) tests were performed according to the LEFM standard for
determining Gc in polymers [12]. The specimens were 6 mm thick and it was found that all
tests failed the linearity criterion with Fmax/F5% calculated to be greater than 1.4. This was
expected because both materials have low quasi static tensile yield stresses, i.e. about 12-20
MPa. It is for this very reason that cutting tests have been developed as a way to measure Gc
when LEFM is violated. In PP the initial sharp crack tip blunts in the test. However, an
estimate of Gc can be made from the F5% point (i.e. the load for a 5% reduction in
compliance) and this gave a value of 4.0 ± 0.1 kJ m-2. For HIPS the rubber toughening of the
polymer leads to crazing around the particles and the absence of shear yielding. Hence, the
initial sharp crack in the HIPS did not blunt. However, after crack initiation, the energy
dissipation associated with crazing leads to a large increase in toughness (i.e. a strongly rising
resistance ‘R’ curve is observed). For HIPS the value at F5% was about 0.4 kJ/m2.
Two points in the data are noteworthy. The values of h (the cut thickness) were measured by
traversing the specimen surface before and after cutting and taking the difference. The
transverse force, Ft, is much less than the cutting force Fc as is usually the case for sharp tool
cutting [6]. It should also be noted that the Gc and Y combinations measured here are not in
accordance with the notion that the toughness measurement arises from ploughing. In that
case, high Gc values would result from materials with high Y values. This is clearly not
observed.
3.2 Blunt Tools
Cutting tests using the range of blunt tools were performed on the two polymers. It was
possible to produce chips for all of the radii except for the 400m tool. This tool failed to
produce chips, with ploughing occurring instead. Figure 9a shows the values recorded for PP
of
Fc
versus h for tests which produced chips. It was noted that only values of h> 0.10mm
b
could be achieved with the 300m tool. However, all the blunt tools gave lines which are
parallel to the “sharp” test data. The values of
Ft
versus h are shown in Figure 9b for all tests
b
which produced chips. These data are again parallel to the sharp tool data but are much
6
higher and increase with tool radius. Similar data were also obtained for HIPS and these
results are shown in Figures 10a and b.
3.3 Discussion of results
The values of the intercepts from the
F
Fc
and t versus h values (i.e. extrapolated to h = 0)
b
b
are plotted against the tool tip radius  in Figures 11a and 11b for PP and HIPS respectively.
There is a linear dependency for the 33 to 300 m values. However, for the sharp tool the
values fall below the linear fits for the
Fc
lines, i.e. the force containing Gc.
b
It is proposed here that the cutting and ploughing processes are additive. Thus the cutting
contribution at h = 0 is given by Equation 6 and is,
 Fc 
 Ft 
   Gc  tan o  
 b o
 b o
The ploughing terms are given by Equation 3, so that the totals are:
 Fc 
 Ft 
    Y  (1  cos  )   sin    Gc  tan o  
 b T
 b 0
 Ft 
 Ft 
    Y  sin    (1  cos  )   
 b T
 b 0
The various parameters may be estimated from the experimental data. From the
Ft
versus h
b
data in Figures 7 it can be seen that there are mostly negative slopes. This arises from the
tool-chip interaction which gives [5]
Ft    tan   Fc


b  1   tan   b
7
where  is the rake angle, (=10o in this case), i.e. tan  = 0.18. Thus  must be less than tan
 to give the negative slope and here we will assume it is approximately zero. The slope and
intercepts of Figures 7, 8 and 11 are given in Table 2. From the slopes of the plots of
intercept versus tool radius, Figures 11a and 11b, we may determine the values of  which
were calculated to be 61o for PP and 52o for HIPS. In addition we may determine Y and this
was found to be 114 MPa for PP and 182 MPa for HIPS using equation 3. These values may
be compared with those derived from the shear plane analysis in the sharp tool data (Figure 8)
which gave 79 MPa and 121 MPa respectively, i.e. a factor of about 1.4 higher. These latter
values are much higher than the tensile yield stresses and arise from the very high strains in
the shear plane and accompanying work hardening. The blunt tools, on the other hand, indent
the surface giving a very high local constraint. The concept that the process works at a
constant yield stress appears to fit the observations but if the indentation gave permanent
deformation the observed original chip thickness h would include the hp term, i.e. (1-cos )
i.e.  0.5 in this case. This would arise because of the measurement method of h and would
not be the true value. The chips are measured after the test to find hc and when this is done
there is no difference between sharp tools and blunt suggesting that hp in these materials is
almost fully recovered elastically. This can be clearly seen in Figure 12a which shows the
forces acting on a sharp and the 200 m tool at a cut depth of 0.13 mm. Figure 12b shows
the subsequent measured forces acting on the tool on a second pass with no additional cut
depth applied. It is seen that there is very little interaction between the tool and workpiece
after a cut with the sharp tool. However, a second ‘non-cutting’ pass with the blunt tool
shows significant forces are measured. It should be noted that there is no material removal
during the second pass suggesting that there is elastic deformation and recovery associated
with the ploughing term in blunt tool cuts. The values of
can be predicted from the slopes of the
F
Fc
and t for the non-cutting pass
b
b
F
Fc
and t intercept lines given in Table 2, which
b
b
were derived from Figure 11b. For Figure 12b, the 200m tool would imply a ploughing
force,
Fc
=70.9 MPa ×0.2 ×10-3 m. For a tool width of 6mm, this would suggest a force, Fc
b
= 85 N would be generated, as shown by the lower dashed line in Figure 12b. This is very
close to the maximum value of Fc attained in the second pass. For the transverse force due to
ploughing,
Ft
=143.4 MPa ×0.2 ×10-3 m, which, for the same tool width, would imply a
b
8
transverse force, Ft = 172 N. This is shown as the upper dashed line in Figure 12b and is
again very close to the maximum value of Ft attained in the second pass. The observation that
both the Fc and Ft values increase with time during the second pass is indicative of viscoelastic effects being present; on the second pass, the far end of the ploughed surface has had
longer to recover and thus higher forces are induced. The value of h was also cross-checked
by comparing with changes in the tool setting. The two values were close but not identical
because of compliance effects.
It would appear that the plastic deformation is very close to the surface and most of hp is
recovered. Very small burrs are seen on surfaces cut with the bluntest tools. It is also of
interest to note that for blunt tools, Figures 9 and 10 indicate that it is only possible to achieve
steady state cutting for values of h > 0.5. The intercepts in Figure 11 at  = 0 are:
 Fc 
 Ft 
   Gc  tan 0  
 b T ,0
 b 0
F 
F 
and  t    t 
 b T ,0  b 0
The analogy here with the wire cutting case is striking since the data for h = 0 do, of course,
F 
only include ploughing plus Gc and a residue resulting from the asymmetry in  t  . The
 b 0
accuracy in Gc values measured using the method of varying tool sharpness is however, lower
than the preferred method of varying h, because a smaller number of blunt tools than
thicknesses are used. This is confirmed by the intercepts given in Table 2. For example
Ft
Ft
 Ft 
  should be the same for b vs h data in the sharp tool cases and in b vs  and the
 b 0
values are;
PP
h, Figure 7
, Figure 11
3.85 ± 0.1 kJ m-2
3.84 ± 0.66 kJ m-2
HIPS 1.38 ± 0.09 kJ m-2 0.87 ± 0.47 kJ m-2
9
i.e. in very good agreement for PP and reasonable agreement for HIPS. The value of tan 0
may be found from the sharp tool data since Gc is known from extrapolation, i.e.
 Fc 
   Gc
 b 0
tan o 
 Ft 
 
 b 0
and we have values of 0.67 for PP and 0.61 for HIPS. Using these values with the intercepts
at  = 0 we may find Gc and these are shown below and compared to the sharp tool values.
PP
h, Figure 8
, Figure 11
Sharp Tool
Blunt Tools
3.13 ± 0.07 kJ m-2 4.8 ± 1.0 kJ m-2
HIPS 0.57 ± 0.11 kJ m-2 2.9 ± 1.0 kJ m-2
As expected, the results from blunt tools, are highly inaccurate especially for HIPS. This
latter value can be improved by noting that the parallel fitting assumed is not the best fit for
Ft
F
in Figure 10b. If the data are fitted for the best lines then the t intercept value at  = 0
b
b
is 3.21 ± 0.67 kJm-2 which gives a Gc value of 1.5 ± 1 kJm-2, i.e. an improvement in accuracy.
This does indicate that the notion of finding Gc by varying the sharpness could work if a
sufficient number of precisely manufactured tools are used. Such a technique is unlikely to
be better than that of varying h since two extrapolations are involved. However, it does
demonstrate that Gc can be measured in a test in which the ploughing term is dominant and
that the Gc measured is not due to ploughing.
4. Conclusions
The notion that the Gc value measured in cutting tests results primarily from the ploughing
term was critically assessed. If such a notion was correct, a high toughness would arise
largely as a result of a high yield stress, i.e. due to the local plasticity induced from
ploughing. In this case, there would then be strong correlation between Gc and the yield
stress. However, a review of some of the published data from the cutting literature found no
10
correlation between values of Gc and the yield stress on the shear plane for several different
materials. In addition, cutting tests performed using a sharp tool (tip radius ~5-10m) on two
polymers, polypropylene and high impact polystyrene gave Gc and Y combinations of (3.14
kJm-2 and 79 MPa) and (0.57 kJm-2 and 121 MPa) respectively, the higher Gc with the lower
yield stress, i.e. the opposite of what would be expected if the measured Gc resulted from
ploughing. It is thus concluded that the cutting test measures a true Gc value, independent of
any ploughing term.
When ploughing is intentionally introduced by using blunt tools, the cutting and transverse
forces were found to increase in proportion to the radius of the tool tip, as expected. A linear
correlation was observed for the larger radii, i.e. between 33 - 300m but the sharp tools give
forces below the line. This was explained by comparing the tool tip radius, to the height
of the fracture process zone,c. For < c, the tip interacts directly with the process zone
and hence the forces drop below the regression line because there is no ploughing (see Figure
2b). For PP,c was 40 m and for HIPS , c was 5 m. The tools of 33 and 41 m radius
give forces on the linear trend suggesting that ploughing is occurring in PP even though
 → c.
The large radii tool data for h = 0 is a “ploughing” test though limited here because of the
number of tools. However the results do show that even here, extrapolation to zero  gives
Gc as in wire cutting. There are, of course, uncertainties in analyses involving extrapolations
to zero for both h and  in this case. The linear fits in Figure 8 using sharp tools are
extremely accurate i.e. R2 = 0.998 and SD’s of 2% for PP and 20% for HIPS with a low Gc
value. The blunt tool data are more limited and hence less accurate but nothing in the data
presented here would suggest a major problem. However, blunt tool tests do not provide a
useful alternative test since the preferred method of using a sharp tool and varying the cut
depth is simple and accurate. Maintaining sufficient sharpness in the test does not appear to
be an issue since repeats over large numbers of tests have shown no trends which suggests
blunting is occurring.
11
5. Acknowledgments
The authors wish to thank Professor Atkins of Reading University and Imperial College
London for helpful discussions on this work. In addition they are grateful to Professor Childs
from Leeds University for raising concerns about tool bluntness in these tests.
6. References
1. I Finnie; “Review of Metal Cutting Theories of the Past Hundred Years”; Mech. Eng.;
1956; 78; 715-21.
2. A. G. Atkins; “Modelling Metal Cutting Using Modern ductile Fracture Mechanics;
Int. J. Mech Sci.; 2003; 45; 373-96.
3. A. G. Atkins; “Toughness and cutting: a new way of simultaneously determining
ductile fracture toughness and strength” Eng Frac Mech.; 2005; 72; 849-60.
4. G. J. Lake & O. H. Yeoh; “Measurement of Rubber Cutting Resistance in the
Absence of Friction”; Int. J. Fract.; 1978; 14; No. 5; 509-26.
5. Y. Patel, B. R. K. Blackman & J. G. Williams; “Measuring Fracture Toughness from
Machining Tests”; Proc. IMechE.; Vol 223 part C 2009; 2861-69.
6. Y. Patel, B. R. K. Blackman & J. G. Williams; “Determining Fracture Toughness
from Cutting Tests on Polymers” J. Eng. Fract. Mech. 2009; 2711-30.
7. J. G. Williams; “The Fracture mechanics of Surface Layer Removal”; Int. Jour. Fract.;
2011; 170; 37-48
8. T. H. L. Childs; “Surface Energy, Cutting Radius and Material Flow Stress Size
Effects in Continuous Chip Formation of Metals”; CIRP J of Man. Sci and Tech.;
2010; 3; 27-39.
9. I. Kamyab, S. Chakrabarti & J. G. Williams; “Cutting Cheese with Wire” J. Mat. Sci.;
1998; 33; 2763-70.
10. A. Kobayashi; “Machining of Plastics”; New York, Mcgraw-Hill; 1967
12
11. D. M. Eggleston, R. “Observations on the Angle Relationships in Metal Cutting”; J.
of Eng. For Industry; 1956; 263-79.
12. ISO 13580-2000 “Plastics-Determination of fracture toughness (GIC and KIC) – Linear
elastic fracture mechanics approach”
13
Table 1: Values of yield stress on the shear plane, Gc (determined from cutting), and the
associated process\s zone height, c, taken from the cutting literature.
Material Y (MPa) Gc (kJm-2) c (m) 
PA 4/6
150
3.7
24
LLDPE
21
3.7
180
HIPS
71
1.7
24
PMMA
250
1.1
4
PE
58
1.5
26
ABS
126
0.6
5
PA
109
1.2
11
PC
129
1.8
14
AC
113
1.9
17
PP
114
0.7
6
Al
650
9.2
14
Steel
740
35
48
-Brass
600
34
57

From [6]
From [10, 5]
From [11, 5]
PA – PolyAmide (Nylon), LLDPE – Linear Low Density Polyethylene; HIPS – High Impact
Polystyrene; PMMA – Polymethyl methacrylate; PC – Polycarbonate; AC – Poly Acetel; PP
– Polypropylene; ABS – Acrylonitrile butadiene styrene.
14
Table 2. Values of slope and intercept obtained from the various plots of F/b versus cut
thickness, h, and additionally from the plots of the intercept values of F/b at h=0 versus tool
radius.
PP
HIPS
PP
HIPS
Plot
Slope (MPa)
Intercept (kJ/m2)
Fc
vs h
b
75.9 ± 0.8
5.71 ± 0.08
Figure 7a
Ft
vs h
b
-21.6 ± 1.0
3.85 ± 0.10
Figure 7a
 Fc Ft

  tan   vs h
 b b

79.0 ± 0.6
3.13 ± 0.07
Figure 8
Fc
vs h
b
146.1 ± 1.3
1.41 ± 0.13
Figure 7b
Ft
vs h
b
-4.9 ± 0.4
1.38 ± 0.04
Figure 7b
 Fc Ft

  tan   vs h
 b b

121.2 ± 0.97
0.57 ± 0.11
Figure 8
Fc
intercepts
b
59.0 ± 1.3
7.43 ± 0.41
Ft
intercepts
b
100.1 ± 4.6
3.84 ± 0.66
Fc
intercepts
b
70.9 ± 4.7
3.48 ± 0.79
Ft
intercepts
b
143.4 ± 2.8
Figure 11a
Figure 11b
0.87 ± 0.47
Note: the right hand column shows the figure number from which the slope and intercept
values are obtained.
15
Figure 1. Elastic Cutting with high rake angle, .
Figure 2. Plastic Shearing. (Thickness h is removed, thickness hp is ploughed, i.e. pressed
down, but recovers to he, the elastic recovery, after the tool has passed.)
16
Figure 3. Ploughing. (hp = he only for full recovery)
Figure 4. Wire cutting.
17
Figure 5. 10m Sharp tool
Figure 6. 200m blunt tool
18
a) PP
b) HIPS
Figure 7. PP and HIPS sharp tool data.
19
Figure 8. Determination of Gc for sharp tool data
20
a) Fc/b vs h
b) Ft/b vs h
Figure 9. PP data for blunt tools
21
a) Fc/b vs h
b) Ft/b vs h
Figure 10. HIPS data for blunt tools
22
a) PP
b) HIPS
Figure 11.
F
Fc
and t at h = 0 as functions or tool radius.
b
b
23
a) Forces on the sharp and the 200 m tool during a cut of 0.13 mm.
b) Forces on the sharp and the 200 m tool during second pass
Figure 12. Forces measured whilst a) cutting HIPS at a depth of 0.13 mm and b) interaction
between tool and workpiece during a second pass with no additional cut depth. (Dashed lines
in 12b were derived from the product of the slope of the regression lines drawn in Figure
11b, and the tool radius.)
24