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 –

10m.

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]

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

some point around the radius at an angle such that the cut depth h is reduced by hp, the

ploughing depth, i.e.

2

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

formation of burrs. The forces per unit width due to ploughing, and may be

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)

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

so that by adding each half we have,

and , from symmetry

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

where > c and, of course, no chip is produced and Ft = 0 because of symmetry.

3

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 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

and were measured as a function of cut thickness, h, for the two polymers using a

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].

Equation 5

Gc was determined by what is referred to in [6] as Method 2*, i.e.

Equation 6

This analysis method requires that the values of be plotted against

. The result is linear and the regression gives Y as the slope and Gc as

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

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 400m tool. This tool failed to

produce chips, with ploughing occurring instead. Figure 9a shows the values recorded for PP

of

versus h for tests which produced chips. It was noted that only values of h> 0.10mm

could be achieved with the 300m tool. However, all the blunt tools gave lines which are

parallel to the “sharp” test data. The values of

versus h are shown in Figure 9b for all tests

6

which produced chips. These data are again parallel to the sharp tool data but are much

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

and versus h values (i.e. extrapolated to h = 0)

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 lines, i.e. the force containing Gc.

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,

The ploughing terms are given by Equation 3, so that the totals are:

The various parameters may be estimated from the experimental data. From the versus h

data in Figures 7 it can be seen that there are mostly negative slopes. This arises from the

tool-chip interaction which gives [5]

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 and for the non-cutting pass

can be predicted from the slopes of the and intercept lines given in Table 2, which

were derived from Figure 11b. For Figure 12b, the 200m tool would imply a ploughing

force, =70.9 MPa ×0.2 ×10-3 m. For a tool width of 6mm, this would suggest a force, Fc

= 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

8

ploughing, =143.4 MPa ×0.2 ×10-3 m, which, for the same tool width, would imply a

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 visco-

elastic 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:

and

The analogy here with the wire cutting case is striking since the data for h = 0 do, of course,

only include ploughing plus Gc and a residue resulting from the asymmetry in . The

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

should be the same for vs h data in the sharp tool cases and in vs and the

values are;

9

h, Figure 7 , Figure 11

PP 3.85 ± 0.1 kJ m-2 3.84 ± 0.66 kJ m-2

HIP

S

1.38 ± 0.09 kJ m-2 0.87 ± 0.47 kJ m-2

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.

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.

h, Figure 8

Sharp Tool

, Figure 11

Blunt Tools

PP 3.13 ± 0.07 kJ m-2 4.8 ± 1.0 kJ m-2

HIP

S

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

in Figure 10b. If the data are fitted for the best lines then the intercept value at = 0

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

10

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

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 ~5-10m) 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 - 300m 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.

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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.

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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

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”

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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.

Materia

l

Y (MPa) Gc (kJm-2) c (m)

PA 4/6 150 3.7 24

From [6]LLDPE 21 3.7 180

HIPS 71 1.7 24

PMMA 250 1.1 4

PE 58 1.5 26

From [10, 5]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

From [11, 5]Steel 740 35 48

-Brass 600 34 57

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.

Plot Slope (MPa) Intercept (kJ/m2)

PP

vs h75.9 ± 0.8 5.71 ± 0.08 Figure 7a

vs h-21.6 ± 1.0 3.85 ± 0.10 Figure 7a

79.0 ± 0.6 3.13 ± 0.07 Figure 8

HIPS

vs h146.1 ± 1.3 1.41 ± 0.13 Figure 7b

vs h-4.9 ± 0.4 1.38 ± 0.04 Figure 7b

121.2 ± 0.97 0.57 ± 0.11 Figure 8

PP intercepts

59.0 ± 1.3 7.43 ± 0.41

Figure 11a

intercepts100.1 ± 4.6 3.84 ± 0.66

HIPS intercepts

70.9 ± 4.7 3.48 ± 0.79

Figure 11b

intercepts143.4 ± 2.8 0.87 ± 0.47

15

Note: the right hand column shows the figure number from which the slope and intercept

values are obtained.

Figure 1. Elastic Cutting with high rake angle, .

16

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.)

17

Figure 3. Ploughing. (hp = he only for full recovery)

Figure 4. Wire cutting.

18

Figure 5. 10m Sharp tool

Figure 6. 200m blunt tool

19

a) PP

b) HIPS

Figure 7. PP and HIPS sharp tool data.

20

Figure 8. Determination of Gc for sharp tool data

21

a) Fc/b vs h

b) Ft/b vs h

Figure 9. PP data for blunt tools

22

a) Fc/b vs h

b) Ft/b vs h

Figure 10. HIPS data for blunt tools

23

a) PP

b) HIPS

Figure 11. and at h = 0 as functions or tool radius.

24

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.)

25