Predicting the Average Lead Angle in Turning

using Non-Dimensional Parameters and considering

the Size Effect

ME590 Research Project by

Gustavo A. Delfino

for

Prof. William J. Endres, Ph.D.

University of Michigan

June 3,1997

Abstract

A new method to calculate the average lead angle (ψ̄) in turning is presented. This method is basedon the integration of forces along the cutting edge taking into account the size effect. The geometry of theturning process changes with the feed (f), depth of cut (d), corner radius (ε) and lead angle (ψr). Hence,the geometry is analyzed for each of the cases that arise.

Unfortunately the integration must be performed numerically. This is a computationally expensive task;therefore, the ultimate goal is to fit an empirical model to the results of the numerical integration to obtaina computationally efficient approximate model. Non-dimensional parameters are introduced to reduce thenumber of independent variables simplifying the eventual curve fitting process.

Chapter 1

Introduction

Many years of research has produced a good model for orthogonal cutting (Figure 1.1). The problem isthat, in practice, orthogonal cutting is rarely found. In cases like turning and boring the tool usually has acorner radius (Figure 1.2). In order to predict forces with the current models an effective lead angle must bemodeled. The effective lead angle is the orientation that and orthogonal cutting process, of same chip load,would require to produce the same thrust force direction as for the non-orthogonal cutting process, which,in this case, exhibits a corner radius. The effective lead angle is very important to successfully represent thecutting process.

Traditionally, the effective lead angle has been calculated using the following methods:

• Colwell’s Method: uses a geometric approach.

• Fu’s Method: integrates a constant force distribution.

• Endres’ Method: geometric approach using width-weighted summations.

• Subramani’s Method: integrates a force proportional to the uncut thickness.

• Size effect Method: incorporates the internal energies inside the integral because they change withthe uncut chip thickness.

The only method that has proven satisfactory over an acceptable range of feed rate and depth of cut is theSize Effect Method. Unfortunately, this accuracy comes at great computational expense because it requirestwo numerical integrations.

The the ultimate goal is to fit an empirical model to the results of this numerical integration to providea computationally fast mechanistic model. The fitting process of a non-linear function with so many inde-pendent variables is not an easy task. In order to facilitate the process it is extremely important to have theproblem well defined and write an analytical model tailored to better curve fitting. This involves—amongother things—selecting the directions in which the differential forces within the integrals are projected. Thispreliminary preparation is the objective of this project.

1

CH

IP L

OA

D

ψr

d

f

h

w

Figure 1.1: The geometry of the turning process, using a tool with zero corner radius.

CH

IP L

OA

D

ψr

d

f

Figure 1.2: The geometry of the turning process, note the presence of the corner radius.

2

Chapter 2

Geometry

In order to perform the integration along the cutting edge, an analytic representation of the uncut chipthickness h(s) is needed. The uncut chip thickness is the perpendicular distance from a point in the cuttingedge contact length, to the profile created in the previous revolution. To write an expression for the uncutchip thickness, the geometry must be known first. The parameters that describe the geometry are thefeed (f), depth of cut (d), corner radius (ε) and lead angle (ψr), as shown in Figure 2.1a.

2.1 Non-dimensional parameters

The geometry of the cutting process can be expressed in terms of non-dimensional parameters. One wayto non-dimensionalize this problem is to divide everything1 by the corner radius. One advantage of thisapproach is that the arc lengths becomes angles. This way the “shape” is going to be a function of 3variables instead of four. These 3 variables are chosen as

Gd = dε Gf = f

ε and ψr (2.1)

as shown in Figure 2.1b, which is unique to the values Gd = 1.2, Gf = 0.5 and ψr = 20◦.

1The angles are already non-dimensional, so they are not affected by the non-dimensionalization.

ψrd

f

Gd

Gf

ψr

1

(a) (b)

ε

Figure 2.1: (a) Geometry for ψr = 20◦, f = 0.5 mmrev , d = 1.2mm and ε = 1mm. (b) Geometry for Gd = 1.2,

Gf = 0.5 and ψr = 20◦

3

(a) (b)

s=

0

s = w

w

S=

0

S = 1

W=w

ε

Figure 2.2: (a) The variable s goes from 0 to the width of cut w. (b) The non-dimensional variable S 6= sε

goes from zero to one regardless of the non-dimensional width of cut W .

1−

√1−

G2f

4

Figure 2.3: Gd must be larger than 1−√

1−G2f /4

The integration is to be performed along the cutting edge contact length. The cutting edge length2 isa function of Gf , Gd and ψr and is defined as follows:

W =w

ε=π

2+ sin−1 Gf

2−{

sin−1 (1−Gd) if Gd < 1− sinψrψr − Gd−1+sinψr

cosψrotherwise (2.2)

To simplify the calculation of integration limits it is convenient to define a non-dimensional variable Sthat describes the position along the cutting edge where 0 ≤ S ≤ 1 (see Figure 2.2).

For all cases, the minimum depth of cut for this model validity is Gd ≥ 1−√

1−G2f /4 (see Figure 2.3).

A depth of cut smaller than this is not typical and would simply produce an effective lead angle of 90◦.

It is important to see the effects Gd on the integrations to be performed. For this reason it is desiredto have Gd as the abscissa in several graphs. It is convenient to have graphs starting at zero rather than1 −

√1−G2

f /4, especially for log-log or semi-log graphics. For this reason Gdx, a new non-dimensional

quantity, is introduced. This is new Gdx is basically the same Gd shifted by 1−√

1−G2f /4 so that it starts

at zero. The definition is

Gdx = Gd − 1 +

√1−

G2f

4. (2.3)

The uncut chip thickness (h) is always measured perpendicular to the cutting edge, as shown in Fig-ure 2.4. The non-dimensional uncut chip thickness is H = h/ε; it is a function of position in the cuttingedge. This function is going to be defined in the following sections.

2from now and on all the lengths are non-dimensional i.e., multiply by ε to get the real length.

4

h

Figure 2.4: Gd must be larger than 1−√

1−G2f /4

1

Gf

Gd

Laf

Laa

Figure 2.5: Case 1

2.2 Case 1

This is the smallest depth of cut case, which is shown in Figure 2.5. The conditions for this case are

Gd ≥ 1−

√1−

G2f

4and [(ψr ≥ 0 and Gd ≤ 1− sinψr) or (ψr ≤ 0 and Gd ≤ 1)] . (2.4)

These conditions determine the boundaries between this case and cases 2, 4 and 7. See Figure 2.6 for details.

The lengths Laf and Laa in Figure 2.5 stand for Larc→free surface and Larc→arc. The equations that describethe geometry of this case are

Laf = tan−1

(1−Gd√

2Gd −G2d −Gf

)− sin−1(1−Gd) (2.5)

5

ψr < 0ψr = 0ψr > 0Case 1

Case 2 Case 4 Case 7

Case 1 Case 1Low

erG

dH

igher

Gd

Gd = 1Gd = 1− sinψrGd = 1− sinψr

Figure 2.6: Transitions from case 1 to cases 2, 4 and 7.

Laa =π

2+ sin−1

(Gf2

)− tan−1

(1−Gd√

2Gd −G2d −Gf

)(2.6)

ψ(S) = SW + sin−1(1−Gd) (2.7)

H(S) =

{1− 1−Gd

sinψ(S) SW ≤ Laf1 +Gf cosψ(S)−

√1−G2

f sin2 ψ(S) otherwise(2.8)

Note that ψ(S) is the lead angle as a function of position along the cutting edge (i.e., ψ(S) 6= ψr) and H(S)is the non-dimensional uncut chip thickness: H(S) = h(s)/ε.

2.3 Case 2

This case is shown in Figure 2.7. The conditions for this case are

ψr > 0 and (1− sinψr < Gd < 1− sinψr +Gf cosψr sinψr) (2.9)

The lengths Llf and Lal in Figure 2.7 stand for Llead edge→free surface and Larc→line. The equations that describethe geometry of this case are

Llf =sinψr +Gd − 1

cosψr(2.10)

Laf = tan−1

((1−Gd) cosψr

1 + (Gd − 1) sinψr −Gf cosψr

)− ψr (2.11)

Lal = tan−1

(sinψr

cosψr −Gf

)− tan−1

((1−Gd) cosψr

1 + (Gd − 1) sinψr −Gf cosψr

)(2.12)

Laa =π

2+ sin−1

(Gf2

)− tan−1

(sinψr

cosψr −Gf

)(2.13)

6

Laa

L af

Llf

L al

(a)

(b)

(c)

Figure 2.7: (a) Case 2. (b) Case 2 close to Gd = 1−sinψr. (c) Case 2 close to Gd = 1−sinψr+Gf cosψr sinψr.

ψ(S) ={ψr SW ≤ LlfSW + ψr − Llf otherwise (2.14)

H(S) =

SWtanψr

SW ≤ Llf1− 1−Gd

sinψ(S) Llf < SW ≤ Llf + Laf

1− 1−Gf cosψrcos(ψ(S)−ψr) Llf + Laf < SW ≤ Llf + Laf + Lal

1 +Gf cosψ(S)−√

1−G2f sin2 ψ(S) SW > Llf + Laf + Lal

(2.15)

2.4 Case 3

This case, shown in Figure 2.8, is the most common. The conditions for this case are

ψr > 0 and Gd ≥ 1− sinψr +Gf cosψr sinψr. (2.16)

The equations are

Llf = Gf sinψr (2.17)

Lll =Gd − 1 + sinψr −Gf sinψr cosψr

cosψr(2.18)

Lal = tan−1

(sinψr

cosψr −Gf

)− ψr (2.19)

Laa =π

2+ sin−1

(Gf2

)− tan−1

(sinψr

cosψr −Gf

)(2.20)

ψ(S) ={ψr SW ≤ Llf + Lllψr + SW − Llf − Lll SW > Llf + Lll

(2.21)

7

Laa

L al

Lll

Llf

Figure 2.8: Case 3

H(S) =

SW

tanψrSW ≤ Llf

Gf cosψr Llf < SW ≤ Llf + Lll1− 1−Gf cosψr

cos(ψ(S)−ψr) Llf + Lll < SW ≤ Llf + Lll + Lal

1 +Gf cosψ(S)−√

1−G2f sin2 ψ(S) Llf + Lll + Lal < SW

. (2.22)

2.5 Case 4

This case could be implemented as a part of case 3. However, for computational efficiency reasons, it ispresented as a separate case. This is one of the easiest cases (see Figure 2.9). The condition for this case is

ψr = 0 and Gd > 1. (2.23)

The equations are

Lll = Gd − 1 (2.24)

Laa =π

2+ sin−1

(Gf2

)(2.25)

ψ(S) ={

0 SW ≤ LllSW − Lll SW < Lll

(2.26)

H(S) =

{Gf SW ≤ Lll1 +Gf cosψ(S)−

√1−G2

f sin2 ψ(S) SW > Lll. (2.27)

2.6 Case 5

This is the negative lead angle version of case 3. There is an important observation to make concerningthe lower triangle of Figure 2.10a. This triangular area is not being taken into account under the standardprocedures that have been used for all previous cases. The point S = 0 is always at the beginning of the

8

L aa

Lll

1

Gd

Gf

Figure 2.9: Case 4

Laa

Lll

Lla

ψr

?✘ ✔

Laa

Lll

Lla

L fl

ψr

S=

0

S=L fl

S=

0(a) (b)

Figure 2.10: Case 5

cutting edge (i.e. the lower left corner); see figure 2.2. If some modification is not introduced, this triangulararea would be neglected. Hence, a smaller chip load would be used with possible adverse effects in theaverage lead angle prediction.

To take this triangular area into account it will be assumed that within this area the direction of thecutting and thrust forces is the same as in the place where the cutting edge starts. For this, and thefollowing cases, the variables S and W will be redefined to include a fictional additional cutting edge. Withthis modification Equation 2.2 is no longer valid. For this case W is defined by Equation 2.33.

The conditions for this case are

ψr < 0 and Gd > 1− sinψr −Gf cosψr sinψr. (2.28)

With this new approach, the equations are

Llf = −Gf sinψr (2.29)

Lll =Gd − 1 + sinψr +Gf cosψr sinψr

cosψr(2.30)

9

Laa

Lla

L fa

L fl

(a)

(b) (c)

Figure 2.11: (a) Case 6. (b) Case 6 close to Gd = 1 − sinψr. (c) Case 6 close to Gd = 1 − sinψr −Gf cosψr sinψr.

Lla = −Gf sinψr (2.31)

Laa =π

2+ sin−1

(Gf2

)− ψr (2.32)

W = Llf + Lll + Lla + Laa (2.33)

ψ(S) ={ψr SW ≤ Lfl + Lll + Llaψr + SW − Lfl − Lll − Lla SW > Lll + Lla

(2.34)

H(S) =

−SWtanψr

SW ≤ LflGf cosψr Lfl < SW ≤ Lfl + Lll1 +Gf cosψr −

√1− (SW − Lll − Lfl)2 Lfl + Lll < SW ≤ Lfl + Lll + Lla

1 +Gf cosψ(S)−√

1−G2f sin2 ψ(S) SW > Lfl + Lll + Lla

. (2.35)

2.7 Case 6

This is the negative lead angle version of case 2. For this case W is defined by Equation 2.40. The conditionsare

ψr < 0 and [1− sinψr < Gd < 1− (1 +Gf cosψr) sinψr] . (2.36)

The equations are

Lla = Lfl =Gd − 1 + sinψr

cosψr(2.37)

Lfa = cos−1√

1− (Gf sinψr + Lla)2 (2.38)

Laa =π

2+ sin−1

(Gf2

)− ψr (2.39)

W = 2Lla + Lfa + Laa (2.40)

ψ(S) ={ψr SW ≤ 2Lla + Lfaψr + SW − 2Lla − Lfa SW > 2Lla + Lfa

(2.41)

10

Laa

Lfa

Figure 2.12: Case 7

H(S) =

− SWtanψr

SW ≤ Lla1− sin(SW−Lla+Lla)

tanψr− cos(SW − Lla) Lla < SW ≤ Lla + Lfa

1 +Gf cosψr −√

1− (SW −Gf sinψr − 2Lla − Lfa)2

{SW ≥ Lla + LfaSW ≤ 2Lla + Lfa

1 +Gf cosψ(S)−√

1−G2f sin2 ψ(S) SW > 2Lla + Lfa .

(2.42)

2.8 Case 7

This is the negative lead angle version of case 1. For this case W is defined by Equation 2.46. The conditionsare

ψr < 0 and [1 < Gd < 1− sinψr] . (2.43)

The equations are

Laa =π

2+ sin−1

(Gf2

)+ sin−1(Gd − 1) (2.44)

Lfa = −Gf sinψ(0) (2.45)W = Laa + Lfa (2.46)

ψ(S) ={− sin−1(Gd − 1) 0 ≤ SW ≤ LfaSW − Lfa − sin−1(Gd − 1) Lfa < SW ≤ Lfa + Laa

(2.47)

H(S) =

{1− SW

tanψ(S) −√

1− (SW )2 0 ≤ SW ≤ Lfa1 +Gf cosψ(S)−

√1−G2

f sin2 ψ(S) Lfa < SW ≤ Lfa + Laa .(2.48)

11

Chapter 3

Integration

3.1 Effective Lead Angle

The effective lead angle is calculated by integrating the in-plane force along the cutting edge. This force,always normal to the cutting edge, is projected into a coordinate system. By taking the inverse tangent ofthe resulting magnitudes, the average direction with respect to this coordinate system can be found. Thedirection of this coordinate system is defined by the angle ψ′. When ψ′ = 0 the magnitude of the projectionsare FRad and FLon and the average lead angle is ψ = tan−1 (FRad/FLon). When ψ′ 6= 0, the magnitudeof the projections is different from FRad and FLon; the resulting magnitudes will be called F ′Rad and F ′Lon.Note that FRad = F ′Rad and FLon = F ′Lon only when ψ′ = 0. In general, the effective lead angle is calculatedwith the following formula

ψ = ψ′ + tan−1

(F ′RadF ′Lon

)(3.1)

The integrals for radial and longitudinal force components are[F ′RadF ′Lon

]=∫ [

dF ′RaddF ′Lon

]=∫

A · dN +∫

B · dP (3.2)

where the normal rake face force (dN) is proportional to the area (da)

dN = K · da (3.3)

and the in-plane (friction/shear) rake force is

dP = µ · dN = µK · da. (3.4)

The rake face force coefficient (K) and the mean equivalent rake face coefficient of friction/shear (µ) aremodeled as

K = ea0h(s)a1V a2 (3.5)µ = eb0h(s)b1V b2 (3.6)

The normal force transformation vector A, though dependent on the characteristic cutting processangles, is constant with the edge location for the flat faced tools considered here. It is defined as

A =[A1

A2

]=[

sin γn sin(ψ − ψ′) + cos γn sinλ cos(ψ − ψ′)sin γn cos(ψ − ψ′)− cos γn sinλ sin(ψ − ψ′)

]. (3.7)

12

(a) (b)ψ′r

Figure 3.1: (a) When ψ′ = 0, the forces are projected into the radial and longitudinal directions. (b) Forψ′ 6= 0 the forces are projected into a different direction. In this case ψ′ = ψ(S=0)

The in-plane force transformation is

B =[B1

B2

]=[

cos ηγ cos γn sin(ψ − ψ′) + (sin ηγ cosλ− cos ηγ sin γn sinλ) cos(ψ − ψ′)cos ηγ cos γn cos(ψ − ψ′)− (sin ηγ cosλ− cos ηγ sin γn sinλ) sin(ψ − ψ′)

](3.8)

where λ is the inclination angle, ηγ is the chip flow angle and γn is the normal rake angle. All these anglesvary with edge orientation angle ψ, which depends on edge location. The chip flow angle is approximatedusing the Stabler’s rule: ηγ = λ. The inclination angle (λ) can be calculated with the formula

λ = tan−1(tan γp cosψ − tan γf sinψ) (3.9)

where γp is the back rake angle and γf is the side rake angle.

From Equation 3.1 is clear that the individual values of F ′Rad and F ′Lon are not essential. What isessential is the ratio F ′Rad/F

′Lon. This is

F ′RadF ′Lon

=∫A1dN +

∫B1dP∫

A2dN +∫B2dP

. (3.10)

Substituting Equation (3.3) and Equation(3.4) into Equation(3.10),

F ′RadF ′Lon

=∫A1Kda+

∫B1µKda∫

A2Kda+∫B2µKda

. (3.11)

Substituting Equation (3.5) and Equation(3.6) into Equation(3.11),

F ′RadF ′Lon

=∫A1e

a0h(s)a1V a2da+∫B1e

b0h(s)b1V b2ea0h(s)a1V a2da∫A2ea0h(s)a1V a2da+

∫B2eb0h(s)b1V b2ea0h(s)a1V a2da

. (3.12)

Grouping similar terms and moving out constants in integrals

F ′RadF ′Lon

=A1e

a0V a2∫h(s)a1da+ ea0+b0V a2+b2

∫B1h(s)a1+b1da

A2ea0V a2∫h(s)a1da+ ea0+b0V a2+b2

∫B2h(s)a1+b1da

(3.13)

Simplifying common terms and substituting ρ = a1 + b1 + 1 and∫h(s)a1da = h̄a1a,

F ′RadF ′Lon

=A1V

a2 h̄(s)a1a+ eb0V a2+b2∫B1h(s)ρ−1da

A2V a2 h̄(s)a1a+ eb0V a2+b2∫B2h(s)ρ−1da

(3.14)

where h̄ = a/w.

13

dS

dθhdS

Figure 3.2: The two different kinds of differentials of area.

3.2 Differential of Area

The differential of area needs to be carefully defined. At the lead edge a differential of area is just a rectangle,but at the curved part it is part of a segment of circle (see Figure 3.2). The area of this segment of circle is

da =12ε2dθ − 1

2(ε− h)2dθ =

12(ε2 − (ε2 − 2εh+ h2)

)dθ =

h

2(2ε− h)

ds

ε=h(s)

2

(2− h(s)

ε

)ds (3.15)

In general, the differential area is

da =

{h(s)ds straight parth(s)

2

(2− h(s)

ε

)ds curved part (3.16)

The position at which the differential area changes from one kind to the other is intrinsic to each of thegeometric cases.

3.3 Non-dimensionalization

In Equation (3.14) the only parts that require numerical integration are the∫B•h(s)ρ−1da where • = 1 or 2.

Hence, these are the terms that require a non-dimensional representation:∫B•h(s)ρ−1da =

ερ−1

ερ−1

∫B•h(s)ρ−1da(s) (3.17)

= ερ−1

∫B•

(h(s)ε

)ρ−1

da(s) (3.18)

= ερ−1

∫B•H

( sw

)ρ−1

da(ws

w

)(3.19)

= ερ−1

∫B•H(S)ρ−1da(wS) (3.20)

14

The differential of area in ερ−1∫B•H(S)ρ−1da(wS) is different for the straight and curved parts. As s = wS,

then ds = wdS. Hence, for the straight part,∫B•h(s)ρ−1da(wS) = ερ−1

∫B•H(S)ρ−1h(wS)wdS (3.21)

= ερ−1εw

∫B•H(S)ρ−1H(S)dS (3.22)

= ερ+1W

∫B•H(S)ρdS (3.23)

For the curved part,∫B•h(s)ρ−1da(wS) = ερ−1

∫B•H(S)ρ−1h(wS)

2

(2− h(wS)

ε

)wdS (3.24)

= ερ−1εw

∫B•H(S)ρ−1H(S)

2(2−H(S)) dS (3.25)

= ερ+1W

∫B•H(S)ρ

(2−H(S)

2

)dS (3.26)

To summarize,

ψ̄ = ψ′ + tan−1

A1Va2 h̄(s)a1a+ eb0V a2+b2ερ+1W

[∫ c0B1H(S)ρdS +

∫ 1

cB1H(S)ρ

(2−H(S)

2

)dS]

A2V a2 h̄(s)a1a+ eb0V a2+b2ερ+1W[∫ c

0B2H(S)ρdS +

∫ 1

cB2H(S)ρ

(2−H(S)

2

)dS] (3.27)

The only parts that require numerical integration are R1 and R2; where

R1 =∫ c

0

B1H(S)ρdS +∫ 1

c

B1H(S)ρ(

2−H(S)2

)dS (3.28)

R2 =∫ c

0

B2H(S)ρdS +∫ 1

c

B2H(S)ρ(

2−H(S)2

)dS (3.29)

But these expressions do not take into account the increase in W as Gd increases. In other words, themagnitude of R1 and R2 reflect a constant width of cut equal to 1 for all values of Gd. This is because theintegration variable S is forced to go from 0 to 1. For this reason, it was found that modeling WR1 andWR2 is sometimes more convenient than modeling R1 and R2. The most appropriate case will be shown foreach integration result.

3.4 Integration Results

In this section, the values of R1 and R2 are calculated for differents values of Gdx to produce a continuousplot1. As the ultimate goal is to fit a function to these results, the plot must be made for all the extremevalues of the independent variables. This is crucial to select a good model to perform the curve fitting.

Each plot overlays three curves; the solid line corresponds to ψr = 45◦, the dashed line corresponds toψr = 0 and the dotted line correspond to ψr = −45◦. Each figure contains four of these graphics to representeach combination of the extreme values of Gf and ρ. The back and side rake angles were set to zero as theireffect is expected to be very small.

1The graphing algorithm in Mathematica automatically select the points to be evaluated to get an appropriate graph

15

3.4.1 Standard Integration (ψ′ = 0)

This is the intuitive first attempt. The results of R1 and WR2 are shown in Figures 3.3 and 3.4 respectively.Figure 3.3 looks smooth, the problem is that for ψr = −45◦, the R1 function becomes negative. This ishighly inconvenient if the curve fitting is to be performed with a log-scale ordinate; hence, a different ψ′ isgoing to be attempted.

Despite the undesired characteristic of R1, there is a very important advantage of using ψ′ = 0. Theimpact of the lead angle in WR2 is minimal as shown in Figure 3.4. The shape of WR2 is very close to astraight line in the log-log space; this is terrific from a curve-fitting point of view. Unfortunately, the sameψ′ must be used for both R1 and R2; hence, if ψ′ = 0 is discarded for R1 it can not be used for R2.

3.4.2 Integration with ψ′ = Colwell’s ψ̄

The idea here is use the Colwell’s estimation of ψ̄ as ψ′. The results are shown in Figures 3.5 and 3.5. Bycomparing this results to the previous ones with ψ′ = 0, it is clear that there is no improvement at all.Now, WR2 is more difficult to model than before because the effect of the lead angle can not be neglected.Also, the problem of the negative values of WR1 is worse and the are sharp corners that arise are difficultto model.

There is an interesting characteristic in Figure 3.5. The values of WR1 converge to a constant despitethe increase in W . This is because R1 goes to zero as W increases; and the multiplication converges to aconstant.

3.4.3 Integration with ψ′ =ψ(S=0)+ψ(S=1)

2

This time the value of ψ′ is selected as the average of the direction at the beginning and end of the contactlength of the cutting edge; that is ψ′ = (ψ(S=0) + ψ(S=1))/2. The results are shown in Figures 3.7 and 3.8.

The average of the aforementioned directions is close the Colwell’s ψ̄; hence, the results of the integrationare similar to those based on Colwell’s ψ̄.

3.4.4 Integration with ψ′ = ψ(S=0)

This time ψ′ is defined in the same direction as the beginning of the cutting edge; just like in Figure 3.1b.The results are shown in Figures 3.9 and 3.10. For this case, R1 is very good because it never gets negativeand it is mainly composed of straight lines in the log-log space. For this kind of behavior it is appropriateto use a model of the form

left · bq(a2 +G2dx)k/2

ak(b2 +G2dx)q/2

(3.30)

where a, b, p and q are constants, and ‘left’ is the value of R1 when Gdx → 0. The situation with R2 is verydifferent. This function is difficult to model due to the sharp points.

16

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-0.1

-0.05

0

0.05

0.1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.5

0

0.5

1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.5

0

0.5

1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.5

0

0.5

1

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

0.49236

0.000633390.1124

-0.49116

0.69128

0.00094766

0.80766

-0.68936

0.023433

0.0000244390.00025692

-0.023388

0.59167 0.00078669

0.57156

-0.57015

Figure 3.3: R1 vs. Gdx with ψ′ = 0

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1210-1110-1010-910-810-710-610-510-410-310-210-1100101

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-810-710-610-510-410-310-210-1100101102

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1110-1010-910-810-710-610-510-410-310-210-1100101102

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-810-710-610-510-410-310-210-1100101102

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

Figure 3.4: WR2 vs. Gdx with ψ′ = 017

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.8

-0.6

-0.4

-0.2

0

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.8

-0.6

-0.4

-0.2

0

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

-0.23347

-0.69818

1.1659·10-8

-0.42305

3.9982·10-8

-0.15387

-0.49304

-0.73843

-0.005651

-0.019711

2.5449·10-11

-0.0022114

1.5127·10-8

-0.043402

-0.016337

-0.022509

Figure 3.5: WR1 vs. Gdx with ψ′ = Colwell angle

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-2

10-1

100

101

102

103

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1

100

101

102

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1

100

101

102

103

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

0.10493

0.75397

5.141·10-6 0.011839

Figure 3.6: WR2 vs. Gdx with ψ′ = Colwell angle

18

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.8

-0.6

-0.4

-0.2

0

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.8

-0.6

-0.4

-0.2

0

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

-1

-0.8

-0.6

-0.4

-0.2

0

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

1.2489·10-8

-0.40777

-0.78112

-0.68636

4.2829·10-8

-0.57235

-0.84518

-0.96324

-0.012831

-0.031038

1.2718·10-9

-0.030642-0.58348

-0.31285

7.5596·10-7

-0.74699

Figure 3.7: R1 vs. Gdx with ψ′ = ψ(S=0)+ψ(S=1)

2

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-6

10-5

10-4

10-3

10-2

10-1

100

101

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-2

10-1

100

101

102

103

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1

100

101

102

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-1

100

101

102

103

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

0.104930.75397

5.141·10-6 0.011839

Figure 3.8: WR2 vs. Gdx with ψ′ = ψ(S=0)+ψ(S=1)

2

19

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-6

10-5

10-4

10-3

10-2

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-4

10-3

10-2

10-1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-4

10-3

10-2

10-1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

10-4

10-3

10-2

10-1

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

1.2489·10-8

4.2829·10-8

1.2718·10-9

7.5596·10-7

Figure 3.9: R1 vs. Gdx with ψ′ = ψ(S=0)

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

0

0.02

0.04

0.06

0.08

0.1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

0

0.2

0.4

0.6

0.8

1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

0

0.2

0.4

0.6

0.8

1

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103

0

0.2

0.4

0.6

0.8

1

Gf = 0.9 and ρ = 0.8 Gf = 0.9 and ρ = 0.05

Gf = 0.02 and ρ = 0.8 Gf = 0.02 and ρ = 0.05

0.10038

0.69613

0.91777

0.695270.72126

0.977250.9932

0.97573

0.033136

0.043701

0.0002569

0.033114

0.821840.80815

0.59164

0.80727

Figure 3.10: R2 vs. Gdx with ψ′ = ψ(S=0)

20

Chapter 4

Conclusion

The prediction of ψ̄ taking into account the size effect is a computationally expensive task. The level of detailneeded to successfully capture the effects of the geometry is substantial. To work around this, a model canbe fitted to the results of the integration to obtain an small and efficient way to estimate ψ̄. After studyingseveral possibilities for ψ′ it was found that the most likely to be be successful is ψ′ = 0. The reason is thesmoothness of the R1 curve and the very easy to fit WR2 curve.

The biggest problem to be faced is to find a good model for R1. It may be possible shift the curve toforce it to be positive and then, if appropriate, use the model presented in Equation 3.30. Once R1 and R2

are modeled, ψ̄ can be calculated using Equation 3.27.

21