Equilibrium between the forces of the nominal and real contact stresses gives
t A
r
s
— = —— — (A3.16a)
kA
n
k
s
n
A
r
p
r
— = —— — (A3.16b)
kA
n
k
According to equation (A3.16a), A
r
/A
n
⇒ t/k as s/k ⇒ 1. From equation (A3.16b), if A
r
/A
n
< (s
n
/k), p
r
/k > 1. However, the slip-line field is not valid if p

n
/k) < 1. In these condi-
tions the ratio of friction to normal stress (the friction coefficient) becomes greater than 1.
A3.6 Friction coefficients greater than unity
In metal machining, and elsewhere, friction coefficients > 1 have been measured in condi-
tions in which asperities have been plastic but (t/k) and (s
n
/k) have been too low for bulk
plastic flow to be a possibility. What could account for this, that has not been considered
in the previous sections?
Work hardening offers two possibilities. First, in the same way as it changes the hydro-
static pressure distribution along the primary shear plane in metal machining (Figures 2.11
and 6.9(b)), it can modify the pressure within a deforming asperity to reduce the mean
value of p
r
to a value less than k. However, there is likely only to be a small effect with the
rake face asperities in machining, already work hardened by previous deformations. A
second possibility imagines a little work hardening and high adhesion conditions, leading
to the interface becoming stronger than the body of the asperity. Unstable asperity flow,
with contact area growth larger than expected for non-hardening materials, has been
observed by Bay and Wanheim (1976).
There is a second type of possibility. In the previous sections it has been assumed that
an asperity is loaded by an amount W by contact with a counterface and that W does not
change as sliding starts. For example, in Section 3.4.2 on junction growth of plastic
contacts, it is written that the addition of a sliding force F to a real contact area creates an
extra shear stress F/A
r
which, if A
r
does not increase, will cause t

Lond. A295, 300–319.
Johnson, K. L. (1985) Contact Mechanics. Cambridge: Cambridge University Press.
Oxley, P. L. B. (1984) A slip line field analysis of the transition from local asperity contact to full
contact in metallic sliding friction. Wear 100, 171–193.
Sutcliffe, M. P. (1988) Surface asperity deformation in metal forming processes. Int. J. Mech. Sci.
30, 847–868.
374 Appendix 3
Fig. A3.10 Qualitative junction load history for zero normal displacement
Childs Part 3 31:3:2000 10:43 am Page 374
Appendix 4
Work material: typical
mechanical and thermal
behaviours
This appendix holds data that support Chapters 3 and 7, in the first instance. In Chapter 3,
reference is made to yield and strain hardening behaviours of aluminium, copper, iron,
nickel and titanium alloys, as determined by room-temperature, low strain rate, compres-
sion testing. Information on this is given in Section A4.1. The thermal conductivity, heat
capacity and diffusivity ranges of these alloys, and their variations with temperature – also
used in Chapter 3 to estimate temperature rises during machining – are tabulated in Section
A4.2. In Chapter 7 the idea was developed that it is not the strain hardening behaviour of
the work materials at room temperature and low strain rates that is needed. What is impor-
tant for predicting chip formation in machining is the strain hardening behaviour at the
temperatures and strain rates that actually occur. Data on this are presented in Section
A4.3. This appendix is also a source for applications studies such as are described after
Chapter 7.
A4.1 Work material: room temperature, low strain rate,
strain hardening behaviours
Figures A4.1 to A4.3 contain representative strain hardening data for commercially pure
samples of aluminium, copper, iron, nickel and titanium, and their alloys. The data have
been obtained either from plane strain compression tests or from measuring the depen-

resistant, use. Commercially they are known as Inconel or Nimonic alloys. They are face
centred cubic, with initial yield stress larger than copper alloys and large amounts of strain
hardening. The titanium alloys (right-hand panel) are hexagonal close packed (h.c.p.) or
mixtures of h.c.p. and body centred cubic. Their initial yield and strain hardening behav-
iours are intermediate between the face centred and body centred cubic materials.
Further elementary reading on metal alloys, their mechanical properties and uses can be
found in Rollason (1973), Cottrell (1975) and Ashby and Jones (1986).
A4.2 Work material: thermal properties
Tables A4.1 to A4.3 contain information on the variation with temperature of the thermal
conductivity, heat capacity and diffusivity of a range of work materials. The main single
376 Appendix 4
Fig. A4.1 Shear stress-strain behaviours of some copper and aluminium alloys
Childs Part 3 31:3:2000 10:43 am Page 376
Thermal properties 377
Fig. A4.2 Shear stress-strain behaviours of some ferritic/pearlitic and austenitic steels
Fig. A4.3 Shear stress-strain behaviours of some nickel and titanium alloys
Childs Part 3 31:3:2000 10:43 am Page 377
378 Appendix 4
Table A4.1 Thermal conductivity [W/mK] of some work material groups
Alloy system Temperature [°C]
0 200 400 600 800
Iron and steel
pure iron* 85 64 50 38 31
0.04–0.25C 52–60 48–54 42–45 35–37 29–30
0.25–0.8C 51–52 46–48 39–42 33–35 29–30
0.8–1.2C 45–51 42–46 37–39 32–33 27–29
low alloy 25–49 30–45 32–40 30–35 27–30
ferritic stainless 21–24 22–25 23–26 24–27 25–29
austenitic stainless 14–17 15–18 17–21 22–25 26–29
high manganese 14 16 19 21 22

Iron and steel
pure iron, C, low alloy 3.5–3.8 4.1–4.3 4.7–5.0 5.6–5.9 6.7–7.1
ferritic stainless 3.5–4.1 3.8–4.3 4.3–5.0 6.0–7.1 5.8–6.2
austenitic stainless 3.5–4.5 4.2–4.7 4.5–4.8 5.2–5.5 5.6–5.9
high manganese 3.9 4.6 – – –
Aluminium
pure, 1000 series 2.4–2.7 2.6–2.8 2.6–2.9 2.6–2.9 –
2000 to 7000 series 2.1–2.8 2.5–3.1 3.2–3.4 – –
Al-Si cast alloys 2.3 2.6 2.8 – –
Copper
pure copper* 3.3 3.7 3.9 – –
Zn, Sn, Ni alloys 3.2–3.4 3.6–3.8 3.8–4.0 – –
Nickel
pure nickel* 4.1 4.3 4.5 4.9 5.5
70Ni–30Cu 3.8 4.0 4.2 4.6 5.2
Superalloys** 3.3–3.5 3.5–3.6 3.7–3.8 4.2–4.3 4.5–4.8
Titanium
pure Ti*,
α
,
α
–
β
,
β
alloys 2.3–2.5 2.55–2.75 2.75–3.05 3.0–3.4 3.3–3.8
*: high and commercial purity; **: including cobalt- and ferrous-base superalloys.
Childs Part 3 31:3:2000 10:43 am Page 378
source of information has been the ASM (1990) Metals Handbook but it has been neces-
sary also to gather information from a range of other data sheets.

)(
——
)(
∫
strain path
(
——
)
de
—
)
(A4.1a)
1000 1000
Strain hardening behaviours at high strain rates 379
Table A4.3 Diffusivity (mm
2
/s) of some work material groups
Alloy system Temperature [°C]
0 200 400 600 800
Iron and steel
pure iron* 23 15 10 6.5 4.5
0.04–0.25C 14–16 11–13 8.6–9.3 6.1–6.4 4.2–4.3
0.25–0.8C 14–15 11–12 8.1–8.7 5.7–6.1 4.2–4.3
0.8–1.2C 12–14 10–11 7.6–8.1 5.6–5.7 3.9–4.2
low alloy 7–13 7–11 6.6–8.2 5.2–6.1 3.9–4.3
ferritic stainless 5.1–6.8 5.1–6.5 4.6–6.0 3.4–4.5 4.0–5.0
austenitic stainless 3.2–3.7 3.5–4.0 3.8–4.4 4.0–4.8 4.4–5.2
high manganese 3.6 3.5 – – –
Aluminium
pure, 1000 series 78–100 75–90 70–80 73–76 –

M+mN
s
—
= A
(
e
T+273
)(
——
)
e
–
N
(A4.1b)
1000
Coefficients A, B, M, m and N for the following annealed metals are as follows.
Metal ABM m N
Aluminium 107 153 0.057 0.064 0.3
a-brass 720 56.7 0.024 0.06 0.5
A4.3.2 Pearlitic carbon and low alloy steels
In early studies, an equation similar to equation (A4.1a) was used but for a changed expo-
nential temperature term and a term dependent on temperature within the strain path inte-
gral. Later, this was developed to
e
—
˘
M
e
—
˘

M
–
s
—
= A
(
——
)
e
—
N
(A4.2b)
1000
A range of measured coefficients is given in Table A4.4, valid for T from 20˚C to 720˚C,
strain rates up to 2000 s
–1
and strains up to 1.
380 Appendix 4
Table A4.4 Flow stress data for annealed or normalized carbon and low alloy steels
Steel Coefficients of equation (A4.2)
0.1C A = 880e
–0.0011T
+ 167e
–0.00007(T–150)
2
+ 108e
–0.00002(T–350)
2
+ 78e
–0.0001(T–650)

–0.0003(T–380)
2
[3]* a = 0.000065 m = 0.0039
0.33C A = 1400e
–0.0012T
+ 177e
–0.000030(T–360)
2
– 107e
–0.001(T–100)
2
–Mn–B M = 0.0375 + 0.000044T N = 0.18e
–0.0012T
+ 0.098e
–0.0002(T–440)
2
[3]* a = 0.000065 m = 0.00039
0.36C A = 1500e
–0.0018T
+ 380e
–0.00001(T–445)
2
+ 160e
–0.0002(T–570)
2
–Cr–Mo M = 0.017 + 0.000068T N = 0.136e
–0.0012T
+ 0.07e
–0.0002(T–465)
2

b
]
where, for e
—
≤ 0.5 a + 0.87, b = 0.8; e
—
≥ 0.5 a = 0.57, b = 0.2
Other forms have been given for a Ti-6Al-4V alloy (Usui et al. 1984) and a Ti-6Al-6V-2Sn
alloy (Maekawa et al. 1994b). For the Ti-6Al-4V alloy:
s
—
= A(e
—
˘
/1000)
M
e
aT
(e
—
˘
/1000)
m
{
c +
[
d +
∫
strain path
e

a = 0.00009 m = 0.0055
References
ASM (1990) Metals Handbook, 10th edn. Ohio: ASM.
Ashby, M. F. and Jones, D. R. H. (1986) Engineering Materials, Vol. 2. Oxford: Pergamon Press.
Childs, T. H. C. and Maekawa, K. (1990) Computer aided simulation and experimental studies of
chip flow and tool wear in turning low alloy steels by cemented carbide tools. Wear 139,
235–250.
Cottrell, A. (1975) An Introduction to Metallurgy, 2nd edn. London: Edward Arnold.
Maekawa, K., Kitagawa, T. and Childs, T. H. C. (1991) Effects of flow stress and friction character-
istics on the machinability of free cutting steels. In: Proc. 2nd Int. Conf. on Behaviour of
Materials in Machining – Inst. Metals London Book 543, pp. 132–145.
Maekawa, K., Kitagawa, T., Shirakashi, T. and Childs, T. H. C. (1993) Finite element simulation of
three-dimensional continuous chip formation processes. In: Proc. ASPE Annual Meeting, Seattle,
pp. 519–522.
Maekawa, K., Ohhata, H. and Kitagawa, T. (1994a) Simulation analysis of cutting performance of a
three-dimensional cut-away tool. In Usui, E. (ed.), Advancement of Intelligent Production.
Tokyo: Elsevier, pp. 378–383.
Maekawa, K., Ohshima, I., Kubo, K. and Kitagawa, T. (1994b) The effects of cutting speed and feed
on chip flow and tool wear in the machining of a titanium alloy. In: Proc. 3rd Int. Conf. on
Behaviour of Materials in Machining, Warwick, 15–17 November pp. 152–167.
References 381
Childs Part 3 31:3:2000 10:43 am Page 381
Maekawa, K., Ohhata, T., Kitagawa, T. and Childs, T. H. C. (1996) Simulation analysis of machin-
ability of leaded Cr-Mo and Mn-B structural steels. J. Matls Proc. Tech. 62, 363–369.
Maekawa, K. (1998) private communication.
Rollason, E. C. (1973) Metallurgy for Engineers, 4th edn. London: Edward Arnold.
Usui, E. and Shirakashi, T. (1982) Mechanics of machining – from descriptive to predictive theory.
ASME Publication PED 7, 13–35.
Usui, E., Obikawa, T. and Shirakashi, S. (1984) Study on chip segmentation in machining titanium
alloy. In: Proc. 5th Int. Conf. on Production Engineering, Tokyo, 9–11 July, pp. 235–239.

Fig. A5.1 Coordinate systems and definitions for the analysis of tool (a) yielding and (b) fracture
(a) (b)
Childs Part 3 31:3:2000 10:43 am Page 383
dt ds
q
r —— + 2t + —— = 0 (A5.2a)
dr dq
At the apex, where r = 0, these become
dt
(s
r
– s
q
) + —— = 0 (A5.1b)
dq
ds
q
2t + —— = 0 (A5.2b)
dq
To avoid yielding of the tool, the shear yield stress of which is k
t
,
1
— (s
q
– s
r
)
2
+ t

s
n
k
t
——— = – 2 ———
∫
0
b
sin 2fdq (A5.6)
k
work
k
work
The largest value of s
n
/k
work
is obtained when the integral takes its largest negative value.
Figure (A5.2) shows the variation of f with q that gives that largest negative value: at q =
b, f = 0; and at q = 0, f is determined by the friction contact stress on the rake face. In
Chapter 3 (Figure 3.18) extreme examples of friction stress were considered, up to k
work
during steady chip creation, but zero at the start of a cut:
f ≡ f
0
= 0, t
f
= 0 (A5.7a)
1k
work

n
, for example
384 Appendix 5
Childs Part 3 31:3:2000 10:43 am Page 384
5k
work
or 2.5k
work
, a minimum ratio of tool to work shear yield stress to avoid yield can be
derived. Taking the tool’s Vickers Hardness HV to equal 5k
t
, relations between tool hard-
ness, k
work
and b to avoid tool yielding can be derived. Thus, the HV/b relations dependent
on k
work
shown in Figure 3.19 are obtained.
A5.2 Tool fracture
Figure A5.1(b) shows a wedge-shaped tool with a line force R per unit length acting at a
friction angle l at a distance d from the apex of the wedge. This force is equivalent to a
force R acting at the apex, with a moment M = Rd. A classical result of stressing a wedge
(Coker and Filon, 1931) is that on the rake face the tensile stress at a distance r from the
apex is
bb bb
cos — sin
(
l + —
)
sin — cos

work
over the contact length l between the work and tool. It is found for this example that
the maximum tensile stress occurs at r ≈ l. To replace the distributed stress by the equiva-
lent line force and moment is only marginally justifiable: the treatment is only approxi-
mate.
Tool fracture 385
Fig. A5.2 Variations of
φ
with
θ
that maximize
σ
n
/
k
work
Childs Part 3 31:3:2000 10:43 am Page 385
References
Coker, E. G. and Filon, L. N. G. (1931) A Treatise on Photoelasticity. London: Cambridge
University Press, pp. 328, 367.
Hill, R. (1954) On the limits set by plastic yielding to the intensity of singularities of stress. J. Mech.
Phys. Solids, 2, 278–285.
386 Appendix 5
Childs Part 3 31:3:2000 10:43 am Page 386
Appendix 6
Tool material properties
More detail is given here than in Chapter 3 of the materials that make up the main tool
groupings.
A6.1 High speed steels
The high speed steels are alloy steels with about 0.75% to 1.5% carbon (C), 4% to 4.5%

from Hoyle (1988), converting from Rockwell to Vickers Hardness, with additional data
from other sources. The data are presented to show the sensitivity of mechanical proper-
ties to composition and heat treatment.
Traditionally, high speed steels have been shaped by hot working. Now, powder metal-
lurgy technology is used to make high speed steel indexable inserts. HV and TRS values
are not much changed but there is evidence that fracture toughness (K
IC
values) can be
higher for powder metallurgy than wrought products. Sheldon and Wronski (1987) give
K
IC
at room temperature for sintered T6 as 30 MP m
1/2
whereas wrought T6 heat treated
in the same way has K
IC
= 15 to 20 MP m
1/2
. This paper also gives the temperature depen-
dence of TRS quoted in Chapter 3 (Figure 3.22).
A6.2 Cemented carbides and cermets
Cemented carbide and cermet cutting tools consist of hard carbide (or carbo-nitride)
grains, bonded or cemented together by up to around 20% by weight of cobalt or nickel,
388 Appendix 6
Fig. A6.1 Variations of room temperature HV and TRS with (a) tempering and (b) austenitizing temperature, for a
range of high speed steels as indicated
(a) (b)
Childs Part 3 31:3:2000 10:44 am Page 388
with minor additions of other metals (such as molybdenum or chromium) possible. The
hardness of the tools reduces and the toughness increases as the proportion of the metal

mise between the two (M). Grade refers to whether the tool material’s mechanical proper-
ties have been optimized for hardness and hence abrasive wear resistance, or for
toughness. Wear resistance is more important than toughness for low feed, finishing cuts.
Toughness is more important for high feed, roughing or interrupted cuts. Grades run from
01 to 50, as properties change from hard to tough.
Different manufacturers achieve a particular tool performance by minor differences of
the processing route, so that there is not a one-to-one relation between a tool’s type and
grade on the one hand and its composition on the other. This is illustrated in Figure A6.2.
Each row of the figure presents data on composition, hardness and transverse rupture stress
(at room temperature) for one manufacturer’s range of tool materials, according to infor-
mation published by Brookes (1992). The first row is data from a German manufacturer,
the second is from a major international company and the third is from a Japanese
producer. Each data point in the left hand column represents the TiC-TaC and Co weight
% of one tool material (the balance is WC). What type and grade is assigned to the mater-
ial is indicated by the solid and dashed lines. The ranges of compositions giving P-,
Cemented carbides and cermets 389
Childs Part 3 31:3:2000 10:44 am Page 389
M- and K-types are slightly different for each producer. So are the ranges of compositions
giving the different grades.
The right-hand column shows the relation between transverse rupture stress and hard-
ness for all the grades. It can be seen that the relation depends on the carbide grain size.
All three manufacturers produce tool materials of 1 to 2 mm grain size. These have the
same relation between transverse rupture stress and hardness, independent of K-, M- and
P-type. However, one set of data, in the first row, is for material of sub-micrometre
390 Appendix 6
Fig. A6.2 Composition and mechanical property differences of cemented carbide cutting tools classified according to
ISO 513 (1991) by three different manufacturers
Childs Part 3 31:3:2000 10:44 am Page 390
grain size: it shows a greater transverse rupture stress for a given hardness than the
coarser grained material. Such a fine grain size is only achievable with WC-Co (K-type)

The cermets are mainly described as P-types, although some manufacturers also recom-
mend them as K-types, but because of their limited toughness (TRS < 2.5 GPa, compared
with up to 4 GPa for fine grained WC-Co materials), none of them are recommended for
heavy duty use, above 30-grade.
Cemented carbides and cermets 391
Fig. A6.3 Hardness dependence on % Co and grain size, for cemented carbides
Childs Part 3 31:3:2000 10:44 am Page 391
392 Appendix 6
Fig. A6.4 Composition dependence of some properties of cemented carbides
Table A6.2 One manufacturer’s range of cermet tool materials
Wt. %
———————
Grain
ISO Ti(C,N) Ni + size
ρ
HV TRS K E
α
e
code + WC Co [
µ
m] [kg/m
3
] [GPa] [GPa] [W/mK] [GPa] [10
–6
K
–1
]
P/K01–05 95 5 1 6800 18.1 1.3 11 410 6.7
P10–P15 86 14 1 7100 15.5 1.65 12 400 7.2
P/K05–15 89 11 <1 7000 16.5 1.65 14 410 7.6