A2.2 Selected problems, with no convection
When u˘
x
= u˘
y
= u˘
z
= 0, and q* = 0 too, equation (A2.4) simplifies further, to
1 ∂T ∂
2
T ∂
2
T ∂
2
T
— —— =
(
—— + —— + ——
)
(A2.5)
k ∂t ∂x
2
∂y
2
∂z
2
where the diffusivity k equals K/rC. In this section, some solutions of equation (A2.5) are
presented that give physical insight into conditions relevant to machining.
A2.2.1 The semi-infinite solid
z
> 0: temperature due to an

0
Equation (A2.6) thus describes the temperature rise caused by releasing a quantity of heat
H per unit area, at z = 0, instantaneously at t = 0; and thereafter preventing flow of heat
across (insulating) the surface z = 0. Figure A2.1(b) shows for different times the dimen-
sionless temperature rC(T – T
0
)/H for a material with k = 10 mm
2
/s, typical of metals. The
increasing extent of the heated region with time is clearly seen.
At every time, the temperature distribution has the property that 84.3% of the associ-
ated heat is contained within the region z/
ǰ˭˭˭
4kt < 1. This result is obtained by integrating
equation (A2.6) from z = 0 to
ǰ˭˭˭
4kt. Values of the error function erf p,
2
p
erf p = ——
∫
e
–u
2
du (A2.8)
Ȉȉ
p
0
that results are tabulated in Carslaw and Jaeger (1959). Physically, one can visualize the
temperature front as travelling a distance ≈

½
The total temperature is obtained by integrating with respect to t′ from 0 to t. The temper-
ature at z = 0 will be found to be of interest. When q is independent of time
2 q
(T – T
0
) = —— —
Ȉȉ
kt (A2.10)
Ȉȉ
p K
The average temperature at z = 0, over the time interval 0 to t, is 2/3rds of this.
A2.2.3 The semi-infinite solid
z
> 0: temperature due to an
instantaneous quantity of heat
H
released into it at the point
x
=
y
=
z
=0
,
at
t
= 0; ambient temperature
T
o

a
,–
b
<
y
<
b
at
z
= 0;
ambient temperature
T
o
Heat flows into the solid over the surface area shown in Figure (A2.2a). In the time inter-
val t′ to t′ + dt′, the quantity of heat dH that enters through the area dA = dx′dy′ at (x′, y′)
is qdAdt′. From equation (A2.11) the contribution of this to the temperature at any point
(x, y, z) in the solid at time t is
(x–x′)
2
+(y–y′)
2
+z
2
qdx′dy′dt′ – —————
d(T – T
0
) = ————————— e
4k(t–t′)
(A2.12)
4rC(pk)

(T – T
0
)
max
= ——
(
sinh
–1
— + — sinh
–1
—
)
pKaab
}
2qa a b b
2 ½
b
2
a
(T – T
0
)
av
= (T – T
0
)
max
–——
[(
— + —

: steady heating at rate
q
per
unit area over the plane
z
= 0 (Figure A2.2b); ambient
temperature
T
o
In the steady state, the form of equation (A2.4) (with q* = 0) to be satisfied is
∂
2
T ∂T
k —— = u˘
z
—— (A2.15)
∂z
2
∂z
The temperature distribution
qq
u˘
z
z
(T – T
0
) = ——— , z ≥ 0; (T – T
0
) = ——— e
——

,
z
= 0 (Figure
A2.2(c)); ambient temperature
T
o
Two extremes exist, depending on the ratio of the time 2a/u˘
x
, for an element of the solid
to pass the heat source of width 2a to the time a
2
/k for heat to conduct the distance 2a
(Section A2.2.1). This ratio, equal to 2k/(u˘
x
a), is the inverse of the more widely known
Peclet number P
e
.
When the ratio is large (P
e
<< 1), the temperature field in the solid is dominated by
conduction and is no different from that in a stationary solid, see Section A2.2.4. Equations
(A2.14) give maximum and average temperatures at the surface within the area of the heat
source. When b/a = 1 and 5, for example,
bqaqa
— = 1:(T – T
o
)
max
= 1.12 —— ; (T – T

less group (qa/K),
qa 2k ½ qa 2k ½
u˘
x
a/(2k) >> 1: (T – T
0
)
max
= 1.13 ——
(
——
)
;(T – T
0
)
av
= 0.75 ——
(
——
)
Ku
x
aKu
x
a
(A2.17b)
356 Appendix 2
Childs Part 3 31:3:2000 10:42 am Page 356
Because these results are derived from a linear heat flow approximation, they depend only
on the dimension a and not on the ratio b/a, in contrast to P

2
∂T
2
∂T
2
I(T) =
∫
V
[
—
{(
——
)
+
(
——
)
+
(
——
)}
2 ∂x ∂y ∂z
∂T
–
∂T
–
∂T
–
–
{

where the temperature gradients ∂T
–
/∂x, ∂T
–
/∂y, ∂T
–
/∂z, are not varied in the minimization
process. The functional does not take into account possible variations of thermal proper-
ties with temperature, nor radiative heat loss conditions.
Equation (A2.18) is the basis of a finite element temperature calculation method if its
volume and surface integrations, which extend over the whole analytical region, are
regarded as the sum of integrations over finite elements:
m
I(T) =
∑
I
e
(T) (A2.19)
e=1
where I
e
(T) means equation (A2.18) applied to an element and m is the total number of
elements. If an element’s internal and surface temperature variations with position can be
written in terms of its nodal temperatures and coordinates, I
e
(T) can be evaluated. Its vari-
ation dI
e
with respect to changes in nodal temperatures can also be evaluated and set to
zero, to produce an element thermal stiffness equation of the form

i
) and so on for the other nodes. Temperature T
e
anywhere in the element is related to the nodal temperatures {T} = {T
i
T
j
T
k
T
l
}
T
by
T
e
= [N
i
N
j
N
k
N
l
]{T} = [N]{T} (A2.21)
where [N] is known as the element’s shape function.
1
N
i
= —— (a

k
|
,
b
i
=–
|
1 y
k
z
k
|
x
l
y
l
z
l
1 y
l
z
l
358 Appendix 2
Fig. A2.3 A tetrahedral finite element
Childs Part 3 31:3:2000 10:42 am Page 358
x
k
1 z
j
x

y
l
1
and
1
1 x
i
y
i
z
i
V
c
= —
|
1 x
j
y
j
z
j
|
(A2.22)
6
1 x
k
y
k
z
k

k
′]{T} = [N′]{T} (A2.23)
where
1
N
i
′ = ——— (a
i
′ + b
i
′x′ + c
i
′y′)
2D
ikj
and
a
i
′ = x
k
′y
j
′ – x
j
′y
k
′; b
i
′ = y
k

′
The other coefficients are obtained by cyclic interchange of the subscripts in the order i, k,
j. x′, y′ are local coordinates defined on the plane ikj. D
ikj
is the area of the element’s trian-
gular face: it may also be written in global coordinates as
1
y
k
– y
i
y
j
– y
j
2
z
k
– z
i
z
j
– z
i
2
x
k
– x
i
x

i
y
k
– y
i
y
j
– y
i
(A2.25)
A2.4.2 Tetrahedral element thermal stiffness equation
Equation (A2.21), after differentiation with respect to x, y and z, and equation (A2.23) are
substituted into I
e
(T) of equation A2.19. The variation of I
e
(T) with respect to T
i
, T
j
, T
k
and T
l
is established by differentiation and set equal to zero. [H]
e
and {F}
e
(equation (A2.20a)) are
[H]

i
+ c
k
c
i
+ d
k
d
i
b
l
b
i
+ c
l
c
i
+ d
l
d
i
——
[
b
i
b
j
+ c
i
c

l
b
j
+ c
l
c
j
+ d
l
d
j
]
36V
e
b
i
b
k
+ c
i
c
k
+ d
i
d
k
b
j
b
k

d
k
b
i
b
l
+ c
i
c
l
+ d
i
d
l
b
j
b
l
+ c
j
c
l
+ d
j
d
l
b
k
b
l

i
+ u˘
z
d
i
u˘
˘x
b
j
+ u˘
y
c
j
+ u˘
z
d
j
u˘
˘x
b
k
+ u˘
y
c
k
+ u˘
˘z
d
k
u˘

y
c
j
+ u˘
˘z
d
j
u˘
x
b
k
+ u˘
y
c
k
+ u˘
˘z
d
k
u˘
x
b
l
+ u˘
y
c
l
+ u˘
z
d

k
+ u˘
˘y
c
k
+ u˘
z
d
k
u˘
x
b
l
+ u˘
˘y
c
l
+ u˘
˘z
d
l
u˘
˘x
b
i
+ u˘
y
c
i
+ u˘

l
+ u˘
y
c
l
+ u˘
z
d
l
hD
ikj
2110
+ ——
[
1210
]
12
1120
0000 (A2.26)
and
11 1
q*V
e
1
qD
ikj
1
hT
0
D

contact length have been carried out, q
s
and the average value of q
f
can be determined as
follows:
q
s
= t
s
V
s
(A2.28a)
q
f
= t
f
V
c
(A2.28b)
where
F
C
cos f – F
T
sin f F
C
sin a + F
T
cos a

/∂t). Then the finite element equation (A2.20a) becomes
360 Appendix 2
Childs Part 3 31:3:2000 10:42 am Page 360
∂T
[C]
e
{
——
}
+ [H]
e
{T} = {F
e
} (A2.30)
∂t
with
rCV
e
2111
[C]
e
= ———
|
1211
|
20
1121
1112
([C] is given here for a four-node tetrahedron).
Numerical (finite element) methods 361

– T
n
{
——
}
=
{
————
}
(A2.31b)
∂t
av
Dt
where q is a fraction varying between 0 and 1 which allows the weight given to the initial
and final values of the rates of change of temperature to be varied. After multiplying equa-
tions (A2.31) by [C], substituting [C]{∂T/∂t}terms in equation (A2.31a) for ({F}–[H]{T})
terms from equation (A2.30), equating equations (A2.31a) and (A2.31b), and rearranging,
an equation is created for temperatures at time t
n+1
in terms of temperatures at time t
n
:in
global assembled form
[C][C]
(
—— + q[K]
)
{T}
n+1
=

This appendix summarizes, in the context of metal machining, understanding of the
stresses that occur at the contacts between sliding bodies. These stresses, with materials’
responses to them, are responsible for materials’ friction (and wear).
All engineering components – for example slideways, gears, bearings, and cutting tools
– have rough surfaces, characteristic of how they are made. When such surfaces are loaded
together, they touch first at their high spots. Figure A3.1 is a schematic view of two rough
surfaces placed in contact under a load W, the top one sliding to the right under the action
of a friction force F.
Figure A3.1(a) shows a contact, the material properties and roughness of which are such
that the surfaces have deformed to bring the direction of sliding into the planes of the real
areas of contact A
r
. Resistance to sliding then comes from the surface shear stresses s.
Friction that arises from shear stresses is called adhesive friction. If the real areas of
contact on average support a normal contact stress p
r
, the adhesive coefficient of friction
m
a
is given by
s
m
a
= — (A3.1)
p
r
Figure A3.1(b) shows surfaces for which the real areas of contact are inclined to the
sliding direction. Each contact is divided into two parts, ahead of (leading) and behind
Fig. A3.1 Friction caused (a) by shear stresses
s

– p
t
A
t
sinq
t
m
d
= —————————— (A3.2a)
p
1
A
1
cosq
1
+ p
t
A
t
cosq
t
Special cases occur. If the contact is symmetrical (p
l
= p
t
; A
l
= A
t
; q

contrast, equation (A3.2b) shows that abrasive deformation friction depends mainly on
surface geometry, insofar as the angle q
l
is the same as the slope of the leading part of the
contact, but this could be modified by material properties if, for example, the real pressure
distribution over A
l
is not uniform.
The main focus of this appendix is to review how the friction coefficient varies with
material properties and contact geometry, in adhesive and deformation friction conditions,
and when both act together.
Two further points can usefully be introduced before proceeding with this review. The
real contact stress p
r
in equation (A3.1) is the natural quantity to be part of a friction law,
but in practice it is the nominal stress, the load divided by the apparent, or nominal, contact
area A
n
, which is set in any given application. In Chapter 2, this stress has been written s
n
.
The first point is that, from load force equilibrium, the ratio of s
n
to p
r
is the same as the
ratio of the real to apparent contact area (A
r
/A
n

——
)
(A3.3b)
(p
r
/k)(s
n
/k) A
n
364 Appendix 3
Childs Part 3 31:3:2000 10:42 am Page 364
In the following sections, a view of how sliding friction depends on material properties,
contact geometry and intensity of loading is developed, by concentrating on how p
r
/k and
A
r
/A
n
vary in adhesive and deformation friction conditions. A more detailed account of
much of the contact mechanics is in the standard text by Johnson (1985). Reference will
be made to this work in the abbreviated form (KLJ Ch.x).
A3.2 The normal contact of a single asperity on an
elastic foundation
As a first step in building up a view of asperity contact, consider the normal loading of a
single asperity against a flat counterface. At the lightest loading, the deformation may be
elastic. At some heavier load, plastic deformation may set in. The purpose of this section
is to establish how transition from an elastic to a plastic state varies with material proper-
ties and asperity shape; and what real contact pressures p
r

, and when a
c
<< R,
a
2
c
a
2
d = d
1
+ d
2
= —— ∝ —— (A3.4)
2R 2R
The surface deformations in the asperity and flat cause sub-surface strains. In the asper-
ity, these are in proportion to the dimensionless ratio d
1
/a and in the flat to d
2
/a. When the
A single asperity on an elastic foundation 365
Fig. A3.2 Models of elastic asperity deformation
Childs Part 3 31:3:2000 10:42 am Page 365
asperity and flat obey Hooke’s law, the mean contact stress p
r
will increase in proportion
to the product of Young’s modulus and strain in each:
from the asperity’s point of view, p
r
∝ E

E* E
1
E
2
and c depends on whether the circular profile of radius R represents a spherically or a
cylindrically capped asperity (Table A3.1).
Similarly, the pressing together of two spherical or two cylindrical asperities with paral-
lel axes, of radii R
1
and R
2
, creates a normal contact stress p
r
:
p
r
= cE*(a/R*) where 1/R*=1/R
1
+1/R
2
(A3.8)
The elastic contact of a wedge or cone on a flat (right-hand part of Figure A3.2(a))
generates a contact pressure p
r
(KLJ Ch. 5):
p
r
= cE* tan b (A3.9)
where c is also given in Table A3.1. The quantities (a/R*) and tan b can be regarded as
representative contact strains. Their interpretation as mean contact slopes will be returned

Conical 0.50 1.6 3.2
Wedge-like 0.50 1.0 2.0
Childs Part 3 31:3:2000 10:43 am Page 366
p
r
= p
2
+ k (A3.10)
Slip-line EDBC is an a-line, so
p
2
= p
1
+ 2ky ≡ k(1 + 2y) (A3.11)
The angle y is chosen to conserve the volume of the flow: material displaced from the
overlap between the flat and the asperity must re-appear in the shoulders of the flow, but
for small values of b, y ≈ p/2. This, with equations (A3.11) and (A3.10), gives
p
r
≈ 2k(1 + p/2) ≈ 5k (A3.12)
A3.2.3 The transition from elastic to plastic contact
The elastic and plastic views of the previous sub-sections are brought together by non-
dimensionalizing the contact pressures p
r
by k. In Figure A3.4(a), the elastic and plastic
model predictions are the dashed lines. The solid line is the actual behaviour. Departure
from elastic behaviour first occurs in the range 1 < p
r
/k < 2.6, at values of (E*/k)(a/R* or
tanb) from 2 to 6.2. The values depend on the asperity shape: they are the last two columns

heights and are not loaded equally. The effect of this on the use of Figure A3.4(b) is the
first point considered in this section. In Figure A3.4(b), predictions are only drawn for
A
r
/A
n
< 0.5: the second point considered in this section is what happens at higher degrees
of contact, when asperity stress fields start to interact.
A3.3.1 Loading of random rough surfaces
Figure A3.5 shows the loading of two rough flat surfaces against a smooth flat counterface.
In case (a) all the asperities on the rough surface are identical, imagined as spherical caps
of radius R, and are shown in contact with the counterface. In case (b), the same asperities
have been shifted in a random manner normal to the surface, so that the peaks have a
random distribution s
s
of heights about their mean height. This situation is the most simple
that can be pictured, to make the point that an increase of load in case (a) causes the load
per contact, the half-width a and the stress severity to increase. However, in case (b), the
number of contacts can also increase, so that the load per contact and the severity of stress
will increase less slowly with load.
The situation of Figure A3.5(b) was considered by Greenwood and Williamson (1966),
supposing the contact stresses to be elastic, and is reproduced in (KLJ Ch. 13). Provided
that the number of asperities in contact is a small fraction of the total available (this means
in practice that A
r
/A
n
≤ 0.5), the number of contacts grows almost in proportion to the load,
so on average the load per contact is almost independent of load. The average real contact
pressure p

r
E*
—— = c —— D
q
(A3.13b)
kk
The severity index (E*/k) D
q
, more commonly called the plasticity index Y, may be used
with Figure A3.4(b) to determine the degree of contact of a rough loaded surface.
A3.3.2 Loading at high degrees of contact
As A
r
/A
n
increases above 0.5, even for a randomly rough surface, the availability of new
contacts becomes exhausted. A load increase will no longer cause a proportional increase
in the number of contacts, but will cause increased deformation of existing contacts. A
r
/A
n
will no longer increase in direct proportion to s
n
/k. Figure A3.6(a) extends Figure A3.4(b)
to higher values of s
n
/k and A
r
/A
n

n
/
k
for different degrees of roughness on an elastic foundation; (b) rigid
plastic compression of ridge-shaped asperities
Childs Part 3 31:3:2000 10:43 am Page 369
A3.4.1 Contact stress regimes under sliding conditions
The stressing of elastic spheres and cylinders loaded against flats, without and with slid-
ing, is reviewed in detail in (KLJ Chs. 4 and 6). Without sliding, the largest shear stress
t
max
occurs from 0.48a to 0.78a below the centre of the contact. With sliding, if m
a
< 0.25
to 0.3, t
max
is not changed in size by sliding, but the position where it occurs moves
towards the surface. For m
a
> 0.25 to 0.3, t
max
occurs at the surface and its size rises
proportionally to m
a
. The constant of proportionality depends on whether a sphere or a
cylinder is being loaded:
t
max
= (1.27 to 1.5)m
a

r
does not increase, result in an increased t
max
. In fact, A
r
grows to prevent
t
max
exceeding k. An alternative view of the cause of this junction growth is that, in the
absence of sliding, the material surrounding a plastic contact helps to support the load by
370 Appendix 3
Fig. A3.7 Asperity state of stress dependence on surface shear and plasticity index
Childs Part 3 31:3:2000 10:43 am Page 370
imposing a hydrostatic pressure on the deviatoric stress field. Its size, from slip line field
modelling, p
2
in equation (A3.11) with 2y ≈ p, is about 4k. The addition of surface shear
on the contact reduces the surrounding’s ability to support the load; in other words, the
hydrostatic pressure component supporting the load reduces.
How the slip-line fields of Figure A3.3 become modified by sliding have been studied
by Johnson, for the case of a soft asperity on a hard flat (KLJ Ch. 7), and by Oxley (1984)
for hard wedges ploughing over a soft flat. The conclusion of both, stemming from the
connection of the plastic flow field beneath the contact to the free surface where p = k,is
that, for s/k close to 1, sliding causes p
r
to fall from around 5k to 1k. For a constant load,
this causes a fivefold increase in real contact, at least while asperities are sufficiently far
apart not to interact with one another.
Figure A3.8 shows Oxley’s prediction of how A
r

hydrostatic pressure within a sliding contact is predicted by slip-line field modelling to be
less the larger is s/k, but while it is controlled by the free surface boundary condition, it
never becomes less than k. As a result, m never becomes greater than 1. (2) The reduction
in hydrostatic pressure, and hence the junction growth and m, is very sensitive to s/k when,
as in Figure A3.8, s/k is large (close to 1). (3) Once A
r
/A
n
= 1, m is no longer independent
of, but becomes inversely proportional to, load.
A3.5 Bulk yielding
When an asperity is supported on an elastic bulk, the only way to accommodate its plastic
distortions is by flow to the free surface. It is this, in the previous section, which ensured
that p
r
never became less than k. When the bulk is plastic, asperity plastic distortion can be
Bulk yielding 371
Fig. A3.8 The dependence of
A
r
/
A
n
and
µ
on
σ
n
/
k

beneath the asperities, has been reduced so much by the influence of p
E
that the stresses
on AD, A′D′ are no longer sufficient to cause a plastic state to extend to the free surface:
region ADEFE′C′ has become rigid. For this to be the case, at least for high values of real
surface shear stress s (s/k ⇒ 1), the results of the previous section suggest p
r
/k < 1. How
p
r
/k – and hence (with s
n
/k) A
r
/A
n
– depends on p
E
and z
bulk
can be determined from the
slip-line field and its limits of validity.
However, it is simpler to consider the overall force balances between the bulk field, the
nominal contact stresses s
n
and t, and the real contact stresses p
r
and s. Force equilibrium
between p
E

1–
(
—
))
(A3.15b)
kk k
In Figure A3.9(b) the dashed lines show combinations of (t/k) and (s
n
/k) consistent with
asperities existing on a bulk plastic flow in which p
E
/k = 0 or 0.5. The region marked ‘elas-
tic bulk’ is that for which (t/k) and (s
n
/k) are associated with an elastic bulk unless p
E
/k <
0. That marked ‘plastic bulk’ is plastic if p
E
/k > 0.5.
372 Appendix 3
Fig. A3.9 (a) Combined asperity and bulk plastic stressing and (b) derived degrees of contact
A
r
/
A
n
, depending on the
apparent contact stresses and the bulk field hydrostatic stress level
P