484 M. Ciavarella, D.A. Hills
following implicit formula px =
1 π A
p
b
2
− x
2
Z
b −b
h
′
t − signtgt √
b
2
− t
2
t − x dt,
10 where the conventional solution, p
x, is obtained by neglecting, as is usual, convection, by approximating h
′
t − signtgt ≃ h
′
t. Equilibrium between the applied load and the pressure distribution gives, according to Shtayermann, 1949
P = − 1
A
Z
b −b
h
′
t − signtgtt dt √
b
2
− t
2
, 11
where convection is correctly allowed for. Now, on substituting gx form Eq. 9, one has, from the chain rule h
′
x − signxgx = h
′
x − βA
Z
x
pξ dξ 1 − signxβApx
. 12
3. Wedge-shaped punch
For a wedge in contact with a half-plane, the function hx is h
′
x = θ,
−b 6 x 6 0, −θ, 0 6 x 6 +b,
13 where θ is the external angle of the wedge, which must remain small, i to maintain the realism of the half-
plane idealization, and ii for the strains in the vicinity of the apex to be within the definition of linear elasticity theory.
This geometry is particulary interesting as the derivative of the profile is constant, i.e. h
′
x − signxgx = −θ 1 − signxβApx
= h
′
x + θ signxβApx, 0 6 x 6 +b
14 and so
p x = −
θ π A
p
b
2
− x
2
Z
b −b
signt dt √
b
2
− t
2
t − x = −
2θ π A
cosh
−1
b x
. 15
Then, considering the first correction, p
1
x = 1
π A
p
b
2
− x
2
Z
b −b
h
′
t − signtgt √
b
2
− t
2
t − x dt
16 = p
x + θβ
π
p
b
2
− x
2
Z
b −b
signtp t
√ b
2
− t
2
t − x dt.
17 Now, substituting the result for p
x, p
1
x = p x −
2θ π A
θβ π
p
b
2
− x
2
Z
b −b
signt cosh
−1
|
b t
| √
b
2
− t
2
t − x dt.
18
A note on convective effects in elastic contact problems 485
By defining the function I
1
s =
p
1 − s
2
Z
1 −1
signt cosh
−1
|
1 t
| √
1 − s
2
t − s dt
19 we can write the correction in dimensionless form as
p
1
s = p s −
2θ π A
θβ π
I
1
s = − 2θ
π A cosh
−1
1 s
+ θβ
π I
1
s .
20 We now calculate the corrected contact area dimension, from Eq. 11. This gives
P = − 1
A
Z
b −b
h
′
t − signtgtt dt √
b
2
− t
2
= − 1
A
Z
b −b
[h
′
t + signtθβApt]t dt √
b
2
− t
2
= 2 θ
A
Z
b
[1 − βApt]t dt √
b
2
− t
2
= 2 θ
A π
2 b − 2θβ
Z
b
ptt dt √
b
2
− t
2
= θ
A π b +
2θ π A
2θβb
Z
1
cosh
−1
|
1 t
|t dt √
1 − t
2
= θ
A b
π + 0.693147 4
π βθ
. 21
Therefore, as b = P
θ A
π , and b = P [
θ A
π1 + 0.693147
4 π
2
βθ ], b
b =
θ A
π
θ A
π1 − 0.693147
4 π
2
βθ =
1 1 − 0.693147
4 π
2
βθ .
It was stated at the beginning of this section that the external wedge angle must be small for the half-plane idealisation to be appropriate. Detailed numerical calculations using ABAQUS have shown that, providing
θ π5, the internal stress state for the elastically similar problem does not differ by more than 0.5 from the exact solution, and so, in order to reduce this influence to a small fraction of the effect under consideration, we
will choose this angle as a representative geometry. Also, we include the maximum amount of elastic mismatch by taking β = −0.5, giving
b b
= 1
1 − 0.693147
2 5π
≃ 1.097, i.e. the maximum correction correction is just less than 10. Figure 1
shows the comparison between the pressure p
o
x obtained neglecting convective effects, and the pressure p
1
x on considering first order correction, with the constants indicated. It is clear that the correction is surprisingly significant at the center of
the contact, where the singularity state of pressure is in effect released. We can draw the following conclusions
• if β 6= 0, for a ‘convex’ indenter, the correction will decrease the pressure, and therefore, the contact area dimension must increase;
• the correction is linearly proportional to β and, for the wedge, also to the external wedge angle, θ ; • the correction produces a change in pressure that is not definable in a pointwise sense: indeed, there are
areas of contact where contact was not predicted, as well as a large reduction of the pressure at the center of the contact area; it may be appreciated that the most significant differences occur at the edge of the contact
area, where pressure was not predicted, and at the centre, where the pressure concentration specifically a logarithmic singularity of the stresses is greatly reduced. Therefore the classical solution, on taking into
account the ‘convective effect’, is conservative with respect to the strength of the contact.
486 M. Ciavarella, D.A. Hills
Figure 1. Pressure distribution for a wedge punch, without ‘convective effect’ p x solid line, and with first order correction p
1
x dashed line. θ = π5, β = −0.5.
• the second correction, for the wedge case, is p
2
s = p s +
θβ π
p
1 − s
2
Z
1 −1
signtp
1
t √
1 − t
2
t − s dt
22 = p
1
s − θβ
π
2
2θ π A
p
1 − s
2
Z
1 −1
signtI
1
t √
1 − t
2
t − s dt.
23 It is clear that the correction has a series representation growing as a power series of θβ. As the first
product, θβ, is already small, there is, in practice, no need for more than one correction, within the context of linear elasticity theory.
• convection produces the effect of making the profile shallower.
4. Parabolic punch
