502 F. Auslender et al.
Application to the sphereplane problem 4 gives pr
= − 3GRδD
h
2
, F
= 2π
Z
∞
prr dr = −
6π GR
2
D δD.
12 Strictly speaking the integral in 12 should be from 0 to r
∞
, radius of the outer lateral surface, but it can be shown that this correction is usually very small. These expressions correspond to the standard Chan and Horn
solution, classically used for the analysis of surface force results Chan and Horn, 1985 either directly or in a straightforward viscoelastic extension. Henceforth, this elastic formulation of the lubrication theory will be
referred to as the Incompressible Model.
Similarly application to the planeplane problem 3 under plane strain condition directly results in p
= − 2EδD
h
3
a
2
− x
2
13 for a thin plane of length 2a in the x
1
= x direction and of width L in the x
2
direction with a ≪ L.
2.3. The compressible case For an isotropic compressible elastic material the constitutive equation 6, 7 has to be replaced with
σ = σ
kk
3 = Kε
kk
, 14
s
ij
= 2Ge
ij
, 15
where σ is the hydrostatic stress, s
ij
and e
ij
the deviatoric stress and strain tensors, while K and G are the bulk and shear elastic modulus which will be considered as constant in the homogeneous case but may also depend
on x
α
and z in case of heterogeneous film. Integration of the nonvanishing volumetric strain ε
ii
then leads to a modified expression for the flow equation
Z
hx
α
ε
ii
dz
= div q + δD 6= 0,
16 where ε
ii
is now given by 14. Further treatment of this equation however requires a better knowledge of the stress tensor structure. To reach this knowledge some simple problems will now be considered.
3. The planeplane problem
Let us first consider the planeplane problem already investigated in the incompressible case 13. A symmetric formulation of the corresponding plane strain boundary value problem is
z = ±h2:
u
x
= 0, u
z
= ±δD2, 17
x = ±a:
σ
xx
= σ
xz
= 0.
Material compressibility effects for the squeeze of very thin films 503
3.1. Variational methods The variational theorems in linear elasticity Germain, 1986 give an estimation of the potential and
complementary energy of the exact solution σ
ij
, u
i
through H
b
σ
ij
6 H σ
ij
= Ku
i
6 K
e
u
i
. 18
The potential energy K
e
u
i
of the kinematically admissible displacement field
e
u
i
is K
e
u
i
= 1
2
Z Z Z e
σ
ij
e
ε
ij
dv −
Z Z Z
f
i
e
u
i
dv −
Z Z
S
f
T
d i
e
u
i
dS, 19
where f
i
and T
d i
respectively denote the body forces and given surface forces on S
f
here x = ±a.
Similarly the complementary energy H
b
σ
ij
of the statically admissible stress field
b
σ
ij
is H
b
σ
ij
=
Z Z
S
u
b
σ
ij
n
j
u
d i
dS −
1 2
Z Z Z b
σ
ij
b
ε
ij
dV , 20
where u
d i
denotes the prescribed displacement on S
u
here z = ±h2.
For the planeplane problem 17 the body forces f
i
and the applied surface forces T
i
= σ
xz
, σ
xz
on the boundaries x
= ±a vanish. The only non vanishing data is the displacement u
z
= ±δD2 on the boundaries z
= ±h2. Accordingly Ku
i
= H σ
ij
= 1
2
Z Z
z =±h2
σ
zz
δD dS =
1 2
kδD
2
so that the basic inequality 18 becomes H
b
σ
ij
= 1
2 ˆkδD
2
6 1
2 kδD
2
6 K
e
u
i
= 1
2 ˜kδD
2
. 21
Any kinematically [resp. statically] admissible field will provide an upper [resp. lower] bound ˜ k [resp. ˆk] of the
contact stiffness k introduced in 2. The kinematically admissible displacement field
e
u
x
=
e
u
y
= 0,
e
u
z
= δD z
h 22
describes the “oedometric” without lateral motion squeeze of the film. The corresponding upper bound ˜k is obtained from 19 as
˜k = 2E1
− ν 1
− 2ν1 + ν L
a h
= λ + 2µ 2La
h = k
∞
, 23
where λ, µ are the usual Lamé elastic constants. This oedometric displacement field is not the solution of the problem because the corresponding uniform
oedometric stress state does not satisfy the boundary condition σ
xx
= 0 for x = ±a. A statistically admissible stress field will be obtained by using this oedometric stress state in the core on the plate, i.e. for
−a
1
6 x a
1
504 F. Auslender et al.
a
1
a and by correcting it near the edges x = ±a in order to satisfy equilibrium and the stress boundary
conditions 17. This can be achieved through the following Airy function φx, z
= − 6q
θ h
3
a − x
3
3 −
a − xθh
2 z
2
2 for x
∈ [a
1
, a ],
24 where θ
= a − a
1
h. The corresponding lower bound is after some computations obtained from 20 as ˆk = k
∞
1 −
h a
4θ +
ν
2 13
6
θ +
3 16θ
3
+
2 +3ν
2θ
51 − ν1 + ν1 − 2ν
. 25
The evaluation of course depends on a
1
, and the next step would be to choose θ to maximise this lower bound and therefore to obtain the optimal size of the considered edge effect. This is however not necessary for the
present purpose which is the thin film limit ha → 0. Indeed if θ is chosen of order of O1, the combination
of these two estimations results in ˆk = k
∞
1 − Oha
6 k 6 ˜
k = k
∞
= 2E1
− ν 1
− 2ν1 + ν La
h .
26 This value 23 of the contact stiffness k
∞
is the thin film limit for any value ν 12. 3.2. Discussion
The evaluation obtained in 26 clearly shows the validity of the oedometric approximation k
∞
for very thin films. However if this limit is obtained for any value ν 12 of the Poisson ratio, it does not hold at the
incompressible limit ν → 12. This is for instance clear on 23 from the 1 − 2ν term in the denominator.
These two limits, thin film and weak compressibility, do not commute. More precisely lim
ν →12
lim
ha →0
k = k
∞
= 2E1
− ν 1
− 2ν1 + ν La
h ,
lim
ha →0
lim
ν →12
k = k
inc
= 8ELa
3
3h
3
, where the second limit has been obtained from Reynolds approximation. We have no proof for that but it is
probably true. The above approximations can be improved through a better choice and optimisation of the kinematically
and statically admissible fields Auslender et al., 1995; Armengaud et al., 1995. For instance the following kinematically admissible field
u
x
= δD
2 g
x a
1 −
4z
2
h
2
, u
y
= 0, u
z
= δD z
h 27
combines the squeeze 22 with the horizontal displacement observed in Reynolds approximation. Here g is an arbitrary function with g0
= 0 which will now be chosen in such a way to minimise the potential energy K
= W . After some computation W
W
∞
=
Z
1
1 + dg
′2
ξ + 2bg
′
ξ + cg
2
ξ dξ ,
W
∞
= 1
2 k
∞
δD
2
, 28
Material compressibility effects for the squeeze of very thin films 505
with ξ =
x a
, d =
8h
2
15a
2
, b =
2 3
λ λ
+2µ h
a
, c =
4 3
µ λ
+2µ
, and where W
∞
is the reference oedometric value. Minimisation of this integral with respect to the function g is a standard variational problem resulting in a
second order differential equation cgξ
− dg
′′
ξ = 0,
g0 = 0,
dg
′
1 + b = 0.
The solution is readily found as
G x
a = −
ν 1
− 2ν
s
5 4
1 − 2ν
1 − ν
sh
x a
a h
q
5 4
1 −2ν
1 −ν
ch
a h
q
5 4
1 −2ν
1 −ν
. 29
Finally, substituting this function in 28, an improved upper bound of the stiffness is obtained ˜k = k
∞
1 −
5 6
ν
2
1 − ν
2
thγ γ
, where
γ =
s
51 − 2ν
1 − ν
a h
. 30
Again the singular incompressible limit ν → 12 is encountered adimensional parameter
√ 1
− 2νah. A similar and more complex analysis can also be performed to improve the lower bound Armengaud et al.,
1995. Asymptotic methods Sanchez-Palencia and Sanchez-Hubert, 1992 can also be used for the thin film limit
ε = ha → 0. However in this particular case, they do not provide any further information with respect to
the variational methods discussed above. Indeed the first order outer solution is found to coincide with the oedometric solution. However, since this solution does not satisfy the boundary solution on x
= ±a, it has to be completed by an inner solution near these boundaries. The classical techniques used for extracting the inner
solution Sanchez-Palencia and Sanchez-Hubert, 1992 unfortunately result in the complete Navier problem. Asymptotic methods therefore only confirm the essential phenomenology of this problem: the oedometric
solution holds true inside the contact and has to be corrected near the edges to account for boundary effects. The influence of these edge effects on the global response of the system however may be neglected for very
thin layer, as follows from the upper and lower bounds 26.
4. Axisymmetric problems
