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
4.1. Axisymmetric planeplane problem The results obtained in Section 3 can be extended to other more realistic geometries. As a first example let us
consider the axisymmetric squeeze of a circular plate of constant thickness h and radius a. The corresponding boundary conditions are, in cylindrical coordinates,
u
z
r, ±
h 2
= ± δD
2
e
z
, σ
rz
a, z = σ
rr
a, z = 0.
31
506 F. Auslender et al.
The kinematically admissible oedometric displacement field is
e
u
r
r, z = 0
e
u
z
r, z =
δD h
z 32
and the associated upper bound for the stiffness is given by ˜k =
E1 − ν
1 + ν1 − 2ν
π a
2
h = k
∞
. 33
The corresponding uniform oedometric stress state is σ
a rr
= σ
a θ θ
= ν
1 − ν
σ
a zz
= νE
1 + ν1 − 2ν
δD h
= q ,
34 σ
a rz
= σ
a θ z
= σ
a θ r
= 0 which do not satisfy the free boundary conditions 31 for r
= a. A statically admissible stress field can then be constructed by connecting this oedometric state inside the contact r a
1
= a − θh to the boundary r = a. The connecting stress field σ
b
, which must satisfy equilibrium, the boundary conditions 31 and continuity with σ
a
on r = a
1
, can be taken as σ
b rr
= σ
b θ θ
= gr, σ
b zz
= g
′′
r + g
′
rr z
2
2, σ
b rz
= −g
′
rz, 35
where gr =
q 2
1 − cos
πa −r
θ h
and r ∈ [a
1
, a ].
The associated stiffness is then ˆk =
πa − θh
2
h E1
− ν 1
− 2ν1 + ν +
2 δD
2
H σ
b
, H σ
b ij
a
2
h = −
π E
2 θ h
a −
θ
2
h
2
a
2
3 4
4 − νh
2
q
2
+ h
2
q
2
π
4
403200θ
4
+ h
2
q
2
π
2
1 + 2ν
212θ
2
− h
2
θ
2
a
2
ν1 − ν
h
2
q
2
π
2
+ h
2
a
2
π
2
q
2
5120θ
2
Z
θ h
a a
− r sin
2
π r θ h
dr .
Finally and after some computation, the correcting terms can be estimated, resulting in the following evaluation for the stiffness k
ˆk = k
∞
1 − Oha
6 k 6 ˜k
= k
∞
= π a
2
h E1
− ν 1
− 2ν1 + ν 36
which is an obvious extension of the result obtained in Section 3. Again the oedometric solution appears as the appropriate approximation for thin layers and the incompressible limit is singular.
4.2. The sphereplane problem The sphere-plane problem is obtained from 1 with hr given by 4. Denoting by r
∞
the lateral dimension of film, the thin film limit is obtained for Dr
∞
→ 0. Further in order to have h = OD, we will also assume r
∞
= O √
RD which is the relevant characteristic lateral length for sphere-plane contact.
Material compressibility effects for the squeeze of very thin films 507
Using standard asymptotic techniques the following reduced variables are introduced ρ
= r
√ RD
, t
= z
D ,
v
i
ρ , t =
u
i
δD ,
ε =
s
D R
. 37
With these variables, Navier equations are obtained as ε
2
λ + 2µ
v
r,ρρ
+ v
r,ρ
ρ −
v
r
ρ
2
+ ε µv
z,ρt
+ λv
r,ρt
+ µv
r,t t
= 0, 38
ε
2
µ v
z,ρρ
+ v
z,ρ
ρ + ελ + µ
v
r,ρt
+ v
r,t
ρ + λ + 2µv
z,t t
= 0 with boundary conditions
t = 0:
v
r
= v
z
= 0, 39
t = 1 +
ρ
2
2 = t
1
ρ: v
r
= 0, v
z
= 1, ρ
= r
∞
√ RD
= ρ
1
: ε
λ + 2µv
r,ρ
+ λ v
r
ρ + λv
z,t
= 0, εv
z,ρ
+ v
r,t
= 0, on S
L
. 40
Substituting the outer asymptotic expansion vρ , t
= v ρ , t
+ εv
1
ρ , t + ε
2
v
2
ρ , t + · · ·
41 in these equations directly gives the following problem at order 0
µv
r,t t
= 0, λ
+ 2µv
z,t t
= 0, 42
t = 0:
v
r
= v
z
= 0, 43
t = t
1
ρ: v
r
= 0, v
z
= 1, ρ
= ρ
1
: v
r,t
= v
z,t
= 0. 44
Equation 42 with the boundary conditions 43 directly gives the oedometric solution v
r
= 0, v
z
= t
t
1
ρ 45
while the lateral boundary conditions 44 cannot be satisfied and therefore requires an inner development near the boundaries ρ
= ρ
1
. The corresponding boundary value problem cannot easily be solved but its contribution will remain small because the corresponding oedometric stress rapidly decreases with increasing ρ.
The oedometric solution
e
u
r
r, z = 0,
e
u
z
r, z =
δD hr
z 46
therefore appears as a reasonable approximation.
508 F. Auslender et al.
5. Oedometric Reynolds model
