Material compressibility effects for the squeeze of very thin films 509
where τ x
α
and u x
α
are two integration constants which are obtained from the boundary conditions 1. A third integration with respect to z finally provides, after some computations
q = R
oed
gradσ
zz
, 50
where R
oed
=
Z
h
z Gz
Z
h
βz
′
dz
′
dz
Z
h
dz Gz
−
Z
h
1 Gz
Z
h
βz
′
dz
′
dz
Z
h
z dz Gz
Z
h
dz Gz
. Combination of the two Eqs 48 and 50 gives the partial differential equation
σ
zz
h K
oed
− div R
oed
gradσ
zz
= δD. 51
This equation defines the oedometric Reynolds model. A much simpler form is obtained in the weakly compressible case GK
≪ 1, ν ∼ 12 for which β ∼ 1 so that 51 becomes
σ h K
− divR grad σ = δD, 52
h K
=
Z
h
dz K
, R
= g
2
− g
2 1
g ,
g
k
=
Z
h
z
k
dz Gx
α
, z ,
k = 0, 1.
This case is important for practical applications. Similarly for an homogeneous layer, Eq. 51 simplifies into
σ h K
+ 4G3 − div
βh
3
12G grad σ
= δD. 53
6. Applications
6.1. The planeplane problem The planeplane problem under plane strain condition, which has been analysed in Section 3 is now
easily solved. Equation 53 becomes an ordinary differential equation which, with the boundary conditions σ
±a = 0 is readily solved as σ
zz
= σ
1 −
ch
2x h
q
3 2
1 −2ν
n 2a
h
q
3 2
1 −2ν
ν
, where
σ =
1 − νE
1 − 2ν1 + ν
δD h
= λ + 2µ δD
h ,
54 where G and K have been replaced by their expression in terms of E and ν. The corresponding stiffness is
found as
k =
21 − νE
1 − 2ν1 + ν
L a
h
1 −
th
2a h
q
3 2
1 −2ν
ν 2a
h
q
3 2
1 −2ν
ν
55
510 F. Auslender et al.
Figure 2. Planeplane problem. Evolution of the reduced stiffness kk
∞
as a function of ν for ha = 10, 30 and 100.
which obviously tends to the oedometric limit 23 for ha → 0.
It can also be verified that for any fixed value x this solution tends to the incompressible solution 13 when ν
→ 12. Again these two limits: incompressibility ν → 12 and thin film ha → 0, do not commute. The appropriate adimensional number for this problem is
α = a
√ 1
− 2νh 56
which combines the film thinness with its compressibility. These results are exemplified in figures 2 and 3. As expected, this solution corresponds to the oedometric stress field inside the film corrected by a boundary
layer near the edge σ
∼ σ 1
− e
−
a −x
l′
, where
l
′
= h
2
s
2 3
ν 1
− 2ν .
57 This oedometric model is consistent with all the information gathered in Sections 3 and 4 about the solution
of some simple problems. It therefore appears reasonable. It remains however empirical and requires further validation, in particular concerning the boundary layers which is the essential new feature it adds with respect
to the simple oedometric approximation.
In order to check this validity some finite elements simulations have been performed. These simulations have been restricted to the boundary layers, i.e. on
= [a
1
= a − θh, a] × [−h2, +h2]
Material compressibility effects for the squeeze of very thin films 511
a
b
Figure 3. Planeplane problem. Evolution of the stress distribution for different values of ν.
with the standard boundary conditions on z = ±h2 and x = a while the oedometric solution is imposed on
x = a − θh. The value of θ has to be chosen large enough with respect with the boundary layer width, which
as shown by 57 increases as ν → 12. Practically θ has been chosen as 5 for ν = 0.3 and 10 for ν = 0.48.
The corresponding results for the σ
xx
stress at different z values are represented in figure 4. Remember that
512 F. Auslender et al.
a
b
Figure 4. Stresses σ
xx
x resulting from a finite element calculation for three different values of z and from the oedometric.
Material compressibility effects for the squeeze of very thin films 513
Figure 5. Sphereplane problem. Compressible correction f α as function of the dimensionless parameter α
= GRKD.
the boundary layer effect essentially results from this σ
xx
component which must decrease from its oedometric value inside the contact to zero.
A correct agreement is obtained except in the vicinity of the corner z = h2, x = a where a stress
singularity exists. The approximation also appears better for larger value of ν. The axisymmetric planeplane problem of Section 4.1 could be treated in the same way but no analytical
solution appears available. 6.2. The sphereplane problem
The axisymmetric problem now has to be solved with hr given by 4. An adimensional formulation is obtained by setting
r =
√ RDρ ,
δD 1
− ν ν
= DU, ¯h =
h D
= 1 + ρ
2
2 ,
σ
zz
ρ =
D RU G
σ
zz
r. 58
The differential equation 53 is α ¯
hσ
zz
− 1
ρ d
dρ ρ
¯h
3
12 dσ
zz
dρ = 1,
where α
= 1
+ ν 3ν
G K
R D
59 which has to be solved with the boundary conditions σ
→ 0, ρ → ρ
∞
or ρ → ∞.
The structure of 59 implies that the solution σ ρ only depends on α. The normal force associated to the normal stress σ
= σ
zz
is then given by F
= 6π GR
2
δD D
1 − ν
ν f α,
where f α
= 1
3
Z
ρ
∞
ρσ ρ , α dρ . 60
514 F. Auslender et al.
Figure 6. Sphereplane problem. Reduced stress distribution σ h2GR as a function of ρ = r
√ DR for α
= 0, 0.8 and 2.0.
This expression for f has been normalised from the incompressible case 12 which corresponds to α = 0
with f 0 = 1. The relevant dimensionless parameter α clearly shows that compressibility cannot be neglected,
even for nearly incompressible materials, i.e. for high values of the ratio KG. Indeed this ratio appears in α as multiplied by the ratio DR which, for thin layers, may be very small. The function f α is plotted in
figure 5. Typically a reduction by half of the force is obtained for α = 0.8, i.e. if for instance ν = 0.4999 for
DR = 1.6 10
−4
. It should be noted that in the mentioned experiments Tonck et al., 1988; Monfort et al., 1991; Georges et al., 1993 DR typically is of order 10
−6
. In fact the limiting case of large α is probably more relevant. This corresponds to the oedometric
approximation 46. The corresponding adimensional stress distribution has been represented in figure 6 for different values of α.
7. Conclusion
