508 F. Auslender et al.
5. Oedometric Reynolds model
5.1. Basic assumptions From a qualitative point of view, the essential features of the displacement fields is clear: oedometric squeeze
corrected by small lateral motion accounting for edge effects. To capture these features the assumptions a and b of the Reynolds model introduced in Section 2.2 are modified in
a
1
The vertical stress σ = σ
zz
does not depend on z, a
2
The horizontal strain ε
αβ
are negligible with respect to ε
zz
, b
′
The terms ∂u
α
∂z are prevailing in ε
αz
. Assumptions a
1
and a
2
correspond to the oedometric squeeze while b
′
is a mere reformulation of assumption b.
It should be noted that these assumptions will result in a constant σ
zz
and ε
αβ
within the contact, which is consistent with static equilibrium and compatibility even in the case of an heterogeneous layered film.
Assumption a
2
with Hooke’s law 14 results in σ
zz
= σ = K
+ 4G
3 ε
zz
, σ
xx
= K
− 2G
3 ε
zz
. 47
Assumption a
2
and the oedometric relations 47 now allow evaluation of the left hand side of the compressible flow equation 16
Z
h
ε
ii
dz =
Z
h
ε
zz
dz =
Z
h
σ dz K
+ 4G3 .
Using assumption a
1
the modified flow equation is finally obtained as σ h
K
oed
− div q = δD,
1 K
oed
= 1
h
Z
h
dz K
+ 43G .
48 Heterogeneity of the film can be taken into account by assuming K and G to depend on x
α
and z resulting in K
oed
x
α
. 5.2. Modified flow equation
Combining assumption b
′
and the oedometric relation 47, the equilibrium equations require ∂
∂z G
∂u
α
∂z = −
∂σ
xx
∂x
α
= −β ∂σ
∂x
α
, 49
βx
α
, z =
Kx
α
, z −
2 3
Gx
α
, z Kx
α
, z +
4 3
Gx
α
, z .
Integrating this equation twice with respect to z u
α
= − ∂σ
∂x
α
Z
z
1 Gz
′
Z
z
′
βz
′′
dz
′′
dz
′
+ τ
Z
z
dz
′
Gz
′
+ u ,
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