Oedometric Reynolds model Directory UMM :Data Elmu:jurnal:E:European Journal of Mechanics A_Solids:Vol18.Issue3.1999:

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