Computational Mechanics

, Volume 49, Issue 6, pp 749–767 | Cite as

A computational homogenization framework for soft elastohydrodynamic lubrication

  • M. Budt
  •  İ. Temizer
  • P. Wriggers
Original Paper


The interaction between microscopically rough surfaces and hydrodynamic thin film lubrication is investigated under the assumption of finite deformations. Within a coupled micro–macro analysis setting, the influence of roughness onto the macroscopic scale is determined using FE 2-type homogenization techniques to reduce the overall computational cost. Exact to within a separation of scales assumption, a computationally efficient two-phase micromechanical test is proposed to identify the macroscopic interface fluid flux from a lubrication analysis performed on the deformed configuration of a representative surface element. Parameter studies show a strong influence of both roughness and surface deformation on the macroscopic response for isotropic and anisotropic surfacial microstructures.


Reynolds equation Surface roughness Homogenization Finite deformation 



Angle of orientation w.r.t. x-axis


Surface quantities


Quantities belonging to the upper-, middle-, lower-surface of the fluid element, respectively (mp, q)

\({\mathcal {L}, \partial \mathcal {L}}\)

Fluid domain and its boundary in current configuration

\({\mathcal {S},\partial \mathcal {S}}\)

Solid domain and its boundary in current configuration


Parameter to penalize the fluid pressure to p a

\({\langle \bullet \rangle}\)

Surface averaged local quantities

\({\bar{\bullet}=\langle \bullet \rangle}\)

Macroscopic quantities


Surface deformation/displacement gradient

\({{\bf n}^{\bullet}}\)

Fluid normal vectors on surfaces +,m,-


Fluid flux per density


Fluid flux Couette term


Fluid flux Poseuille term


Deformed surface area


Undeformed surface area


Fluid pressure


Bearing surrounding ambient pressure


Fluid acceleration


Pressure–viscosity coefficient


Test function w.r.t. p, g or u

\({\bar {v}}\)

Rel. surface velocity


Dynamic viscosity




Body force


Grad[p] pressure gradient


Fluid velocity


Gap height


Sample height z


Gap/sample length x, y


Poison’s ratio


Strain energy function


Young’s modulus


Shear modulus


Bulk modulus


Volumetric part of strain energy function


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Continuum MechanicsLeibniz University of HanoverHannoverGermany
  2. 2.Department of Mechanical EngineeringBilkent UniversityAnkaraTurkey

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