Computational Mechanics

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

A computational homogenization framework for soft elastohydrodynamic lubrication

Original Paper

Abstract

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 FE2-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.

Keywords

Reynolds equation Surface roughness Homogenization Finite deformation 

Nomenclature

β

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

\({\bullet^S}\)

Surface quantities

\({\bullet^{+/m/-}}\)

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

\({\epsilon_{C}}\)

Parameter to penalize the fluid pressure to pa

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

Surface averaged local quantities

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

Macroscopic quantities

F/H

Surface deformation/displacement gradient

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

Fluid normal vectors on surfaces +,m,-

q

Fluid flux per density

qc

Fluid flux Couette term

qp

Fluid flux Poseuille term

a

Deformed surface area

A0

Undeformed surface area

p

Fluid pressure

pa

Bearing surrounding ambient pressure

\({\dot{v}}\)

Fluid acceleration

α

Pressure–viscosity coefficient

\({\eta^p,\eta^{g/u}}\)

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

\({\bar {v}}\)

Rel. surface velocity

μ0

Dynamic viscosity

ρ

Density

b

Body force

g

Grad[p] pressure gradient

v

Fluid velocity

h

Gap height

lz

Sample height z

lx,y

Gap/sample length x, y

ν

Poison’s ratio

\({\Psi}\)

Strain energy function

E

Young’s modulus

G

Shear modulus

K

Bulk modulus

U

Volumetric part of strain energy function

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References

  1. 1.
    Almqvist A, Dasht J (2006) The homogenization process of the Reynolds equation describing compressible liquid flow. Tribol Int 39: 994–1002CrossRefGoogle Scholar
  2. 2.
    Almqvist A, Essel E, Fabricius J, Wall P (2008) Reiterated homogenization applied in hydrodynamic lubrication. Proc Inst Mech Eng Part J J Eng Tribol 222(7): 827–841CrossRefGoogle Scholar
  3. 3.
    Almqvist A, Essel E, Persson L, Wall P (2007) Homogenization of the unstationary incompressible Reynolds equation. Tribol Int 40(9): 1344–1350CrossRefGoogle Scholar
  4. 4.
    Almqvist A, Lukkassen D, Meidell A, Wall P (2007) New concepts of homogenization applied in rough surface hydrodynamic lubrication. Int J Eng Sci 45(1): 139–154MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bakhvalov N, Panasenko G (1989) Homogenisation: averaging processes in periodic media. Kluwer, DordrechtMATHCrossRefGoogle Scholar
  6. 6.
    Bayada G, Chambat M (1989) Homogenization of the Stokes system in a thin film flow with rapidly varying thickness. Modélisation mathématique et analyse numérique 23(2): 205–234MathSciNetMATHGoogle Scholar
  7. 7.
    Bayada G, Martin S, Vázquez C (2005) An average flow model of the Reynolds roughness including a mass-flow preserving cavitation model. J Tribol 127: 793–802CrossRefGoogle Scholar
  8. 8.
    Bayada G, Martin S, Vázquez C (2006) Micro-roughness effects in (elasto) hydrodynamic lubrication including a mass-flow preserving cavitation model. Tribol Int 39(12): 1707–1718CrossRefGoogle Scholar
  9. 9.
    Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. North-Holland, AmsterdamMATHGoogle Scholar
  10. 10.
    Bohan M, Fox I, Claypole T, Gethin D (2003) Influence of non-Newtonian fluids on the performance of a soft elastohydrodynamic lubrication contact with surface roughness. Proc Inst Mech Eng Part J J Eng Tribol 217(6): 447–459CrossRefGoogle Scholar
  11. 11.
    Budt M (2012) Computational homogenization framework for soft elasto-hydrodynamic lubrication. PhD thesis, Institut für Kontinuumsmechanik, Gottfried Wilhelm Leibniz Universität Hannover, Hannover (Germany)Google Scholar
  12. 12.
    Buscaglia G, Ciuperca I, Jai M (2007) On the optimization of surface textures for lubricated contacts. J Math Anal Appl 335(2): 1309–1327MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Charnes A, Osterle F, Saibel E (1952) On the energy equation for fluid-film lubrication. Proc R Soc Lond Ser A Math Phys Sci 214: 133–136MATHCrossRefGoogle Scholar
  14. 14.
    Cope W (1949) The hydrodynamical theory of film lubrication. Proc R Soc Lond Ser A Math Phys Sci 197(1049): 201–217MATHCrossRefGoogle Scholar
  15. 15.
    Curnier A, Taylor RL (1982) A thermomechanical formulation and solution of lubricated contacts between deformable solids. J Lubr Technol 104: 109–117CrossRefGoogle Scholar
  16. 16.
    Kraker A, Ostayen RAJ, Rixen DJ (2010) Development of a texture averaged Reynolds equation. Tribol Int 43: 2100–2109CrossRefGoogle Scholar
  17. 17.
    Dowson D (1995) Elastohydrodynamic and micro-elastohydrodynamic lubrication. Wear 190(2): 125–138CrossRefGoogle Scholar
  18. 18.
    Elrod H (1979) A general theory for laminar lubrication with Reynolds roughness. ASME Trans J Lubr Technol 101: 8–14CrossRefGoogle Scholar
  19. 19.
    Ervin RD, Balderas L (1990) Hydroplaning with lightly-loaded truck tires. Technical Report UMTRI-90-6, Transportation Research Institute, The University of MichiganGoogle Scholar
  20. 20.
    Fabricius J (2008) Homogenization theory with applications in tribology. PhD thesis, Luleå University of TechnologyGoogle Scholar
  21. 21.
    Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, New YorkGoogle Scholar
  22. 22.
    Hamrock B, Schmid S, Jacobson B (2004) Fundamentals of fluid film lubrication. CRC Press, Boca RatonCrossRefGoogle Scholar
  23. 23.
    Huebner K (1975) The finite element method for engineers. Wiley-Interscience, New YorkGoogle Scholar
  24. 24.
    Jackson RL (2010) A scale dependent simulation of liquid lubricated textured surfaces. J Tribol 132: 022001CrossRefGoogle Scholar
  25. 25.
    Jaffar M (2000) A numerical solution for a soft line elastohydrodynamic lubrication contact problem with sinusoidal roughness using the Chebyshev polynomials. Proc Inst Mech Eng Part C J Mech Eng Sci 214(5): 711–718CrossRefGoogle Scholar
  26. 26.
    Jai M, Bou-Said B (2002) A comparison of homogenization and averaging techniques for the treatment of roughness in slip- flow-modified Reynolds equation. J Tribol 124: 327CrossRefGoogle Scholar
  27. 27.
    Kane M, Bou-Said B (2004) Comparison of homogenization and direct techniques for the treatment of roughness in incompressible lubrication. J Tribol 126: 733CrossRefGoogle Scholar
  28. 28.
    Kane M, Bou-Said B (2005) A study of roughness and non-newtonian effects in lubricated contacts. J Tribol 127: 575MATHCrossRefGoogle Scholar
  29. 29.
    Kane M, Do T (2006) A contribution of elastohydrodynamic lubrication for estimation of tire-road friction in wet conditions. In: Proceedings of the International Conference on Tribology, Parma, Italy, September 20–22 2006: AITC-AIT 2006Google Scholar
  30. 30.
    Larsson R (2009) Modelling the effect of surface roughness on lubrication in all regimes. Tribol Int 42(4): 512–516CrossRefGoogle Scholar
  31. 31.
    Lewis R, Gallardo-Hernandez EA, Hilton T, Armitage T (2009) Effect of oil and water mixtures on adhesion in the wheel/rail contact. Proc IMechE Part F J Rail Rapid Transit 223: 275–283CrossRefGoogle Scholar
  32. 32.
    Lukkassen D, Meidell A, Wall P (2007) Bounds on the effective behavior of a homogenized generalized Reynolds equation. J Funct Spaces Appl 5: 133–150MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Mitsuya Y, Fukui S (1986) Stokes roughness effects on hydrodynamic lubrication. Part I—comparison between incompressible and compressible lubricating films. J Tribol 108: 151CrossRefGoogle Scholar
  34. 34.
    Patir N, Cheng H (1978) An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. ASME Trans J Lubr Technol 100: 12–17CrossRefGoogle Scholar
  35. 35.
    Patir N, Cheng H (1979) Application of average flow model to lubrication between rough sliding surfaces. ASME J Lubr Technol 101(2): 220–230CrossRefGoogle Scholar
  36. 36.
    Persson B (2000) Sliding friction: physical principles and applications, vol 1. Springer, BerlinGoogle Scholar
  37. 37.
    Persson BNJ (2010) Fluid dynamics at the interface between contacting elastic solids with randomly rough surfaces. J Phys Condens Matter 22: 265004CrossRefGoogle Scholar
  38. 38.
    Rabinowicz E (1995) Friction and wear of materials, 2nd edn. Wiley, New YorkGoogle Scholar
  39. 39.
    Rajagopal K, Szeri A (2003) On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc R Soc Lond Ser A Math Phys Eng Sci 459(2039): 2771MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Reynolds O (1886) On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil. Philos Trans R Soc Lond 177: 157–234CrossRefGoogle Scholar
  41. 41.
    Sahlin F, Almqvist A, Larsson R, Glavatskih S (2007) Rough surface flow factors in full film lubrication based on a homogenization technique. Tribol Int 40(7): 1025–1034CrossRefGoogle Scholar
  42. 42.
    Sahlin F, Larsson R, Almqvist A, Lugt P, Marklund P (2010) A mixed lubrication model incorporating measured surface topography. Part 1: theory of flow factors. Proc Inst Mech Eng Part J J Eng Tribol. 224(4): 335–351CrossRefGoogle Scholar
  43. 43.
    Sahlin F, Larsson R, Marklund P, Almqvist A, Lugt P (2010) A mixed lubrication model incorporating measured surface topography. Part 2: roughness treatment, model validation, and simulation. Proc Inst Mech Eng Part J J Eng Tribol 224(4): 353–365CrossRefGoogle Scholar
  44. 44.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, BerlinMATHGoogle Scholar
  45. 45.
    Shi F, Salant R (2000) A mixed soft elastohydrodynamic lubrication model with interasperity cavitation and surface shear deformation. J Tribol 122(1): 308–316CrossRefGoogle Scholar
  46. 46.
    Shinkarenko A, Kligerman Y, Etsion I (2009) The validity of linear elasticity in analyzing surface texturing effect for elastohydrodynamic lubrication. J Tribol 131: 021503CrossRefGoogle Scholar
  47. 47.
    Shukla J (1978) A new theory of lubrication for rough surfaces. Wear 49(1): 33–42CrossRefGoogle Scholar
  48. 48.
    Stachowiak G, Batchelor A (2005) Engineering tribology. Butterworth-Heinemann, BostonGoogle Scholar
  49. 49.
    Stupkiewicz S (2007) Micromechanics of contact and interphase layers micromechanics of contact and interphase layers. Springer, BerlinMATHGoogle Scholar
  50. 50.
    Stupkiewicz S, Maciniszyn A (2004) Modelling of asperity deformation in the thin-film hydrodynamic lubrication regime. In: Proceedings of the 2nd international conference on tribology in manufacturing processes, Nyborg, Denmark, June 15–18, 2004: ICTMP2004, 695pGoogle Scholar
  51. 51.
    Szeri AZ (2011) Fluid film lubrication, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  52. 52.
    Tala-Ighil N, Fillon M, Maspeyrot P (2011) Effect of textured area on the performances of a hydrodynamic journal bearing. Tribol Int 44: 211–219CrossRefGoogle Scholar
  53. 53.
    Temizer İ (2011) Thermomechanical contact homogenization with random rough surfaces and microscopic contact resistance. Tribol Int 44(2): 114–124MathSciNetCrossRefGoogle Scholar
  54. 54.
    Temizer İ, Wriggers P (2010) Thermal contact conductance characterization via computational contact homogenization: a finite deformation theory framework. Int J Numer Methods Eng 83(1): 27–58MathSciNetMATHGoogle Scholar
  55. 55.
    Temizer İ, Wriggers P (2011) Homogenization in finite thermoelasticity. J Mech Phys Solids 59(2): 344–372MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties. Springer, BerlinMATHGoogle Scholar
  57. 57.
    Tripp J (1983) Surface roughness effects in hydrodynamic lubrication: the flow factor method. J Lubr Technol 105(3): 458–465CrossRefGoogle Scholar
  58. 58.
    Wagner W, Gruttmann F (1994) A simple finite rotation formulation for composite shell elements. Eng Comput 11(2): 153–155MathSciNetGoogle Scholar
  59. 59.
    Walowit J, Anno J (1975) Modern developments in lubrication mechanics. Applied Science Publishers, LondonGoogle Scholar
  60. 60.
    Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, BerlinMATHCrossRefGoogle Scholar
  61. 61.
    Zohdi TI, Wriggers P (2008) An introduction to computational micromechanics, vol 20 of Lecture Notes in Applied and Computational Mechanics. Springer, BerlinGoogle Scholar

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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