Winkler Support Model and Nonlinear Boundary Conditions Applied to 3D Elastic Contact Problem Using the Boundary Element Method

  • J. Vallepuga-Espinosa
  • Lidia Sánchez-González
  • Iván Ubero-MartínezEmail author


This work presents a numerical methodology for modeling the Winkler supports and nonlinear conditions by proposing new boundary conditions. For the boundary conditions of Winkler support model, the surface tractions and the displacements normal to the surface of the solid are unknown, but their relationship is known by means of the ballast coefficient, whereas for nonlinear boundary conditions, the displacements normal to the boundary of the solid are zero in the positive direction but are allowed in the negative direction. In those zones, detachments of nodes might appear, leading to a nonlinearity, because the number of nodes that remain fixed or of the detached ones (under tensile tractions) is unknown. The proposed methodology is applied to the 3D elastic receding contact problem using the boundary element method. The surface tractions and the displacements of the common interface between the two solids in contact under the influence of different supports are calculated as well as the boundary zone of the solid where the new boundary conditions are applied. The problem is solved by a double-iterative method, so in the final solution, there are no tractions or penetrations between the two solids or at the boundary of the solid where the nonlinear boundary conditions are simulated. The effectiveness of the proposed method is verified by examples.


Boundary element method Elastic contact problem Winkler support model Nonlinear boundary conditions 


  1. 1.
    Selvadurai APS. Elastic analysis of soil-foundation interaction. Amsterdam: Elsevier; 2013.Google Scholar
  2. 2.
    Winkler E. Die Lehre von der Elasticitaet und Festigkeit: mit besonderer Rücksicht auf ihre Anwendung in der Technik, für polytechnische Schulen, Bauakademien, Ingenieure, Maschinenbauer, Architecten, etc. Dominicius; 1868.Google Scholar
  3. 3.
    Katz C, Werner H. Implementation of nonlinear boundary conditions in finite elements analysis. Comput Struct. 1982;15(3):299–304.CrossRefzbMATHGoogle Scholar
  4. 4.
    Al-Hosani K. A non-singular fundamental solution for boundary element analysis of thick plates on Winkler foundation under generalized loading. Comput Struct. 2001;31(79):2767–80.CrossRefGoogle Scholar
  5. 5.
    Costa JA Jr, Brebbia CA. The boundary element method applied to plates on elastic foundations. Eng Anal. 1985;4:2.Google Scholar
  6. 6.
    Costa JA, Brebbia CA. On the reduction of domain integrals to the boundary for the BEM formulation of plates on elastic foundation. Eng Anal. 1986;2(3):123–6.CrossRefGoogle Scholar
  7. 7.
    Xiao JR. Boundary element analysis of unilateral supported Reissner plates on elastic foundations. Comput Mech. 2001;1(27):1–10.CrossRefzbMATHGoogle Scholar
  8. 8.
    Rashed YF, Aliabadi MH, Brebbia CA. The boundary element method for thick plates on a winkler foundation. Int J Numer Methods Eng. 1998;41:1435–62.CrossRefzbMATHGoogle Scholar
  9. 9.
    Tsai N, Westmann RA. Beam on tensionless foundation. J Eng Mech. 1967;93:1–12.Google Scholar
  10. 10.
    Weitsman Y. On foundations that react in compression only. J Appl Mech. 1970;4:37.zbMATHGoogle Scholar
  11. 11.
    Couchaux M, Hjiaj M, Ryan I. Enriched beam model for slender prismatic solids in contact with a rigid foundation. Eng Anal. 2015;93:181–90.Google Scholar
  12. 12.
    Ioakimidis NI. Derivation of conditions of complete contact for a beam on a tensionless Winkler elastic foundation with Mathematica. Mech Res Commun. 2016;72:64–73.CrossRefGoogle Scholar
  13. 13.
    Zhang L, Zhao M. New method for a beam resting on a tensionless and elastic–plastic foundation subjected to arbitrarily complex loads. Int J Geomech. 2015;4:16.Google Scholar
  14. 14.
    Zhong J, Fu Y, Chen Y, Li Y. Analysis of nonlinear dynamic responses for functionally graded beams resting on tensionless elastic foundation under thermal shock. Compos Struct. 2016;142:272–7.CrossRefGoogle Scholar
  15. 15.
    Panahandeh-Shahraki D, Shahidi AR, Mirdamadi HR, Vaseghi O. Nonlinear analysis of uni-lateral buckling for cylindrical panels on tensionless foundation. Thin-Walled Struct. 2013;62:109–17.CrossRefGoogle Scholar
  16. 16.
    Li YS. Buckling analysis of magnetoelectroelastic plate resting on Pasternak elastic foundation. Mech Res Commun. 2014;56:104–14.CrossRefGoogle Scholar
  17. 17.
    Attar M, Karrech A, Regenauer-Lieb K. Non-linear analysis of beam-like structures on unilateral foundations: a lattice spring model. Int J Solids Struct. 2016;88–89:192–214.CrossRefzbMATHGoogle Scholar
  18. 18.
    Rodríguez-Tembleque L, Abascal R, Aliabadi MH. A boundary element formulation for wear modeling on 3D and rolling-contact problems. Int J Solids Struct. 2010;47:2600–12.CrossRefzbMATHGoogle Scholar
  19. 19.
    Sfantos GK, Aliabadi MH. Application of BEM and optimization technique to wear problems. Int J Solids Struct. 2006;43:3626–42.CrossRefzbMATHGoogle Scholar
  20. 20.
    Sfantos GK, Aliabadi MH. Wear simulation using an incremental sliding boundary element method. Wear. 2006;260:1119–28.CrossRefGoogle Scholar
  21. 21.
    Dargush GF, Soom A. Contact modeling in boundary element analysis including the simulation of thermomechanical wear. Tribol Int. 2016;100:360–70.CrossRefGoogle Scholar
  22. 22.
    Polonsky IA, Keer LM. A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques. Wear. 1999;231:206–19.CrossRefGoogle Scholar
  23. 23.
    Stanley HM, Kato T. An FFT-based method for rough surface contact. J Tribol. 1997;119(3):481–5.CrossRefGoogle Scholar
  24. 24.
    Yastrebov VA, Anciaux G, Molinari J-F. From infinitesimal to full contact between rough surfaces: evolution of the contact area. Int J Solids Struct. 2015;52:83–102.CrossRefGoogle Scholar
  25. 25.
    Tian X, Bhushan B. A numerical three-dimensional model for the contact of rough surfaces by variational principle. J Tribol. 1996;118(1):33–42.CrossRefGoogle Scholar
  26. 26.
    Kalker JJ, Van Randen Y. A minimum principle for frictionless elastic contact with application to non-Hertzian half-space contact problems. J Eng Math. 1972;6(2):193–206.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kalker JJ. Numerical calculation of the elastic field in a half-space. Commun Appl Numer Methods. 1986;2:401–10.CrossRefzbMATHGoogle Scholar
  28. 28.
    Wayne Chen W, Jane Wang Q. A numerical model for the point contact of dissimilar materials considering tangential tractions. Mech Mater. 2008;40:936–48.CrossRefGoogle Scholar
  29. 29.
    Zhang S, Li X, Ran R. Self-adaptive projection and boundary element methods for contact problems with Tresca friction. Commun Nonlinear Sci Numer Simul. 2018;68:72.MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhong B-D, Yan F, Lv J-H. Continuous–discontinuous hybrid boundary node method for frictional contact problems. Eng Anal Bound Elem. 2018;87:19–26.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang S, Li X. A self-adaptive projection method for contact problems with the BEM. Appl Math Model. 2018;55:145–59.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shu X, Zhang J, Han L, Dong Y. A surface-to-surface scheme for 3D contact problems by boundary face method. Eng Anal Bound Elem. 2016;70:23–30.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Balci MN, Dag S. Dynamic frictional contact problems involving elastic coatings. Tribol Int. 2018;124:70–92.CrossRefGoogle Scholar
  34. 34.
    Gimperlein H, Meyer F, Özdemir C, Stephan EP. Time domain boundary elements for dynamic contact problems. Comput Methods Appl Mech Eng. 2018;333:147–75.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Alonso P, Garrido García JA. BEM applied to 2D thermoelastic contact problems including conduction and forced convection in interstitial zones. Eng Anal Bound Elem. 1995;3(15):249–59.CrossRefGoogle Scholar
  36. 36.
    Mediavilla AF. Tridimensional elastic contact problem by boundary integrals. Ph.D., Universidad de Valladolid; 1991.Google Scholar
  37. 37.
    Garrido JA, Lorenzana A. Enriched beam model for slender prismatic solids in contact with a rigid foundation. Eng Anal Bound Elem. 1998;4(21):295–303.CrossRefGoogle Scholar
  38. 38.
    Garrido JA. The contact problem in elasticity by integral equations. Ph.D., Universidad Politécnica de las Palmas; 1986.Google Scholar
  39. 39.
    Garrido JA, Foces A, Paris F. BEM applied to receding contact problems with friction. Math Comput Model. 1991;3–5(15):143–53.CrossRefzbMATHGoogle Scholar
  40. 40.
    Paris F, Foces A, Garrido JA. Application of boundary element method to solve three-dimensional elastic contact problems without friction. Comput Struct. 1992;43:19–30.CrossRefzbMATHGoogle Scholar
  41. 41.
    Espinosa JV. 3D thermoelastic contact problem without friction analysis using BEM. Microelectronic application, Ph.D., Universidad de Valladolid; 2009.Google Scholar
  42. 42.
    Espinosa JV, Mediavilla AF. Boundary element method applied to three dimensional thermoelastic contact. Eng Anal Bound Elem. 2012;6(36):928–33.CrossRefzbMATHGoogle Scholar
  43. 43.
    González R, Sánchez L, Vallepuga J, Alfonso J. Solving linear systems of equations from BEM codes. In: International joint conference SOCO’13-CISIS’13-ICEUTE’13. Springer; 2014. p. 81–90Google Scholar
  44. 44.
    González R, Sánchez-González L, Vallepuga J, Ubero I. Parallel performance of the boundary element method in thermoelastic contact problems. In: International joint conference SOCO’17-CISIS’17-ICEUTE’17. Springer; 2017. p. 524–32Google Scholar
  45. 45.
    Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques. Berlin: Springer; 1984. p. 177–236.CrossRefzbMATHGoogle Scholar
  46. 46.
    Hartmann F. The Somigliana identity on piecewise smooth surfaces. J Elast. 1981;11(4):403–23.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    TE DB SE-C, Seguridad Estructural Cimientos. BOE 25/01/2008.Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics 2019

Authors and Affiliations

  1. 1.Departamento de Tecnología Minera, Topografía y de EstructurasUniversidad de LeónLeónSpain
  2. 2.Departamento de Ingenierías Mecánica, Informática y AeroespacialUniversidad de LeónLeónSpain

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