Skip to main content
SpringerLink
Log in

A Numerical Lagrangian Approach for Analysis of Contact Viscoelastic Problems

  • Published:
Computational Mathematics and Modeling Aims and scope Submit manuscript

The objective of this paper is to develop a finite-element formulation associated with an incremental adaptive procedure established for analysis of frictional contact problems in viscoelastic solids. A generalized Maxwell model has been used to model the viscoelastic constitutive equations in which bulk and shear relaxation functions are represented by the sum of a series of decaying exponential functions of time. A generalized finite-element approach, based on the principle of virtual work, has been developed using an incremental relaxation procedure. The application of the finite element method to contact viscoelastic problems is investigated. The contact treatment between bodies has been studied through an augmented Lagrangian approach. Finally, an example is presented to evaluate the computational solution procedure presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Wriggers, Computational Contact Mechanics, John Wiley and Sons Ltd., Chichester (2002).

    Google Scholar 

  2. Z.-H. Zhong, Finite Element Procedures for Contact — Impact Problems, Oxford University Press, New York (1993).

    Google Scholar 

  3. E. H. Lee and J. R. M. Radok, “The Contact Problem for Viscoelastic Bodies,” J. Appl. Mech., 27, 438–444 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  4. S. C. Hunter, “The Hertz problem for a rigid spherical indenter and a viscoelastic half-space,” J. Mech. Phys. Solids, 8, 219–234 (1960).

    Article  MATH  MathSciNet  Google Scholar 

  5. K. L. Johnson, Contact Mechanics, Cambridge University Press, New York (1985).

    Book  MATH  Google Scholar 

  6. W. H. Chen, C. M. Chang, and J. T. Yeh, “An incremental relaxation finite element analysis of viscoelastic problems with contact and friction,” Computer Methods Appl. Mech. and Eng., 109, 315–329 (1993).

    Article  MATH  Google Scholar 

  7. A. Amassad and C. Fabre, “Analysis of a viscoelastic unilateral contact problem involving the Coulomb friction law,” J. Optim. Theory and Appl., 116, No. 3, 465–483 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  8. J. R. Fernandez and M. Sofonea, “Numerical analysis of a frictionless viscoelastic contact problem with normal damped response,” Comput. Math. Appl., 47, 549–568 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Ashrafi, “An augmented Lagrangian treatment for viscoelastic contact formulation,” Proceedings of the 2009 Joint ASCE–ASME — SES Conference on Mechanics and Materials, Blacksburg, Virginia, 24–27 June (2009), pp. 122–123.

  10. H. Ashrafi and M. Farid, “A finite element formulation of contact problems for viscoelastic structures based on the generalized maxwell relaxation model,” Mech. Aerospace Eng. J., 5, 11–21 (2009).

    Google Scholar 

  11. F. F. Mahmoud, A. G. El-Shafei, and A. A. Mohamed, “An incremental adaptive procedure for viscoelastic contact problems,” J. Tribol., 129, 305–313 (2007).

    Article  Google Scholar 

  12. H. Ashrafi and M. Shariyat, “A nanoindentation modeling of viscoelastic creep and relaxation behaviors of ligaments,” Proceedings of the IEEE 17th Iranian Conference on Biomedical Engineering, Medical University of Esfahan, Esfahan, 3–4 November (2010), pp. 1–5.

  13. J. C. Simo and T. J. R. Hughes, Computational Inelasticity, Springer, New York (1998).

    MATH  Google Scholar 

  14. J. T. Oden and N. Kikuchi, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Publications, Philadelphia (1988).

    MATH  Google Scholar 

  15. J. C. Simo and T. A. Laursen, “An augmented Lagrangian treatment of contact problems involving friction,” Comput. Struct., 42, No. 1, 97–116 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Papadopoulos and R. L. Taylor, “A simple algorithm for three-dimensional finite element analysis of contact problems,” Comput. Struct., 46, No. 6, 1107–1118 (1993).

    Article  MATH  Google Scholar 

  17. G. Zavarise, P. Wriggers, and B. A. Schrefler, “On augmented Lagrangian algorithms for thermomechanical contact problems with friction,” Int. J. Numer. Methods Eng., 38, 2929–2949 (1995).

    Article  MATH  Google Scholar 

  18. A. R. Mijar and J. S. Arora, “An augmented Lagrangian optimization method for contact analysis problems, 1: Formulation and algorithm,” Struct. Multidiscipl. Optimiz., 28, 99–112 (2004).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    H. Ashrafi & M. Shariyat

Authors
  1. H. Ashrafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Shariyat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Ashrafi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ashrafi, H., Shariyat, M. A Numerical Lagrangian Approach for Analysis of Contact Viscoelastic Problems. Comput Math Model 25, 416–422 (2014). https://doi.org/10.1007/s10598-014-9236-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10598-014-9236-z

Keywords

Search

Navigation