Skip to main content

Advertisement

SpringerLink
Log in

Advances in computational contact mechanics

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

Abstract

Great strides have recently been made in the application of computational mechanics to the design of highly complex engineering systems. It has now become abundantly clear that advanced modelling techniques are central to the competitiveness of the industrialised nations. Excellent examples of this assertion are the computer-integrated design of the recent Boeing 777 aircraft, the collapsible foam-filled structures for the car of the next century and new prosthetic implants for Rheumatoid Arthritis. It is with this in mind that the author focuses his attention to a class of problems where contact mechanics plays a major role in dictating the mechanical integrity of the component/system. Three aspects of the current study are accordingly examined. The first is concerned with the development of the appropriate dynamic variational inequalities expressions, which are capable of the accurate and consistent representation of contact problems. The second is concerned with the development of robust solution algorithms that guarantee the accurate imposition of the kinematic contact constraint and avoid interpenetration. The third is concerned with the application of the developed algorithms to realistic design problems involving intricate mechanical and biomechanical systems.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26

Similar content being viewed by others

References

  • Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  • Aliabadi, M.H., Brebbia, C.A.: Advanced Formulations in Boundary Element Methods. Southampton Computational Mechanics Publications (1993)

  • ANSYS version 5.5, User’s Manual. Ansys Engineering Systems, Houston (1999)

  • Balinski, M.L., Wolfe, P.: Mathematical Programming Study 3: Nondifferentiable Optimization. North-Holland Publishing Company, Amsterdam (1975)

    Google Scholar 

  • Bathe, K.J.: Finite Element Procedures. Prentice-Hall, New Jersey (1996)

    Google Scholar 

  • Bischoff, D.: Indirect optimization algorithms for nonlinear contact problems. Proc. Mathematics of Finite Elements and Applications V, pp. 533–545. Uxbridge, England (1984)

  • Bogomolny, A.: Variational formulation of the roller contact problem. Math. Methods Appl. Sci. 6, 84–96 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Böhm, J.: A comparison of different contact algorithms with applications. Comput. Struct. 26, 207–221 (1987)

    Article  MATH  Google Scholar 

  • Chaudhary, A.B., Bathe, K.J.: A solution method for planar and axisymmetric contact problems. Int. J. Numer. Methods Eng. 21, 65–88 (1985)

    Article  MATH  Google Scholar 

  • Chaudhary, A.B., Bathe, K.J.: A solution method for static and dynamic analysis of three-dimensional contact problems with friction. Comput. Struct. 24, 855–873 (1986)

    Article  MATH  Google Scholar 

  • Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. J. Appl. Mech. 60, 371–375 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Crisfield, M.A.: Non-linear Finite Element Analysis of Solids and Structures. John Wiley & Sons, New York (1997)

    Google Scholar 

  • Czekanski, A., Meguid, S.A.: Analysis of dynamic frictional contact problems using variational inequalities. Finite Elem. Anal. Design, 37, 861–879 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Czekanski, A., Meguid, S.A.: Solution of dynamic frictional contact problems using nondifferentiable optimization. Int. J. Mech. Sci. 43, 1369–1386 (2001)

    Article  MATH  Google Scholar 

  • Czekanski, A., Meguid, S.A., El-Abbasi, N., Refaat, M.H.: On the elastodynamic solution of frictional contact problems using variational inequalities. Int. J. Numer. Methods Eng. 50, 611–627 (2001)

    Article  MATH  Google Scholar 

  • Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer-Verlag, Berlin Heilberg (1976)

    MATH  Google Scholar 

  • El-Abbasi, N.: A new strategy for treating frictional contact in shell structures using variational inequalities, Ph.D. Thesis, University of Toronto (1999)

  • Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design—A Practical Guide. Academic Press, Toronto (1997)

    Google Scholar 

  • Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarily problems. SIEM Rev. 39, 669–713 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Fletcher, R.: Practical Methods of Optimization. John Wiley and Sons, New York (1982)

    Google Scholar 

  • Gladwell, G.M.L.: Contact Problems in the Classical Theory of Elasticity. Sijhoff and Noordhoff, Alphen aan den Rijn (1980)

  • Glowinski, R., Lions, T.C., Tremolieres, R.: Numerical Analysis of Variational Inequalities. North Holland Publishing Company, Amsterdam (1981)

    Book  MATH  Google Scholar 

  • Hertz, H.: Über die Berührung Fester Elastischer Körper (On the Contact of Elastic Solids). Journal für die Reine und Angewandte Mathematik 92, 156–171 (1882)

    Article  Google Scholar 

  • Hlavacek, I., Haslinger, J., Necas, J., Lovisek, J.: Solution of Variational Inequalities in Mechanics. Springer-Verlag, New York (1988)

    MATH  Google Scholar 

  • Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, New Jersey (1987)

    MATH  Google Scholar 

  • Hughes, T.J.R., Taylor, R.L. Sackman, J.L., Curnier, A., Kanoknukulchai, W.: A finite element method for a class of contact-impact problems. Comput. Methods Appl. Mech. Eng. 8, 249–276 (1976)

    Article  MATH  Google Scholar 

  • Johnson, K.L.: Contact Mechanics. Prentice-Hall, Cambridge University Press, New York (1985)

    MATH  Google Scholar 

  • Kenny, B., Patterson, E.A., Said, M., Aradhya, K.S.S.: Contact stress distributions in a turbine disc dovetail type joint—a comparison of photoelastic and finite element results. Strain 27, 21–24 (1991)

    Article  Google Scholar 

  • Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Method. SIAM, Elsevier Science, Amsterdam (1988)

    Google Scholar 

  • Kinderlehrer, D., Stampacchia, G.: An Introduction of Variational Inequalities and Their Applications. Academic Press, New York (1980)

    Google Scholar 

  • Klarbring, A.: On discrete and discretized non-linear elastic structures in unilateral contact (stability, uniqueness and variational principles). Int. J. Solids Struct. 24, 459–4789 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Laursen, T.A., Simo, J.C.: A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int. J. Numer. Methods Eng. 36, 3451–3485 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Lemarechal, C.: Nondifferentiable optimization; subgradient and ɛ-subgrandient methods. Math. Program. Study 7, 384–387 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  • MARC, Theoretical Manual. MARC Software International Inc., Palo Alto, California (1993)

  • Mifflin, R.: A modification and an extension of lemarechal’s algorithm for nonsmooth minimization. Math. Program. Study 17, 77–90 (1982)

    MATH  MathSciNet  Google Scholar 

  • Newmark, N.M.: A method of computation for structural dynamics. ASCE J. Eng. Mech. Div. 13, 67–94 (1959)

    Google Scholar 

  • Oden, J.T., Martins, J.A.C.: A numerical analysis of class of problems in elastodynamics with friction. Comput. Methods Appl. Mech. Eng. 40, 327–360 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • Oden, J.T., Martins, J.A.C.: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Parisch, H.: Consistent tangent stiffness matrix for three-dimensional non-linear contact analysis. Int. J. Numer. Methods Eng. 28, 1803–1812 (1989)

    Article  MATH  Google Scholar 

  • Qin, Q.H., He, X.Q.: Variational principles, FE and MPT for analysis of non-linear impact-contact problems. Comput. Methods Appl. Mech. Eng. 122, 205–222 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Rabier, P., Martins, J.A.C., Oden, J.T., Campos, L.: Existence and local uniqueness of solutions to contact problems in elasticity with nonlinear friction law. Int. J. Mech. Sci. 24, 1755–1768 (1986)

    MATH  MathSciNet  Google Scholar 

  • Refaat, M.H., Meguid, S.A.: On the elastic solution of frictional contact problems using variational inequalities. Int. J. Mech. Sci. 36, 329–342 (1994)

    Article  MATH  Google Scholar 

  • Refaat, M.H., Meguid, S.A.: A novel finite element approach to frictional contact problems. Int. J. Numer. Methods Eng. 39, 3889–3902 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Shimizu, K., Ishizuka, Y., Barol, J.F.: Nondifferentiable and Two-level Mathematical Programming. Kluwer Academic Publishers, Boston (1997)

    MATH  Google Scholar 

  • Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer-Verlag, New York (1985)

    MATH  Google Scholar 

  • Wierzbicki, A.P.: Lagrangian functions and nondifferentiable optimization. In: Nurminski, E.A. (ed.) Progress in Nondifferentiable Optimization, pp. 97–124. International Institute for Applied System Analysis, Luxemburg (1982)

    Google Scholar 

  • Wriggers, P., Simo, J.C.: A note on tangent stiffness for fully nonlinear contact problems. Commun. Appl. Numer. Methods 1, 199–203 (1985)

    Article  MATH  Google Scholar 

  • Wriggers, P., Vu Van, T., Stein, E.: Finite element formulation of large deformation impact-contact problems with friction. Comput. Struct. 37, 319–331 (1990)

    Article  MATH  Google Scholar 

  • Zhong, Z.H.: Finite Element Procedures for Contact-Impact Problems. Oxford University Press, Oxford (1993)

    Google Scholar 

  • Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. McGraw-Hill, London (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, Canada, M5S 3G8

    S. A. Meguid & A. Czekanski

Authors
  1. S. A. Meguid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Czekanski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Meguid.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meguid, S.A., Czekanski, A. Advances in computational contact mechanics. Int J Mech Mater Des 4, 419–443 (2008). https://doi.org/10.1007/s10999-008-9077-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-008-9077-z

Keywords

Search

Navigation