Abstract
When two or more bodies collide, contact occurs between two surfaces of the bodies so that they cannot overlap in space. Metal formation, vehicle crash, projectile penetration, various seal designs, and bushing and gear systems are only a few examples of contact phenomena. This chapter is organized as follows. In Sect. 5.2, simple one-point contact examples are presented in order to show the characteristics of contact phenomena and possible solution strategies. In Sect. 5.3, a general formulation of contact is presented based on the variational formulation. Sect. 5.4 focuses on finite element discretization and numerical integration of the contact variational form. Three-dimensional contact formulation is presented in Sect. 5.5 . From the finite element point of view, all formulations involve use of some form of a constraint equation. Because of the highly nonlinear and discontinuous nature of contact problems, great care and trial-and-error are necessary to obtain solutions to practical problems. Section 5.6 presents modeling issues related to contact analysis, such as selecting slave and master bodies, removing rigid-body motions, etc.
The original version of this chapter was revised. An erratum to this chapter can be found at https://doi.org/10.1007/978-1-4419-1746-1_6
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Rigorous discussions on this topic with variational inequality and its equivalence to the constrained optimization can be found in J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization, Springer-Verlag, Berlin, 1991. A brief summary will be presented in Sect. 5.3.
- 2.
It will be clear later that the contact force is equivalent to the Lagrange multiplier in the constrained optimization, which is the reason to use the Greek symbol λ.
- 3.
It is interesting to note that the physical interpretation of the negative Lagrange multiplier is the force that is required to apply at the tip of the beam in order to close the gap.
- 4.
This is the Rayleigh-Ritz method. For details, readers are referred to N. H. Kim and B. V. Sankar, Introduction to Finite Element Analysis and Design, Wiley & Sons, NY, 2008.
References
Duvaut G, Lions JL. Inequalities in mechanics and physics. Berlin: Spring-Verlag; 1976.
Kikuchi N, Oden JT. Contact problems in elasticity: a study of variational inequalities and finite element method. Philadelphia: SIAM; 1988.
Ciarlet PG. The finite element method for elliptic problems. New York: North-Holland; 1978.
Klarbring A. A mathematical programming approach to three-dimensional contact problems with friction. Comput Methods Appl Mech Eng. 1986;58:175–200.
Kwak BM. Complementarity-problem formulation of 3-dimensional frictional contact. J Appl Mech Trans ASME. 1991;58(1):134–40.
Luenberger DG. Linear and nonlinear programming. Boston: Addison-Wesley; 1984.
Barthold FJ, Bischoff D. Generalization of Newton type methods to contact problems with friction. J Mec Theor Appl. 1988;7 Suppl 1:97–110.
Arora JS. Introduction to optimum design. 2nd ed. San Diego: Elsevier; 2004.
Wriggers P, Van TV, Stein E. Finite element formulation of large deformation impact-contact problems with friction. Comput Struct. 1990;37:319–31.
Kim NH, Park YH, Choi KK. An optimization of a hyperelastic structure with multibody contact using continuum-based shape design sensitivity analysis. Struct Multidiscip Optim. 2001;21(3):196–208.
DeSalvo GJ, Swanson JA. ANSYS engineering analysis system, user’s manual, vols. I and II. Houston: Swanson Analysis Systems Inc.; 1989.
Laursen TA, Simo JC. A continuum-based finite element formulation for the implicit solution of multibody large deformation frictional contact problems. Int J Numer Methods Eng. 1993;36:2451–3485.
Kim NH, Yi KY, Choi KK. A material derivative approach in design sensitivity analysis of 3-D contact problems. Int J Solids Struct. 2002;39(8):2087–108.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kim, NH. (2015). Finite Element Analysis for Contact Problems. In: Introduction to Nonlinear Finite Element Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1746-1_5
Download citation
DOI: https://doi.org/10.1007/978-1-4419-1746-1_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-1745-4
Online ISBN: 978-1-4419-1746-1
eBook Packages: EngineeringEngineering (R0)