# An Analysis of the Unilateral Contact Problem with Friction of Beams and Plates on an Elastic Half-Space

• L. Ascione
• D. Bruno
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 304)

## Summary

In this paper we analyze the unilateral contact problem with friction of some one-dimensional models, like beams and circular plates, constrained on an elastic half-space. Firstly, the main features of the continuous model, which has been developed in a previous work, are discussed. Then a finite element model is presented and some numerical results, which show the influence of the friction on the nature of the soil-foundation interaction, are given.

## Keywords

Contact Problem Circular Plate Auxiliary Problem Finite Element Approximation Unilateral Contact

## Sommario

In questo lavoro si sviluppa una analisi del problema di contatto con attrito di alcuni modelli monodimensionali di travi e piastre circolari vincolati monolateralmente su un semispazio elastico. Vengono innanzitutto richiamati gli aspetti fonda-mentali del modello continuo, sviluppato in un precedente lavoro. Si fornisce quindi un modello agli elementi finiti e si presenta-no alcuni risultati numerici che mostrano l’influenza del fenomeno di attrito sulla natura dell’interazione suolo-struttura.

## References

1. 1.
Duvaut, G. and Lions, J.L.: Les Inéquations en Mécanique et en Phisique, Dunod, Paris, 1–972.Google Scholar
2. 2.
Necas, J., Jaruâec, J. and Haslinger, J.: On the solution of the variational inequality to the Si-gnorini problem with small friction, Boll. U.M.I. (5), 17-B, 1980, 736–811.Google Scholar
3. 3.
Duvaut, G.: Problèmes mathématiques de la Mécanique-Équilibre d’un solide élastique avec contact unilateral et frottement de Coulomb, C.R. Acad. Sc., Paris, t. 290, Série A, 1980, 263, 265.Google Scholar
4. 4.
Oden, J.T. and Pires, E.: Contact problems in elastostatics with non-local friction laws, TICOM Report 81–12, November, 1981.Google Scholar
5. 5.
Oden, J.T. and Pires, E.: Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity, TICOM Report 82–3, April, 1982.Google Scholar
6. 6.
Panagiotopoulos, P.D.: A nonlinear programming approach to the unilateral contact and friction-boundary value problem in the theory of elasticity, Ingenieur-Archiv 44, 1975, 421–432.Google Scholar
7. 7.
Ascione, L. and Grimaldi, A.: Unilateral contact between a plate and an elastic foundation, Meccanica, 1984, 19, 223–233.Google Scholar
8. 8.
Ascione, L. and Olivito, R.S.: Unbonded contact of a Mindlin plate on an elastic half-space, Meccanica, 20, 1985, 49–58.
9. 9.
Ascione, L. and Bruno, D.: The unilateral contact problem with friction of a plate resting on an elastic half-space, Rep. no. 71, Dept. of Structures, University of Calabria, 1984.Google Scholar
10. 10.
Mindlin, R.D.: Influence of rotatory inertia on shear and flexural motions of isotropic elastic plates, J. Appl. Mech., 18, 1951, 31–88.
11. 11.
Ascione, L. and Olivito, R.S.: Some topics about the force-displacement relationship of an elastic obstacle in view of contact problem solution, Rep. no. 31, Dept. of Structures, University of Calabria, 1981.Google Scholar
12. 12.
Gladwell, G.M.L.: Contact Problems in the Theory of Elasticity, Sijthoff & Noordhoff, 1980.
13. 13.
Dunford, N. and Schwartz, J.: Linear Operators, Part I, Interscience Public, 1958.Google Scholar
14. 14.
Ascione, L. and Grimaldi, A.: Penalty Formulations of the Unilateral Contact Problem Between Plates and an Elastic Half-space, in Penalty — Finite Element Methods in Mechanics (ed. J.N. Reddy), American Society of Mechanical Engineers, 1982.Google Scholar
15. 15.
Brezzi, F., Hager, W.W., Raviart, P.A.: Error estimates of the finite element solution of variational inequalities, Numer. Math. 28, 1977, 431–443.
16. 16.
Oden J.T. Mixed finite element approximations via interior and exterior penalties for contact problems in elasticity, Int. Sym. on Hibrid and Mixed Finite Element Methods, Georgia Institute of Technology, Atlanta, Georgia, April, 8–10, 1981.Google Scholar