Previous |  Up |  Next


evolution equation; existence and uniqueness; Faedo-Galerkin method; friction; elasticity; dynamic process
In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.
[1] V.  Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leyden, 1976. MR 0390843 | Zbl 0328.47035
[2] H.  Brézis: Analyse fonctionnelle. Théorie et Application. Masson, Paris, 1983. MR 0697382
[3] F. E. Browder: On non-linear wave equations. Math.  Z. 80 (1962), 249–264. DOI 10.1007/BF01162382 | MR 0147769 | Zbl 0109.32102
[4] T. Cazenave, A. Haraux: An Introduction to Semilinear Evolution Equations. Clarendon Press, Oxford, 1998. MR 1691574
[5] O.  Chau, J. R.  Fernández, W.  Han, and M.  Sofonea: Variational and numerical analysis of a dynamic frictionless contact problem with adhesion. J.  Comput. Appl. Math. 156 (2003), 127–157. DOI 10.1016/S0377-0427(02)00909-3 | MR 1982938
[6] P. G. Ciarlet: Mathematical Elasticity, Vol.  I: Three-dimmensional Elasticity. North Holland, Amsterdam, 1988. MR 0936420
[7] R. Dautray, J.-L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology, Tome  5. Springer-Verlag, Berlin, 2000.
[8] G.  Duvaut, J.-L.  Lions: Inequalities in Mechanics and Physics. Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 0521262
[9] J.  Jarušek: Dynamic contact problems with given friction for viscoelastic bodies. Czech. Math. J. 46(121) (1996), 475–487. MR 1408299
[10] J.  Jarušek, C.  Eck: Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Models Methods Appl. Sci. 9 (1999), 11–34. DOI 10.1142/S0218202599000038 | MR 1671535
[11] J.-L.  Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod et Gauthier-Villars, Paris, 1969. MR 0259693 | Zbl 0189.40603
[12] T. Kato: Linear and quasi-linear equations of evolution of hyperbolic type. Hyperbolicity, C.I.M.E., II. ciclo, Cortona 1976, Liguori, Napoli, 1977, pp. 125–191. Zbl 0456.35052
[13] N.  Kikuchi, J. T.  Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM publications, Philadelphia, 1988. MR 0961258
[14] J.-L.  Lions, E.  Magenes: Problèmes aux limites non homogènes et applications, Vol.  1. Dunod, Paris, 1968. MR 0247243
[15] J. A. C.  Martins, T. J. Oden: Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52 (1985), 527–634. DOI 10.1016/0045-7825(85)90009-X | MR 0822757
[16] J.  Nečas, I.  Hlaváček: Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Elsevier, Amsterdam, 1981. MR 0600655
[17] H. Tanabe: Equations of Evolution. Pitman, London, 1979. MR 0533824 | Zbl 0417.35003
[18] L.  Trabucho, J. M.  Viaño: Mathematical modelling of rods. In: Handbook of Numerical Analysis, Vol.  IV, P. G.  Ciarlet, J.-L. Lions (eds.), North Holland, Amsterdam, 1996, pp. 487–974. MR 1422507
[19] E.  Zeidler: Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators. Springer-Verlag, New York, 1990. MR 1033497 | Zbl 0684.47028
[20] E.  Zeidler: Nonlinear Functional Analysis and Its Applications, II/B: Non-linear Monotone Operators. Springer-Verlag, New York, 1990. MR 1033498
Partner of
EuDML logo