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On an elasto-dynamic evolution equation with non dead load and friction. (English) Zbl 1164.34460
Summary: 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 the Faedo-Galerkin method.
34G20 Nonlinear differential equations in abstract spaces
34K07 Theoretical approximation of solutions to functional-differential equations
34K10 Boundary value problems for functional-differential equations
47H05 Monotone operators and generalizations
74B20 Nonlinear elasticity
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[1] V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leyden, 1976. · Zbl 0328.47035
[2] H. Brézis: Analyse fonctionnelle. Théorie et Application. Masson, Paris, 1983.
[3] F. E. Browder: On non-linear wave equations. Math. Z. 80 (1962), 249–264. · Zbl 0109.32102
[4] T. Cazenave, A. Haraux: An Introduction to Semilinear Evolution Equations. Clarendon Press, Oxford, 1998. · Zbl 0926.35049
[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. · Zbl 1021.74029
[6] P. G. Ciarlet: Mathematical Elasticity, Vol. I: Three-dimmensional Elasticity. North Holland, Amsterdam, 1988.
[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.
[9] J. Jarušek: Dynamic contact problems with given friction for viscoelastic bodies. Czech. Math. J. 46(121) (1996), 475–487. · Zbl 0879.73022
[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. · Zbl 0938.74048
[11] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod et Gauthier-Villars, Paris, 1969.
[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.
[13] N. Kikuchi, J. T. Oden: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM publications, Philadelphia, 1988. · Zbl 0685.73002
[14] J.-L. Lions, E. Magenes: Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris, 1968.
[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. · Zbl 0567.73122
[16] J. Nečas, I. Hlaváček: Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Elsevier, Amsterdam, 1981.
[17] H. Tanabe: Equations of Evolution. Pitman, London, 1979. · 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. · Zbl 0873.73041
[19] E. Zeidler: Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators. Springer-Verlag, New York, 1990. · Zbl 0684.47028
[20] E. Zeidler: Nonlinear Functional Analysis and Its Applications, II/B: Non-linear Monotone Operators. Springer-Verlag, New York, 1990. · Zbl 0684.47029
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