Previous |  Up |  Next


von Kármán system; viscoelastic plate; integro-differential variational inequality; semidiscretization; Banach fixed point theorem
We deal with the system describing moderately large deflections of thin viscoelastic plates with an inner obstacle. In the case of a long memory the system consists of an integro-differential 4th order variational inequality for the deflection and an equation with a biharmonic left-hand side and an integro-differential right-hand side for the Airy stress function. The existence of a solution in a special case of the Dirichlet-Prony series is verified by transforming the problem into a sequence of stationary variational inequalities of von Kármán type. We derive conditions for applying the Banach fixed point theorem enabling us to solve the biharmonic variational inequalities for each time step.
[1] I. Bock: On large deflections of viscoelastic plates. Math. Comput. Simul. 50 (1999), 135–143. DOI 10.1016/S0378-4754(99)00066-X | MR 1717642 | Zbl 1053.74560
[2] I. Bock, J. Lovíšek: On unilaterally supported viscoelastic von Kármán plates with a long memory. Math. Comput. Simul. 61 (2003), 399–407. DOI 10.1016/S0378-4754(02)00095-2 | MR 1984140
[3] I. Bock, J. Lovíšek: On a contact problem for a viscoelastic plate with geometrical nonlinearities. In: IMET 2004 Proceedings of the conference dedicated to the jubilee of Owe Axelsson, Prague, May 25-28, 2004, J.  Blaheta, J.  Starý (eds.), Institute of Geonics AS CR, Praha, pp. 38–41.
[4] P. G. Ciarlet, P. Rabier: Les équations de von Kármán. Springer Verlag, Berlin, 1980. MR 0595326
[5] G. Duvaut, J.-L.  Lions: Les inéquations en mécanique et en physique. Dunod, Paris, 1972. MR 0464857
[6] O. John: On Signorini problem for von Kármán equations. Apl. Mat. 22 (1977), 52–68. MR 0454337 | Zbl 0387.35030
[7] J. Kačur: Application of Rothe’s method to evolution integrodifferential equations. J.  Reine Angew. Math. 388 (1988), 73–105. MR 0944184
[8] E. Muñoz Rivera, G. Perla Menzala: Decay rates of solutions to a von Kármán system for viscoelastic plates with memory. Q.  Appl. Math. 57 (1999), 181–200. DOI 10.1090/qam/1672191 | MR 1672191
[9] J. Naumann: On some unilateral boundary value problem for the von Kármán equations. Apl. Mat. 20 (1975), 96–125. MR 0437916
Partner of
EuDML logo