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On a contact problem for a viscoelastic von Kármán plate and its semidiscretization. (English) Zbl 1099.49003
Summary: 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.

MSC:
49J40 Variational inequalities
65R20 Numerical methods for integral equations
74D10 Nonlinear constitutive equations for materials with memory
74K20 Plates
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References:
[1] I. Bock: On large deflections of viscoelastic plates. Math. Comput. Simul. 50 (1999), 135–143. · Zbl 1053.74560
[2] I. Bock, J. Lovisek: On unilaterally supported viscoelastic von Karman plates with a long memory. Math. Comput. Simul. 61 (2003), 399–407. · Zbl 1043.74029
[3] I. Bock, J. Lovisek: 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. Stary, eds.). Institute of Geonics AS CR, Praha, pp. 38–41.
[4] P. G. Ciarlet, P. Rabier: Les equations de von Karman. Springer Verlag, Berlin, 1980.
[5] G. Duvaut, J.-L. Lions: Les inequations en mecanique et en physique. Dunod, Paris, 1972.
[6] O. John: On Signorini problem for von Karman equations. Apl. Mat. 22 (1977), 52–68. · Zbl 0387.35030
[7] J. Kacur: Application of Rothe’s method to evolution integrodifferential equations. J. Reine Angew. Math. 388 (1988), 73–105. · Zbl 0638.65098
[8] E. Munoz Rivera, G. Perla Menzala: Decay rates of solutions to a von Karman system for viscoelastic plates with memory. Q. Appl. Math. 57 (1999), 181–200. · Zbl 1021.35127
[9] J. Naumann: On some unilateral boundary value problem for the von Karman equations. Apl. Mat. 20 (1975), 96–125. · Zbl 0311.73029
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