On a contact problem for a viscoelastic von Kármán plate and its semidiscretization

Igor Bock; Ján Lovíšek

Applications of Mathematics (2005)

  • Volume: 50, Issue: 3, page 203-217
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Bock, Igor, and Lovíšek, Ján. "On a contact problem for a viscoelastic von Kármán plate and its semidiscretization." Applications of Mathematics 50.3 (2005): 203-217. <http://eudml.org/doc/33218>.

@article{Bock2005,
abstract = {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.},
author = {Bock, Igor, Lovíšek, Ján},
journal = {Applications of Mathematics},
keywords = {von Kármán system; viscoelastic plate; integro-differential variational inequality; semidiscretization; Banach fixed point theorem; von Kármán system; viscoelastic plate; integro-differential variational inequality; semidiscretization; Banach fixed point theorem},
language = {eng},
number = {3},
pages = {203-217},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a contact problem for a viscoelastic von Kármán plate and its semidiscretization},
url = {http://eudml.org/doc/33218},
volume = {50},
year = {2005},
}

TY - JOUR
AU - Bock, Igor
AU - Lovíšek, Ján
TI - On a contact problem for a viscoelastic von Kármán plate and its semidiscretization
JO - Applications of Mathematics
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 203
EP - 217
AB - 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.
LA - eng
KW - von Kármán system; viscoelastic plate; integro-differential variational inequality; semidiscretization; Banach fixed point theorem; von Kármán system; viscoelastic plate; integro-differential variational inequality; semidiscretization; Banach fixed point theorem
UR - http://eudml.org/doc/33218
ER -

References

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  2. 10.1016/S0378-4754(02)00095-2, Math. Comput. Simul. 61 (2003), 399–407. (2003) MR1984140DOI10.1016/S0378-4754(02)00095-2
  3. 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. Les équations de von Kármán, Springer Verlag, Berlin, 1980. (1980) MR0595326
  5. Les inéquations en mécanique et en physique, Dunod, Paris, 1972. (1972) MR0464857
  6. On Signorini problem for von Kármán equations, Apl. Mat. 22 (1977), 52–68. (1977) Zbl0387.35030MR0454337
  7. Application of Rothe’s method to evolution integrodifferential equations, J.  Reine Angew. Math. 388 (1988), 73–105. (1988) MR0944184
  8. 10.1090/qam/1672191, Q.  Appl. Math. 57 (1999), 181–200. (1999) MR1672191DOI10.1090/qam/1672191
  9. On some unilateral boundary value problem for the von Kármán equations, Apl. Mat. 20 (1975), 96–125. (1975) MR0437916

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