Abstract
This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k → ∞, the authors obtain the damped von Kármán model with associated energy exponentially decaying to zero as well.
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This paper is partially supported by INCTMat, FAPESQ-PB, CNPq (Brazil) under Grant Nos. 308150/2008-2 and 620108/2008-8, the MICINN (Spain) under Grant No. MTM2008-03541, the Advanced Grant FP7-246775 NUMERIWAVES of the ERC, and the Project PI2010-04 of the Basque Government, also dedicated to Prof. D. L. Russell on his 70th birthday.
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Araruna, F.D., Braz E Silva, P. & Zuazua, E. Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system. J Syst Sci Complex 23, 414–430 (2010). https://doi.org/10.1007/s11424-010-0137-8
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DOI: https://doi.org/10.1007/s11424-010-0137-8