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We consider a dynamical one-dimensional
nonlinear von Kármán model for beams
depending on a parameter ε > 0 and study
its asymptotic behavior for t large, as ε → 0. Introducing appropriate damping
mechanisms we show that the energy of solutions
of the corresponding damped models decay
exponentially uniformly with respect to the
parameter ε. In order for this to be true the
damping mechanism has to have the appropriate
scale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam models
for which the energy tends to zero exponentially
as well. This is done both in the case of
internal and boundary damping. We address the same
problem for plates with internal damping.
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