On spatial behavior of harmonic vibrations in viscoelastic Reissner–Mindlin plates

This paper studies the spatial behavior of steady-state solutions in bending problems of linear viscoelastic plates described by the Reissner–Mindlin theory. A rectangular viscoelastic plate for which one of the sides is subjected to a displacement which varies harmonically in time and the remainder boundary is clamped is considered. It is proven that the dissipation mechanism gives rise ultimately to a steady-state vibration and provides for the amplitude a cross-sectional line-integral measure which decays spatially faster that an exponential of the distance from the loaded side of the plate. The estimate holds for every value of the frequency of vibrations and for the entire class of viscoelastic materials whose relaxation functions are compatible with thermodynamics. The case of a semi-infinite plate is also studied and an alternative of Phragmén–Lindelöf type is established.