The plane‐strain, axisymmetric, thickness‐stretch modes of thick, isotropic and orthotropic elastic shells embedded in a thin elastic casing are investigated. The steady‐state dynamic response of their viscoelastic counterparts is obtained and compared with the response of the elastic shells. Use is made a of a higher‐order shell theory and its accuracy for the lower modes is demonstrated. Finally, the effect of orthotropy on the rapidity of the transition from the free vibrations of a plate to that of a solid elastic circular cylinder is investigated.

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