Low-frequency vibrations of strongly inhomogeneous multicomponent elastic structures

• Main Author: Sergushova, Olga
• Published: Keele University 2018
• Subjects: QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.762191

The thesis deals with 1D and 2D scalar equations governing dynamic behaviour of strongly inhomogeneous layered structures. Harmonic vibrations of a composite rod and antiplane shear motions of a cylindrical body consisting of several components are studied paying particular attention to the lowest frequencies. The main focus is on a strong contrast between the parameters characterising structure components, including their sizes, material stiffness, and densities. We start with a multi-parametric analysis of the near-rigid body motions of rods and cylindrical bodies with piecewise uniform properties. The listed problems allow exact analytical solutions demonstrating that the values of all lowest eigenfrequencies tend to zero at large/small ratios of material and geometric parameters. The low-frequency behaviour is considered for so-called global and local regimes, and simple explicit conditions on the problem parameters, underlying each of the regimes, are derived. Further, we present a perturbation procedure for a more general setup based on the evaluation of the almost rigid body motions of “stronger” components assuming a high contrast of material parameters. The proposed approach is extended to structures of arbitrary shape, with variable material parameters, as well as to multi-component structures. We obtain asymptotic formulae for the lowest natural frequencies and also present illustrative examples for each of the studied problems. Many of asymptotic estimations are compared with exact solutions. The results of the thesis are applicable to a mathematical justification of shear deformation theories for multi-layered plates and shells with a strong transverse inhomogeneity.