Dynamic Plate and Shell Equations Using Power Series Expansions - Applications Including Elasticity, Poroelasticity and Fluid Loading

Doktorsavhandling, 2007

Thin structures are used in a wide range of engineering applications. The subject of this thesis is the derivation of approximate equations for the dynamics of thin plates and shells using series expansion methods. Approximate boundary conditions that simulate the behaviour of fluid-loaded elastic or poroelastic plates by eliminating the internal field variables are developed, using series expansions of the fields in the thickness coordinate. The resulting boundary conditions contain the sums and differences of the pressure fields and their partial derivatives in the thickness direction. The equations have been validated using transmission coefficients and dispersion curves for different combinations of fluids and elastic or poroelastic solids by comparison with the exact three-dimensional solutions as well as with classical plate models. Another approach studied in this thesis uses power series expansions of the displacement field in the thickness coordinate to obtain approximate dynamic equations for a cylindrical shell. The expansions are inserted into the equations of motion, creating a set of recursion relations expressing higher order displacement coefficients in terms of lower order ones. Using the recursion relations, equations with only the two lowest order coefficients are derived from the boundary conditions. By truncating the resulting shell equations to include terms up to a certain power of the thickness, a hierarchy of approximate shell equations is formed. The shell equations are presented explicitly with terms including up to the square of the thickness. The natural frequencies in the two-dimensional case (the ring) and the dispersion curves for propagating modes in the three-dimensional case are presented for different truncation orders, and compared with exact three-dimensional theory as well as with classical theories. The two seemingly different approaches used for plates and shells are in fact equivalent. One important factor of the approaches is that they are believed to yield asymptotically correct equations to any order, as the thickness tends to zero. Due to the complexity of higher order equations, only those of low order seem to be useful in an engineering context.