# Dynamic Equations for Spherical, Orthotropic and Piezoelectric Cylindrical Shells Using the Power Series Method

[Doktorsavhandling]

Shells are commonly used in many branches of engineering, and have therefore been investigated for a number of different types of shells. A shell is considered to be curved a plate with small thickness compared both to the other geometrical dimensions as well as to the wavelengths of importance. The most important superiority of shells in comparison to plates is that the membrane stiffness of shell structures enables them to provide high strength and low weight. Spherical and cylindrical shells appear in some applications and some dynamic shell theories have thus been developed for these cases. These theories seem to depend somewhat ad hoc kinematical assumptions and/or other approximations. In the present thesis, dynamic equations for isotropic spherical and orthotropic and piezoelectric cylindrical shells are derived using a method developed during the last decade for bars, plates, and beams. The main advantage of this method is that it is very systematic and can be developed to any order. The resulting structural equations also appear to be asymptotically correct. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using these expansions, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions. Applying the boundary conditions on the surfaces of the shells and eliminating all but some of the lowest order expansion functions gives the shell equations as a power series in the shell thickness. In principle, the equations can be truncated to any order in the shell thickness, which leads to very complicated expressions. For all cases, results are compared to exact theory and other results from the literature. The computations of eigenfrequencies from the power series approximation are in excellent agreement with results from the exact solution. In Paper A dynamic equations are derived for a spherical shell made of a homogeneous, isotropic material. Surface differential operators are introduced to reduce the length of all expressions. The exact 3D solution is given in a general vector format. Paper B uses the same technique to derive dynamic equations for an orthotropic cylindrical shell. For a transversely isotropic shell the exact 3D solution is also given. Paper C extends the work in Paper B to quite general end boundary conditions by employing a generalized Hamilton’s method. Tables with eigenfrequencies for various end boundary conditions are supplied. Shell equations for a radially polarized piezoelectric cylindrical shell are derived in Paper D.

**Nyckelord: **Shells, spherical shell, cylindrical shell, shell equations, surface differential operators, orthotropic, recursion relations, Hamilton’s principle, piezoelectricity, dynamics, eigenfrequency.

Denna post skapades 2015-09-07.

CPL Pubid: 222015