Recent Developments in Anisotropic Heterogeneous Shell Theory

A wide class of problems on natural vibrations of anisotropic inhomogeneous shells is solved by using the refined model. Shells with constructional (with variable thickness) and structural inhomogeneity (made of functionally gradient materials) are considered. Initial boundary-value, eigenvalue, and partial derivative problems with variable coefficients are solved by spline-collocation, discrete-orthogonalization, and incremental search methods. In the case of hinged shells, the results obtained by means of analytical and proposed numerical methods are compared and analyzed. It is studied how the geometrical and mechanical parameters as well as the type of boundary conditions influence the distribution of dynamical characteristics of the shells under consideration. The frequencies and modes of natural vibrations of an orthotropic shallow shell of double curvature with variable thickness and various values of a radius of curvature are determined. The dynamical characteristics have been calculated for the example of cylindrical shells made of a functionally gradient material with thickness varying differently in circumferential direction. The values of natural frequencies obtained for this class of shells under some boundary conditions are compared with the data calculated by means of three-dimensional theory of elasticity.

In the present chapter models of three-dimensional theory of elasticity are used in order to study the stationary deformation of hollow and solid anisotropic inhomogeneous cylinders of finite length. Solutions for the stress–strain state and natural vibrations of hollow inhomogeneous finite-length cylinders are presented, which were obtained by making use of spline-collocation and discrete-orthogonalization methods. The influence of geometrical and mechanical parameters, of the boundary conditions, of the loading character on the distributions of stress and displacement fields, and of the dynamical characteristics in such cylinders are analyzed. In some cases the results obtained by three-dimensional and shell theory are compared. When solving dynamical problems for orthotropic hollow cylinders with different boundary conditions at the ends the method of straight-line methods in combination with the discrete-orthogonalization method was applied as well. Computations for solid anisotropic cylinders of finite length with different boundary conditions were carried out by using the semi-analytical finite element method. In the case of free ends the results of the calculations for the natural frequencies were compared with those determined experimentally. The results of calculations of the mechanical behavior of anisotropic inhomogeneous circular cylinders demonstrate the efficiency of the discrete-continuous approaches proposed in this monograph for solving shell problems when using three-dimensional models of elasticity theory.