Recent Developments in Anisotropic Heterogeneous Shell Theory

Elastic bodies in the form of thin and thick-walled anisotropic shells are considered. The shells may be made both of homogeneous and inhomogeneous materials with discrete (multilayer) structure or of continuously inhomogeneous materials (functionally gradient materials). The stationary deformation of such shells is analyzed by using various mechanical models. The basic relations of the theory of elasticity, which include equilibrium equations of motion, geometrical, and physical relations, are presented. By using classical and refined shell theories, the original three-dimensional problem is reduced to a two-dimensional one. The basic equations of the classical (Kirchhoff-Love) shell theory, which are based on the hypothesis of undeformed normals, are presented. It is assumed that all of the shell layers are stiffly joined and operate mutually without sliding and separation. Moreover, geometrical and mechanical parameters of the shells and mechanical loads applied to them are such that, considering the shell as a unit stack, the hypothesis of undeformed normals is valid. In the case of laminated shells made of new composite materials with low shearing stiffness, where anisotropy and inhomogeneity of the mechanical properties of the layers vary considerably, the refined model based on the straight-line hypothesis is used. The basic equations of the model are presented and various physically consistent boundary conditions at the bonded surfaces of the shells are specified.

Analytical-numerical methods for solving boundary-value and boundary-value eigenvalue problems for systems of ordinary differential equations and partial differential equations with variable coefficients are presented. In order to solve one-dimensional problems, the discrete-orthogonalization method is proposed. This approach is based on reducing the boundary-value problem to a number of Cauchy problems followed by their orthogonalization at some points of the integration interval which provides stability of calculations. In the case of the boundary-value eigenvalue problems, this approach is employed in combination with the incremental search method. In order to solve two-dimensional problems, the original system of partial differential system is reduced to systems of ordinary differential equations while making use of spline approximation and solving them by the discrete-orthogonalization method. Employing spline-functions is favorable, first, because of stability with respect to local disturbances, i.e., in contrast to polynomial approximation the spline behavior in the vicinity of a point does not influence the spline behavior as a whole; second, more satisfactory convergence is achieved, in contrast to the case of polynomials being applied as approximation functions; third, simplicity and convenience in calculation and implementation of spline-functions with the help of modern computers results. Besides, a nontraditional approach to solving problems of that class is proposed. It makes use of discrete Fourier series, i.e., Fourier series for functions specified on the discrete set of points. The two-dimensional boundary-value problem is solved by reducing it to a one-dimensional one after introducing auxiliary functions and separation of variables by using discrete Fourier series. Taking into account the calculation possibilities of modern computers, which make it possible to calculate a large number of series terms, the problem can be solved with high accuracy.

Results for stationary deformation of anisotropic inhomogeneous shells of various classes are presented by using classical Kirchhoff-Love theory and the numerical approaches outlined in Chap. 2 of this book. The stress-strain problems for shallow, noncircular cylindrical shells and shells of revolution are solved. Various types of boundary conditions and loadings are considered. Distributions of stress and displacement fields in shells of the aforementioned type are analyzed for various geometrical and mechanical parameters. The practically important stress problem of a high-pressure glass-reinforced balloon is solved. Dynamical characteristics of an inhomogeneous orthotropic plate under various boundary conditions are studied. The problem of free vibrations of a circumferential inhomogeneous truncated conical shell is solved. The effect of variation in thicknesses, mechanical parameters, and boundary conditions on the behavior of natural frequencies and vibration modes of a plate and cone is analyzed. Much attention is given to the validation of the reliability of the results obtained by numerical calculations.