An asymptotic theory for dynamic response of anisotropic inhomogeneous and laminated cylindrical shells
This study treats the dynamic response of anisotropic inhomogeneous laminated cylindrical shells in a new refined theory based on an asymptotic expansion, originally three-dimensional (3-D), and with multiple time scales. First it is observed that the classical laminated shell theory does not give accurate stress distributions through the thickness, hence the method of asymptotic expansion. Moreover, an expansion with a single time scale will not result in a convergent solution. Hence again the introduction of multiple time scale which proved to be effective. The 3-D equilibrium equations are written in cylindrical coordinates, which through an integration with respect to the thickness gives three 2-D linear differential equations in the three displacements unknowns. These three dimensionless equations are written as functions of a small parameter \(\varepsilon\), i.e. the thickness. Then the multiple time scale is introduced. The asymptotic expansion provides equations of different orders, so that the order zero (slower time order) corresponds to secular solutions. At each order, the differential equations depend on forcing functions issued from the preceding order solution, in particular, the secular terms. Now the discussion leads to the elimination of such secular terms that the expansion remains uniformly valid, regardless of the time scale. (The reason is that they would give, for instance, an order one term with the same frequency as the order zero, then giving resonance or infinitely increasing solutions with the time). These terms must be eliminated. This elimination leads to a linear differential equation from which the dependence of the solution at order \(n\) can be determined. Application of the theory is performed on a cross-ply cylindrical shell problem treated as a free vibration problem. Harmonic shape-like solutions are assumed including a phase shift function of different scales. The solution in the angular pulsation is found by a boundary equation. The equation at order \(n\) provides similar eigensolutions. The resulting modes are coupled. Higher order terms reveal to be more and more hard to handle. This important study treated in the case of a cylindrical shell, seemingly hints some deeper mathematical phenomena which deserve more general reflexions.
scientific article; zbMATH DE number 711816
scientific article