Extended tolerance modelling of dynamic problems for thin transversally graded cylindrical shells

The objects of consideration are thin, linearly elastic, Kirchhoff–Love-type, open, circular cylindrical shells having a functionally (transversally) graded macrostructure and a tolerance-periodic microstructure in the circumferential direction. At the same time, the shells have constant geometrical, elastic and inertial properties in the axial direction. The aim of this contribution is to formulate and discuss a new, mathematical, averaged, non-asymptotic model for the analysis of selected dynamic problems for such shells. This, so-called, general (or extended) tolerance model will be derived by applying a certain extended version of the well-known tolerance modelling technique. The starting equations are the well-known governing equations of linear Kirchhoff–Love theory of thin, elastic cylindrical shells. For the functionally graded shells under consideration, the starting equations have highly oscillating, non-continuous and tolerance-periodic coefficients in the circumferential direction, whereas equations of the proposed model have continuous and slowly changing coefficients in this direction. Moreover, some of these coefficients depend on a microstructure size. It means that the proposed model makes it possible to study the effect of a microstructure size on the shell dynamics (the length-scale effect). The main similarities and differences between the general tolerance model derived here and the corresponding, known, standard tolerance model derived by means of the classical tolerance modelling technique will be discussed. It will be shown that in the framework of both the general and standard tolerance models not only the fundamental cell-independent, but also the new, additional, cell-dependent free vibration frequencies can be derived and analysed.