The combined asymptotic-tolerance modelling of dynamic problems for thin longitudinally graded cylindrical shells

The objects of considerations are thin, linearly elastic, Kirchhoff–Love-type, circular, cylindrical shells of a heterogeneous microstructure, which is periodic in the circumferential direction and continuously slowly changing along the axial coordinate (composite shells with a space-varying periodic microstructure). Since macroscopic (averaged) properties of such shells are constant in the circumferential direction but smoothly slowly varying in the axial one (i.e. in the direction parallel to interfaces between constituents), then we deal with shells of a functionally longitudinally graded macrostructure. The aim of the contribution is to formulate and discuss a new, mathematical, averaged, non-asymptotic model for the analysis of selected dynamic problems for these shells. This, so-called, combined asymptotic-tolerance model is derived by applying both the consistent asymptotic and the tolerance non-asymptotic procedures, which are coupled together into a new modelling technique. The combined model equations derived here include coefficients constant in periodicity direction and continuously slowly changing along the axial coordinate. Since some of these coefficients depend on a cell size, then the model can be applied to study the effect of a microstructure size on the shells dynamics. Moreover, it makes it possible to analyse micro-dynamics of the shells independently of their macro-dynamics.