Combined asymptotic-tolerance modelling of dynamic problems for functionally graded shells

Abstract

The objects of consideration are thin linearly elastic Kirchhoff-Love-type open circular cylindrical shells having a functionally graded macrostructure and a tolerance-periodic microstructure in circumferential direction. The aim of this contribution is to formulate and discuss a new mathematical averaged model for the analysis of selected dynamic problems for these shells. This, so-called, combined asymptotic-tolerance model is derived by applying the combined modelling technique which includes both the asymptotic and tolerance non-asymptotic procedures. Contrary to the starting exact shell equations with highly oscillating, non-continuous and tolerance-periodic coefficients, governing equations of the proposed combined model have continuous and slowly varying coefficients depending also on a cell size. Hence, this model can be applied to study the effect of a microstructure size on dynamic behaviour of the shells (the length-scale effect). An important advantage of this model is that it makes it possible to analyse micro-dynamics of tolerance-periodic shells independently of their macro-dynamics.