Influence of coupling effects on analytical solutions of functionally graded (FG) spherical shells of revolution

Justyna Flis,Aleksander Muc

doi:10.1515/rams-2021-0064

Justyna Flis, Aleksander Muc

Open Access

https://doi.org/10.1515/rams-2021-0064

Copy DOI

Abstract

Abstract Due to the lack of commercially available finite elements packages allowing us to analyse the behaviour of porous functionally graded (FG) structures in this paper, axisymmetric deformations of coupled FG spherical shells are studied. The analytical solution is presented by using complex hypergeometric polynomial series. The results presented agree closely with the reference results for isotropic spherical shells of revolution. The influence of the effects of material properties is characterized by a multiplier characterizing an unsymmetric shell wall construction (stiffness coupling). The results can be easily adopted in design procedures. The present results can be treated as the benchmark for finite element investigations.

Highlights

The analytical description of axisymmetric isotropic shells of revolution is demonstrated in monographs [1,2,3]
Considering the structural deformations in only the elastic range, the analysis of composite constructions can be divided into three categories: – Unidirectional laminates – the potential abrupt variation of mechanical properties between laminae. – Porous functionally graded materials (FGM) – a smooth variation of the properties from the bottom to the top surface. – Nanostructures reinforcing isotropic matrix – they are made of a polymer matrix reinforced with nanoplatelets or carbon nanotubes; the material properties are derived with the use of homogenization theories, e.g. the Mori–Tanaka method
It has been proved analytically that for FGM shells the influence of the unsymmetric shell wall construction can be described by only one parameter and is independent of the form of boundary value problems

Summary

Introduction

The analytical description of axisymmetric isotropic shells of revolution is demonstrated in monographs [1,2,3]. Solutions are functions of two geometrical variables x and y for shells with unsymmetric loading and boundary conditions and shell panels (Figure 1) and shells having unsymmetric wall configurations; see Muc and MucWierzgoń [10] This class deals with buckling and free vibrations problems where the expansion along the circumferential direction is required. It has been proved analytically that for FGM shells the influence of the unsymmetric shell wall construction (the non-zero terms of the stiffness matrix B, Figure 1c) can be described by only one parameter (function – multiplier) and is independent of the form of boundary value problems

Formulation of the problem

Solution of governing relation for axisymmetric spherical shells

Spherical shells under a uniform internal pressure

Concluding remarks