Axisymmetric large deformation problems of thin shallow shells with different moduli in tension and compression

Abstract

When materials are in tension or in compression, their mechanical properties are essentially different, however, in existing studies it is generally ignored because of the complexity of the analysis. In this paper, a theoretical study of the problem of axisymmetric large deformation of thin shallow shells with different moduli in tension and compression (bimodular thin shallow shells) under uniformly vertical loading is presented. Based on the Föppl–von Kármán equations of flexible thin plates, the governing equation of the axisymmetric large deformation of bimodular thin shallow shells are established by introducing an appropriate curvature modification term from thin shallow shells and a bending stiffness term from bimodular plates. By selecting the central deflection as a perturbation parameter, the perturbation solution of the axisymmetric large deformation problem is derived using the perturbation method. The applications of variation method as well as the implementation of numerical simulation verify, from the theoretical and numerical perspectives, the correctness of the perturbation solution obtained. This study shows that the introduction of different moduli in tension and compression not only affects the relationship of loads vs. central deflection, but also affects the critical values of the rise-thickness ratio of bimodular thin shallow shells when the snapping phenomenon occurs. In particular, the bimodular effect may accelerate or slow down the occurrence of the snapping phenomenon of thin shallow spherical shells, according to the relative magnitude of tensile and compressive moduli. The work presented here will be helpful to analyze the mechanical response of heavily loaded thin shallow shells with obvious bimodular effect.