Second-order two-scale asymptotic analysis for axisymmetric and spherical symmetric structure with periodic configurations

A new second-order two-scale (SOTS) analysis finite element algorithm is developed for the axisymmetric and spherical symmetric elastic problems with small periodic configurations. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction, and the spherical symmetric structure is only periodic in radial direction and homogeneous in other two directions. The SOTS asymptotic expansions for the space problem, plane axisymmetric problem, and spherical symmetric problem are presented, and the main feature is that the anisotropic material is obtained by the homogenization. The analytical expressions of the cell functions and homogenized solutions for plane axisymmetric and spherical symmetric problems are obtained, and the error estimations of the expansions are established. The second-order asymptotic analysis finite-element algorithm is presented and the numerical examples are solved including the hollow cylinder, rotating disk and hollow sphere composed of periodic composite materials. The computational results demonstrate the effectiveness and accuracy of the SOTS asymptotic analysis algorithm, and the converging behavior of the asymptotic analysis algorithm agrees well with the theoretical prediction. It is also indicated that the stress distributions can be correctly computed only by adding the second-order correctors.