Quadrilateral scaled boundary spectral shell elements for static and dynamic analyses of functionally graded plates and shells

Abstract

This paper presents new shell elements for static and dynamic analyses of functionally graded plates and shells with the material properties varying through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. In the formulation, a shell element is treated as a three-dimensional linear elastic body and its middle surface is represented with a quadrilateral spectral element. Along the thickness direction, the shell geometry is described by scaling the middle surface while the displacements are approximated by interpolating the displacements on the top, middle and bottom surfaces using quadratic Lagrange shape functions. The assumed natural strain method is applied to eliminate transverse shear locking, membrane locking and curvature thickness locking. The developed shell elements are featured by requiring only the shell mid-surface to be discretized, containing no rotational degree of freedom, adopting the full three-dimensional constitutive law and involving only the in-plane numerical integration. The validity and performance of the formulation are demonstrated through benchmark examples covering plates, cylindrical shells and spherical shells with varying span-to-thickness ratio and material gradation index. Static solutions of two functionally graded shell problems under gravity loads are also provided.