Are “higher-order” and “layer-wise zig-zag” plate & shell theories necessary for functionally graded materials and structures?

Similar to the very vast prior literature on analyzing laminated composite structures, “higher-order” and “layer-wise higher-order” plate and shell theories for functionally-graded (FG) materials and structures are also widely popularized in the literature of the past two decades. However, such higher-order theories involve (1) postulating very complex assumptions for plate/shell kinematics in the thickness direction, (2) defining generalized variables of displacements, strains, and stresses, and (3) developing very complex governing equilibrium, compatibility, and constitutive equations in terms of newly-defined generalized kinematic and generalized kinetic variables. Their industrial applications are thus hindered by their inherent complexity, and the fact that it is difficult for end-users (front-line structural engineers) to completely understand all the newly-defined generalized DOFs for FEM in the higher-order and layer-wise theories. In an entirely different way, very simple 20-node and 27-node 3-D continuum solid-shell elements are developed in this paper, based on the simple theory of 3D solid mechanics, for static and dynamic analyses of functionally-graded plates and shells. A simple Over-Integration (a 4-point Gauss integration in the thickness direction) is used to evaluate the stiffness matrices of each element, while only a single element is used in the thickness direction without increasing the number of degrees of freedom. A stress-recovery approach is used to compute the distribution of transverse stresses by considering the equations of 3D elasticity in Cartesian as well as cylindrical polar coordinates. Comprehensive numerical results are presented for static and dynamic analyses of FG plates and shells, which agree well, either with the existing solutions in the published literature, or with the computationally very expensive solutions obtained by using simple 3D isoparametric elements (with standard Gauss Quadrature) available in NASTRAN (wherein many 3D elements are used in the thickness direction to capture the varying material properties). The effects of the material gradient index, the span-to-thickness ratio, the aspect ratio and the boundary conditions are also studied in the solutions of FG structures. Because the proposed methodology merely involves: (2) standard displacement DOFs at each node, (2) involves a simple 4-point Gaussian over-integration in the thickness direction, (3) relies only on the simple theory of solid mechanics, and (4) is capable of accurately and efficiently predicting the static and dynamical behavior of FG structures in a very simple and cost-effective manner, it is thus believed by the authors that the painstaking and cumbersome development of “higher-order” or “layer-wise higher-order” theories is not entirely necessary for the analyses of FG plates and shells.