Three-dimensional variational theory of laminated composite plates and its implementation with Bernstein basis functions

Abstract A new three-dimensional (3-D) variational theory aimed at the stress analysis of thick laminated rectangular plates with anisotropic layers is presented. The developed theory is versatile , it allows one to use various types of basis functions and apply them independently to each of the three coordinate directions. Also, the theory is material-adaptive , enabling us to impose the conditions of displacement continuity between all of the discretization elements (3-D bricks) within the body and, in addition, the conditions of stress continuity between adjacent bricks made from the same material. The theory can be applied, in principle, to any boundary value problem of linear elasticity, considering arbitrarily distributed static surface forces and external kinematic boundary conditions. One specific realization of the theory, elaborated in this work, applies Bernstein basis functions of an arbitrary degree for the displacement approximation in the three coordinate directions. This type of basis function, which are a rarity in the structural analysis, deserve more attention. They provide computational efficiency and unique analytical elegance to the mathematical algorithms. The developed theory can be also used for generating new types of higher-order hexahedral finite element, as illustrated here on the example of a novel 8-node “Bernstein finite element”. Numerical examples given in this work show that, following the concept of material-adaptive inter-element continuity conditions, a higher smoothness and accuracy of the computed stresses can be achieved by incorporating additional inter-element constraints between the same material bricks. However, it is also easy to spoil the solution by adding the same type constraints between adjacent bricks having distinct material properties. Finally, numerical comparison for the benchmark 3-D problem of transverse bending of simply supported 3-layer cross-ply laminated plate validates that the present analysis is significantly more accurate and computationally efficient than ANSYS SOLID 46 element.