Application of triangular element invariants for geometrically nonlinear analysis of functionally graded shells

This paper is an attempt to construct a computationally effective curved triangular finite element for geometrically nonlinear analysis of elastic shear deformable shells fabricated from functionally graded materials. The focus is on the concise finite-element formulation under the demand of accuracy-simplicity trade-off. To this end, a nonconventional approach based on the invariants of the natural strains of fibers parallel to the element edges is used. The approach allows one to obtain algorithmic formulas for computing the stiffness matrix, gradient, and Hessian of the total strain energy of the finite element. Transverse shear deformation effects are taken into account using the first order shear deformation theory with the shear correction factor dependent on the material property distribution across the shell thickness. The performance of the proposed finite element is demonstrated using problems of functionally graded plates and shells under mechanical and thermal loads.