Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner-Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.