A unified Kirchhoff-extended higher-order shear deformation theory in surface-tensor form for functionally graded shells

This paper develops a unified Kirchhoff extended higher-order shear deformation theory (Ke-HSDT) for static bending and free vibration analyses of functionally graded material (FGM) shells. By generalizing the classical Kirchhoff hypothesis with an arbitrary higher-order thickness function, a robust five-variable framework is established. This model successfully recovers the classical Kirchhoff-Love theory in the limit of a vanishing higher-order function and reduces to standard plate HSDT when the curvature tensor is null. Developed within a consistent surface tensor framework, the kinematic, strain and stress relations are rigorously expressed via covariant differentiation, ensuring geometric exactness over arbitrarily curved configurations. Consequently, the same governing equations universally describe flat plates as well as cylindrical, spherical, conical, toroidal and hyperbolic paraboloidal shells without geometry-specific modifications, requiring only the corresponding mid-surface mapping. Due to second-order derivatives in the strain field, the formulation requires C 1 continuity, which is naturally satisfied by NURBS-based isogeometric analysis (IGA) without auxiliary degrees of freedom. The IGA framework enables exact representation of the curved mid-surface, completely eliminating geometric approximation errors typical of conventional finite element methods. Numerical benchmarks confirm that the proposed five-variable tensor-based formulation provides a highly accurate and computationally efficient platform for analyzing thin to moderately thick shells.