Mathematical analysis

Tangent

Materials science

Linearization

Virtual work

Composite material

Geometry

Lagrange multiplier

Physics

Finite element method

Nonlinear system

Curvilinear coordinates

Shell (structure)

Mathematics

Mathematical optimization

Quantum mechanics

Applied mathematics

Thermodynamics

Mechanics of Advanced Materials and Structures

In this work, a unified formulation of full geometrically nonlinear refined shell theory is developed for the accurate analysis of highly flexible shell structures. The tensor calculus is utilized to explicitly derive the linear and nonlinear differential operator matrices of the geometrical relation in the orthogonal parallel curvilinear coordinate system. By employing the Carrera Unified Formulation (CUF), various kinematics of two-dimensional shell structures are consistently formulated via an appropriate index notation and a generalized expansion of the primary variables by arbitrary functions in the thickness direction, leading to lower- to higher-order shell models with only pure displacement variables. Furthermore, the principle of virtual work and a finite element approximation are exploited to straightforwardly formulate the nonlinear governing equations in a total Lagrangian approach. Particularly, the path-following Newton–Raphson linearization method based on the arc-length constraint is used to deal with the full geometrically nonlinear problem. Independent of the theory approximation order, the forms of the fundamental nuclei of the secant and tangent stiffness matrices of the unified shell element are formulated via the CUF and the three-dimensional Green-Lagrange strain components. Numerical assessments and comparisons of the present results with those provided in the literature for popular benchmark problems involving different metallic shell structures are found to be excellent and demonstrate the capabilities of the developed CUF shell model to predict the post-buckling, large-deflection, snap-through and snap-back nonlinear responses with high accuracy.

Geometrically nonlinear refined shell theories by Carrera Unified Formulation