Comparison of single-field and three-field fem in nonlinear shell calculations

On the basis of physical equations of deformation theory of plasticity using Kirchhoff–Lava hypothesis, matrix dependences between columns of forces and moments and columns of deformations and curvatures of the shell midface are determined at the loading step. As a finite element, a quadrilateral fragment of the shell midface with nodal unknowns in the form of: increments of displacements and their derivatives; increments of deformations and increments of curvatures; increments of forces and increments of moments were used. To approximate the required quantities, the following expressions are adopted: bicubic functions with elements of Hermite polynomials of the third degree for displacements; bilinear functions for deformation and force parameters. To obtain the stiffness matrix of the finite element, the nonlinear Lagrangian functional on the loading step was used with an additional condition: the real work of the difference of forces determined using their direct approximation and using approximating expressions for displacements, on deformations and curvatures of the loading step must be equal to zero. Minimisation of the functional by nodal unknowns provides three systems of equations, the solution of which determines the stiffness matrix of the finite element used to calculate the displacement fields. The force and deformation parameters at the discretisation nodes of the shell are determined from the displacements found. Case studies show the effectiveness of using a three-field finite element method (FEM) technique compared to using FEM in the displacement method formulation (single-field technique).