Physically nonlinear deformation of the shell using a three-field fem

Mikhail Yu. Klochkov ; Anatoly P. Nikolaev ; Valeria A. Pshenichkina ; Olga V. Vakhnina ; Aleksandr S. Andreev ; Yuri V. Klochkov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 18 (2025) no. 2, pp. 209-217

Résumé

A method has been developed for implementing an algorithm for determining the stress-strain state (SSS) of a thin shell based on the finite element method (FEM) in a three-field formulation under step loading. A quadrangular fragment of the median surface of the thin shell is accepted as the finite element. Nodal unknowns at the loading step used: increments of kinematic quantities (increments of displacements and their derivatives); increments of deformation quantities (increments of deformations and curvatures of the median surface); increments of force values (increments of forces and moments). The approximation of kinematic quantities was carried out using bicubic shape functions based on Hermite polynomials of the third degree, and force and deformation quantities using bilinear functions. To account for the physical nonlinearity of the shell material, the defining equations are used in two versions: the first is the defining equations of the theory of plastic flow and the second is the defining equations based on the proposed hypothesis of proportionality a component of deviators of strain increments and stress increments. The stiffness matrix of the finite element is formed on the basis of a nonlinear Lagrange functional for the loading step, expressing the equality of possible and actual work of given loads and internal forces, with the complementary condition that the actual work of the increments of internal forces is equal to zero on the difference in increments of deformation quantities determined by geometric relations and using approximating expressions. An example of calculation is given using the resulting finite element stiffness matrix.

Détail

Keywords: finite element in the three-field formulation, physical nonlinearity of the material, variants of the governing equations, nonlinear Lagrange functional with condition.