ELASTIC-PLASTIC DEFORMATION OF THIN-WALLED STRUCTURES IN TWO-DIMENSIONAL STATEMENT BASED ON MIXED FEM

An algorithm for obtaining the stiffness matrix of a mixed finite element in the form of a quadrangular fragment of the middle surface of a thin-walled structure is presented. The following are accepted as nodal unknowns: kinematic quantities (displacement increments and their first derivatives) and deformation quantities (strain increments and curvature increments of the middle surface). To approximate the kinematic quantities of an internal point of a finite element, bicubic shape functions with Hermite polynomials of the third degree were accepted. Strain increments and curvature increments were approximated through nodal unknowns by bilinear shape functions. Physical nonlinearity is implemented based on the theory of plastic flow. The components of the elastic strain increment tensor were determined through the components of the stress increment tensor in a curvilinear coordinate system based on the relations of continuum mechanics, taking into account the Kirchhoff – Love hypothesis. The components of the plastic strain increment tensor are obtained in a curvilinear coordinate system based on the hypothesis of the plastic flow theory about their proportionality to the components of the stress deviator with a straight normal to the middle surface. To obtain the stiffness matrix of the developed finite element, a nonlinear functional was used based on the equality of possible and actual works of specified loads and arising internal forces at the loading step, with the replacement of the actual work of internal forces by the difference between the total and additional work of internal forces at the loading step. After substituting the approximating expressions into the nonlinear mixed functional and performing minimization over the nodal unknowns, two systems of algebraic equations are obtained with respect to the sought nodal unknowns. Based on the obtained systems of algebraic equations, a 36x36 finite element stiffness matrix is obtained. The structure stiffness matrix is formed in the usual way for FEM. The efficiency of the developed finite element is shown in a numerical example.