Constitutive Equations of a Plastically Deformable Body with FEM-Based Implementation in the Calculation of a Shell Considering Shear Strain

In the calculation of a thin shell taking into account the transverse shear deformation based on the Timoshenko hypothesis, the results of the elastic–plastic stress state are compared using the constitutive relations in two variants with implementation of the step loading method. In the first variant, the constitutive equations are obtained by differentiating the relations of the plasticity deformation theory with the unchanged metric of the deformation process. The cumbersomeness of expressions is emphasized even with an unchanged metric of the deformation process. In the second variant, the constitutive equations at the loading step are obtained using the hypothesis of a proportional relationship between the components of deviators of the stress and strain increments without dividing the strain increments into elastic and plastic parts. A quadrangular fragment of the middle surface of the shell with kinematic nodal unknowns in the form of displacements and their first derivatives is adopted as a finite element. The efficiency of the developed constitutive equations for taking into account physical nonlinearity is shown using the example of calculation of the shell.