A shell finite element for large strain elastoplasticity with anisotropies—: Part I: Shell theory and variational principle

Abstract

A shell model for finite elastic and finite plastic strains is derived taking into account initial and induced anisotropies. A corresponding eight-node C 0 shell element with three displacement and three director degrees-of-freedom at each node is developed, which combines the advantages of an isoparametric description of geometry and deformation with an effective plane stress formulation. The element accounts for isochoric or approximately isochoric deformation due to finite plastic strains. Because of the three displacement and three director degrees-of-freedom at each node, it is easily possible to link different parts of a composed irregular shell structure or to connect the derived shell element with solid (brick) elements. This paper presents the shell theory based on the kinematics of finite elastoplasticity proposed in Schieck and Stumpf (1995) and the special geometric concept of the derived shell model. The Lagrange multiplier method is applied to introduce into the virtual work principle the transverse normality constraint and the condition of isochoric deformation, where the Lagrange multipliers can be condensed inside the element procedure. Various assumed strain techniques designed to avoid the membrane locking are compared with known methods in the literature. According to the numerical experience so far the proposed shell finite element is free of locking effects and spurious modes. Part II presents the constitutive equations for finite elastic–plastic strains accounting for initial and induced anisotropies and the implementation of the model into the FE-code. A comprehensive set of numerical examples is provided, involving the tension of a plane specimen with necking and shear-band localization, the elastic–plastic response of a simply supported plate with a localization of the plastic bending strains in the four corner zones, the elastic–plastic deformation mode of the so-called Scordelis-Lo roof, and the elastic–plastic buckling of a cylindrical shell showing an essential influence of the anisotropic material behavior. The results illustrate the performance of the proposed shell finite element for a wide range of engineering applications.