A 6-node co-rotational triangular elasto-plastic shell element

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A 6-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element is defined by the vectors directing from one vertex to the other two vertices and their cross product. Based on such a co-rotational framework, the element rigid-body rotations are excluded in calculating the local nodal variables from the global nodal variables. The two smallest components of each nodal orientation vector are defined as rotational variables, resulting in the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite element formulations, both the element tangent stiffness matrices in the local and in the global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of the strain energy with respect to the local nodal variables and, through chain differentiation, with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell---Huber---Hencky---von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is employed. To overcome locking problems, the assumed linear membrane strains and shear strains are obtained by using the line integration method proposed by MacNeal, and the assumed higher-order membrane strains are obtained by enforcing the stationarity of the mixed displacement-strain canonical functional, these assumed strains are then employed to replace the corresponding conforming strains. The reliability and convergence of the present 6-node triangular shell element formulation are verified through two elastic plate patch tests as well as two elastic and five elasto-plastic plate/shell problems undergoing large displacements and large rotations.