Computational Shell Mechanics by Helicoidal Modeling, II: Shell Element

The virtual work of stresses developed in Part I for the helicoidal shell model and then reduced to the material surface is taken as one term of a variational principle stated on a two-dimensional domain. The other terms related to the external loads and to the boundary constraints are added here and include a weak-form treatment of the constraints, which becomes necessary in the context of helicoidal modeling. All terms are cast in incremental form and yield a linearized variational principle of the virtual work type for two-dimensional continua, endowed with an internal constraint conjugate to an extra stress field that is able to control the drilling degree of freedom. The virtual functional and the virtual tangent functional are approximated by the finite element method, using helicoidal interpolation for the kinematic field (which ensures objectivity and path independence) and a uniform representation for the extra stress field. A low-order four-node shell element is obtained, with 6 degrees of freedom per node and a unique stress-vector discrete unknown per element. Several test cases demonstrate the performance of the element and its outstanding locking-free behavior.