Geometrically exact assumed stress–strain multilayered solid-shell elements based on the 3D analytical integration

This paper presents a family of geometrically exact assumed stress–strain four-node curved solid-shell elements with six displacement degrees of freedom per node by using the first-order equivalent single-layer theory. The proposed finite element formulation is based on the new strain–displacement relationships written in general reference surface coordinates, which are objective, i.e., invariant under rigid-body motions. This is possible because displacement vectors of the bottom and top surfaces of the shell are introduced and resolved in the reference surface frame. To overcome shear and membrane locking and have no spurious zero energy modes, the assumed strain and stress resultant fields are invoked. In order to circumvent thickness locking, three types of the modified material stiffness matrix extracted from the literature are employed and compared. All three elemental stiffness matrices have six zero eigenvalues and require only direct substitutions. Besides, they are evaluated by applying the 3D analytical integration that is very economical and allows using extremely coarse meshes.