Non-linear geometrically exact assumed stress–strain four-node solid-shell element with high coarse-mesh accuracy

This paper presents a family of geometrically exact assumed stress–strain four-node solid-shell elements with six displacement degrees of freedom per node based on the finite rotation first-order multilayered shell theory. The proposed formulation is based on the new objective non-linear strain–displacement relationships, which are invariant under arbitrarily large rigid-body motions. To improve a non-linear shell response, the modified assumed natural strain method is applied. This enhanced non-linear solid-shell element formulation allows using coarser meshes. To avoid 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, the ad hoc modified laminate constitutive stiffness matrix corresponding to the generalized plane stress condition is employed. The fundamental unknowns consist of six displacements and nine strains of the face and middle surfaces of the shell, and nine conjugate stress resultants. For the analytical description of surface geometry, an effective numerical algorithm of smoothing the data by cubic spline functions developed by the first author as early as 1981 is used. To demonstrate the efficiency and accuracy of the developed non-linear geometrically exact solid-shell element and to compare its performance with isoparametric elements, extensive numerical studies are presented.