In this work, first, a computational approach suitable for combined material and geometrically nonlinear analysis for 2D quadrilateral elements is explained. Its main advantage is reuse: once a finite element has been developed with good performance in linear analysis, extension to material and geometrically nonlinear problems is simplified. Extension to geometrically nonlinear problems is enabled by a corotational kinematic description, and that to material nonlinear problems by an optimization-based solution algorithm. The approach thus comprises three ingredients—the development of high performance linear finite element using the Assumed natural deviatoric strain (ANDES) concept, a corotational kinematic description for quadrilateral element, and an optimization algorithm. The work illustrates the realization of the three ingredients on plane stress problems that exhibit elastoplastic material behavior. Numerical examples are presented to illustrate the effectiveness of the approach.

Second, an eight-node solid-shell element based on ANS, ANDES and EAS concepts is presented. The mechanical response of the element is split into three parts: (1) In-plane response, which is also decomposed into membrane and bending, (2) Thickness response or normal strains in thickness direction; and (3) Transverse shear response. This separation gives the liberty of using any type of membrane quadrilateral formulation for the in-plane response. In the present work, ANDES membrane element is used for the in-plane response. ANS concept is implemented to account for the transverse shear and thickness strains, which has proven to circumvent the curvature thickness and transverse shear locking problems. EAS approach with one degree-of-freedom is applied on the thickness strain so as to alleviate the Poisson thickness locking. The formulation yields exact solution for both membrane and bending patch tests.

Third, an eight-node solid-shell element based on ANS and EAS concepts is presented. Five enhanced degrees-of-freedom are used to improve the in-plane response of the element and one to alleviate the Poisson's thickness locking problem.

Numerical results for some benchmarks show the robustness of both solid-shell formulations in geometrically linear problems.

With the proposed linear element at hand, the corotational kinematic description is used to add geometric nonlinearity to this work. Problems with small strains are addressed in this work, however, EICR could be extended to large deformations. The Corotated frame is defined such that it is independent of whether the mid-surface is warped or not. Numerical results for geometric nonlinear solid-shell and the comparisons with other solid-shell and shell formulations are presented in the end.