Finite-strain low order shell using least-squares strains and two-parameter thickness extensibility

We present a thickness-extensible finite strain quadrilateral element based on least-squares in-plane shear strains and assumed transverse-shear strains. At each node, two thickness parameters are connected to the constitutive laws by a linear system. The zero out-of-plane normal stress condition is satisfied at the constitutive level using the normal strain as unknown in all integration points. Assumed in-plane strains based on least-squares are introduced as an alternative to the enhanced-assumed-strain (EAS) formulations and, contrasting with these, the result is an element satisfying ab-initio both the in-plane and the transverse Patch tests. There are no additional degrees-of-freedom, as it is the case with EAS, even by means of static condensation. Least-squares fit allows the derivation of invariant finite strain elements which are shear-locking free and amenable to be incorporated in commercial codes. With that goal, we use automatically generated code produced by AceGen and Mathematica. Full assessment of the element formulation and the two-parameter thickness variation methodology is accomplished. Alternative thickness variation algorithms are tested. All benchmarks show very competitive results, similar to the best available enriched shell elements.