Finite strain quadrilateral shell using least-squares fit of relative Lagrangian in-plane strains

• Simplified finite strain element formulation satisfying a priori the Patch test.
• Least-squares determination of in-plane assumed strains, compatible with the three-field variational principle.
• Absence of additional degrees-of-freedom (as is the case of EAS).
• Elastic and inelastic benchmark tests solved without hourglass or locking.

This work presents a finite strain quadrilateral element with least-squares assumed in-plane shear strains (in covariant/contravariant coordinates) and classical transverse shear assumed strains. It is an alternative to enhanced-assumed-strain (EAS) formulation and, in contrast to this, produces an element satisfying ab initio the Patch-test. No additional degrees-of-freedom are present, unlike EAS. Least-squares fit allows the derivation of invariant finite strain elements which are both in-plane and out-of-plane shear-locking free and amenable to standardization in commercial codes. With that goal, we use automatically generated code produced by AceGen and Mathematica to obtain novel finite element formulations. The corresponding exact linearization of the internal forces was, until recently, a insurmountable task. We use the tangent modulus in the least-squares fit to ensure that stress modes are obtained from a five-parameter strain fitting. This reproduces exactly the in-plane bending modes. The discrete equations are obtained by establishing a four-field variational principle (a direct extension of the Hu–Washizu variational principle). The main achieved goal is coarse-mesh accuracy for distorted meshes, which is adequate for being used in crack propagation problems. In addition, as an alternative to spherical interpolation, a consistent director normalization is performed. Metric components are fully deduced and exact linearization of the shell element is performed. Full linear and nonlinear assessment of the element is performed, showing similar performance to more costly approaches, often on-par with the best available shell elements.