A finite strain quadrilateral based on least-squares assumed strains

• Least-squares approach to assumed strains.
• Mixed quadrilateral with straightforward application to finite strain plasticity and localization problems.
• Competitive coarse-mesh accuracy for linear bending problems.
• Absence of locking and hourglassing with nearly incompressible problems.

When compared with advanced triangle formulations (e.g. Allman triangle and Arnold MINI), specially formulated low order quadrilateral elements still present performance advantages for bending-dominated and quasi-incompressible problems. However, simultaneous mesh distortion insensitivity and satisfaction of the Patch test is difficult. In addition, many enhanced-assumed (EAS) formulations show hourglass patterns in finite strains for large values of compression or tension; EAS elements often present convergence difficulties in Newton iteration, particularly in the presence of high bulk modulus or nearly-incompressible plasticity. Alternatively, we discuss the adequacy of a new assumed-strain   4-node quadrilateral for problems where high strain gradients are present. Specifically, we use relative strain projections to obtain three versions of a selectively-reduced integrated formulation complying a priori with the patch test. Assumed bending behavior is directly introduced in the higher-order strain term. Elements make use of least-square fitting and are generalization of classical B‾ and F‾ techniques. We avoid ANS (assumed natural strains) by defining the higher-order strain in contravariant/contravariant coordinates with a fixed frame. The kinematical part of the constitutive updating is based on quadratic incremental Green–Lagrange strains. Linear tests and both hyperelastic and elasto-plastic constitutive laws are used to test the element in realistic cases.