A finite strain quadrilateral based on least-squares assumed strains. (1st October 2015) - Areias, P.; Rabczuk, T.; César de Sá, J.
Highlights: 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. Abstract: 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.