Investigating the F-bar method as a remedy for volumetric locking in Finite Element Analysis with the total Lagrange formulation

In finite element analysis (FEA) of deformation problems, volumetric locking is a common issue in nearly incompressible materials. Standard low-order elements (such as linear quadrilaterals or hexahedra) can become overly stiff under volumetric constraints, leading to inaccurate deformation predictions, checkerboard patterns in stress distributions, or, in some cases, divergence. Several methods are commonly used to address this issue, including selective reduced integration (e.g., the B-bar method and the F-bar method), mixed formulations, enhanced assumed strain (EAS) methods, higher-order elements, and polygonal/polyhedral elements. The F-bar method is specifically designed for large deformation problems and typically employs the incremental formulations of FEM for finite strain. This study derives an F-bar method for the total Lagrangian formulation. The derived linearized discretized weak form of the momentum balance equation resembles that of the B-bar method, adopting a concise and compact form. The proposed algorithms are verified using several classic large deformation examples, which exhibit volumetric locking in solutions obtained with standard FEA.