Large deformation analysis of plane‑stress hyperelastic problems via triangular membrane finite elements

A finite-element formulation based on triangular membranes of any order is proposed to analyze problems involving highly deformable hyperelastic materials under plane-stress conditions. The element kinematics is based on positional description and the degrees of freedom are the current plane coordinates of the nodes. Two isotropic and nonlinear hyperelastic models have been selected: the compressible neo-Hookean model and the incompressible Rivlin–Saunders model. The constitutive relations and the consistent tangent operator are condensed to the compact 2D forms imposing plane-stress conditions. The resultant algorithm is implemented in a computer code. Three benchmark problems are numerically solved to assess the formulation proposed: the Cook’s membrane, involving bending, shear, and a singularity point; a partially loaded membrane, which presents severe mesh distortion and large compression levels; and a rubber sealing, which is a more realistic problem. Convergence analysis in terms of displacements, applied forces, and stresses is performed for each problem. It is demonstrated that mesh refinement avoids locking problems associated with incompressibility condition, bending-dominated problems, stress concentration, and mesh distortion. The processing times are relatively small even for fifth-order elements.