Analytical and Numerical Evaluation of Hookean and Neo-Hookean Material Formulations in One- and Two-Dimensional Problems

Abstract A one-dimensional linearized Neo-Hookean material (LNHM) model that allows developing closed-form analytical solutions is derived and used to shed light on the accuracy of the nonlinear Neo-Hookean material (NHM) models in case of zero shear strains. The accuracy of the LNHM model is verified in case of small strains by comparing its results with the results of geometrically nonlinear finite-element (FE) NHM models. The analytical and numerical results obtained demonstrate that the LNHM and nonlinear NHM models do not reduce to the one-dimensional tension-test constitutive model, and both models lead to overly stiff behavior regardless of whether or not the transverse deformation is constrained. The conditions under which the LNHM model reduces to the tension-test model are presented. By using kinematics- and material-consistent linear Hookean material (LHM) models, excellent agreement with the analytical longitudinal-vibration (ALV) solution is obtained using one finite element. Such an agreement cannot be achieved using NHM models for the simple motion scenario considered even when transverse-deformation constraints are relaxed to emulate tension-test experiment.