Behavior of Neo-Hookean Material Models in the Analysis of Non-Shear Deformable Beams

Nonlinear neo-Hookean material (NHM) models are written in terms of the shear modulus, which is the second Lame’s constant. Nonetheless, the effect of the shear modulus when using NHM models can remain dominant even in longitudinal-vibration problems. The NHM strain-energy term that depends on the shear modulus does not approach zero as the shear strain approaches zero. This paper demonstrates the effect of this strain-energy term in longitudinal-vibration problems, characterized by zero shear and normal strains at the integration points. To this end, fully parameterized finite-element (FE) beam models with complete sets of gradient vectors are used to allow implementing general NHM models. In developing such models, distinction is made between constraint-free end conditions (CFEC) and stress-free end conditions (SFEC). The latter requires enforcing natural boundary conditions, which are not required for the CFEC models. The results show that the NHM model leads to stiff behavior and oscillation frequencies that are different from the analytical solution. It is shown that the strain-energy term that depends on the shear modulus can absorb approximately 50% of the total strain energy in longitudinal-vibration problems. The longitudinal and transvers element spatial coordinates ensure zero transverse-normal and shear strains at all integration points in the axial-vibration problem considered.