Large deformation of hyperelastic modified Timoshenko–Ehrenfest beams under different types of loads

Abstract

In this paper, a nonlinear finite element formulation for a hyperelastic, modified Timoshenko–Ehrenfest beam with geometrical and material nonlinearities is developed for the first time. A new five-parameter beam element is introduced. The parameters contain displacement, values of difference vector and a through-the-thickness scalar value. Moreover, a new procedure is employed to apply the moment to the beam. The constitutive formulation is derived for Saint Venant–Kirchhoff (SVK) as well as the compressible neo-Hookean (n-H) hyperelastic model. The beam is subjected to different loads such as dead load, point and distributed loads, and follower pressure. To demonstrate the applicability of the formulations, several examples are solved. The results reveal that this formulation can capture the previous results reported in the literature. Furthermore, a comparative study is done between two hyperelastic models. It is demonstrated that in the case of large rotation of the beam, both Saint Venant–Kirchhoff and neo-Hookean models show the same behavior. However, in case of large deformation with large strains of the beam, the Saint Venant–Kirchhoff model behaves stiffer than the neo-Hookean hyperelastic model.