Nonlinear Modelling of Axially Deformable Elastica based on Hyperelasticity

Abstract

Considering the cross-sectional dimension of the elastica much smaller than the wavelength of its global deformation, the three-dimensional hyperelasticity is rigorously split to a one-dimensional macroscopic global analysis and a two-dimensional cross-sectional analysis. These two analyses are featuring both geometric and material nonlinearities. The two-dimensional nonlinear cross-sectional analysis utilizes the three-dimensional hyperelastic material models to homogenize the nonlinear beam cross-sectional constitutive relations, which are input as material properties for the global one-dimensional geometric exact beam analysis. Restricted to small strains, analytical solutions can be obtained and proves that the sectional analysis of the Euler-Bernoulli type results in an effect that the bending stiffness will vary with the axial deformation. Without ad hoc assumptions, exact solution of such extension effect for the strain energy up to the third order is given analytically for isotropic beams with neo-Hookean model. In addition, the present theory without the small strain restriction is implemented using the finite element method in VABS, a general-purpose cross-sectional analysis tool, of which the solutions are compared with the analytical results. The geometrical exact beam theory is extended to handle one-dimensional nonlinear material model so that both geometry and material nonlinearities can be handled. Finally, the present theory is validated using the three-dimensional analysis in commercial finite element software.