Exact solutions for the large axially symmetric deformations of a neo-Hookean rod subjected to static loads

In this paper, we study the problem of the large axially symmetric deformations of a neo-Hookean rod with one end fixed and the other end subjected to a static load. By using a reasonable assumption, we derive a one-dimensional rod equation as the model equation by taking integrations to the field equations. Then, we manage to solve exactly this nonlinear equation together with the boundary conditions. Solutions expressed in terms of the elliptic integrals of the first and third kinds are obtained. Graphic results are presented and some interesting phenomena are observed. For example, it is found that in the middle portion of the rod the azimuthal stretch is almost constant when the radius-length ratio is relatively small. An asymptotic analysis is conducted to deduce simple asymptotic formulae. Then, we are able to offer satisfactory analytical explanations for those interesting physical phenomena. Comparisons between the analytical results given in this paper and existing results are presented at the end.