A theory for rubber-like rods

A theory for incompressible rubber-like straight rods undergoing finite strains and finite rotations is presented. Strains are expanded asymptotically for transverse coordinate of undeformed rod. The equations of equilibrium and corresponding boundary conditions are derived by implementing minimum total potential energy principle. Necessary conditions for the satisfaction of the stress-free boundary conditions on the top and bottom free surfaces of the rubber-like rods are derived. For the illustration and test of the proposed theory, the flexural buckling problem of Mooney-Rivlin rods under axial compressive loads is considered. Exact solutions corresponding to (i) various alternatives about the perturbation terms of the strain components, (ii) a very rigorous rod theory developed previously, and (iii) the three dimensional elasticity are obtained and compared. Degree of accuracy of the aforementioned approaches is discussed basing on the three dimensional elasticity solution. It is observed that considering all of the second order permutation terms yields very appealing results, almost coinciding with the results corresponding to the three dimensional elasticity for thin and quite thick rods.