Thermal post-buckling of a heated elastic rod with pinned-fixed ends

On the basis of Kirchhoff's assumption and geometrically nonlinear theory for axially extensible rods, this article presents a mathematical model for the post-buckling of an elastic rod with pinned-fixed ends when it is applied by a static increasing temperature. In order to describe accurately the post-buckling of the heated rod with axial extension, we introduce an axial stretching, mu (x), and take the are length, s(x), of the deformed central axis as one of the basic unknown functions in the mathematical model. Using the shooting method in conjunction with the concept of analytical continuation, the nonlinear boundary value problem of ordinary differential equations of the model is solved numerically. We get some quantitative results of thermal post-buckled configurations and secondary equilibrium paths of the problem with some specific parameters, for example, buckling temperature and the slenderness ratio. The numerical results show that both the critical buckling temperature and the post-buckled temperature of the rod are sensitively influenced by the slenderness ratio, which is distinctively different from that of the post-buckling of the inextensible rod subjected to mechanical compressive loads. In addition, it is found that the magnitude of the axially constrained force reaches a maximum value at the onset of the buckling state and then decreases as the temperature increases in the post-buckling regime.