Modeling and steady state analysis of the extensible thermoelastic beam

In this work we formulate a nonlinear mathematical model for the thermoelastic beam assuming the Fourier heat conduction law. Boundary conditions for the temperature are imposed on the ending cross sections of the beam. A careful analysis of the resulting steady states is addressed and the dependence of the Euler buckling load on the beam mean temperature, besides the applied axial load, is also discussed. Finally, under some simplifying assumptions, we deduce the model for the bending of an extensible thermoelastic beam with fixed ends. The behavior of the resulting dissipative system accounts for both the elongation of the beam and the Fourier heat conduction. The nonlinear term enters the motion equation, only, while the dissipation is entirely contributed by the heat equation, ruling the thermal evolution.