Analytical solutions for the thermal vibration of strain gradient beams with elastic boundary conditions

A strain gradient Euler beam described by a sixth-order differential equation is used to investigate the thermal vibrations of beams made of strain gradient elastic materials. The sixth-order differential equation of motion and elastic boundary conditions are determined simultaneously by a variation formulation based on Hamilton's principle. Analytical solutions for the free vibration of the elastic constraint strain gradient beams subjected to axial thermal stress are obtained. The effects of the thermal stress, nonlocal effect parameter, and boundary spring stiffness on the vibration behaviors of the strain gradient beams are investigated. The results show that the natural frequencies obtained by the strain gradient Euler beam model with the thermal stress decrease while the temperature is rising. The thermal effects are sensitive to the boundary spring stiffness at a certain stiffness range. In addition, numerical results also show the importance of the nonlocal effect parameter on the vibration of the strain gradient beams.