Size-dependent functionally graded beam model based on an improved third-order shear deformation theory

• This model contains both microscale and higher-order shear deformation effects.
• Interaction between the elastic foundations and the FG microbeam is considered.
• Length scale parameters are viewed as the function of material mixture ratio.
• Analytical solutions are provided for simply supported boundary conditions.

In this study, a size-dependent beam model made of functionally graded materials (FGMs) is developed. This model contains both microscale and shear deformation effects. The microscale effect is captured using the strain gradient elasticity theory, while the shear deformation effect is included using an improved third-order shear deformation theory which is based on a more rigorous kinematics of displacements. In addition, interaction between the Winkler–Pasternak elastic foundation and the FG microbeam is considered. The material properties of the FG microbeams are assumed to vary in the thickness direction and estimated through the Mori-Tanaka homogenization technique. Material length scale parameters are viewed as the function of material mixture ratio rather than a constant. The equations of motion and boundary conditions are derived from Hamilton's principle. Analytical solutions are obtained using the Navier method for bending, free vibration, and buckling problems of FG microbeams with simply supported boundary conditions. The effects of material length scale parameter, aspect ratio, various material compositions, elastic foundation parameters and shear deformation on mechanical responses of the FG microbeam are investigated in detail. Some of the present results are validated by comparing the present results to those available in literature. The results indicate that the microscale effect, elastic foundation and material compositions greatly affect the mechanical behavior of FG microbeams. The new results can be used as benchmark solutions for future researches.

Size-dependent FG microbeam resting on a Winkler-Pasternak elastic foundation.Figure optionsDownload as PowerPoint slide