Static and dynamic modeling of functionally graded Euler–Bernoulli microbeams based on reformulated strain gradient elasticity theory using isogeometric analysis

Abstract

Strain Gradient Theories (SGTs) are often considered to capture the intrinsic microstructural behaviors of the microbeams. In the present work, the recent Reformulated Strain Gradient Theory (RSGT), which accounts the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively, is extended for the static and dynamic modeling of Functionally Graded (FG) Euler–Bernoulli microbeams to account for the effect of the strain gradient, couple stress (rotation gradients) and velocity gradients collectively. The present formulation also considers the shift of neutral axis due to material inhomogeneity in the FG microbeam. The problems of static deflection and free vibrations with different boundary conditions are addressed. Numerical solutions are obtained using isogeometric analysis. Analytical solutions of the present formulation for simply supported FG microbeam for static deflection and free vibration problems are also obtained and employed to validate the numerical results. The influence of the material index, different length scale parameters on the static deflection and free vibration responses of the FG microbeams are discussed. The present FG RSGT microbeam shows the prominent size scale effect in thin beams and predicts stiffer response than the FG microbeam based on modified couple stress theory.