Size-dependent nonlinear bending of microbeams based on a third-order shear deformation theory

Abstract

In this paper, the size-dependent nonlinear bending of microbeams subjected to mechanical loading is studied using a finite element formulation. Based on the von Kármán nonlinear relationship and the third-order shear deformation theory, a size-dependent nonlinear beam element is derived by using the modified couple stress theory (MCST) to capture the microstructural size effect. The element with explicit expressions for the element vector of internal forces and tangent stiffness matrix is derived by employing the transverse shear rotation as a variable. Nonlinear bending of microbeams under different mechanical loading is predicted with the aid of Newton–Raphson iterative method. Numerical investigation shows that the derived element is efficient, and it is capable of giving accurate results by several elements. The obtained results reveal the importance of the micro-size effect on the nonlinear behavior of the microbeams, and the deflections are overestimated when the microstructural effect is ignored. The effects of the material length scale parameter, boundary conditions and loading type on the bending response of the microbeams are studied and highlighted.