Mechanical behavior analysis of size-dependent micro-scaled functionally graded Timoshenko beams by strain gradient elasticity theory

In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGMs). The developed formulation is based on the strain gradient theory; a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Five new equivalent length scale parameters are introduced as functions of the constituents’ length scale parameters. It is shown that the size-dependent static and dynamic behavior of FG micro-beams can be described using these equivalent length scales. The governing differential equations of motion and both classical and non-classical sets of boundary conditions are derived for the proposed strain gradient FG Timoshenko beam using a variational approach. As case studies, the static bending deformation and free vibration of the new model are investigated for an FG simply supported beam made of tungsten/copper (W/Cu). The results of the new model are compared to those of the modified couple stress and the classical theories where the two latter theories are special cases of the strain gradient theory.