Finite element formulation of metal foam microbeams via modified strain gradient theory

Size-dependent behaviours of metal foam microbeams with three different porosity distribution models are studied in this paper. Based on the finite element model, a normal and shear deformation theory has been employed for the first time to investigate their structural behaviours by using modified strain gradient theory and considering the effects of variable material length scale parameter. The equations of motion and boundary conditions of system are derived from Hamilton's principle. Finite element models are presented for the computation of deflections, vibration frequencies and buckling loads of the metal foam microbeams. The verification of proposed models is carried out with a comparison of the numerical results available in the literature. Calculations using the different parameters reveal the effects of the porosity parameters (distribution and coefficient), small size, boundary conditions and Poisson's ratio on the displacements, frequencies and buckling loads of metal foam microbeams. Some benchmark results of these structures for both models (modified couple stress theory and modified strain gradient theory with constant and variable material length scale parameter) and with/without Poison's effect are provided for future study.