Complementary energy principle associated with modified couple stress theory for Euler micro-beams considering support movements and negative Poisson's ratio

The complementary energy principle associated with the modified couple stress theory (MCST) is proposed and proved for analyzing the size effects of Euler micro-beams considering support movements and negative Poisson's ratio. The analytical solutions with the size effects are obtained and analyzed for the support reactions, internal forces, curvatures, rotations and deflections separately based on the principle. The results show that the support reactions and internal forces are greatly affected by the size effects when the support displacements are considered. It means that the size effects exist in piezoresistive pressure sensors that are sensitive to stresses. Moreover, the direction changes to the opposite for the total moment when the ratio of the material scale parameter l to the height of the micro-beam h (l/h) goes from 0 to 1. The deflections, curvatures and rotations have nothing to do with the size effects, respectively, if the support movements are only considered. Negative Poisson's ratio materials may be a good choice to verify the controversial issue of whether Poisson's ratio should be considered in the MCST beam model for the micro-scale experiments. The main reason is that the influence of negative Poisson's ratio on the size effects is larger than that of positive Poisson's ratio in the theoretical analysis. The analytical solutions agree with the published data for the deflections and rotations without the support movements.