A nonclassical formulation for torsion of variable cross section functionally graded microbars

This paper investigates the size-dependent static and dynamic behavior of functionally graded variable cross section microbars. To this goal, considering the formulation of the modified couple stress theory and utilizing Hamilton's principle, the equation of motion and boundary conditions of the microbar are derived. To simplify the equation of motion, equivalent quantities including equivalent length scale parameters are proposed for the functionally graded tapered microbar. Using a semi-analytical approach, the equation of motion is solved and the static and dynamic responses of the system are calculated in general form. For a tapered functionally graded microbar made of aluminum and alumina, static torsion and natural frequencies are obtained and the effects of size dependency and volume fractions of material constituents on the static and dynamic behavior of the system are studied. As a special case of this study, the static torsion and natural frequencies of the uniform cross section microbars are calculated and compared to those from the literature. Furthermore, the convergence of the proposed semi-analytical method is investigated and the accuracy of this approach is confirmed.