A non-classical Kirchhoff rod model based on the modified couple stress theory

Abstract

A new non-classical Kirchhoff rod model is developed using the modified couple stress theory, which contains one material length scale parameter and can account for microstructure-dependent size effects. The governing equations and boundary conditions are determined simultaneously by a variational formulation based on the principle of minimum total potential energy. The newly developed model recovers its classical elasticity-based counterpart as a special case when the microstructure effect is not considered. To illustrate the new non-classical Kirchhoff rod model, two sample problems are analytically solved by directly applying the general formulas derived. One problem is the equilibrium analysis of a helical rod of circular cross section deformed from a straight rod, and the other is the buckling of a straight rod of circular cross section induced by an axial compressive force. In the former, the rod undergoes a twisting-dominated deformation, while in the latter the rod deformation is bending dominated. Two closed-form expressions are obtained for the force and couple needed in deforming the helical rod, and an analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, with the microstructure effect incorporated in each case. These formulas reduce to those based on classical elasticity when the microstructure effect is suppressed. For the helical rod problem, the numerical results show that the couple predicted by the current non-classical rod model is significantly larger than that predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase in the rod radius. For the buckling problem, it is found that the critical buckling load based on the new non-classical Kirchhoff rod model is always higher than that given by the classical elasticity-based model, with the difference being significant for a very thin rod.