A simplified strain gradient Kirchhoff rod model and its applications on microsprings and microcolumns

The equilibrium equations and boundary conditions of elastic Kirchhoff rods are presented within the theoretical framework of the simplified strain gradient theory. The newly developed Cosserat rod model contains only one intrinsic material length squared parameter to account for the effects of microstructures. Applications of the theory are also presented in this paper. Examples include the equilibrium analysis of a microspring and the buckling behavior of a microcolumn. The first application focuses on estimating the restoring force of a microspring that is deformed from an originally straight rod with uniform cross- sectional area. Semianalytical results show that the restoring force of the microspring predicted by the new strain gradient rod model is always larger than that of its classical counterpart. The restoring force is found to increase with both the intrinsic material length squared parameter and the rod radius. For the stability analysis of a microcolumn, an analytical expression is derived for the critical buckling load. It is found that the critical force predicted by the developed nonclassical Kirchhoff rod model depends linearly on the intrinsic material length squared parameter, quantitatively indicating the significance of strain gradient effects.