An asymptotic analysis of large deflections and rotations of elastic rods

Abstract The distortions within beams and rods undergoing large displacements and rotations are here derived from three-dimensional elasticity by an asymptotic procedure. This procedure, based on the premise that strains vary more gradually along the rod than in transverse directions, takes full account of the shape of the cross section, the traction conditions on the lateral boundary and of any material anisotropy. The manner in which it generates a hierarchy of sets of rod equations is outlined. In the fundamental set, the distortions over each cross section are the anti-clastic curvature and warping associated with the St.-Venant semi-inverse solutions for bending and torsion, suitably corrected to allow for large rotations. The corresponding equations governing the rod configuration are Kirchhoff's equations, with bending and torsional rigidities computed from the St.-Venant distortions. The procedure gives some three-dimensional substance to elastica theory, relating the constitutive assumptions to three-dimensional elasticity. It gives also a logical procedure for obtaining higher order corrections to the theory, and shows how St-Venant's hypothesis concerning details of end loading arises naturally.