An intrinsic formulation for nonlinear elastic rods

Using the theory of a Cosserat rod with two directors, special constrained Bernoulli- Euler type theory is developed for large spatial deformations which omits the effects of normal cross-sectional extension, tangential shear deformation, and the normal cross-sectional shear deformation but allows the nonlinear elastic strain energy to be a general function of the extension, curvature and twist of the rod. The resulting equilibrium equations are written in an intrinsic form in terms of the extension, curvature, twist, and the geometric torsion of the rodʹs reference curve. It is known that exact integrals of the equilibrium equations exist for the simple case when the rod is loaded only by forces and moments applied to its ends. Here, similar exact integrals yield implicit algebraic functions of curvature for the extension, twist and geometric torsion. The remaining equilibrium equation becomes a second order equation for curvature alone which can be analyzed completely. An example is presented which shows the influence of extensional deformation on axial force and bending moment. © 1997 Elsevier Science Ltd