Consistent tangent operator for an exact Kirchhoff rod model

In the paper, it is considered an exact spatial Kirchhoff rod structural model. The configuration space for this model that has dimension 4 is obtained considering an ad hoc split of the rotation operator that implicitly enforces the constraints on the directors. The tangent stiffness operator, essential for the nonlinear numerical simulations, has been studied. It has been obtained as second covariant gradient of the internal energy functional for the considered structural model that preserves symmetry for any configuration, either equilibrated or not. The result has been reached evaluating the Levi-Civita connection for the tangent space of the configuration manifold. The results obtained extend to the case of Kirchoff -Love rods those presented by Simo (Comput Methods Appl Mech Eng 49:55-70, 1985) for Timoshenko rods. Given the different structure of the tangent spaces in this case, it has been necessary to introduce a specific metric that accounts for the rotation of the intrinsic triad due to the change of the position of the centroid axis of the rod.