Elastic rods are thin flexible objects typically undergoing large non-linear deformations that cannot be modeled with linear methods. They are used in a number of research fields, e.g, to represent hair or ropes in animations, or catheters or needles in medical simulations. In this paper, we propose a deformation model for inextensible elastic rods. The method of Lagrange multipliers is employed to enforce the inextensibility of the rod, and to couple the material frames with the centerline. The resulting system is banded, allowing for an efficient linear time solution. We also propose a manifold projection method to incorporate the non-penetration constraints resulting from contact handling into our constrained Lagrangian mechanics problem. We further augment the contact model by treating torsional friction. This allows to reproduce friction effects such as dynamic rolling and twisting of rods. Various examples underline the benefits and applicability of our model.