Euler-Lagrange Equations for Nonlinearly Elastic Rods with Self-Contact

We derive the Euler-Lagrange equations for nonlinear elastic rods with self-contact. The excluded volume constraint is formulated in terms of an upper bound on the global curvature of the centreline. This condition is shown to guarantee the global injectivity of the deformation mapping of the elastic rod. Topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed, e.g., in DNA-molecules, are included using the notion of isotopy and Gaussian linking number. The global curvature as a nonsmooth side condition requires the use of Clarke’s generalized gradients to obtain the explicit structure of the contact forces, which appear naturally as Lagrange multipliers in the Euler-Lagrange equations. Transversality conditions are discussed and higher regularity for the strains, moments, the centreline and the directors is shown. Mathematics Subject Classification: 49K15, 53A04, 57M25, 74B20, 74G40, 74G65, 74K10, 74M15 92C40 ∗[email protected] ∗∗[email protected]