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Euler-Lagrange equations for nonlinearly elastic rods with self-contact. (English) Zbl 1030.74029
Summary: We derive the Euler-Lagrange equations for nonlinearly elastic rods with self-contact. The excluded-volume constraint is formulated in terms of an upper bound on the global curvature of the centre line. This condition is shown to guarantee the global injectivity of the deformation 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 by using the notion of isotopy and Gaussian linking number. The bound on the global curvature as a nonsmooth side condition requires the use of Clarke’s generalized gradients to obtain the explicit structure of contact forces which appear naturally as Lagrange multipliers in Euler-Lagrange equations. Transversality conditions are discussed, and higher regularity for strains, moments, the centre line and directors is shown.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G45 Bounds for solutions of equilibrium problems in solid mechanics
74M15 Contact in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
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