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Statics of curved rods on account of torsion and flexion. (English) Zbl 0938.74040
Summary: We adress the problem of a thin curved rod in linear elasticity for small displacements. We use an asymptotic two-scale method based on the small parameter \(\varepsilon\), ratio of the thickness to the overall lengths. The two leading-order terms have the Bernoulli’s structure, and the corresponding displacement is inextensional. The constitutive law involves torsion effects of the same order as flexion effects, so that the description of the kinematics involves an angle \(\theta\) which is the rotation of the sections. The Lagrange multiplier, associated with the constraint of inextensibility, is discussed. We give a variational formulation of the problem in the subspace of the inextensional displacements, and discuss the equations involving the Lagrange multiplier.

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B05 Classical linear elasticity
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
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