An asymptotic linear thin-walled rod model coupling twist and bending

Abstract

A linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods is presented. A dimensional analysis of the linear three-dimensional equilibrium equations yields dimensionless numbers that reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions. In the case of low force levels, we obtain a one-dimensional model whose kinematics, traction, and twist equations correspond to the Vlassov ones. However, this model couples twist and bending effects in the bending equations, unlike the Vlassov model where the twist angle and the bending displacement are uncoupled Keywords: thin-walled rod model, linear elasticity, asymptotic methods 1. Introduction. Thin and thin-walled structures (plates, shells, rods and thin-walled rods) are widely used in industry because they provide maximum stiffness with minimum weight. However, there exists many different models in the literature. Therefore, engineers must know a priori their respective domain of validity and what model to use in function of the given data of the problem (geometry of the structure, applied loads, boundary conditions).Classical models (the Kirchhoff-Love, Koiter, Bernouilli, Vlassov, etc.) are generally obtained from three-dimensional equilibrium equations by making a priori (kinematic and static) assumptions on the unknowns of the problem. Therefore, the domain of validity of these classical models with respect to the given data of the problem is difficult to specify rigorously.Asymptotic methods enable to deduce rigorously plate, shell, and rod models from the three-dimensional equations without making any a priori assumption. In linear plate and shell theory, since the pioneering work of Goldenveizer [11], there exists a large literature on the subject [2,7,[38][39][40][41].In the linear theory of rods, the first works on the subject are due to Rigolot [33]. More recently, other justifications of linear and nonlinear rod models by asymptotic expansion were developed in [3,[20][21][22]42]. Let us also cite the synthesis [46] of previous works [44,45], which recall the different possible approaches in the linear theory of elastic rods (displacement formulation and mixed formulation in stress-displacements).These results then have been extended to thin-walled rods. The approach used is based on the asymptotic behavior of the Poisson equation in a thin domain when the thickness tends to zero [34,35,46]. This way, Rodriguez and Viaño [36] have justified a linear elastic model of Vlassov for a thin-walled rod by asymptotic method similar to the Vlassov one. However, their approach uses "a priori" scaling assumptions on the displacement field, which is an unknown of the problem. Moreover, it is based on an expansion at the second order of the equations with respect to the diameter e and then the relative thickness h is assumed to tend to zero. Th...

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