Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case

Abstract

The modelling of ordinary beams and thin-walled beams is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity. In the case of isotropic homogeneous elasticity, ordinary beams yield the Navier-Bernoulli beam model, thin-walled beams with open profile yield the Vlassov beam model and thin-walled beams with closed profile the Navier-Bernoulli beam model. The formal asymptotic analysis is also extensively performed in the case of the most general anisotropic transversely heterogeneous material (meaning the heterogeneity is the same in every cross-section), delivering the same qualitative results. We prove, in particular, the non-intuitive fact that the warping function appearing in the Vlassov model for general anisotropic transversely heterogeneous material, is the same as the one appearing in the isotropic homogeneous case. In the general case of anisotropic transversely heterogeneous material, the analysis provides a rigorous and systematic constructive procedure for calculating the reduced elastic moduli, both in Navier-Bernoulli and Vlassov theories.

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability

“…On the side of slender bodies, besides the classical rod models, in the literature are often adopted a number of refinements, many of them having the scope to go beyond the limits of the standard rod models, such as Timoshenko's or Euler's. A full description of the huge literature on the subject is unattainable; we here quote [5][6][7][8] and, among the most recent works [9][10][11][12][13][14][15][16]. In particular, in [10], a model for rods and thin-walled rods is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity.…”

“…A full description of the huge literature on the subject is unattainable; we here quote [5][6][7][8] and, among the most recent works [9][10][11][12][13][14][15][16]. In particular, in [10], a model for rods and thin-walled rods is rigorously obtained from a formal asymptotic analysis of three-dimensional linear elasticity. In [16], a general method for deriving one-dimensional models for nonlinear structures has been proposed; the models capture the contribution to the strain energy arising not only from the macroscopic elastic strain but also from the strain gradient.…”

“…10) Using(3.3) 1 and (3.7) 2 , this choice suffices to describe all the components of the membrane stress tensorN xx = ∂ yy ϕ = f (x)ψ (y), N yy = ∂ xx ϕ = f (x)ψ(y), N xy = −∂ xy ϕ = −f (x)ψ (y). (3.11)Moreover, conditions (3.9) allow to satisfy the boundary conditions N yy (x, y = ±ε/2) = 0 and N xy (x, y = ±ε/2) = 0 along the lateral sides of the shell.…”