Mathematical modelling of rods

Concerning thin structures, such as plates and shells, the idea of reducing the equations of elasticity to two-dimensional models defined on the midsurface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the midsurface. As nowadays, the asymptotic expansion of the displacement solution of the three-dimensional linear model is fully known at least for plates and clamped elliptic shells, we start from a description of these expansions in order to introduce the two-dimensional models known as hierarchical models: These models extend the classical models, and presuppose the displacement to be polynomial in the thickness variable, transverse to the midsurface. Because of the singularly perturbed character of the elasticity problem as the thickness approaches zero, boundary, or internal layers may appear in the displacements and stresses, and so may numerical locking effects. The use of hierarchical models, discretized by higher degree polynomials (p-version of finite elements) may help overcome these severe difficulties.


Keywords:

shells;
plates;
hierarchical models;
asymptotic expansion;
modal analysis;
eigen-frequencies;
finite elements

Plates and Shells: Asymptotic Expansions and Hierarchic Models

Modeling and justification of eigenvalue problems for junctions between elastic structures

Asymptotic derivation of a general linear model for thin-walled elastic rods

In this work we obtain a generalization and a justification of Saint Venant's torsion theory in beams, by applying the asymptotic expansion method to the Hellinger-Reissner variational formulation of the three dimensional linearized elasticity model

A derivation of generalized saint venant’s torsion theory from three-dimensional elasticity by asymptotic expansion methods

This work is a continuation of an earlier work by Bermudez and Viano (1984) on the same subject. In fact, using the same asymptotic expansion in linear elastic beams we give a complete characterization of displacements, bending moments and shear forces of orders 0, 1 and 2. These results include a characterization of the stress field of order 0 and of the axial and shear stresses of order 1. An appropriate physical interpretation of these results, which is considered elsewhere, will allow us to derive and to justify, from a mathematical point of view, the most well-known classical extension, bending and torsion theories for linear elastic beams, including the Bernoulli-Navier, Saint Venant, Timoshenko and Vlasov models.

J.M. Viaño

Papers

Existence and characterization of higher-order terms in an asymptotic expansion method for linearized elastic beams