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Three‐point bending tests—Part I: Mathematical study and asymptotic analysis
Author(s) -
Quintela P.,
Sánchez M. T.
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1434
Subject(s) - uniqueness , mathematics , compatibility (geochemistry) , boundary value problem , mathematical analysis , beam (structure) , bending stiffness , von mises yield criterion , bending , structural engineering , finite element method , engineering , chemical engineering
The goal of this work is to study the static behaviour of a three‐dimensional elastic beam when is subjected to a three‐point bending test. In the first part, under suitable compatibility conditions on the applied forces and on the geometry of the beam, we will prove the existence of a unique solution for the associated contact elastic problem; these conditions of compatibility on the data come from the absence of a Dirichlet condition on the beam boundary. In the second part, we will study the asymptotic behaviour of this problem; in particular, we will deduce the one‐dimensional models associated with the displacement components, and we will give the existence and uniqueness of solution for them. Moreover, we will give an expression for the normal axial stress in the beam which is related to the modulus of rupture of brittle materials. In the final part of the work, we will deal with the regularity of the solution for the bending problem and we will prove some properties of the coincidence set. Copyright © 2011 John Wiley & Sons, Ltd.