Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities

Details Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities

2004-07-27

arxiv.org/abs/math/0407459v1

Comptes Rendus de l'Academie des Sciences. Serie 1, Mathematique 338 (2004) pp. 975-980

Mathematics

Analysis of PDEs

Preliminary version of a Note to be published in a slightly abbreviated form in C. R. Acad. Sci. Paris, Ser. I, 338 (2004), pp

Scientific paper

We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter $\epsilon$ tends to zero. The cylinder is assumed to be fixed (homogeneous Dirichlet boundary condition) on the whole of one of its extremities, but only on a small part (of size $\epsilon r^\epsilon$) of the second one; the Neumann boundary condition is assumed on the remainder of the boundary. We show that the result depends on $r^\epsilon$, and that there are 3 critical sizes, namely $r^\epsilon=\epsilon^3$, $r^\epsilon=\epsilon$, and $r^\epsilon=\epsilon^{1/3}$, and in total 7 different regimes. We also prove a corrector result for each behavior of $r^\epsilon$.

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