Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities
Juan Casado‐Díaz, Manuel Luna‐Laynez, François Murat · Comptes Rendus Mathématique · 2004
We study the asymptotic behavior of the solution of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder whose diameter ε 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 εr ε ) of the second one; the Neumann boundary condition is imposed on the remainder of the boundary. We show that the result depends on r ε , and that there are 3 critical sizes, namely r ϵ = ϵ 3 , r ϵ = ϵ , and r ε = ε 1/3 , and in total 7 different regimes. We also prove a corrector result for each behavior of r ε .