Asymptotic behavior of an elastic beam fixed on a small part of one of its extremities
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 assumed 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 $r^ε$.
math.AP↗