  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 $r^\epsilon$.
