  We investigate Lam\'e systems in periodically perforated domains, and
establish quantitative homogenization results in the setting where the domain
is clamped at the boundary of the holes. Our method is based on layer
potentials and it provides a unified proof for various regimes of hole-cell
ratios (the ratio between the size of the holes and the size of the periodic
cells), and, more importantly, it yields natural correctors that facilitate
error estimates. A key ingredient is the asymptotic analysis for the rescaled
cell problems, and this is studied by exploring the convergence of the periodic
layer potentials for the Lam\'e system to those in the whole space when the
period tends to infinity.
