  We show how bilateral, linear, elastic foundations (i.e. Winkler foundations)
often regarded as heuristic, phenomenological models, emerge asymptotically
from standard, linear, three-dimensional elasticity. We study the parametric
asymptotics of a non-homogeneous linearly elastic bi-layer attached to a rigid
substrate as its thickness vanishes, for varying thickness and stiffness
ratios. By using rigorous arguments based on energy estimates, we provide a
first rational and constructive justification of reduced foundation models. We
establish the variational weak convergence of the three-dimensional elasticity
problem to a two-dimensional one, of either a "membrane over in-plane elastic
foundation", or a "plate over transverse elastic foundation". These two regimes
are function of the only two parameters of the system, and a phase diagram
synthesizes their domains of validity. Moreover, we derive explicit formulae
relating the effective coefficients of the elastic foundation to the elastic
and geometric parameters of the original three-dimensional system.
