  We argue that the standard classification of isometric deformations into
infinitesimal v.s. finite is inadequate for the study of compliant shell
mechanisms. Indeed, many compliant shells, particularly ones that are
periodically corrugated, exhibit low-energy deformations that are far too large
to be infinitesimally isometric and far too rich to be finitely isometric.
Here, rather than abandon the geometric standpoint in favor of a full theory of
elastic shells, we introduce the concept of \emph{effective isometric
deformations} defined as deformations that are first-order isometric in a small
scale separation parameter given by the ratio of the size of the corrugation to
the size of the shell. The main result then states that effective isometries
are solutions to a quasilinear second-order PDE whose type is function of an
effective geometric Poisson's ratio. The result is based on a self-adjointness
property of the differential operator of infinitesimal isometries; it holds for
periodic surfaces that are smooth, piecewise smooth or polyhedral, i.e.,
periodic surfaces with or without straight and curved creases. In particular,
it unifies and generalizes a series of previous results regarding the effective
Poisson's ratio of parallelogram-based origami tessellations. Numerical
simulations illustrate and validate the conclusions.
