  The principles of origami design have proven useful in a number of
technological applications. Origami tessellations in particular constitute a
class of morphing metamaterials with unusual geometric and elastic properties.
Although inextensible in principle, fine creases allow origami metamaterials to
effectively deform non-isometrically. Determining the strains that are
compatible with coarse-grained origami kinematics as well as the corresponding
elasticity functionals is paramount to understanding and controlling the
morphing paths of origami metamaterials. Here, within a unified theory, we
solve this problem for a wide array of well-known origami tessellations
including the Miura-ori as well as its more formidable oblique, non-developable
and non-flat-foldable variants. We find that these patterns exhibit two
universal properties. On one hand, they all admit equal but opposite in-plane
and out-of-plane Poisson's ratios. On the other hand, their bending energy
detaches from their in-plane strain and depends instead on the strain gradient.
The results are illustrated over a case study of the self-equilibrium geometry
of origami pillars.
