  A large family of periodic planar non-linear bimode metamaterials are
constructed from rigid bars and pivots. They are affine materials in the sense
that their macroscopic deformations are only affine deformations: at large
distances any deformation must be close to an affine deformation. Bimode means
that the paths of all possible deformations of Bravais lattices that preserve
the periodicity of the lattice lie on a two dimensional surface in the three
dimensional space of invariants describing the deformation (excluding
translations and rotations). By adding two actuators inside a single
microscopic cell one can control the macroscopic deformation, which may be
useful for the design of adaptive structures. The design of adaptable nonlinear
affine trimode metamaterials (for which the macroscopic deformations lie within
a three-dimensional region in the space of invariants) is discussed, although
their realization remains an open problem. Examples are given of non-affine
unimode and non-affine bimode materials. In these materials the deformation can
vary from cell to cell in such a way that the macroscopic deformation is
non-affine.
