  The Poynting effect generically manifests itself as the extension of the
material in the direction perpendicular to an applied shear deformation
(torsion) and is a material parameter hard to design. Unlike isotropic solids,
in designed structures, peculiar couplings between shear and normal
deformations can be achieved and exploited for practical applications. Here, we
engineer a metamaterial that can be programmed to contract or extend under
torsion and undergo nonlinear twist under compression. First, we show that our
system exhibits a novel type of inverted Poynting effect, where axial
compression induces a nonlinear torsion. Then the Poynting modulus of the
structure is programmed from initial negative values to zero and positive
values via a pre-compression applied prior to torsion. Our work opens avenues
for programming nonlinear elastic moduli of materials and tuning the couplings
between shear and normal responses by rational design. Obtaining inverted and
programmable Poynting effects in metamaterials inspires diverse applications
from designing machine materials, soft robots and actuators to engineering
biological tissues, implants and prosthetic devices functioning under
compression and torsion.
