  When tensioned, ordinary materials expand along the direction of the applied
force. Here, we explore network concepts to design metamaterials exhibiting
negative compressibility transitions, during which a material undergoes
contraction when tensioned (or expansion when pressured). Continuous
contraction of a material in the same direction of an applied tension, and in
response to this tension, is inherently unstable. The conceptually similar
effect we demonstrate can be achieved, however, through destabilisations of
(meta)stable equilibria of the constituents. These destabilisations give rise
to a stress-induced solid-solid phase transition associated with a twisted
hysteresis curve for the stress-strain relationship. The strain-driven
counterpart of negative compressibility transitions is a force amplification
phenomenon, where an increase in deformation induces a discontinuous increase
in response force. We suggest that the proposed materials could be useful for
the design of actuators, force amplifiers, micro-mechanical controls, and
protective devices.
