  We study the effect of geometric frustration on dilational mechanical
metamaterial membranes. While shape frustrated elastic plates can only
accommodate non-zero Gaussian curvature up to size scales that ultimately
vanish with their elastic thickness, we show that frustrated {\it
metamembranes} accumulate hyperbolic curvatures up to mesoscopic length scales
that are ultimately independent of the size of their microscopic constituents.
A continuum elastic theory and discrete numerical model describe the
size-dependent shape and internal stresses of axisymmetric, trumpet-like
frustrated metamembranes, revealing a non-trivial crossover to a much weaker
power-law growth in elastic strain energy with size than in frustrated elastic
membranes. We study a consequence of this for the self-limiting assembly
thermodynamics of frustrated trumpets, showing a several-fold increase the size
range of self-limitation of metamembranes relative to elastic membranes.
