  Most granular packings possess shear moduli ($G$) that increase with the
applied external pressure, and bulk moduli ($B$) that increase or remain
constant with pressure. This paper presents "tessellated" granular
metamaterials for which both $G$ and the ratio $G/B$ decrease with increasing
pressure. The granular metamaterials are made from flexible tessellations
forming a ring of closed cells, each containing a small number of solid
particles. For under-constrained tessellations, the dominant contributions to
$G$ and $B$ are the particle-particle and particle-cell interactions. With
specific particle configurations in the cells, we limit the number of possible
particle rearrangements to achieve decreasing $G$ as we increase the pressure
difference between the inside and outside of the tessellation, leading to $G/B
\ll 1$ at large pressures. We further study tessellated granular metamaterials
with cells containing a single particle and many particles to determine the
variables that control the mechanical response of particle-filled tessellations
as a function of pressure.
