  We report a three-dimensional mechanical metamaterial that simultaneously
possesses negative stiffness, negative bulk modulus, and negative Poisson's
ratio. This metamaterial is a periodic arrangement of binder-shell elements.
Under compression, the spherical shells dent inwards which cause the material
to contract in the lateral directions. At larger deformations, snap through
instability occurs and the material exhibits negative incremental stiffness.
Interestingly, both incremental stiffness and incremental Poisson's ratio
approach negative infinity (under displacement control) when snap-back is
observed. We further showed that the multi-negative index metamaterial
satisfies the strong ellipticity condition, therefore, a block of the
metamaterial with many unit cells is stable under displacement constraint.
