  In an unconstrained elastic body, emergence of zero natural frequencies is an
expectable outcome on account of the body's ability to purely translate or
rotate with no structural deformation. Recent advances in literature have
pushed such conventional definition and demonstrated properties transcending
typical zero-frequency modes, such as localization of deformation at a
structural edge or corner. In this paper, a spring-mass honeycomb lattice with
an elastic foundation, referred to here as mechanical graphene, is designed to
exhibit zero-frequency corner modes. A central element in the proposed design
is the elastic foundation, and the zero-frequency corner modes are enabled by
intricate modulation of the elastic-foundation's stiffness. These modes are
proven to have their origins from the dynamics of a diatomic chain, made from a
single strip of the mechanical graphene with free boundaries. Different shapes
of finite mechanical graphene with free boundaries are considered and
conditions leading to the manifestation of corner modes are correlated with the
angle of corners and stiffness of springs supporting them. Finally, the effect
of defects on zero-frequency corner modes is briefly discussed, demonstrating
robustness against structural defects that are distant from corners.
