  Elastic metamaterials may exhibit band gaps at wavelengths far exceeding
feature sizes. This is attributed to local resonances of embedded or branching
substructures. In branched configurations, such as a pillared plate, waves
propagating in the base medium-e.g., the plate portion-experience attenuation
at band-gap frequencies. Considering a simplified lumped-parameter model for a
branched medium, we present a theoretical treatment for a periodic unit cell
comprising a base mass-spring chain with a multi-degree-of-freedom,
mono-coupled branch. Bloch's theorem is applied, combined with a
sub-structuring approach where the resonating branch is modelled separately and
condensed into its effective dynamic stiffness. Thus, the treatment is
generally applicable to an arbitrary branch regardless of its size and
properties. We provide an analysis-with guiding graphical illustrations-that
yields an identification of fundamental bounds for the band-gap edges as
dictated by the dynamical characteristics of the branch. Analytical sensitivity
functions are also derived for the dependence of these bounds on the degree of
coupling between the base and the branch. The sensitivity analysis reveals
further novel findings including the role of the frequency derivative of the
branch dynamic stiffness in providing a direct relation between the band-gap
edge locations and variation in the coupling parameters-the mass and stiffness
ratios between the base chain and the branch root. In additional analysis,
sub-Bragg bounds of an exact model comprising a one-dimensional (1D) continuous
base-modelled as a rod-and a discrete branch are derived and shown to be
tighter than those of the all-discrete model. Finally, the applicability of the
derived bounds and sensitivity functions are shown to be valid for a
corresponding full 2D finite-element model of a pillared waveguide admitting
out-of-plane shear waves.
