  In this paper we report metamaterial properties including negative and
singular effective properties for what would traditionally be considered non
locally resonant 2-phase phononic unit cells. The negative effective material
properties reported here occur well below the homogenization limit and are,
therefore, acceptable descriptions of overall behavior. The material property
combinations which make this possible were first revealed by a novel level set
based topology optimization process which we describe. The optimization process
revealed that a 2-phase unit cell in which one of the phases is simultaneously
lighter and stiffer than the other results in dynamic behavior which has all
the attendant characteristics of a locally resonant composite including
negative effective properties far below the homogenization limit. We
investigate this further using the Craig-Bampton decomposition and clarify that
these properties emerge through an interplay between the fundamental internal
modeshape of the unit cell and a rigid body mode. Through explicit numerical
calculations on 1-D, 2-phase unit cells, we show that negative effective
properties only appear for the specific material property combination mentioned
above. Furthermore, we provide a proof which supports this conclusion. The
concept is also shown to hold for 2-D unit cells where we show that an
appropriately designed hexagonal unit cell made of 2 material phases exhibits
negative effective shear modulus and density in an appropriate frequency regime
in which it also exhibits negative refraction. An important conclusion of this
paper is that the class of unit cells expected to result in negative properties
can be expanded beyond the classic unit cell (three-phase unit cells with an
explicit locally resonant phase) to include topologically simpler 2-phase unit
cells as well.
