The creation of negative stiffness (NS) metamaterials is of interest for damping treatments, vibration isolation, and even more exotic applications such as acoustic lenses displaying negative refraction. This work will present ongoing efforts to produce NS metamaterial elements that rely on a bistable microscale geometry that leads to NS under quasistatic loading conditions. A candidate bistable microstructure employing thermal mismatch will be introduced along with finite element simulations predicting its full stiffness tensor. Effective medium modeling will show the broadband utility of these elements to enhance acoustic absorption for low volume fraction bi-material composites. The special case of a bi-material periodic composite containing NS inclusions will then be explored. It will be shown that a periodic composite containing NS inclusions permits the simultaneous elimination of the acoustic branch and reduction of the lower frequency of the optical branch passband to arbitrarily low frequencies. Physical interpretations of these results and potential applications will be discussed. This material is based upon work supported by the U. S. Army Research Office under grant number W911NF-11-1-0032.