Elastic Composite Materials Having a Negative Stiffness Phase Can Be Stable

We prove that composite materials containing an isotropic phase having negative bulk and Young's moduli (hence being unstable by itself) can be stable overall, under merely applied traction boundary conditions, if the stable encapsulating phase is sufficiently stiff. We derive specific quantitative requirements on the elastic moduli of the constituent materials that ensure composite stability for two fundamental composite geometries. These results legitimize the concept of negative-stiffness-phase composites, thus dramatically expanding the parameter landscape in which novel and optimal overall material properties may be sought.