A negative-stiffness phase in elastic composites can produce stable extreme effective dynamic but not static stiffness

We investigate the effective elastic properties and overall stability of four specific two-phase elastic composite systems having a non-positive-definite phase (often referred to as a negative-stiffness phase) to determine whether or not the presence of the negative-stiffness phase can lead to stable extreme overall stiffness. We start with an instructive spring-mass model to illustrate the underlying physical mechanisms before proceeding to the two- and three-dimensional two-phase solids of coated cylindrical and coated spherical inclusions, and we finally study a general particle-matrix composite. For all examples, we correlate effective stiffness with overall stability to demonstrate that the static effective stiffness measures can never reach extreme values due to the inclusion of a negative-stiffness phase in a stable manner, while dynamic loading indeed permits resonance-induced extreme effective stiffness.