Stability criteria for continuous and discrete elastic composites and the influence of geometry on the stability of a negative-stiffness phase
Recent experimental findings and theoretical analyses have confirmed the bound-exceeding performance of composite materials with one constituent of so-called negative stiffness (i.e., with non-positive-definite elastic moduli): the overall elastic properties greatly exceed those of the composite constituents, when the negative-stiffness phase's properties are appropriately tuned. However, the stability of such composite materials has remained a key open question. It has been shown, e.g., that a spherical particle of a negative-stiffness material can be stabilized when embedded in a sufficiently stiff and thick coating to impose a geometrical constraint on the negative-stiffness phase. For general composite geometries (as those arising from actual manufacturing processes), no such investigation has been reported. We review the classical stability conditions for homogeneous linear elastic solids and outline methods to determine the sufficient stability conditions for elastic composites. In addition, a numerical technique to obtain the stability restrictions on the elastic moduli of a composite with, in principle, arbitrary geometry is presented. Based on this method, we investigate the stability of simple elastic two-phase composites consisting of an inclusion (having non-positive-definite elastic moduli) embedded in a different coating material. In particular, the influence of the geometry of the encapsulated particles and the surrounding matrix is shown to considerably affect the overall stability. Our results compare the stability limits for two- (2D) and three-dimensional (3D) composite arrangements and provide design guidelines for optimal stability.