Stability of extreme static and dynamic bulk moduli of an elastic two-phase composite due to a non-positive-definite phase

Abstract

Elastic composite materials having phases that violate elastic positive-definiteness (so-called negative-stiffness phases) have been reported to, in principle, realize extreme values of overall elastic moduli such as the effective bulk modulus, if the composite geometry and the constituents’ elastic properties are appropriately tuned. In addition, previous studies have confirmed the stabilizing effect of the geometric constraints on the individual phases within a composite material: a non-positive-definite phase can be stabilized by a constraining matrix that is sufficiently stiff and sufficiently thick. However, to date no analysis has correlated the predicted extreme elastic response to the regime of stability. Therefore, it has remained an open question whether or not the inclusion of a non-positive-definite phase in an elastic composite can lead to stable extreme overall moduli. In this contribution we aim to close this gap by investigating the effective static and time-harmonic dynamic response of the simple elastic two-phase system of a coated spherical inclusion, and we report the results of its stability analysis. We show that the predicted extreme effective static bulk modulus of the two-phase system cannot be stable, whereas time-harmonic dynamic conditions indicate potential for considerable increases of the effective dynamic bulk stiffness in a resonant-like fashion due to the negative-stiffness induced low-frequency resonance.