Infinitely stiff composite via a rotation-stabilized negative-stiffness phase

Combining several materials (“phases”) into a composite material permits creation of new materials with tunable overall response, e.g., elastic properties, which can be related to the elastic moduli, arrangement, and concentrations of the composite’s phases. A recent advance predicts use of one phase having correctly tuned non-positive-definite elastic moduli (i.e., negative stiffness) in a two-phase composite can produce theoretically infinite composite stiffness. Employing this theory, the Lakes group recently made such a composite (inclusion negative-stiffness produced by a constrained phase transformation), which exhibited momentary (unstable) extreme composite stiffness ten times that of diamond. Although a homogeneous, free-standing body of negative stiffness is unstable by itself, we have recently shown that a negative-stiffness phase can be stabilized if it is an inclusion completely encapsulated by a sufficiently stiff matrix material. However, the range of negative inclusion stiffness so stabilized is not sufficient to allow for the tuning required to produce stable theoretically infinite overall composite stiffness. We therefore seek strategies to enlarge the stability regime of composite materials containing a negative-stiffness phase. Here we illustrate one such strategy: use of dynamic excitation. It is known that appropriate dynamic excitation can broaden a system’s stability regime; a famous example is the inverted pendulum. (Inertia can also do this.) We analyze here the instructive case of a rotating composite (whose nonrotating dynamic stability regime is known). We employ a full dynamic analysis to derive the restrictions on the elastic moduli of the phases in the rotating composite that guarantee composite stability. We show that for sufficient rates of rotation, the composite will remain stable even when the inclusion stiffness is sufficiently negative to produce theoretically positive-infinite composite stiffness. The composite to be analyzed is comprised of two homogeneous, isotropic, linear elastic phases: an infinitely long, circular cylinder (radius a, Lamé moduli k1, l1), embedded in a concentric coating (outer radius b, Lamé moduli k2, l2). The composite is in a state of plane strain with deformations confined to the x,y-plane, and it rotates with angular velocity X about the z-axis with the center of rotation at the origin (see Fig. 1). For simplicity and clarity, we have chosen the simplest realistic solid composite geometry to illustrate the phenomenon described herein; clearly, this phenomenon can be made to occur in more geometrically complex composite materials when subjected to appropriate dynamic excitation. The dynamic governing equations for a homogeneous, isotropic, linear elastic solid with Lamé moduli k, l, when no body forces act, are the Navier equations