Using nonlinearity and spatiotemporal property modulation to control effective structural properties: dynamic rods

What are the effective properties of a generally nonlinear material or structure, whose local properties are modulated in both space and time? It has been suggested to use spatiotemporal modulation of structural properties to create materials and structures with adjustable effective properties, and to call these dynamic materials or spatiotemporal composites. Also, according to theoretical predictions, structural nonlinearity enhances the possibilities of achieving specific effective properties. For example, with an elastic rod having cubical elastic nonlinearities, it seems possible to control the effective propagation speed of long waves by varying the amplitude of a superimposed high-frequency standing wave. It should be possible to change most properties related to structural stiffness, energy dissipation, and equilibrium states this way, by exploiting the general effects of stiffening, biasing, and smoothening that characterize mechanically high-frequency excited structures. This work explores fundamental issues for spatiotemporally modulated nonlinear materials. First the effective stress-strain relation of a generally nonlinear material with a prescribed HF strain field is derived, discussed, and exemplified. Then simple approximate analytical expressions are derived for the effective wave speed and natural frequencies for one-dimensional wave propagation in a nonlinear elastic rod, where the spatiotemporal modulation is imposed as a high-frequency standing wave, supposed to be given. Finally the more realistic case is briefly considered, where the HF standing wave is not given, but is a solution to given external HF input.