Modeling heterogeneous media with microstructures of different scales

The objective of this paper is to extend the framework of the continuum theory so that it can capture the properties that are embedded in the microstructure or nanostructure and still keep its simplicity and efficiency The model thus developed is capable of accounting for local deformation of microstructures in solids especially their micro- (local) inertia effect. The essence underlying this approach is the introduction of a set of bridging functions that relate the local deformation of microstructures to the macrokinematic variables. Once the solution of the macroscopically homogeneous continuum is obtained, the solutions in the microstructures are obtained through the use of these bridging functions. Propagation of waves of different wavelengths is considered and the dispersion curve is used to evaluate the accuracy of the model. The model is also employed to study wave reflection and transmission at the boundary of two media with microstructures Of very different scales.