Abstract :
With recent improvements in fabrication processes, many structural components and solid materials are being designed at microstructural scales to provide specific macroscopic response characteristics. Optimal macroscopic properties and decreased susceptibilities to failure may be achieved by designing media with strictly periodic microstructures. Averaging and homogenization techniques, used to estimate the macroscopic properties of structured media, also are formulated on the basis of assumed microstructural periodicity. Few structures or materials, however, possess perfectly periodic microstructures. In the present work, the influence of perturbations in microstructural periodicity on the macroscopic response of structured media is investigated and quantified by examining the behavior of discrete media with both periodic and nearly periodic microstructures. The idealized macroscopic response of media with perfectly periodic microstructures is compared to the response obtained after perturbations in geometry and material properties are introduced into the models at the microstructural scale. Analysis shows that, for specific, well-defined classes of discrete media, the macroscopic properties are influenced only to second order in the perturbation amplitude parameter. In certain circumstances, however, the effects of these perturbations can become substantially more pronounced – demonstrating the limits of applicability of analysis techniques that assume that an underlying periodic microstructure exists for the discrete media under consideration.
