Scale Effects in Media With Periodic and Nearly Periodic Microstructures, Part I: Macroscopic Properties

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Abstract
Traditional averaging and homogenization techniques, developed to predict the macroscopic properties of heterogeneous media, typically ignore microstructure related scale effects—that is, the influence of the size of the representative volume, relative to the size of the unit cell. This issue is presently investigated by exploring the behavior of a nonlinearly elastic, planar, lattice model, which is subjected to general macroscopic deformations. For these materials, scale effects may be due to nonuniformities in the macroscopic strain field throughout the specimen, or alternatively, to the presence of microstructural imperfections that may be either geometric or constitutive in nature. For the case of macroscopic strain nonuniformities, it is shown that the microstructure related scale effects can be accounted for by the presence of higher order gradient terms in the macroscopic strain energy density of the model. For the case of microstructural imperfections, the difference between the respective macroscopic properties of the perfect and imperfect models are shown to depend on the relative size of the specimen, and on the imperfection amplitude and wavelength, while being nearly insensitive to the imposed macroscopic strain. For all of the cases considered, several analytical approximations are proposed to predict the influence of scale on the macroscopic properties, and the accuracy of each method is examined.