Effective shear modulus of solids reinforced by randomly oriented-/aligned-elliptic nanofibers in couple stress elasticity

Nowadays, by adding a small amount (about 0.5-5% by weight) of a desired nanomaterial to a matrix having certain properties one may design a multifunctional nanocomposites with a remarkably improved macroscopic properties of interest. The capability of conventional continuum theories in treating the problems of embedded ultra-small inhomogeneity with any of its dimensions comparable to the characteristic lengths of the involved constituent phases is questioned, mainly, on the grounds of the accuracy and the size effect. The micromechanical framework based on the Eshelby's ellipsoidal inclusion theory [1] which has been widely used to estimate the overall behavior of composites falls under the same category, as is size insensitive. In this work, effort is directed at the prediction of the macroscopic shear modulus of composites consisting of nano-/micro-size fibers of elliptic cross-sections via couple stress theory, a physically realistic theory that encompasses the size effect. To this end, the fundamental equations of couple stress elasticity in elliptic coordinates are derived and several fundamental elliptic inhomogeneity problems in plane couple stress elasticity are solved analytically. For the purpose of the application of these results to the study of the effective properties of the composites of interest, Mori and Tanaka theory [2] is first reformulated in the mathematical framework of couple stress theory. Subsequently, the overall shear modulus of solids reinforced by aligned as well as randomly oriented elliptic nanofibers will be predicted. The influences of the size, shape, orientation, rigidity, and intrinsic length of the reinforcing nanofibers as well as the effects of the characteristic length of the matrix on the effective shear modulus of the composite are addressed.