Effective Shear Modulus of Functionally Graded Fibrous Composites in Second Strain Gradient Elasticity

In this study, Mindlin's second strain gradient elasticity is employed to extract an estimate of the effective shear modulus of a functionally graded (FG) unidirectional fibrous composite subjected to an anti-plane mode of loading. The composite is supposed to be a periodically microstructured medium reinforced with infinitely long fibers, each of them surrounded by a coating made of a FG material. To determine the elastic field of the composite, an extended version of the equivalent inclusion method adapted with second strain gradient elasticity is utilized and, then, a unified procedure for the solution to the associated consistency equations is implemented that is applicable, in a similar manner, to any kind of two-phase, multi-phase, or FG composites. Subsequently, within a homogenization scheme, the effective shear modulus of the composite is calculated. The obtained results reflect well the size effect on the overall properties of the composite and demonstrate that the displacement field predicted by second strain gradient elasticity turns out to be a smooth function over the entire medium.