Bounds and estimates for linear composites with strain gradient effects

Overall mechanical properties are studied for linear composites demonstrating a size effect. Variational principles of Hashin-Shtrikman type are formulated for incompressible composites involving the gradient of strain in their constitutive description. These variational principles are applied to linear, statistically homogeneous and isotropic two-phase composites. Upper and lower bounds of Hashin-Shtrikman type for the effective shear modulus and related self-consistent estimates are derived in terms of volume fraction and a two-point correlation function accounting for the scale of microstructure. An alternative selfconsistent scheme for matrix-inclusion strain-gradient composites is also proposed by a development of the approach laid down by Budiansky and Hill. Some numerical results are given to demonstrate the size effect.