Scale effects on the elastic behavior of periodic and hierarchical two-dimensional composites

The apparent stiffness tensors of two-dimensional elastic composite samples smaller than the representative volume element (RVE) are studied as a function of system size. Numerical experiments are used to investigate how the apparent properties of the composite converge with increasing scale factor n, defined to be the ratio between the linear size of the composite and the linear size of the unit cell. Under affine (Dirichlet-type) or homogeneous stress (Neumann-type) boundary conditions, the apparent elastic moduli overestimate or underestimate, respectively, the effective elastic moduli of the infinitely periodic system. The results show that the difference between the Dirichlet, Neumann and the effective stiffness tensors depends strongly on the phase stiffness contrast ratio. Dirichlet boundary conditions provide a more accurate estimate of the effective elastic properties of stiff matrix composites, whereas Neumann boundary conditions provide a more accurate estimate for compliant matrix structures. It is shown that the apparent bulk and shear moduli may lie outside of the Hashin-Shtrikman bounds. However, these bounds provide good upper and lower estimates for the apparent bulk and shear moduli of structures with a scale factor n≥2. A similar approach is used to study hierarchical composites containing two distinct structural levels with a finite separation of length scales. It is shown, numerically, that the error associated with replacing the smallest-scale regions by an equivalent homogeneous medium is very small, even when the ratio between the length scales is as low as three.