Homogenization for Periodic Heterogeneous Materials with Arbitrary Position-Dependent Material Properties

Abstract

We present a rigorous homogenization approach for efficient computation of a class of physical problems in a one-dimensional periodic heterogeneous material. This material is represented by a spatially periodic array of unit cells with a length of ε. More specifically, the method is applied to the diffusion, heat conduction, and wave propagation problems. Heterogeneous materials can have arbitrary position-dependent continuous or discontinuous materials properties (for example heat conductivity) within the unit cell. The final effective model includes both effective properties at the leading order and high-order contributions due to the microscopic heterogeneity. A dimensionless heterogeneity parameter β is defined to represent high-order contributions, shown to be in the range of [-1/12, 0], and has a universal expression for all three problems. Both effective properties and heterogeneity parameter β are independent of ε, the microscopic scale of heterogeneity. The homogenized solution describing macroscopic variations can be obtained from the effective model. Solution with sub-unit-cell accuracy can be constructed based on the homogenized solution and its spatial derivatives. The paper represents a general approach to obtain the effective model for arbitrary periodic heterogeneous materials with position-dependent properties.