A Locally Exact Homogenization Theory for Periodic Microstructures With Isotropic Phases

Elements of the homogenization theory are utilized to develop a new micromechanics approach for unit cells of periodic heterogeneous materials based on locally exact elasticity solutions. The interior inclusion problem is exactly solved by using Fourier series representation of the local displacement field. The exterior unit cell periodic boundary-value problem is tackled by using a new variational principle for this class of nonseparable elasticity problems, which leads to exceptionally fast and well-behaved convergence of the Fourier series coefficients. Closed-form expressions for the homogenized moduli of unidirectionally reinforced heterogeneous materials are obtained in terms of Hill’s strain concentration matrices valid under arbitrary combined loading, which yield homogenized Hooke’s law. Homogenized engineering moduli and local displacement and stress fields of unit cells with offset fibers, which require the use of periodic boundary conditions, are compared to corresponding finite-element results demonstrating excellent correlation.