Locally-exact homogenization theory for periodic materials with unidirectional reinforcements

A new locally-exact homogenization theory has been developed for the analysis of materials with periodic microstructures characterized by unit cells containing unidirectional reinforcement. The theory models the interior inclusion problem exactly and satisfies the periodic boundary conditions using a new variational principle developed for this class of problems. A multiscale displacement representation is employed in the solution of the unit cell problem, which is the combination of macroscopic strain terms superposed on the locally fluctuating displacement field. For the fluctuating displacement field, a Fourier series representation is assumed in the matrix and fiber regions, which satisfies the stress equilibrium equations a priori, while also satisfying the fiber-matrix continuity equations exactly. The use of a Fourier series eliminates the spatial discretization present in other numerical methods (e.g., finite element, finite difference) in favor of a function space discretization of the unit cell. Previous research into analytical models for periodic microstructures focused on symmetric unit cells, which simplified the application of the periodic boundary conditions. In the proposed model, the unit cell can be asymmetric, which requires the application of the full periodic boundary conditions on the surface displacements and tractions.; The capability of the locally-exact homogenization theory is demonstrated by first calculating the effective engineering moduli for unit cells with offset fibers. The convergence of the solution was examined for various fiber volume fractions as a function of the number of harmonics used in the series representation of the displacement field. It is shown that relatively few harmonics are required for macroscopic convergence even at extreme fiber volume fractions. Closed-form expressions for the homogenized moduli were obtained in terms of Hill's strain concentration matrices valid under arbitrary combined loading, which yielded the homogenized Hooke's law. Converged effective moduli and microscopic field quantities (i.e., displacements and stresses) were then compared to the corresponding results from finite-element analysis demonstrating excellent correlation. The need for the proposed variational principle is demonstrated by implementing the periodic boundary conditions using three alternative boundary methods commonly used in the literature. The results show that the new principle exhibits faster and more stable convergence compared to these methods.; Finally, the locally-exact homogenization theory was extended to enable the analysis of unit cells with multiple inclusions, demonstrating that the developed theoretical framework can be employed to analyze more complex microstructures.