Asymptotic Homogenization via Locally Exact Elasticity and Finite Volume Methods

Asymptotic homogenization, also known as mathematical homogenization, is a rigorous theory developed for the analysis of heterogeneous materials with periodicity. It involves two steps called homogenization and localization. The homogenization step replaces the periodic heterogeneous microstructures by a homogeneous material with equivalent elastic moduli that are employed in the analysis of a composite structure, thereby avoiding the prohibitively large computational cost of explicitly accounting for every microstructural detail. In the localization step, the microlevel displacement, strain and stress fields are calculated based on the knowledge of the homogenized fields, and the response of the basic building block of the periodic microstructure under uniform and gradient strain fields, determined independently of the structural analysis. Compared to the widely used classical homogenization which assumes the microstructural details representative of the material-at-large are taken from an infinite array and strain gradients are negligible, asymptotic homogenization explicitly accounts for the microstructural size effect and the macroscale strain gradients which are important in structural problems with finite-sized microstructures. This is accomplished through a two-scale asymptotic expansion of the local displacement fields in powers of a parameter that represents the scale ratio of the microstructural characteristic size to the structural size, in contrast to a two-term expansion devoid of microstructural information used in classical homogenization.
Two novel implementations of the asymptotic homogenization theory have been developed in this dissertation that contribute to the state-of-the-art. The first is an elasticity-based analytical approach named locally exact asymptotic homogenization theory (LEAH), which is the only existing analytical solution for periodic structures with two-dimensional unit cell architectures within the framework of asymptotic homogenization. Compared to the mainstream numerical implementations, the constructed analytical theory possesses several advantages, including simple input data construction, mesh-free unit cell analyses, high accuracy, and rapid solution convergence and execution. Hence it is ideally suited for the verification of existing and developing asymptotic homogenization approaches, as well as efficient multiscale analysis of materials with hierarchical microstructures. The second is a semi-analytical approach named finite volume based asymptotic homogenization theory (FVBAHT), which is unique among the existing numerical implementation due to the use of finite volume approach in the solution of unit cell problems at different asymptotic expansion orders. Both asymptotic homogenization theories were developed for unidirectionally reinforced composite structures with finite-sized microstructures, with FVBAHT offering greater range of applicability than LEAH at present.
LEAH leverages elements of the previously developed locally exact homogenization theory for the determination of homogenized elastic moduli under macroscopically uniform strain fields, and is based on an asymptotic displacement field expansion that simulates anti-plane shear loading. The theory has been developed up to the 2nd order in the asymptotic displacement and stress field expansions in the scale ratio, which involves the solutions to local interior unit cell problems up to the 3rd order using Fourier series representations of the microfluctuation functions. The exterior unit cell periodic boundary value problems at different orders are tackled by an asymptotic extension of the previously introduced balanced variational principle for periodic materials. The effectiveness of the theory is verified by solving a structural problem with varying number of inclusions directly, and comparing the local fields with those reconstructed by the locally exact asymptotic homogenization approach. Linearly viscoelastic constituent phases have also been incorporated via the elastic-viscoelastic correspondence principle to simulate the time-dependent response of polymeric matrix composite widely used in structural applications.
FVBAHT builds upon the previously developed finite-volume direct averaging micromechanics theory applicable under uniform strain fields, and extends it to account for strain gradients and nonvanishing microstructural scale relative to structural dimensions up to the 2nd order of asymptotic displacement and stress field expansions. This involves the unit cell problems solved up to the 3rd order by satisfying local equilibrium equations in each subvolume of the discretized microstructure in a surface-averaged sense. The FVBAHT’s ability to accurately recover local fields is illustrated through comparison with the direct numerical solution of a periodic structure with varying number of inclusions under gradient loading. FVBAHT is more versatile than LEAH due to its ability to tackle composite structure problems under three-dimensional loadings, with unit cells comprised of multiple arbitrarily shaped inclusions.
Among many applications of the developed theories, a major contribution is the quantification of errors stemming from the use of classical homogenization in the local stress recovery in periodic composite structures subjected to loading that produces strain gradients at the structural scale. These errors have been quantified using both LEAH and FVBAHT for selected structural problems and provide important information for the analysis and design of composite structures with microlevel stress fields that do not exceed critical values.