Generalized Locally Exact Homogenization Theory for Unidirectionally Reinforced Composites

This dissertation describes the construction, validation and applications of a stable and quickly converging elasticity-based locally exact homogenization theory for unidirectionally- reinforced composites. Elasticity-based homogenization approaches offer a number of advantages relative to finite-element, finite-difference or finite-volume homogenization schemes, including extremely fast input data construction, ability to investigate composites with very thin coatings or interphases without experiencing convergence issues common to finite-element analyses, and ability to efficiently accommodate phases with cylindrically orthotropic constituents without much effort. The constructed homogenization theory enables efficient analysis of the elastic and viscoelastic response of unidirectional composite materials with rectangular, square, hexagonal and tetragonal periodic microstructures comprised of isotropic, transversely isotropic, and (cylindrically or circumferentially) orthotropic phase constituents, and third phases such as coatings or interphases.
The success of this homogenization theory is rooted in the balanced variational principle which plays a key role in the implementation of non-separable periodic boundary conditions, leading to quickly-converging homogenized moduli and stable local stress distributions. This variational principle, originally proposed by Drago and Pindera (2008) for rectangular unit cell architectures, was extended herein to hexagonal and tetragonal unit cells and demonstrated to produce quickly converging homogenized moduli and local stress fields regardless of phase modulus contrast, orthotropy type or viscoelasticity effects.
The constructed homogenization theory has been validated upon comparison with known elasticity solutions and micromechanics models. These include the solution to the Eshelby problem which was used as a benchmark to demonstrate the robustness and stability of the developed unit cell solution approach, and the finite volume direct averaging micromechanics (FVDAM) theory which produces high-fidelity results comparable to the finite-element method. Comparison with the classical composite cylinder assemblage (CCA) and Mori-Tanaka micromechanics models establishes applicability and limitations of these approaches based on simplified geometric representation of composite material microstructures. Selected numerical results are generated to provide insight into the efficiency and robustness of the theory. To demonstrate its advantage, the key component of the theory, namely the balanced variational principle, is compared with recently adopted approaches based on a derivative variational principle proposed originally in the context of locally-exact finite-element solutions. Finally, the elastic problem has been extended to viscoelastic domain via the elastic-viscoelastic correspondence principle, validated at the homogenized and local field levels at different times, and employed to investigate thus-far undocumented features of time-dependent response of polymeric matrix composites. The significant findings include the effect of array type on the creep response which increases dramatically with increasing time for certain loading directions. The theory’s utility in support of constructing homogenized response functions of polymeric matrix composites from experimental data was also demonstrated.
Because of its analytical nature, the constructed theory may easily be incorporated into larger structural analysis algorithms in a multi-scale computational setting. This capability is illustrated herein in the context of laminated plate and functionally graded tube analyses, wherein local homogenized elastic moduli of the investigated structural components are generated on the fly for use in the governing equations at the structural level.
The theory’s efficiency and stability in generating homogenized moduli and stress fields with very simple input data construction make it readily accessible to professionals and non-professionals alike. Hence it is expected that this approach will quickly gain popularity and become not only a design and research tool used by diverse communities involved in materials characterization, design and development, but also a comparison standard for bench mark purposes.