Generalized Finite-volume Micromechanics Theory for Heterogeneous Materials

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. Micromechanics of heterogeneous materials plays an important role in the development of these materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem generic to the analysis of periodic heterogeneous media is explored in this dissertation, with emphasis initially on the well-established 0th order version of the finite-volume method called finite-volume direct averaging micromechanics (FVDAM) theory. Differences and similarities with the finite element method are highlighted using newly introduced stress measures, and the resulting tangible advantages of the finite-volume approach are discussed and illustrated. A recent attempt to develop an alternative version of this technique is also discussed, illustrating shortcomings intrinsic to the 0th order and alternative versions, setting the stage for further development of this theory in order to enhance its predictive capability and efficiency.
Towards this end, a generalized finite-volume theory is constructed for two-dimensional linear elasticity problems on rectangular domains. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in the 0th order theory. The higher-order displacement field is expressed in terms of elasticity-based surface-averaged kinematic variables which are subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. Satisfaction of subvolume equilibrium equations in an integral sense, a defining feature of finite-volume techniques, provides the required additional equations for the local stiffness matrix construction. The theory is constructed in a manner which enables systematic specialization through reductions to lower-order versions. Comparison of predictions by the generalized theory with its predecessor, analytical and finite element results illustrates substantial improvement in the satisfaction of interfacial continuity conditions at adjacent subvolume faces, producing smoother stress distributions and good interfacial conformability.
Given these very promising results, the generalized finite-volume theory is further extended to accommodate finite deformations of periodic materials with complex microstructures. This is accomplished by embedding the generalized finite-volume theory with newly incorporated finite-deformation features into the 0th order homogenization framework, and introducing parametric mapping to enable efficient mimicking of complex microstructural details, producing a parametric version of the new approach named generalized finite-volume direct averaging micromechanics (FVDAM) theory. As in the case of the linear theory, the higher-order fluctuating displacement field representation within subvolumes of the discretized unit cell microstructure, expressed in terms of elasticity-based surface-averaged kinematic variables, substantially improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes. This improvement is particularly important in the finite-deformation domain wherein large differences in adjacent subvolume face rotations may lead to the loss of mesh integrity. The nonlinear theory is also constructed in a manner which enables systematic specialization through reductions to lower-order versions with the 0th order corresponding to the standard FVDAM theory. The advantages of the generalized FVDAM theory are illustrated through examples based on a known analytical solution and finite element results generated with a recently constructed finite element formulation that mimics the generalized theory's framework. An application of the generalized FVDAM theory involving the response of wavy multilayers confirms previously generated results with the 0th order theory that revealed microstructural effects in this class of materials applicable to bio-inspired material architectures that mimic certain biological tissues.