A New Hybrid Homogenization Theory for Heterogeneous Materials with Random Elastic and Inelastic Phases

Unidirectional composites are the basic building blocks of laminated composite plates, shells and tubes, as well as woven composites, employed to fabricate large-scale composite structures. Homogenization techniques for traditional unidirectional composites based on simplified microstructural representations have been available for a long time, but they cannot be implemented to accurately analyze the response of statistically homogeneous materials, nor periodic materials with locally random microstructures. To address this shortcoming, a new hybrid homogenization theory, named HHT, has been constructed to analyze the homogenized response and micromechanical fields of unidirectional composites reinforced or weakened by locally random inclusion phases in the elastic and elastic-plastic regions across multiple scales. The theory incorporates ideas from two previously developed homogenization approaches, the finite-volume method, and the exact elasticity solution of the unit cell problem, i.e., known in the literature as FVDAM and LEHT, respectively. The novelty of the present approach lies in the coupling of continuous function description of the displacement fields in the inclusion phase with the discrete simple polynomial description in the discretized matrix phase, proposed herein for the first time in the solution of non-separable boundary-value problems. The quadratic convergence of stress fields with decreasing matrix discretization relative to exact elasticity solution is obtained, yielding accurate homogenized moduli with relatively coarse matrix meshes. Fourfold and greater reductions in execution times relative to the finite-volume and finite-element micromechanics justify the new theory’s construction.

Two versions of the theory have been developed to analyze the response of periodic composites with locally random microstructures characterized by the smallest building block called the Repeating Unit Cell (RUC) subject to periodic boundary conditions, as well as statistically homogeneous composites characterized by the smallest building block called the Representative Volume Element (RVE). The latter structural version of HHT has been developed to understand the influence of boundary conditions on the homogenized moduli and local stress fields of locally random composites, and to address the question of the RVE size in accurately predicting the response of composites characterized by multiple inclusion/porosity RVEs. Elastic-plastic, and energetic surface capabilities based on the Gurtin-Murdoch model, have been incorporated into the new theory to facilitate the analysis of a wide range of composite materials across multiple scales, including ceramic, metal, and polymeric matrix composites, as well as emerging composites with nanoscale inclusions or porosities. Finally, a reduced-order model called transformation field analysis (TFA) was constructed, based on HHT simulations capabilities, for the homogenized elastic-plastic response of locally random metal-matrix composites. HHT-based TFA eliminates the need for the solution of the entire RUC/RVE boundary-value problem through pre-calculation of elastic strain concentration and plastic influence matrices used in generating the homogenized response. This capability may be used in a stand-alone fashion to further accelerate the elastic-plastic analysis of complex microstructures, or as a subroutine in a large structural analysis algorithm.

The capabilities of HHT have been demonstrated through applications to the elastic-plastic homogenization and localization analysis of metal-matrix composites with locally random fiber distributions, which have not been extensively investigated due to high computational demand; quantification of the size of RVEs with multiple inclusions for accurate determination of homogenized moduli; study of the effect of surface energies on local stress fields and homogenized moduli of nanoporous materials; and investigation into the efficiency of TFA as a function of inclusion number. The contributions include quantification of the amount of scatter in the elastic-plastic response of locally random metal matrix composites as a function of loading direction; convergence of homogenized moduli of locally random composites, including mean values and associated ranges, as a function of RVE size relative to uniformly spaced composites; range of the magnitudes of homogenized moduli of locally random, nanoporous materials with surface energies; and determination of the RUC size below which TFA is more efficient than the full-scale analysis.

Overall, HHT is well suited for addressing fundamental questions regarding determination of homogenized and localized response of locally random composites with elastic-plastic phases, associated errors caused by erroneous microstructural characterization, and moreover it has the potential to be a rapid and accurate computational technique in structural analysis, advanced material development and quality assurance in the design and manufacturing of novel materials.