All Mean Field Homogenization Methods Are Approximate: Some Might Be Useful
摘要
A fundamental problem in mechanics of materials is homogenization which involves the calculation of the effective response of heterogenous materials. Mean field homogenization (MFH) methods are easy to setup, computationally cheap, and reasonably accurate making them quite attractive. However, there are persistent questions about their accuracy, physically admissibility, and versatility. In this paper, the experience of the co-authors over past 3 years is presented. MFH methods are studied from a wide range of perspectives from predictive abilities to confirming physically admissibility for a range of composites with varied fiber architectures. Algorithms are developed for microstructure generation with complex fiber architectures and finite element (FE) models are created by automation using python scripts. The MFH methods are benchmarked against the full FE solutions with comparisons at several length scales—effective modulus, stresses in individual inclusions and stresses in the interphase. Using this modus operandi, a wide range of MFH schemes is studied from full Mori–Tanaka (MT) formulation to pseudo-grain discretized version of MT to various multi-step MT formulations. It is concluded that each of the MFH variants is approximate with reasonable predictions for some microstructure and large errors for some microstructures. The accuracy of these methods decrease as the length scale of comparison is reduced. MFH methods are extended to model composites with discontinuous curved fibers which involve transformation of fibers to equivalent assembly of inclusions. An unbiased comparison formulation methodology is developed to validate the MFH methods involving transformation of fibers to equivalent assembly of inclusions.