Variational asymptotic method for unit cell homogenization of periodically heterogeneous materials

Abstract

A new micromechanics model, namely, the variational asymptotic method for unit cell homogenization (VAMUCH), is developed to predict the effective properties of periodically heterogeneous materials and recover the local fields. Considering the periodicity as a small parameter, we can formulate a variational statement of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. In comparison to other approaches, VAMUCH does not rely on ad hoc assumptions, has the same rigor as MHT, has a straightforward numerical implementation, and can calculate the complete set of properties simultaneously without using multiple loadings. This theory is implemented using the finite element method and an engineering program, VAMUCH, is developed for micromechanical analysis of unit cells. Many examples of binary composites, fiber reinforced composites, and particle reinforced composites are used to demonstrate the application, power, and accuracy of the theory and the code of VAMUCH.

A Critical Evaluation of the Predictive Capabilities of Various Advanced Micromechanics Models

“…Recently, a new micromechanics model, the Variational Ssymptotic Method for Unit Cell Homogenization (VAMUCH), [15][16][17] has been developed by invoking three assumptions: 1) size of the microstructure is much smaller than the macroscopic size of the material; 2) exact solutions of the field variables have volume averages over the UC; 3) effective material properties are independent of macroscopic geometry, boundary and loading conditions of the structure.…”

“…14 A recently developed variant of the asymptotic homogenization approach is the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH). [15][16][17] In contrast to conventional asymptotic methods, VAMUCH carries out an asymptotic analysis of the variational statement, synthesizing the merits of both variational methods and asymptotic methods. Finally, there are a number of purely numerical approaches, such as finite element analyses, 18,19 and particle-in-cell methods, 20 that have been used to model the micromechanical response of heterogeneous materials.…”

Determination of Size of Representative Volume Element for Coke Using Homogenization Method Based on Digital Image of Actual Microstructure

“…Under this periodicity assumption, the coke structure was idealized as a periodic assembly of many unit cells. 18,19) Figure 5 shows the image of periodic tiled microstructure unit cells.…”

A variational asymptotic micromechanics model for predicting conductivities of composite materials

“…These models include simple rules of mixtures, self consistent scheme [Hashin 1968], generalized self consistent scheme [Lee et al 2006], finite element method [Ramani and Vaidyanathan 1995;Islam and Pramila 1999;Xu and Yagi 2004;Kumlutas and Tavman 2006], effective unit cell approach [Ganapathy et al 2005] and variational bounds [Hashin and Shtrikman 1962]. Very recently, a new framework for micromechanics modeling, namely variational asymptotic method for unit cell homogenization (VAMUCH) [Yu and Tang 2007a], has been introduced using two essential assumptions in the context of micromechanics for composites with an identifiable unit cell.…”

“…This new approach to micromechanical modeling has been successfully applied to predict thermomechanical properties including elastic properties, coefficients of thermal expansion, and specific heats [Yu and Tang 2007a;2007b]. In this work, we will use this approach to construct micromechanics models for effective thermal conductivity and the corresponding local fields such as temperature and heat flux within a unit cell.…”