A two-dimensional, higher-order, elasticity-based micromechanics model

Abstract

A new, two-dimensional (2D) homogenization theory is proposed. The theory utilizes a higher-order, elasticitybased cell model (ECM) analysis. The material microstructure is modeled as a 2D periodic array of unit cells where each unit cell is discretized into four subregions (or subcells). The analysis utilizes a (truncated) eigenfunction expansion of up to fifth order for the displacement field in each subcell. The governing equations for the theory are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through an order consistent with the order of the subcell displacement field. The formulation is carried out independently of any specified constitutive models for the behavior of the individual phases (in the sense that the general governing equations hold for any constitutive model). The fifth order theory is subsequently specialized to a third order theory. Additionally, the higher order analyzes reduce to a theory equivalent to the original 2D method of cells (MOC) theory when all higher order terms are eliminated. The proposed 2D theory is the companion theory to an equivalent 3D theory [T.O. Williams, A three-dimensional, higher-order, elasticity-based micromechanics model, Int. J. Solids Struc., in press].Comparison of the predicted bulk and local responses with published results indicates that the theory accurately predicts both types of responses. The high degree of agreement between the current theory results and published results is due to the correct incorporation of the coupling effects between the local fields.The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, two-dimensional microstructures.

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

“…The Elasticity-Based Cell Model (ECM) 5,11,12 starts with the same type of microstructural discretization as used by the original GMC theory, Fig. 1.…”

“…All of these global quantities can be expressed in a closed-form manner in terms of the mechanical and thermal concentration tensors which appear in Eqn. (11) together with the inelastic term D I(α,β,γ) , which are given in Ref. 30 as follows…”

Refined modeling of composite plates with in‐plane heterogeneity

“…With these effective properties, the analyst can replace the original heterogeneous structure with a homogeneous one and carry out structural analysis for global behavior. Referring to the existing literature, numerous approaches have been suggested, such as the self-consistent model [3,11,13], the variational approach [12,24], the method of cells [1,2,26,32], recursive cell method [5], mathematical homogenization theories [6,25,28], finite element approaches using conventional stress analysis of a representative volume element [30], variational asymptotic method for unit cell homogenization (VAMUCH) [34,36], and many others. See [15,16,19] for a review.…”

A three-dimensional, higher-order, elasticity-based micromechanics model

“…More recently, Williams (2004) has developed a elasticity based cell model (ECM) approach based on the use of eigenfunctions to describe the displacement field within a subcell which are part of a repeating unit cell. This work is based on a strong solution approach, i.e.…”

“…The purpose of the current work is to extend the 2D analysis of Williams (2004) to three dimensions. In particular, the formulation for a new three-dimensional micromechanical theory utilizing an higher-order, elasticity-based cell model (ECM) analysis is presented.…”