De-homogenization

approach, there are several shortcomings, which, if overcome, may provide increasingly accurate predictions of ultimate properties. The most apparent shortcomings of a homogenized analysis are: the modeling of fictitious interfaces; stresses and strains in the homogenized continuum exist in neither the fiber phase nor the matrix phase and the loss of the residual micromechanical thermal stress field due to a temperature change. The current chapter will focus on the latter two shortcomings by predicting the strain state within the fiber and matrix phases using a process referred to as micromechanical enhancement (Gosse & Christensen, 2001; Buchanan et al., 2009).

The role of micromechanical enhancement is to provide a computationally efficient micromechanics analysis that includes congruent homogenization and de-homogenization steps. The current approach uses a single finite element model subjected to canonical states of deformation to provide the information needed for both homogenization (micromechanics) and de-homogenization (micromechanical enhancement) and is thus considered to be a self-consistent approach. This chapter is primarily focused on building a general framework required to obtain self-consistent results and transferring information between micro and macro scale composite models. Through the use of a simple example problem we will address the process used to recover strains at the micro-scale resulting from both mechanical loading and residual thermal stresses.

First, consider a representative volume element subjected to an arbitrary state of average mechanical strain, {є – aAT}, where є is the average total strain and a, the vector of effective coefficients of thermal expansion for the homogenized medium. As shown in Equation 7, the state of strain at a prescribed point within the representative volume element is calculated using an influence function formulation.

єІ – aiAT = Mij ( – ajAT)+ AikAT (i, j = 1- 6) (7)

The matrix Mkj and the vector Ak are the influence function matrix and the thermal superposition vector of strain for the kth point in the representative volume element, respectively (Gosse & Christensen, 2001; Buchanan et al., 2009).

The components of the influence function matrix can be determined uniquely, in a fashion similar to determining the stiffness matrix for the effective homogeneous medium, by prescribing a canonical state of deformation in the representative volume element and carrying out three-dimensional finite element analyses to determine the components of the strain tensor at the specified point, k. For example, let Є1 Ф 0 , in the absence of the other five strain components and with no thermal loading. Shown in Equation 8, the first column of the influence matrix can be determined by relating the local strain to the average axial strain by using Equation 7.

) = мЄ or Mn = Ує (i = 1 – 6) (8)

Note that a single finite-element analysis with boundary conditions that meet the condition, є1 Ф 0 with Є2-6 = 0, yields six of the 36 coefficients in the influence function matrix at any point within the representative volume element. A total of six finite-element analyses are required to completely determine terms of Mikj at any point within the domain for a given representative volume element geometry.

Calculation of the thermal superposition vector requires an additional finite element analysis in which the unit cell is subjected to a temperature change with the constraint that the average mechanical strain vanishes, i. e. {є-аАТ} = 0. Equation 9 gives the thermal

superposition vector obtained by inserting this constraint into Equation 7.

Ak = ei – aiAT (i=1 6)

AT v ’

(9)