Statistical model for characterizing random microstructure of inclusion–matrix composites

Abstract

The variation of arrangement of micro-structural entities (i.e. inclusions) influences local properties of composites. Thus, there is a need to classify and quantify different micro-structural arrangements. In other words, it is necessary to identify descriptors that characterize the spatial dispersion of inclusions in random composites. On the other hand, Delaunay triangulation associated with an arbitrary set of points in a plane is unique which makes it a good candidate for generating such descriptors. This paper presents a framework for establishing a methodology for characterizing microstructure morphology in random composites and correlating that to local stress field. More specifically, in this paper we address three main issues: correlating microstructure morphology to local stress fields, effect of clustering of inclusions on statistical descriptors identified in the paper, and effect of number of realizations of statistical volume elements (SVEs) on statistical descriptors.

Appendices

Appendix 1

Table 4 Mean and standard deviation of probability distributions

Appendix 2

Table 5 Probability distribution library

Appendix 3: Self Consistent method [62]

Consider a system of aligned fibers along the x ₁ direction. Media of this type have symmetry properties in the plane normal to the fiber direction (x ₂ x ₃ directions). Such media are characterized as transversely isotropic materials. Such media have five independent constants (i.e. (i.e. ${E_{11}, \nu_{12}, K_{23},\mu_{12}, \mu_{23}}$). Where E, K and μ are the Young’s, bulk and shear modulii respectively and v is the Poisson’s ratio.

$$ E_{11}=(1-f)E_{m}+fE_{f}+\frac{4f(1-f)(\nu_f -\nu_m)^2 \mu_m} {\left[(1-f)\mu_m/(K_f+\mu_f/3)\right]+\left[f\mu_m/(K_m+\mu_m/3)\right]+1} $$

(A1)

$$ \mu_{12}=(1-f)\nu_{m}+f\nu_{f}+\frac{f(1-f)(\nu_f -\nu_m)\left[\mu_m/ (K_m+\mu_m/3)-\mu_m/(K_f+\mu_f/3)\right]}{\left[(1-f)\mu_m/(K_f+\mu_f/3\right] +\left[f\mu_m/(K_m+\mu_m/3)\right]+1} $$

(A2)

$$ K_{23} =K_m+\frac{\mu_m}{3}+\frac{f} {1/\left[K_f-K_m+\frac{1}{3}(\mu_f-\mu_m)\right]+(1-f)/\left[K_m+\frac{4 \mu_m}{3}\right]} $$

(A3)

$$\frac{\mu_{12}}{\mu_{m}}=\frac{\mu_f(1+f)+\mu_m(1-f)} {\mu_f(1-f)+\mu_m(1+f)} $$

(A4)

$$A\left(\frac{\mu_{23}}{\mu_m}\right)^2+2B\left(\frac{\mu 23}{\mu_m}\right) +C=0 $$

(A5)

where

$$ \begin{array}{ll} &A=3f(1-f)^2\left(\frac{\mu_f}{\mu_m}-1\right) \left(\frac{\mu_f}{\mu_m} =\eta_f\right)+\left[\frac{\mu_f}{\mu_m}\eta_m+\eta_f\eta_m=\left( \frac{\mu_f}{\mu_m}-\eta_f\right)f^3\right]\\ &*\left[f\eta_m\left(\frac{\mu_f}{\mu_m}-1\right)-\left(\frac{\mu_f}{\mu_m} \eta_m+1\right)\right]\\ &B=3f(1-f)^2\left(\frac{\mu_f}{\mu_m}-1\right) \left(\frac{\mu_f}{\mu_m} =\eta_f\right)+\frac{1}{2}\left[\frac{\mu_f}{\mu_m}\eta_m+ \left( \frac{\mu_f}{\mu_m}-1\right)f+1\right]\\ &*\left[(\eta_m-1)\left( \frac{\mu_f}{\mu_m}\eta_m+\eta_f\right)-2 \left(\frac{\mu_f}{\mu_m} \eta_m-\eta_f\right)f^3\right]\\ &+\frac{f}{2}(\eta_m+1)\left( \frac{\mu_f}{\mu_m}-1\right) \left[\left( \frac{\mu_f}{\mu_m}\eta_m+\eta_f\right)+ \left(\frac{\mu_f}{\mu_m} \eta_m-\eta_f\right)f^3\right]\\ &C=3f(1-f)^2\left(\frac{\mu_f}{\mu_m}-1\right) \left(\frac{\mu_f}{\mu_m} +\eta_f\right)+\left[\frac{\mu_f}{\mu_m}\eta_m+\left( \frac{\mu_f}{\mu_m}-1\right)f+1\right]\\ &*\left[\frac{\mu_f}{\mu_m}\eta_m+\left(\frac{\mu_f}{\mu_m} \eta_m-\eta_f\right)f^3\right]\\ \end{array} $$

(A6)

where f = Volume fraction η = 3–4ν.