Conductivity of periodic composites made of matrix and polydispersed aggregates

We present closed-form expressions of the effective conductivity of composites made of matrix and impenetrable polydispersed particles. The approach developed in a recent work has been generalized to treat periodic multiphase composite materials. Particular cases including orthorhombic arrangements of spherical or spheroidal inclusions and random distributions of spheres are considered where simple closed-form expressions are obtained. The case of random distributions allows us to recover Mori-Tanaka estimation after introducing partial structure factors as a statistical information on the microstructure.

Figure 1

2D images of RVEs with polydispersed hard aggregates (left: orthorhombic body centered arrangement, right: random distribution).

Figure 2

Grid of $\mathit{\xi }$ in 2D. The lattice sum ${\sum }_{\mathit{\xi }\in D}{I}_{\alpha }\left(\mathit{\xi }\right)I_{\beta }(-\mathit{\xi })$ in the volume element $D$ is approximated by $\rho {\int }_{D}P\left(\xi \right)dV.$

Figure 3

Comparison between present estimation [Eq. (16)] and exact solution (Fourier transform method with resolution $128\times 128\times 128)$. The microstructure is body-centered cubic with $R_1=2R_2,k_1=10k_0$, and $k_2=0.1k_0$. The maximal volume fraction is 0.62 when the spheres touch each other.

Figure 4

Example 1: Effective conductivity along two different directions $x_1,x_2$ estimated by Eq. (16). The unit cell is parallelepedic box of dimension $1\times 2\times 2$ with the spheres of properties $R_1=2R_2,k_1=10k_0$, and $k_2=0.1k_0$.

Figure 5

Example 2: Effective conductivity along two different directions $x_1,x_2$ estimated by Eq. (16). The unit cell is parallelepedic box of dimension $2\times 1\times 1$ with ellipsoidal inclusions. The ellipsoids are identical in size and shape with the semiaxes $2R,R$, and $R$ (i.e., $R_1=R_2=R,\chi =0.5)$ but different in properties $k_1=10k_0$ and $k_2=0.1k_0$.