Electrostatic and magnetostatic properties of random materials

Scale dependence of electrostatic and magnetostatic properties is investigated in the setting of spatially random linear lossless materials with statistically homogeneous and spatially ergodic random microstructures. First, from the Hill-Mandel homogenization conditions adapted to electric and magnetic fields, uniform boundary conditions are formulated for a statistical volume element (SVE). From these conditions, there follow upper and lower mesoscale bounds on the macroscale (effective) electrical permittivity and magnetic permeability. Using computational electromagnetics methods, these bounds are obtained through numerical simulations for composites of two types: (i) two-dimensional (2D) random checkerboard (two-phase) microstructures and (ii) analogous 3D random (two-phase) media. The simulation results demonstrate a scale-dependent trend of these bounds toward the properties of a representative volume element (RVE). This transition from SVE to RVE is described using a scaling function dependent on the mesoscale $\delta$, the volume fraction $v_f$, and the property contrast $k$ between two phases. The scaling function is calibrated through fitting the data obtained from extensive simulations ($\sim 10000$) conducted over the aforementioned parameter space. The RVE size of a given microstructure can be estimated down to within any desired accuracy using this scaling function as parametrized by the contrast and the volume fraction of two phases.

Figure 1

Mesoscale bounds on effective (i.e., macroscopic) permittivity of random checkerboards with increasing volume fraction of second phase (a) $v_f=0.4$, (b) $v_f=0.5$, (c) $v_f=0.6$, and (d) $v_f=0.8$ for ${\epsilon }_1=1$ and ${\epsilon }_2=5$.

Figure 2

Mesoscale bounds on effective (i.e., macroscopic) permeability of a random checkerboard with increasing volume fraction of second phase (a) $v_f=0.5$ and (b) $v_f=0.8$ for ${\mu }_1=1$ and ${\mu }_2=3$.

Figure 3

Mesoscale bounds on effective (i.e., macroscopic) permeability of a random checkerboard with increasing volume fraction of second phase (a) $v_f=0.5$ and (b) $v_f=0.8$ for ${\mu }_1=1$ and ${\mu }_2=7$.

Figure 4

(a) Realization of a two phase random checkerboard for $L=16$. (b) Magnetic flux norm density (T). (c) Magnetic field norm (A/m).

Figure 5

Mesoscale bounds on effective (i.e., macroscopic) permittivity of a random checkerboard in (a) $x$ direction and (b) $y$ direction with $v_f=0.5$.

Figure 6

Two-phase random checkerboard for $\epsilon =\{1,10\}$ and $v_f=0.5$. (a) 3D two-phase random checkerboard on the $L\times L\times L$ lattices with $L=10$. (b) Electric flux density norm ($\mathrm{C}/{\mathrm{m}}^2$) for 3D two-phase random checkerboard, taken from the attached animation. (c) Comparison of scaling effects for $v_f=0.5$ in 2D and 3D random two-phase checkerboard.

Figure 7

Normalized scaling function $g\left(\delta \right)$ as a function of $\delta$ showing a good collapse of the stretched exponential function for (a) $v_f=0.5$ and (b) $k=7$.