MIT10_626S11_lec33

# MIT10_626S11_lec33 - VII Porous Media Lecture 33...

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Unformatted text preview: VII. Porous Media Lecture 33: Macroscopic Conductivity of Composites MIT Student An authoritative reference for effective properties of composite materials is Random Heterogeneous Materials by S. Torquato (Springer, 2002) 1. Conductivity of Composite Media We consider a volume of a composite material subjected to a 1-D applied electric field, and giving rise to a potential difference, Δ ϕ : Figure 1: Macroscopic composite volume element subjected to a 1-D electric field. By Ohm’s Law, V = IR and the average (macroscopic) current density is J = I / A . Substituting for I , these can be combined to give: J = L V L Δ φ = σ E (1) = AR L AR L where σ ≡ L is the mean macroscopic conductivity of the volume, and E = Δ φ is the mean AR L electric field. We are interested in how we can relate σ to the microscopic conductivity, σ ( x ) . Consider steady conduction in composite media consisting of N “phases” with isotropic constant conductivities σ i (i = 1,2,…N) and volume fractions Φ i (Figure 2), where each domain σ − 1 − 1 ≤ σ ≤ σ (3) where generally, N c = ∑ Φ i c i . (4) i = 1 Lecture 33: Conductivity of composite media 10.626 (2011) Bazant Figure 2: Piecewise composite material with domain conductivities σ i . satisfies Poisson’s equation and suitable boundary conditions enforcing continuity of normal current between domains i and j: ∇ 2 φ = (2) Boundary conditions: n ˆ ⋅ j i = n ˆ ⋅ j j (We can justify the use of Poisson’s equation by noting the definition of microscopic current density, j i = − σ i ∇ φ . Since conservation of charge requires ∇ ⋅ j i = , this implies that ∇ 2 φ = ). 2. Anisotropic Composites We will show in this section that the effective conductivity of an anisotropic composite, σ , lies within the Wiener bounds: The left-hand term − 1 σ − 1 is the harmonic mean of conductivity, while the right-hand term σ is the arithmetic mean....
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MIT10_626S11_lec33 - VII Porous Media Lecture 33...

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