Composite material

A composite material contains distinct constituent phases arranged in a microstructure. Its macroscopic response depends on both phase properties and geometry, not just volume fractions. Anisotropic effective conductivity can arise even when every constituent has isotropic thermal conductivity.

An elastic laminate has planar perfectly bonded layers with a common normal. Its piecewise constant strains differ by $\mathrm{sym}(a\otimes n)$, while its normal traction is common to every layer. For positive elastic stiffness tensors, these conditions and the mean strain uniquely determine the layer strains: the difference of two solutions has zero mean and zero total elastic work, and positivity forces it to vanish.

The effective elastic energy is the minimum averaged strain energy density under prescribed affine boundary displacement. For differentiable stable elastic response without body force, varying the prescribed mean strain gives $D{W}^{\mathrm{eff}}(E)=⟨\sigma ⟩$. The result follows by integration by parts, because the interior variation with zero boundary data does no work against an equilibrated stress tensor.

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

The Reuss elastic approximation assumes uniform stress. The local strain is the derivative of the convex conjugate $W^{\ast }(S,x)$; its mean is $D⟨W^{\ast }⟩(S)$. Under invertible differentiable duality, this mean determines $S=D{W}_R(E)$. For positive linear elasticity, $C_R=⟨C^{-1}{⟩}^{-1}$. Uniform stress is generally not accompanied by a compatible local displacement field, so this approximation is distinguished from an exact heterogeneous solution.

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is $C_V=⟨C⟩$. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.

For symmetric stress in static elastic equilibrium without body force, write $E=⟨\epsilon ⟩$ and $S=⟨\sigma ⟩$. The difference between microscopic and macroscopic work is $|\Omega {|}^{-1}{\int }_{\partial \Omega }(\sigma -S)n\cdot (u-Ex)$. It vanishes for affine boundary displacement, uniform boundary traction, or matching periodic displacement fluctuations and opposite tractions on paired faces.

The volume fraction of a constituent is its occupied volume divided by total sample volume. Fractions sum to one for a partition into phases. In a statistically homogeneous medium the phase fraction also equals the probability that a typical point lies in that phase, under the appropriate spatial or ensemble averaging assumptions.

The indicator correlation gives the probability of specified phases at two points. Statistical homogeneity makes it depend on separation; isotropy makes it depend only on its length. In particular $p_{11}(0)=p_1$, so the connected correlation at zero separation is $p_1-p_1^2=p_1{p}_2$.

Effective conductivity describes the averaged response of a microscopic conductivity field to a macroscopic temperature or electric-potential gradient. Here the positive conductivity flux is $a\nabla u$; physical heat flux has the opposite sign. For periodic media it is computed by cell correctors, and can be a tensor even for isotropic constituents.

A self-consistent conductivity approximation sets the homogeneous comparison effective conductivity equal to the predicted response. For a uniformly oriented two-dimensional polycrystal with principal conductivities $a,b>0$, the comparison formula gives $C_{\mathrm{SC}}^2=ab$, so its positive solution is $\sqrt{ab}$. This is an approximation defined by a fixed-point closure; exactness requires additional geometric or statistical hypotheses.

For a scalar reference $b$, choose a trial polarization $q$ and solve $b\Delta v+\nabla \cdot q=0$ with zero Dirichlet data. The trial functional isA reference above all phase conductivities gives $J(u)\le {\mathcal{F}}_b(q)$; one below all of them gives $J(u)\ge {\mathcal{F}}_b(q)$. If the reference equals a phase conductivity, admissible polarization vanishes in that phase and the inverse is taken only on the contrast subspace.

For uniformly distributed rotations of a two-dimensional conductivity tensor with positive principal values $a\le b$, the Hashin-Shtrikman conductivity variational principle with isotropic comparison $cI$ yields $H(c)I$. Averaging a rotated diagonal tensor replaces it by half its trace times the identity. Since $H^{\prime }(c)=(a-b{)}^2/(a+b+2c{)}^2$, the admissible comparison endpoints give $a(a+3b)/(3a+b)\le C_{\mathrm{eff}}\le b(3a+b)/(a+3b)$ within the statistically isotropic polycrystal model.

These are the optimal volume-fraction bounds for three-dimensional statistically isotropic two-phase conductivity with $0<a_1<a_2$. Constant phase polarization trial fields reduce the variational principle to a scalar quadratic optimization. The upper trial functional is minimized; the lower one is maximized.

An affine trial temperature in the minimum-energy principle gives the upper tensor bound by the volume-averaged local conductivity. For an isotropic two-phase composite, $a^{\ast }\le p_1{a}_1+p_2{a}_2$. The same uniform-field variational idea supplies the familiar corresponding elastic upper bound.

For isotropic phases with planar interfaces normal to $n$, normal conductivity is the weighted harmonic mean $a_H=(\sum p_i/a_i{)}^{-1}$ and tangential conductivity is the weighted arithmetic mean $a_A=\sum p_i{a}_i$. Continuity of normal flux and tangential temperature gradient gives the two averages directly.

Conductivity homogenization replaces rapidly varying positive coefficients in a divergence-form elliptic equation by a macroscopic effective conductivity. Periodic media admit cell problems; statistically homogeneous media can instead be described by ensemble averages. The small-scale assumptions and boundary regime must be specified.

Relative to a uniform reference conductivity $a_0$, this polarization is the excess positive conductivity flux. The equation becomes $a_0\Delta u+\nabla \cdot P=0$. For a prescribed mean gradient $E$, averaging gives $a^{\ast }E=a_{0}E+⟨P⟩$. This sign convention differs from the physical heat-flux sign.

The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre $y$. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.

A sphere of conductivity $a_1$ in a host $a_0$ under a uniform incident gradient $E$ has interior gradient $3a_{0}E/(a_1+2a_0)$. Outside a sphere of radius $R$, its potential correction is $-R^3(a_1-a_0)(E\cdot x)/[(a_1+2a_0)|x{|}^3]$. Its polarization is $\beta E$ inside and zero outside. Temperature and normal-flux continuity determine these coefficients.

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is $a_0+p\beta +p^2{\beta }^2/(3a_0)+O(p^3)$. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.

If the first exciting-field correction at inclusion $i$ is $-\beta {\Lambda }_{i}E/p_i$, its revised polarization is $\beta {\chi }_i(E-\beta {\Lambda }_{i}E/p_i)$. Averaging weights it by $p_i$ and gives the displayed second-reflection approximation. Further dipole reflections and finite-size multipoles are neglected.

The periodic zero-mean corrector $w_j$ adjusts a unit imposed gradient so microscopic flux is divergence-free. For bounded uniformly elliptic coefficients, the weak problem has a unique solution modulo constants. The averaged flux is $a^{\ast }{e}_j=⟨a(e_j+\nabla w_j)⟩$. For symmetric coefficients this also gives a symmetric positive-definite effective tensor.

A periodic expansion with $y=x/ϵ$ separates the macroscopic variable from the cell variable. Its leading cell equation forces $u_0$ to be independent of $y$, the next equation gives $u_1=w_j(y){\partial }_{x_j}{u}_0$, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.

A bulk cell expansion generally leaves an oscillatory Dirichlet mismatch. A boundary corrector can solve the homogeneous oscillatory conductivity equation with boundary data cancelling that mismatch. It restores the prescribed physical boundary trace while leaving the leading bulk cell formula unchanged. Estimates depend on boundary regularity and on the coefficient assumptions.