Imperfect interfaces in a laminate, and the size effect
A planar interface is the curvature-free case of the spherical one, so the four interface models of LayeredSpheres carry over unchanged — and the algebra collapses to two additive terms whose effects are exactly complementary:
primal (
SpringInterface,KapitzaInterface) — a jump of the field. Moves the out-of-plane response and leaves the in-plane one untouched.dual (
MembraneInterface,SurfaceConductiveInterface) — a surface stiffness. The interfaces being planar,
Both enter with the weight 1/L, an interface density: at fixed volume fractions, doubling the period halves the correction. That is why a laminate stores thicknesses and not just fractions.
using MeanFieldHomogenization
using TensND
using Printf
using LinearAlgebra
using Plots
default(; left_margin = 5Plots.mm, bottom_margin = 5Plots.mm)
k₁, μ₁, k₂, μ₂ = 2.0, 0.8, 0.5, 0.2
C₁ = TensISO{3}(3k₁, 2μ₁)
C₂ = TensISO{3}(3k₂, 2μ₂)
K₁, K₂ = TensISO{3}(2.0), TensISO{3}(0.3)((2.0) 𝟏, (0.3) 𝟏)A bilayer parameterized by its interfaces and by an overall length scale.
function bilayer(; itf = PerfectInterface(), itf2 = PerfectInterface(), L = 1.0)
lam = Laminate(; normal = (0, 0, 1))
add_layer!(lam, :A, Dict(:C => C₁, :K => K₁); thickness = 0.3L, interface = itf)
add_layer!(lam, :B, Dict(:C => C₂, :K => K₂); thickness = 0.7L, interface = itf2)
return lam
end
C₃₃(lam) = Matrix(KM(homogenize(lam, Laminated(), :C)))[3, 3]
C₁₂(lam) = Matrix(KM(homogenize(lam, Laminated(), :C)))[6, 6] / 2
k_perp(lam) = Matrix(components(homogenize(lam, Laminated(), :K)))[3, 3]
k_par(lam) = Matrix(components(homogenize(lam, Laminated(), :K)))[1, 1]
ref = bilayer()
@printf "perfect bonding : C₃₃₃₃ = %.6f C₁₂₁₂ = %.6f\n" C₃₃(ref) C₁₂(ref)perfect bonding : C₃₃₃₃ = 0.989247 C₁₂₁₂ = 0.380000The primal interface acts out of plane only
SpringInterface(kn, kt) takes the interface stiffnesses; the keyword form SpringInterface(; sn, st) takes the matching compliances, which is what the type stores. sn = st = 0 is perfect bonding, and growing the compliance decouples the layers.
println("\nspring interface (kn = kt)")
println(" kn C₃₃₃₃ C₁₂₁₂")
println("─"^42)
for kn in (0.0, 1.0e-3, 1.0e-2, 1.0e-1, 1.0e0, 1.0e2)
cell = bilayer(; itf = SpringInterface(; sn = kn, st = kn))
@printf " %8.1e %10.6f %10.6f\n" kn C₃₃(cell) C₁₂(cell)
end
println(" → C₁₂₁₂ never moves: a spring is invisible in the plane.")
spring interface (kn = kt)
kn C₃₃₃₃ C₁₂₁₂
──────────────────────────────────────────
0.0e+00 0.989247 0.380000
1.0e-03 0.988270 0.380000
1.0e-02 0.979557 0.380000
1.0e-01 0.900196 0.380000
1.0e+00 0.497297 0.380000
1.0e+02 0.009900 0.380000
→ C₁₂₁₂ never moves: a spring is invisible in the plane.The out-of-plane law stays exact, with the compliance simply added to the series:
kn = 5.0e-2
lam = bilayer(; itf = SpringInterface(; sn = kn, st = kn))
λ(k, μ) = k - 2μ / 3
series = 0.3 / (λ(k₁, μ₁) + 2μ₁) + 0.7 / (λ(k₂, μ₂) + 2μ₂)
@printf "\nexact check : 1/C₃₃₃₃ = %.10f series + kn/L = %.10f\n" 1 / C₃₃(lam) (series + kn)
exact check : 1/C₃₃₃₃ = 1.0608695652 series + kn/L = 1.0608695652The dual interface acts in plane only
A Gurtin-Murdoch membrane of surface moduli (κs, μs) adds μs/L to C₁₂₁₂ — exactly, additively — and leaves C₃₃₃₃ alone.
println("\nmembrane interface (κs, μs = κs/2)")
println(" κs C₃₃₃₃ C₁₂₁₂ ΔC₁₂₁₂")
println("─"^54)
for κs in (0.0, 0.02, 0.05, 0.10, 0.20)
cell = bilayer(; itf = MembraneInterface(κs, κs / 2))
@printf " %8.3f %10.6f %10.6f %10.6f\n" κs C₃₃(cell) C₁₂(cell) (C₁₂(cell) - C₁₂(ref))
end
println(" → ΔC₁₂₁₂ = μs/L exactly, and C₃₃₃₃ never moves.")
membrane interface (κs, μs = κs/2)
κs C₃₃₃₃ C₁₂₁₂ ΔC₁₂₁₂
──────────────────────────────────────────────────────
0.000 0.989247 0.380000 0.000000
0.020 0.989247 0.390000 0.010000
0.050 0.989247 0.405000 0.025000
0.100 0.989247 0.430000 0.050000
0.200 0.989247 0.480000 0.100000
→ ΔC₁₂₁₂ = μs/L exactly, and C₃₃₃₃ never moves.The size effect
Keep the volume fractions fixed and scale the whole cell. With perfect bonding nothing happens — the classical result depends on fractions alone. With an interface, the correction decays like 1/L.
Ls = 10 .^ range(-1.5, 2.5; length = 60)
kn_fixed = 5.0e-2
c33_spring = [C₃₃(bilayer(; itf = SpringInterface(; sn = kn_fixed, st = kn_fixed), L = L)) for L in Ls]
c33_perf = [C₃₃(bilayer(; L = L)) for L in Ls]
p1 = plot(
Ls, c33_spring; xscale = :log10, lw = 2, label = "spring, kn = $(kn_fixed)",
xlabel = "period L", ylabel = "C₃₃₃₃", legend = :bottomright
)
plot!(p1, Ls, c33_perf; lw = 2, ls = :dash, label = "perfect bonding")
κs_fixed = 0.05
c12_memb = [C₁₂(bilayer(; itf = MembraneInterface(κs_fixed, κs_fixed / 2), L = L)) for L in Ls]
c12_perf = [C₁₂(bilayer(; L = L)) for L in Ls]
p2 = plot(
Ls, c12_memb; xscale = :log10, lw = 2, label = "membrane, κs = $(κs_fixed)",
xlabel = "period L", ylabel = "C₁₂₁₂", legend = :topright
)
plot!(p2, Ls, c12_perf; lw = 2, ls = :dash, label = "perfect bonding")
A thin cell is dominated by its interfaces; a thick one forgets them. The spring softens (compliance added), the membrane stiffens (stiffness added).
plot(p1, p2; layout = (1, 2), left_margin = 8Plots.mm, bottom_margin = 8Plots.mm, size = (900, 350))
Transport: Kapitza and the conductive surface layer
The same two roles, in closed form. An interfacial resistance adds to the series law; a conductive surface layer adds to the parallel one.
println("\ntransport")
println(" ρ (Kapitza) 1/k⊥ k∥")
println("─"^48)
for ρ in (0.0, 0.05, 0.2, 1.0)
cell = bilayer(; itf = KapitzaInterface(ρ))
@printf " %10.3f %10.6f %10.6f\n" ρ 1 / k_perp(cell) k_par(cell)
end
@printf " (series alone = %.6f)\n" (0.3 / 2.0 + 0.7 / 0.3)
println("\n ks (surface) 1/k⊥ k∥")
println("─"^48)
for ks in (0.0, 0.05, 0.2, 1.0)
cell = bilayer(; itf = SurfaceConductiveInterface(ks))
@printf " %10.3f %10.6f %10.6f\n" ks 1 / k_perp(cell) k_par(cell)
end
@printf " (parallel alone = %.6f)\n" (0.3 * 2.0 + 0.7 * 0.3)
transport
ρ (Kapitza) 1/k⊥ k∥
────────────────────────────────────────────────
0.000 2.483333 0.810000
0.050 2.533333 0.810000
0.200 2.683333 0.810000
1.000 3.483333 0.810000
(series alone = 2.483333)
ks (surface) 1/k⊥ k∥
────────────────────────────────────────────────
0.000 2.483333 0.810000
0.050 2.483333 0.860000
0.200 2.483333 1.010000
1.000 2.483333 1.810000
(parallel alone = 0.810000)The displacement jump
What distinguishes a spring interface from a softer layer: the compliance shows up as a discontinuity, not as a strain.
E = Tens([0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 1.0e-3]) # pure normal strain
lam = bilayer(; itf = SpringInterface(; sn = 1.0e-2, st = 0.0))
jump = interface_jump(lam, 1, E)
Σ33 = Matrix(components(homogenize(lam, Laminated(), :C) ⊡ E))[3, 3]
@printf "\nunder E₃₃ = 1e-3 : Σ₃₃ = %.6e, [u] = (%.2e, %.2e, %.3e)\n" Σ33 jump[1] jump[2] jump[3]
@printf " [u]₃ / (kn Σ₃₃) = %.12f\n" jump[3] / (1.0e-2 * Σ33)
under E₃₃ = 1e-3 : Σ₃₃ = 9.795571e-04, [u] = (0.00e+00, 0.00e+00, 9.796e-06)
[u]₃ / (kn Σ₃₃) = 1.000000000000This page was generated using Literate.jl.