Cracks

Before this page

Crack opening displacement and compliance, where the tensors and computed below are derived, and Thermal cracks for their conduction counterparts.

A crack is an inclusion of zero volume: the   limit of an ellipsoid. Two consequences run through this page — the amount of cracking is a density, not a volume fraction, and everything is written in the crack's own frame , whose third vector is the normal.

plotly_scene(shape_traces(PennyCrack(1.0)); uid = "man-crack-penny", height = 400,
    title = "Penny-shaped crack: a = b, c = 0, normal <u><i>n</i></u>")

plotly_scene(shape_traces(RibbonCrack(0.5)); uid = "man-crack-ribbon", height = 400,
    title = "Ribbon crack: half-width b, unbounded along <u><i>ℓ</i></u>")

The full set of shapes, tilted cracks included, is in The inclusion zoo; the geometry and the symbols are defined in Crack opening displacement.

using MeanFieldHomogenization, TensND
E, ν = 210.0, 0.3
k = E/(3*(1-2ν)); μ = E/(2*(1+ν))
C₀ = TensISO{3}(3k, 2μ)

# Penny-shaped crack — size-independent COD tensor B
pc = PennyCrack(1.0)
B  = cod_tensor(pc, C₀)

# Size-independent compliance contribution tensor H = (3/4) n̂ ⊗ˢ B ⊗ˢ n̂
H  = inverse_property_contribution(pc, C₀)

# Dilute compliance correction ΔS from the Budiansky density ε³ᵈ = N a b²
ε³ᵈ = 0.05
ΔS  = delta_inverse_property(pc, H, ε³ᵈ)      # = (4π/3) ε³ᵈ H

# Ribbon crack — same pattern, ε²ᵈ = N b² and ΔS = π ε²ᵈ H
r   = RibbonCrack(0.5)
H_r = inverse_property_contribution(r, C₀)    # H = (2/π) n̂ ⊗ˢ B ⊗ˢ n̂
ΔS_r = delta_inverse_property(r, H_r, 0.05)

# Thermal / conductivity — scalar COD b and rank-1 resistivity tensor R
K₀ = TensISO{3}(1.0)
b  = cod_tensor(pc, K₀)                  # scalar
R  = inverse_property_contribution(pc, K₀)     # R = (3/4) b (ŵ⊗ŵ)
ΔR = delta_inverse_property(pc, R, 0.05)      # = (4π/3) ε³ᵈ R

Cracks with finite interface stiffness (Sevostianov)

A flat crack carrying a spring-like interface elasticity with stiffness tensor (order 2,   symmetric — e.g. isotropic with a normal stiffness and a tangential one ) modifies the COD tensor via

where is the in-plane half-width, semi_minor(crack). The two limits are the familiar ones:   gives a traction-free crack (recovering ), and   a rigid bond ( , i.e. no crack at all).

# Elasticity : iso interface stiffness K = 5·𝟏
B_eff = cod_tensor(pc, C₀; K_interface = TensISO{3}(5.0))
H_eff = inverse_property_contribution(pc, C₀; K_interface = TensISO{3}(5.0))

# Conductivity (Kapitza scalar interface conductance α)
b_eff = cod_tensor(pc, K₀; α_interface = 1.0)
R_eff = inverse_property_contribution(pc, K₀; α_interface = 1.0)

When building an RVE for a homogenize call, attach the interface data as phase properties so the dispatcher picks them up automatically :

rve = RVE()
add_phase!(rve, :M, Ellipsoid(1.0), Dict(:C => C₀, :K => K₀); fraction = :rest)
add_phase!(rve, :CRACK, PennyCrack(1.0),
            Dict(:C => C₀,
                  :K_interface => TensISO{3}(5.0),  # elastic interface
                  :K => K₀,
                  :α_interface => 1.0);             # Kapitza scalar
            density = 0.10, symmetrize = :iso)

C_eff = homogenize(rve, MoriTanaka(), :C)
K_eff = homogenize(rve, MoriTanaka(), :K)

For SC-type schemes on cracked RVEs, which form applies depends on the orientation distribution:

• single orientation — the symmetric SelfConsistent raises a SingularException: its strain-concentration tensor degenerates for a phase with no volume and no orientation average to smooth it. Use AsymmetricSelfConsistent.

• isotropic or TI distribution (symmetrize = IsoSymmetrize() or TISymmetrize(axis)) — both forms run, and they solve different fixed points: the symmetric one iterates on the stiffness, the asymmetric one on the compliance. Both percolate, but not at the same crack density: for randomly oriented penny cracks the compliance form reaches zero at the classical Budiansky–O'Connell value   exactly, the stiffness one at   — both independent of . Between the two thresholds they therefore disagree qualitatively, not just numerically. Which one to pick is a modeling decision, worked out with numbers in Crack distributions: isotropic or parallel.

For the time-dependent (ALV) version with Rn(t,t') and Rt(t,t') aging interface kernels, see the Viscoelasticity manual. References: Sevostianov and Kachanov (2002), Barthélémy et al. (2019).

Where to go next

Layered inclusions leaves the ellipsoid family for composite patterns, which have no Hill tensor. The tutorial Cracks and crack density drives the calls above through the schemes, and Crack distributions: isotropic or parallel quantifies the choice between the two self-consistent forms discussed above.