Thermal cracks — COD scalar and resistivity contribution

Before this page

Crack opening displacement and compliance, whose elastic derivation is transposed here step by step, and The Eshelby inclusion problem §3 for the transport form of the inclusion problem under the convention   .

This page transposes Crack opening displacement and compliance tensors to the 2nd-order (conductivity / diffusion / Darcy) problem, where the driving field is a vector rather than a symmetric 2-tensor.

1. What is specific to a crack

The general elasticity ↔ conductivity dictionary — fields, fluxes, moduli, Hill and concentration tensors — is stated once in Extension to conductivity, together with the    convention that makes every row of it sign-free. Only the rows below are proper to the flat-inclusion limit, and they are the reason this chapter exists at all.

Elasticity (4-tensor problem)	Conductivity (2-tensor problem)
COD tensor — 2-tensor (6 components)	COD scalar — scalar (1 component)
Kachanov factorization     ,   (elliptic),   (ribbon)	Rank-1 factorization     , same
Dilute contribution   (elliptic), (ribbon)	Dilute contribution   (elliptic), (ribbon)
Sextic acoustic polynomial (Masson, 2008)	Quadratic acoustic form    → analytical
Stress intensity factors	Heat-flux intensity factor — scalar (mode I analog only)
Displacement intensity factor	Temperature intensity factor — scalar

A scalar suffices because is a scalar and only   produces a jump — there are no sliding or shear modes to resolve, unlike the 6-component . The direction is always : the null space of     is spanned by for any (derivation below), so carries all the matrix anisotropy.

2. Crack geometry

Same geometric families as in the elasticity chapter:

• Elliptic cracks of aspect ratio    (EllipticCrack), including the circular penny   (PennyCrack).

• Ribbon cracks (tunnel cracks) of half-width (RibbonCrack), infinite along .

3. Hill tensor Taylor expansion and block-matrix limit

Mirror of the elasticity derivation of (Barthélémy, 2009). For a flat ellipsoidal inclusion of aspect ratio  , the 2nd-order Hill tensor is computed via the formula     (Giraud et al., 2019), where is assembled in the eigenbasis of   (right singular vectors of   ). As  :

• The null vector of   is   ,    .

• The corresponding Newton potential   while  , so

which is rank-1 along . The acoustic block is then

with      . Its null space is spanned by , since           (one-dimensional in the 2-tensor case — to be contrasted with the 3-dimensional null space in the elasticity problem). The limit

is rank-1 along , with    .

4. From the Green operator to

The section above reached by flattening an ellipsoid. This one reaches from the Fourier kernel instead — the transport twin of From the Green operator to — and it is the route that substantiates the row "Quadratic acoustic form → analytical" of the opening table: the order-2 problem has a closed form for every anisotropy, where elasticity needs its sextic to factorize. scripts/16_cod_symbolic_thermal.jl runs the whole chain with the six components of as free symbols.

The reduced kernel

The acoustic object    is a scalar, not a 3×3 matrix — that single fact is what separates this page from its elastic sibling.

The integral is elementary, at any anisotropy

Write the three contractions

so that     and     , with   . Then every power of cancels in the numerator:

a constant over a quadratic. Completing the square,     with    because   is the Gram determinant of restricted to and is positive definite. One elementary integral,   , closes it:

the second form following from              , valid for a symmetric , with    . So is a norm of   in the metric  , manifestly homogeneous of degree 1. No Stroh roots, no residues, no cubature — at any anisotropy. The elastic page's closing section, "what breaks in the general case", has no counterpart here.

The crack-plane integral: an effective ellipse

On the contour      of Barthélémy (2026) one has      , so with  

Diagonalizing this 2×2 form — a quadratic, so closed form — with eigenvalues    and rotating to its principal axes turns the integral into the isotropic one:

An arbitrarily anisotropic conductor therefore behaves, around a flat crack, as an isotropic one of conductivity containing a crack of effective aspect ratio . In general  : even a circular crack acquires an effective ellipticity from the anisotropy (  for   on the sample of the script). The specializations:

(crack frame)
isotropic
TI aligned,
any, ribbon ( ,  )	—	—

The aligned-TI entry is the geometric mean announced below, and the ribbon entry involves only the transverse 2×2 block — both structures the existing formulas already have.

Relation to the route

The two are equivalent, but they cost differently. The square-root transform needs the eigendecomposition of the 3×3 and the singular values of  ; the adjugate form needs one 2×2 eigenvalue problem. That is why the adjugate route is evaluable on symbolic scalars, where eigen and svdvals are not.

5. Closed-form COD scalar

All of these are the two boxed formulas above, evaluated on a more symmetric . Implementation: src/Cracks/cod_analytical_thermal.jl.

Anisotropic matrix — the general case

arbitrary,  , and     ,      and      the in-plane entries of  , so that  :

Being a 2×2 eigenvalue problem, this is closed form for every anisotropy — no symmetry assumption, and no need for the numerical route below.

Isotropic matrix

   gives   and  :

Penny limit  :  , which is exactly the surface average of the textbook jump of an insulating circular crack,  , divided by . Ribbon:  .

Transversely isotropic matrix aligned with the crack normal

  in the crack frame gives   and  , so the effective conductivity is the geometric mean:

Ribbon crack — 2D formula

Only the transverse block of enters, since  :

which reduces to   for an isotropic matrix.

The route

Historically these formulas were obtained by the square-root change of variable    (Giraud et al., 2019), which maps the problem to an isotropic one with a transformed crack shape: with    and its singular values    , the transformed aspect ratio is  . It is equivalent to the adjugate form —   — but it needs the eigendecomposition of the 3×3 and a singular-value decomposition, so it is Float64 only. The package uses the adjugate form, which is why the anisotropic thermal COD flows through automatic differentiation and symbolic scalars.

These prefactors changed in v0.4.0

Up to v0.3.2 the formulas above read and — too small by and . Every thermal , , and thermal DIF therefore changes. The elastic branch is unaffected, and so is fracture_permeability, whose conduction side does not go through these formulas.

6. Resistivity contribution and dilute correction

The size-independent crack resistivity contribution tensor is assembled from the scalar and the crack normal :

The geometric prefactors and are the same as in the elasticity case (they come from evaluated on the ellipsoidal and ribbon geometries — see Crack compliance and COD tensor). The rank-1 direction is always the crack normal , for any conductivity tensor .

inverse_property_contribution(crack, K₀) returns directly. The dilute resistivity correction to the effective resistivity of the cracked conductor is obtained via delta_inverse_property(crack, R, ε):

These reduce to the expressions of Sevostianov and Kachanov (2002), Kachanov and Sevostianov (2018).

7. Intensity factors

Thermal analogs of the elastic stress / displacement intensity factors:

Both are driven by   , the transport twin of the remote stress — the convention of the table above — so each formula is the elastic one with the symbols substituted.

• Heat-flux intensity factor (mode I analog, scalar): the singular crack-tip field scales as . For a ribbon crack of half-width ,    . For an elliptic crack the formula involves the tangent-ribbon COD ratio exactly as in the elasticity case ( replaces  ).

• Temperature intensity factor (scalar):    .

See sif and dif for the full signatures (dispatched on  ).

Where to go next

The next page, Degenerate shapes and extreme aspect ratios, closes the chapter: it evaluates the closed forms of this chapter and of the previous one at the shapes where they degenerate, the disc of a crack among them. The dictionary invoked in §1 is stated in full in Extension to conductivity, which lists what carries over from elasticity to the order-2 problem and the two points where the latter is simpler. The scalar and the resistivity contribution are called as described in the manual page Conductivity.