Concentration and contribution tensors

Before this page

The Eshelby inclusion problem, whose inhomogeneity problem is solved here, and Hill polarization tensors, which supplies the the pivot formula is built on.

The Hill tensor determines the uniform strain inside an ellipsoidal inhomogeneity under a remote load, and from it MeanFieldHomogenization builds the four dilute concentration tensors of the Eshelby problem, together with the size-independent stiffness and compliance contribution tensors of Kachanov–Sevostianov.

Everything on this page is one chain, and each link is a separate entry point — which is what makes it possible to plug a morphology in halfway through (custom inclusions):

1. Pivot formula

For an inclusion of shape tensor and stiffness embedded in an infinite matrix of stiffness , the strain–strain concentration tensor is

where is the Hill polarization tensor (hill_tensor) and is the symmetric identity 4-tensor. The three other concentration tensors follow algebraically:

with  . The four functions exposed by MeanFieldHomogenization are:

Function	Return value
gradient_gradient_concentration(incl, C₁, C₀)
flux_gradient_concentration(incl, C₁, C₀)
gradient_flux_concentration(incl, C₁, C₀)
flux_flux_concentration(incl, C₁, C₀)

For phase of an RVE, computed in the reference medium is the dilute concentration tensor written on the pages of the schemes, and the subscript is appended after a comma when the four are needed for one phase, as in .

2. Contribution tensors

The stiffness contribution tensor (Kachanov and Sevostianov, 2018) is

and its dilute-scheme volume average is

for a dilute family of volume fraction . The dual compliance contribution tensor is

Functions: property_contribution, inverse_property_contribution, with density helpers delta_property and delta_inverse_property.

All of the above is one-site: the inclusion feels its neighbors only through the reference medium. Resolving the pairwise interaction explicitly replaces by the two-inclusion interaction tensor , of which is the self term — see the N-body schemes.

3. Cracks (Kachanov convention)

For flat cracks the crack-density convention (Budiansky–O'Connell) is used instead of a volume fraction. The same entry points apply, with the density replacing , and the compliance contribution built from the crack-opening-displacement tensor and the crack normal (Crack opening displacement):

• inverse_property_contribution(crack, C₀) returns the size-independent ;

• property_contribution(crack, C₀) returns (first order in the density, provided for API symmetry);

• delta_inverse_property(crack, H, ε) and delta_property(crack, N, ε) apply the appropriate or geometric prefactor.

4. Conductivity (2nd-order transport)

Every routine above serves conduction as well, under the same name: the methods on ::AbstractTens{2,3} properties are the elastic formulas with replaced by , by and by . Each routine also answers to its name in elasticity, and the contributions to their names in conduction; these accept one order only:

Generic	Elasticity	Conduction
gradient_gradient_concentration	strain_strain_concentration
flux_gradient_concentration	stress_strain_concentration
gradient_flux_concentration	strain_stress_concentration
flux_flux_concentration	stress_stress_concentration
property_contribution	stiffness_contribution	conductivity_contribution
inverse_property_contribution	compliance_contribution	resistivity_contribution
delta_property	delta_stiffness	delta_conductivity
delta_inverse_property	delta_compliance	delta_resistivity

5. Type-genericity

All four concentration and both contribution tensors are generic in the element type; the only requirement is that hill_tensor supports it.

6. Extending to user-defined inclusions

A concrete subtype of AbstractInclusion inherits the four concentration and the contribution tensors as soon as it provides hill_tensor. When has no convenient closed form (e.g. LayeredSphere), override gradient_gradient_concentration instead — the rest is derived algebraically.

See the developer guide Adding a new inclusion for a step-by-step recipe.

Where to go next

Everything on this page concerns one inclusion, alone in its reference medium. The next page, Two-inclusion interaction tensors, asks what one inclusion induces in another, the ingredient of the two schemes that resolve interactions pair by pair. The contribution tensors of §2 are what the Homogenization schemes add up, the crack form of §3 is derived in Crack opening displacement and compliance, and the gates through which a user-defined morphology enters the chain of this page are described in Custom inclusions.