Here the vectors and represent the ``prime'' and ``dual'' fields, respectively, joined by the tensor of material's properties . These fields are constant if the composite is uniformly loaded or if the width of layers does to zero.

Examples

- For conducting composite,
the constitutive relations is the Ohm's law:

where is the current (dual variable), is the electrical field (prime variable), and is the conductivity tensor. Of course, we may reverse the Ohm's law:

then is a dual and is a prime variable. - For elastic materials, the constitutive relation is the Hook's law

where is the symmetric second-rank stress tensor, is the symmetric second-rank strain tensor, is the fourth-rank tensor of elastic stiffness, (:) is the scalar product in the tensor space.

Let us derive the effective properties of a two-component laminate. It is characterized by a normal and by volume fractions and of subdomains and that are occupied by the materials with tensor properties and . The composite is submerged into a uniform field .

The fields in the laminates are
piecewise constant:

where = constant().

The effective tensor of laminate
joins the averaged fields

where is the effective tensor of laminates which we are determining here.