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Equilibrium

Effective properties of a laminate are defined by the type of equilibrium in the medium. Here we derive these properties. Assume that the materials are linear, that is that the constitutive relation in th material is

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

1. 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.

2. 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

 (1)

as follows:
 (2)

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

Next: Continuity of field components Up: Simple laminates Previous: Simple laminates
Andre Cherkaev
2001-07-31