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# Differential scheme

Developing the idea of laminates of a high rank, we come to a special class of laminates of infinite rank. Consider the following differential scheme: An infinitesimal portion of a pure material is added to the composite at each infinitesimal step. Such materials are described in many works, starting from [#!Bruggemann:1935:BVP!#]. They were rediscovered and systematically used in [#!Norris:1985:DSE!#,#!Lurie:1985:OPM!#,#!Avellaneda:1987:IHD!#], among other papers.

Consider the process of formation of a laminate composite. Suppose that a portion of material is added to the composite with the effective tensor and that the periodic cell has volume . The material is added in a infinitesimal thin periodic layers with the normal . The resulting composite cell has the volume and an effective properties tensor denoted by .

Differential scheme

Let us compute by using 8, where we set

The relation 8 becomes

where

The function depends on through , because is determined by .

As tends to zero, we obtain the differential equation

 (13)

This equation shows the rate of change of effective properties. It is integrated with respect to . The functions and determine the structure of the composite. We assume that different materials with properties are added to the composite in different times'' . We also assume that : The direction of laminates is generally changed during formation of the composite.

This scheme is effectively used in our Workshop to the Workshop to compute effective properties of the Hedgehogs, Pipe-brushes, end similar structures. It is flexible and robust.

Next: Effective medium theory Up: Modeling the structures Previous: Laminates of a rank
Andre Cherkaev
2001-07-31