Effective properties of a composite are kind of averages of properties of their constituencies. Although these averages depend on structure, they vary in a certain range. It is highly desirable to know the range of variation of effective properties of the bounds. The bounds are independent of the geometry of the mixture.
Bounds on effective propertiesBesides these practical needs, bounds are expressed as new integral inequalities that use and highlight convexity and account for differential properties of the fields in materials, such as divergence- of curlfree nature of them.
Bounds are needed for structural optimization; they point to optimal structures. Bounds show the limits of possible improvements; they also challenge to attain the permitted values of effective parameters. Bounds allow to parametrize the range of variation of effective parameters. Bounds help in numerical procedures.
This problem has been studied for decades and a lot of results is obtained.
I am afraid that the technique of the derivation of the bounds and even the results themselves are too technical to be exposed at a web site.
For the detailed discussion, see VMSO.
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