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Bounds on effective properties

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 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.
  • Besides 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.



     

    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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