Hedgehogs

The structure is obtained by the differential scheme. The mixed materials are called N (for nucleus) and C (for core). The total volume fractions of these materials N and C, respectively, are:
m1 = nN and m2 = mC =1- mN.

mN =

mC =

The structural element consists of three parts: The inner part - nucleus, the exterior part - core and the intermediate part: -- needles. The nucleus and the core are made of pure materials N and C, respectively, and the intermediate part is made of their laminate mixtures. The structure is formed as follows.

First, the initial nucleus is formed from the material N. The relative size (the percent of the total volume) of this nucleus is FN. Of course, it is assumed that FN < mN .

Second, the differential scheme is used. Each step of the scheme consists of adding four infinitesimal layers to the existing structure. Layers are added to the structure from four directions in equal quantities, The normals are oriented by the angles of 0, 45, 90 and 135 degrees; these arrangements preserve the isotropy of the structure.

Each layer is an anisotropic laminate from the original materials N and C, and the inside layers are orthogonal to the exterior normal of the added layer (the "tiger tail structure, " see fig). Since the layers are tangent to the growing structure, the directions of the inner layers made from the core materials point to the center. They form the "needles" of the "hedgehog." The volume fraction FN of the nucleus material N decreases from one to zero and then stays equal to zero (needles become thinner and then disappear) .

The third (external) part of the structure corresponds to the pure core from material C . This part defines another structural parameter FC:-- the amount of the core material C used for the pure exterior envelope. Of course, it is assumed that FC < mC .

Insert structural parameters:

FN -- The mass fraction of the initial nucleus.
Insert FN = (default value FN = 0)
FC:-- the mass fraction of the pure exterior envelope.
Insert FC = (default value FC = 0).

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Comments

The dependency FC(M): The assumed rate of increase of the volume fraction of the material C with the increase of the mass M of the obtained structure, is linear:

FC(M) = 0    if M < FN *f1,

FC(M) = 1    if M > 1- f2* FC,

FC(M) =   FN *f1 + M ( )

Alternatively, one may use the scheme of effective medium theory. In this approach, one solves the problem for the "coated circles" geometry by separation of variables. The results coincide?