Self-repeating structures

Here we describe an elegant yet unexpected approach to the calculation of polycrystal structures suggested in [#!Nesi:1991:PCM!#].

Consider an anisotropic material $D$. Let us laminate this material with some anisotropic material $A$ in a laminate with a normal $n$ and a volume fraction $m_D$ of $D$. Denote the resulting composite by $L( D, A, m_D, n)$. Suppose that it is possible to choose the parameters $m_D \in (0,1)$, $n$, and the unknown material $A$ so that

$$L( D, A, m_D, n) = \Phi A \Phi ^T,$$ (23)

where $\Phi$ is the tensor of rotation through an angle. The formula 23 states that lamination of the given anisotropic material $D$ with a material with properties $A$ leads to the composite with the properties equal to $A$ rotated on some angle.

If the relation 23 is satisfied, then $A$ is a polycrystal assembled of variously oriented fragments of $D$.

To show this, we again use the process with an infinite number of steps. At the first step, we obtain the composite 23. At the second step, we laminate the composite $L( D, A, m_D, n) = \Phi A \Phi ^T$ with the rotated material $\Phi D \Phi ^T$ in the same way as in the first step. Obviously, the result of this lamination is $\Phi ^2 A ( \Phi ^2)^T$, i.e., the material $A$ rotated on the double angle. This step can be repeated infinitely many times; still, the resulting material remains equal to the rotated material $A$.

Let us calculate the volume fractions of materials in a composite obtained at the $i$th step. The material $A$ is used only on the first step of the procedure. Its volume fraction is equal to $1-m_D$ in the composite obtained after the first step. After the second step this fraction becomes $(1-m_D)^2$, because the material $D$ has been added to the mixture in the volume fraction $m_D$. Similarly, after the $i$th iteration the fraction of the initial ``seed'' $A$ is equal to $(1- m_D)^i$. When $i$ increases, the volume fraction of the ``seed'' $A$ becomes arbitrarily small, and almost all the volume is occupied with differently oriented fragments of material $D$. In the limit $i~ \rightarrow~ \infty$, the described composite becomes a polycrystal of $D$.

To find this polycrystal, we solve equation 23 for the unknown tensor $A$. The parameter $m_D$ and the rotation tensor $\Phi$ can be arbitrarily assigned. Any solution represents a polycrystal of $D$. The set of solutions $A(\Phi, m_D, n)$ represents a class of polycrystals of $D$ that can be obtained by the described method.

Here, an animated graph will show the process of formation of a repeated structure