Theoretical limits to specific strength with hierarchical structures Specific strength (measured in units of pressure/density or speed$^2$ which in MKS there is a proposal for labelling it as the Yuri (meter/second)$^2$) is defined as tensile strength divided by material density
Hierarchical structures are on the other hand, using self-similar hierarchies of shapes (fiber bundles, honeycombs lattices, etc.) in order to improve the specific strength of structural elements made with certain material
I'm wondering, is there a theoretical limit to how much I can improve the specific strength of a given material by using 'clever' structural lattices? Is this theoretical limit given by the specific strength of a defect-free microscopic element, or can a hierarchical structure exceed the specific strength of the underlying defect-free material?
Take for example, carbon nanotubes, with specific strengths of $4.6 \times 10^7$ Yuri: Is theoretically possible make a structural cable with carbon nanotubes with the above given specific strength, and reorganize and structure the threads in such a way that the resulting element has higher specific strength? If not, is there a theorem forbidding this?
 A: No, it's not possible.
The general rule of strength in foams is $\rho^3\propto\sigma^2$ so if you cut the density by a factor of 4, then your strength would be cut by a factor of 8.
From a conceptual standpoint slicing a organized shape under a tensile load by a plane perpendicular to the load will show that there are $n$ discrete elements, all under a tensile load $\sigma_i$ in a direction $\theta_i$ from the normal of the plane, each with an area $a_i$
The total tensile force across the plane would be $\sum\limits_{i=1}^n \sigma_i a_i cos(\theta_i)$ which is trivially maximized when $\sigma_i=\sigma_y$ and $\theta_i=0$. So the force is limited to $\sigma_y\sum\limits_{i=1}^n a_i$
If the planar slice is a representative slice, then the porosity or density ratio would be:
$$\frac{\sum\limits_{i=1}^n a_i}A$$
So the maximum effective specific strength would be:
$$\frac{F}{\rho_{effective} A} = \frac{\sigma_y\sum\limits_{i=1}^n a_i}{\rho \frac{\sum\limits_{i=1}^n a_i}A A} = \frac{\sigma_y}{\rho}$$
The same as the original.
Now of course it's much more likely to get defect free materials the smaller and thinner they are, so in practice the yield strength (and certainly the fatigue strength) of one element in the structured material may be much much higher than the typical bulk material strength. So in practice it is probably possible.
