Light Like a Feather: A Fibrous Natural Composite with a Shape Changing from Round to Square

Only seldom are square/rectangular shapes found in nature. One notable exception is the bird feather rachis, which raises the question: why is the proximal base round but the distal end square? Herein, it is uncovered that, given the same area, square cross sections show higher bending rigidity and are superior in maintaining the original shape, whereas circular sections ovalize upon flexing. This circular‐to‐square shape change increases the ability of the flight feathers to resist flexure while minimizes the weight along the shaft length. The walls are themselves a heterogeneous composite with the fiber arrangements adjusted to the local stress requirements: the dorsal and ventral regions are composed of longitudinal and circumferential fibers, while lateral walls consist of crossed fibers. This natural avian design is ready to be reproduced, and it is anticipated that the knowledge gained from this work will inspire new materials and structures for, e.g., manned/unmanned aerial vehicles.

For three-point bending, the flexural rigidity is calculated as [1] : where E is the flexural modulus, F, δ and L are the flexural load, flexural deflection and supporting span. The flexural modulus is thus: where I is the initial area moment of inertia of the tubes ( II. Ovalization and pure bending of thin polymeric circular tubes. Figure S1. Pure bending of straws and the Brazier effect: uniform bending moment is applied by loading on the two ends of the thin hollow circular tubes, and the original circular cross section (dashed circles) at the middle of the tube deforms into an oval shape. The degree of ovalization, ζ, is characterized by the ratio of δ over r.
III. Theoretical derivation of bending curvature as a function of deflection for three-point bending. Figure S2. Free body diagram of a simply loaded beam with concentrated load at center.
From a simply supported beam (length L) with concentrated load (P) at center, as shown in Figure S2, the vertical deflection, y(x) is: We assume flexural rigidity (EI) constant, then the bending curvature is: At x=L/2, P = δ . Plugging these two into Eq. (S4), we obtain the bending curvature at the center of the beam as: For each measured δ, we can calculate the curvature; then, using the theory of ovalization (Eqns. (3), (4) in main text), we obtain the degree of ovalization and corresponding area moment of inertia, , . At the center of beam, the load is related to deflection δ by, We can calculate the force for each measured deflection , and therefore obtain the theoretical flexural load-deflection curves incorporating ovalization.
IV. Fibers at micrometer scale in the feather shaft cortex usually observed under a scanning electron microscope.

VI. Nanoindentation results and analysis
The differences in nanoindentation modulus and hardness on the feather cortex are the results of changes in fiber orientation. In the simple case of a composite with uniaxially aligned fibers, the modulus with loading parallel to the fiber orientation is higher than that with loading perpendicular to them: loading parallel to fibers will allow the force endured by fibers and place them under compression ( Figure S6a). Fiber buckling is impeded by the surrounding amorphous phase. When loading is applied, it tends to separate the fibers and a portion of the force goes into the softer matrix ( Figure S6b).
Nanoidentation measurements made on the transverse section along dorsal cortex at the calamus (#2) and the distal rachis (#6) are shown in Figure S6c  VII. Calculation of the reduced Young's modulus and hardness from nanoindentation.
The initial unloading curve (90~50% data) is fitted by a power law, and the contact stiffness, S, is given by the slope of this line: where P is the indentation load and H the indentation depth. The reduced Young's modulus E r is defined as where is β a constant depending on geometry of the indenter ( [2] ; 0.3 for fingernails [3] ; 0.37-0.48 for hair keratin [4] . The hardness is determined by Eqn. (S10): = (S10) where P max and A c are the maximum load and the projected contact area. An average of five consistent measurements for each position and orientation was used for analysis.