Feather measurements
The mechanical forces involved in diving, plunging, and alighting are not accessible to direct measurement by current technologies in any reliable or representative way. Any such data would not be meaningfully correlated to the resulting yield or flexure of barbs and vanes during forceful interaction with water. However, the bending and flexing of materials of different shapes and sizes have been well described in engineering physics and it is from these considerations that a number of conclusions in relation to our hypothesis can be drawn.
When a force F is applied over the length of a single barb, the barb will bend in the direction of the applied force with its tip flexing over a distance S . This relates to the barb lengthl and barb radius r as
S = F . l3/2π .r4 . E (1)
where E stands for the Young’s elastic modulus of the feather keratin (Bonser and Purslow 1995, Greenwold et al. 2014). For the purpose of modeling, barbs are here assumed to be cylindrically shaped. When the force is applied to the vane, the flexural displacement of the tips of the vane per repeating unit 2(r + d ) can be written as
Sv = Fv .l3 . 2(r + d)/2π .r4 . E (2)
where the subscript v refers to the repeating unit of the vane. Rearrangement of Eqn 2 then yields
\(\pi\).E.Sv/Fv = (l/r)3 . (r + d)/r (2a)
Apart from π and the elastic modulus E , the left-hand side of Eqn 2a represents the extent of flexing of the tips of barbs per unit of force applied over the lengths of the barbs and measured over a distance 2(r + d ). For the bending of the entire vane,Fv needs to be considered for the number of repeating units per vane. Note that the right-hand side of the equation is made up of the feather variables l , r and d , which, unlike Sv and Fv , are easily and directly accessible to measurement. These considerations allow us to predict semi-quantitatively the bending of the vane under an applied force from the dimensions and spacing of the barbs alone.
The role of the barbules in resisting bending of the vane is to be considered in the light of their primary function, i.e., keeping the barbs from separating under an applied force and doing so by their hooks sliding in the flanges of the barbule next more distal. For this reason, but mostly for their small size, barbules are assumed to make only a minimal, if any, contribution to the over-all resistance to bending.
According to Eqn 2a, the bending of the vane of the contour feather under the impact of forces associated with diving or alighting - here referred to as the deflection parameter - consists of two factors: (1) the ratio of the length to the thickness of the barbs expressed asl/r and (2) the wettability parameter (r + d)/r . The first factor indicates that short and thick barbs make the vane stiff resisting bending, whereas long and thin barbs favor flexibility that promotes bending. The appearance of the wettability parameter in the deflection parameter shows that feathers resistant to water penetration also help prevent their bending, whereas highly water repellent feathers do not. Note that l/r enters the equation in the form of a third power which markedly enhances its contribution to the deflection parameter and dwarfs that of the other factor: over its range of 2.5 to 7 or higher, (r + d)/r increases by only a factor of 3 or 4, whereas (l/r )3 does so by about three orders of magnitude.