Curvature stiffens fish fins [with video]

A fish fin is thin, and yet not floppy when it pushes on the surrounding water for propulsion. We show that fish could modulate fin stiffness by changing its curvature. The rays contribute to its bending stiffness in the way an elastic beam would, but it is the curvature that substantially increases its stiffness by engaging the intervening membrane. Curvature, however, could be embedded within the fin structure and need not be externally visible.

HFSP Young Investigator Grant holders Mahesh Bandi, Shreyas Mandre and Madhusudhan Venkadesan and colleagues
authored on Thu, 01 June 2017

The most diverse group of fish, the Actinopterygii, use rayed fins for propulsion, comprised of bony rays with intervening membranes. The fin's overall stiffness underlies its ability to maintain shape under hydrodynamic load, and is therefore central to its function in propulsion. A floppy fin would deform too much and fail to push sufficiently hard on the surrounding water [1]. Imagine trying to stir a cup of water with a soft and floppy paddle instead of a stiff stirrer! The topic of this study is to uncover the biomechanical basis for how fish fins meet this challenging demand.

Figure: Illustration of morphological and functional curvature. (a) Schematic representation of the cross-section of a fin with a morphological curvature. The rectangles represent cross-section of rays, the connecting segments represent the membranes, and the arrows denote the ray bending directions. The fin when curved transverse to the rays, misaligns the ray bending directions. (b) Schematic representation of a fin with a functional curvature. Note that this fin is morphologically flat yet the directions of the soft bending axes of the cross section are misaligned, constituting the functional curvature. (c) View of the internal structure of an isolated mackerel pectoral fin held geometrically flat (obtained using X-ray imaging). The bony rays appear bright, whereas the soft elastic membranes appear dark. (d) Ray cross-sections taken at SS’ shown in (c). The cross-hair depict approximations to the principal axes and magnitudes of the ray bending rigidities. The difference in angles between any two adjacent rays constitutes the functional curvature in this sample fin.

The current understanding is that the bony rays are the girders that are primarily responsible for the fin's stiffness [2]. We show that upon curling in a direction transverse to the length of the rays, the fin stiffens beyond the value determined by the rays alone. Such stiffening shares it mechanism with a common observation: a sheet of paper that droops under its own weight stiffens when curled along the transverse direction. Both the fin and the paper sheet stiffen because curvature couples the out-of-plane bending and in-plane stretching deformations. However, the details differ because the paper sheet is uniform in composition, whereas the fin is made of bony rays connected by a membrane. The cross-section of these rays are anisotropic; they are thinner and more flexible in one specific direction. When this flexible direction is predominantly perpendicular to the fin's surface, transversally curling the fin would cause the flexible direction of adjacent rays to be mismatched. Consequently, the rays have a tendency to splay apart when loaded, thereby engaging the interconnecting membrane and stiffening the fin. (See video below for an illustration.)

The essential property of the rays is that their softer bending direction be misaligned with that of their neighbors. Clearly, this could be achieved by transversally curling the fin. But, that is not the only way to achieve the stiffening behavior. X-ray CT imaging shows that such misalignment exists (e.g. in a mackerel pectoral fin, see figure) even when it is held flat. In other words, the essential principle that makes a curved fin stiffer is also realized by virtue of the structure embedded within a geometrically flat fin. This possibility of a “functional curvature” is distinct from that of a “morphological curvature” as shown in the figure. Therefore, the thin membrane that connects the fin rays is an equally important structural element, even when the fin is morphologically flat.

Fish fins are fascinating organs. Although their primary purpose is propulsion, they are mechanically passive, i.e. they have no biological motors (muscles) embedded within them. And yet this fin structure is found in more than 99% of fish. A prevalent hypothesis [3] is that ray-finned fish owe their ecological success to their ability to alter fin stiffness dynamically, which modulates the force they generate on the water, which in turn may underlie their maneuverability. However, without any embedded muscles within the fin, this hypothesis lacked a credible mechanism for remotely altering the fin's stiffness. Curvature-induced stiffening offers such a mechanism, and thereby opens the way for testing the hypothesis on the dominance of ray-finned fishes. These results also inspire design of robotic appendages for maneuverable aquatic propulsion, where the mechanical principle, rather than external appearance, is mimicked for functionality.

Video caption: An illustration of the mechanism underlying the curvature-induced stiffening of a fish fin using a physical mimic. The curvature couples the out-of-plane bending deformation of the fin rays to the in-plane stretching of the membrane, thereby stiffening the fin.

Reference

Curvature-induced stiffening of a fish fin. Khoi Nguyen, Ning Yu, Mahesh M. Bandi, Madhusudhan Venkadesan & Shreyas Mandre, Journal of Royal Society Interface, 20170247. http://dx.doi.org/10.1098/rsif.2017.0247

Other references

[1] Kinematic condition for maximizing the thrust of a robotic fish using a compliant caudal fin. Park, Y.-J., Jeong, U., Lee, J., Kwon, S.-R., Kim, H.-Y., and Cho, K.-J. 2012. IEEE Transactions on Robotics 28, pp. 1216–1227. doi: http://dx.doi.org/10.1109/TRO.2012.2205490.

[2] The trout tail fin: a self-cambering hydrofoil.  McCutchen, C. 1970. Journal of Biomechanics 3,pp. 271–281. doi: http://dx.doi.org/10.1016/0021-9290(70)90029-1.

[3] Mechanical properties of fish tail joints.  Videler, J. 1977. Fortschritte der Zoologie 24.2-3, pp. 183–194.

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