Magic Angles and Force Transmission in Helically Wrapped Worms

Many animal structures and appendages are pressurized, cylindrical, and helically wrapped with fibers. Squid tentacles, elephant trunks, echinoderm tube feet, notochords, arteries, and the bodies of sharks, nematodes, and annelids are all helically wrapped and their structural function depends on force transmission by this wrapping. Classical understanding of helically-wrapped cylinders in biology originates with calculations and concepts developed in the context of worm bodies, particularly nemerteans, turbellarians, and nematodes. This work recognized the geometric effects of the fiber angle on the cylinder volume and the fiber stretch. Subsequent work on tongues, tentacles, trunks, and polychaete worms used cylinder geometry to infer mechanical advantage. However, these studies did not explicitly consider forces and hence are limited in developing a general understanding for the mechanics of soft skeletons. Recently, a more precise theory was developed that incorporates force transmission, enhancing an understanding of mechanical and displacement advantage in these biological hydrostats. Some general insights are derivable from this foundation. A pressurized cylinder has a hoop stress that is twice the longitudinal stress and a crossed-helical wrapping of fibers can carry all of those stresses, if the fibers are at the magic angle of 54.7°. In a variable-volume cylinder with inextensible fibers, the magic angle corresponds to the maximum enclosed volume, but in a constant-volume cylinder, shape change necessitates stretching of the helical fibers. Strain and the length-to-radius ratio (aspect ratio) are functions of fiber angle, but aspect ratio also depends on the number of fiber wrappings. A constant-volume cylinder at any other angle will generate higher pressure, stretch helical fibers, store energy, cause shape changes, and possibly generate axial or radial external forces. Due to geometry, the mechanical advantage of force transmission from longitudinal muscles to radial output forces is not the inverse of the mechanical advantage of circumferential muscles transmitting force to the axial direction. Additionally, the mechanical advantage of short, wide cylinders is higher in extension than that of longer, thinner cylinders, suggesting that short and wide earthworm and polychaete segments have a higher mechanical advantage in generating axial forces during burrowing. Helical fibers can reduce the mechanical advantage during extension and retraction because energy is stored in the fibers; stiffer fibers reduce the mechanical advantage more; and some worms have helical muscles that might allow behavioral modulation of mechanical advantage. These inferences demonstrate insights that may be gleaned from explicit considerations of the mechanical principles of hydrostatic skeletons.