Elastic cylinder with helicoidal orthotropy: Theory and applications

The elasticity equations are derived for a helicoidally symmetric cylinder. An isotropic cylinder in the center is wrapped by a thick layer of a material with helicoidal anisotropy formed by spiraling of fibers around the core. Such structures are seen in springs, armored cables, trusses, composites, and biostructures. Structure under axial loading and pure bending are considered. The analytical expressions for displacements are obtained from equations of three-dimensional elasticity. The results are verified numerically via the finite element method. The stiffness matrix for an equivalent simplified model that links the displacements of the ends with the applied force and momentum is formulated. A coupling effect is revealed: the bar twists and elongates when axially loaded.The developed technique is used to explain an evolution of structure in nature. A pine trunk with spiralling grain is investigated from an optimization of a mechanical construction viewpoint. We model the trunk as an anisotropic cylinder with helicoidal symmetry and compute the displacements and stresses using nonlinear finite element model. An optimized spiralling angle of the grains accounts for a composite failure, transverse deflection, and fluid transportation. We suggest a combined criterion that could explain this spiralling morphology.