Equivalent stiffness and compliance of curvilinear elastic fibers

In this paper the mechanical response of curvilinear, anisotropic and inhomogeneous elastic fibers is addressed. A general model accounting for fiber three-dimensional geometry, as well as for shear and torsional effects together with the extensional ones is rationally deduced from the three-dimensional elasticity by means of a constrained approach. Closed-form relationships describing chord-referred equivalent tangent stiffnesses and compliances are obtained, generalizing a number of classical results when non-conventional effects are included. Proposed numerical applications highlight the influence of fiber geometric parameters and shear deformability, enabling also to put in evidence some limits of usually-employed assumptions. Finally, the fiber mechanical response in a large-displacement regime is modelled through an incremental formulation. In this context, explicit relationships for tangent and secant equivalent along-the-chord elastic moduli are deduced in the case of planar fibers, leading to powerful and direct relationships, useful for the analysis and the design of advanced composite materials reinforced by curvilinear elastic fibers.