Extension or Compression Induced Twisting in Fiber-Reinforced Nonlinearly Elastic Circular Cylinders

The problem of pure extension or compression of a solid circular fiber-reinforced cylinder is considered in the context of the theory of nonlinear elasticity. Here we are interested in a particular feature of this problem which only occurs for certain classes of incompressible anisotropic hyperelastic materials and is completely absent in the isotropic case. We show that pure extension or compression of a transversely isotropic fiber-reinforced cylinder can induce twisting of the cylinder. The transverse isotropy is due to a single family of helically wound extensible fibers distributed throughout the cylinder. For general pitch angles, it is shown that in the absence of an applied torsional moment, the cylinder subjected to an applied tensile or compressive force exhibits an induced twisting due to mechanical anisotropy. Such an induced twist does not occur when the fibers are oriented longitudinally or azimuthally. It is also shown that at a special pitch angle (a “magic” angle), the initial slope of the axial force and induced moment with respect to stretch is independent of the Young’s modulus of the fibers. The results on induced twist are described in detail for three specific strain-energy densities. These models are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and are consistent with the linearized theory of elasticity for fiber-reinforced materials. The results are illustrated on using elastic moduli for muscles based on experimental findings of other authors.