Response of a helix made of a fractional viscoelastic material

Under investigation is the effective response of a helical strand (helix) made of a viscoelastic material governed by a constitutive relation with fractional-order (i.e., not integer-order) derivatives. The relation involves a 5-parameter model, which is well known to represent a real response much better than the conventional, integer-order models with the same number of parameters. We employ the correspondence principle of viscoelasticity to pass from the level of the strand's material to that of an effective, coupled axial-torsional response of the helix. The resulting fractional-order differential equation is more complex (i.e., it involves higher derivatives) than the constitutive equation governing the material per se. Also, the use of a fractional-order model results in more complexity of the helix' effective viscoelastic response than does an integer-order model with the same number of parameters. It is shown that shear deformations are more important than dilatational deformations. Lastly, a standard relaxation test is studied and an analytic solution is derived.