On elastic and viscoelastic helices

Abstract

Under consideration are helices made of linear elastic and viscoelastic materials, henceforth called elastic and viscoelastic helices. While the effective (macro level) mechanical response of any helix exhibits coupling of axial with torsional responses, of interest in this study is the derivation of that response from mechanics of a single helical strand (micro level). First, using the earlier results of Costello [Theory of Wire Rope (Springer, New York, 1997)], we develop explicit forms of all the effective constitutive coefficients of a linear elastic helix and plot their dependencies on the geometry and material constants of the strand. Next, using the correspondence principle of viscoelasticity, we derive differential equations of a helix at the macro level by considering three types of viscoelastic models of the strand: Kelvin, Maxwell, and Zener. In general, the helix macro level linear viscoelastic response is different in type (and more complex) from that of the viscoelastic material at the micro level. It is only in the singular case of the strand material's Poisson ratio equal to zero that the type of viscoelastic response is qualitatively the same (i.e. governed by the same order differential equation) as the viscoelastic response of the strand at the micro level. Consequently, direct viscoelastic generalizations of effective constitutive equations of helices, not based on analyses such as those presented here, are likely to be invalid.