## Geometry and mechanics of uniform *n*-plies: from engineering ropes to biological filaments

S. Neukirch & G.H.M. van der Heijden

We study the mechanics of uniform *n*-plies, correcting and extending
previous work in the literature. An *n*-ply is the structure formed when
*n* pretwisted strands coil around one another in helical fashion. Such
structures are encountered widely in engineering (mooring ropes, power
lines) and biology (DNA, proteins). We first show that the well-known lock-up
phenomenon for *n=2*, described by a pitchfork bifurcation, gets unfolded
for higher *n*. Geometrically, *n*-plies with *n>2* are all
found to behave qualitatively the same. Next, using elastic rod theory, we
consider the mechanics of *n*-plies, allowing for axial end forces and
end moments while ignoring friction. An exact expression for the interstrand
pressure force is derived, which is used to investigate the onset of strand
separation in plied structures. After defining suitable displacements we also
give an alternative variational formulation and derive (nonlinear)
constitutive relationships for torsion and extension (including their
coupling) of the overall ply. For a realistic loading problem in which the
ends are not free to rotate one needs to consider the topological
conservation law, and we show how the concepts of link and writhe can be
extended to *n*-plies.
keywords: multi-strand plies, rod mechanics, end loads, constitutive
relations, twist-stretch coupling, strand separation, birdcaging, helix,
link, writhe, wire rope, DNA, proteins

*Journal of Elasticity* **69**, 41-72 (2002)