Origin of Unusually High Rigidity in Selected Helical Coil Structures

Origin of Unusually High Rigidity in Selected Helical Coil Structures David Tománek1, ∗ and Arthur G. Every2 1Physics and Astronomy Department, Michigan State University, East Lansing, Michigan 48824, USA 2School of Physics, University of the Witwatersrand, Private Bag 3, 2050 Johannesburg, South Africa (Dated: February 18, 2018) Using continuum elasticity theory, we describe the elastic behavior of helical coils with an asymmetric double-helix structure and identify conditions, under which they become very rigid. Theo- retical insight gained for macro-structures including a stretched telephone cord and an unsupported helical staircase is universal and of interest for the elastic behavior of helical structures on the micro- and nanometer scale. PACS numbers: 63.22.-m, 62.20.de, 62.25.Jk I. INTRODUCTION (a) (b) (c) Helical coil structures, ranging from a stretched telephone cord in Fig. 1(a) and an unsupported spiral staircase in Fig. 1(b) on the macro-scale to DNA and proteins L = on the micro-scale abound in Nature. Since their elastic inner [H2+(4πR)2]½ behavior is governed by the same laws of Physics inde- stringer pendent of scale, insight obtained on the macro-scale will H axis H benefit the understanding of helical micro- and nanos- R tructures. An intriguing example of unusual high rigidity i on the macro-scale, which has remained unexplained to outer date, is the unsupported all-wooden spiral staircase in the stringer Loretto Chapel1 in Santa Fe, New Mexico, constructed Ro around 1878 and shown in Fig. 1(b). In the following we explore the elastic behavior of this structure using continuum elasticity theory in order to identify the reason φ for its rigidity2{4. Since continuum elasticity theory applies from nanometer-sized fullerenes and nanotubes5{7 4πR Figure 1 to the macro-scale, we expect our approach to be useful to explore the rigidity of helical structures on the micro- FIG. 1. (a) Photograph of a coiled telephone cord with the same topology as an unsupported spiral staircase. (b) Re- and nanometer scale. touched photograph of the unsupported spiral staircase at the The use of continuum elasticity theory rather than the Loretto Chapel as constructed. (c) Trace of the helical inner case-specific finite-element method3 in this case is mo- or outer stringers of the staircase on the surface of a cylinder, tivated by our objective to identify the universal origin which can be unwrapped into a rectangle. of the high rigidity of the Loretto spiral staircase and related helical coils with an asymmetric double-helix structure. Theoretical insight gained for macro-structures in- can be equivalently described as a helicoid, or a “filled- cluding a stretched telephone cord and an unsupported in" helix, with finite nonzero inner and outer radii. helical staircase is universal and of interest for the elastic There is an extensive literature on the properties of he- behavior of helical structures on the micro- and nanome- 2 ter scale. lical springs, which dates back to Love's treatise , as well as the more recent Refs. [3,4] and literature cited therein. arXiv:1706.03288v1 [cond-mat.soft] 10 Jun 2017 Yet the compound helical structure of the Loretto staircase and related helical coils appears to have escaped II. ELASTIC BEHAVIOR OF A HELICAL COIL attention in publications so far. The rigidity of the con- necting steps provides the spiral staircase with a remark- From a Physics viewpoint, the spiral staircase of able degree of stiffness, as we show below. We propose Fig. 1(b) is a compression coil or a helical spring with a that this property is shared by similar helical structures rather high pitch. It can be characterized as an asymmet- independent of their scale. ric double-helix structure consisting of an inner stringer Any coil of radius R and total height H, such as the coil of radius Ri and an outer stringer coil of radius Ro, inner and the outer stringer of the staircase, lies within and spans two turns in total. The two stringer coils are a cylindrical wall of the same height, which can be un- connected by rigid steps of width Ro − Ri. The staircase wrapped onto a triangle, as shown in Fig. 1(c). Let us 2 2 2 2 first consider the case of incompressible inner and outer Ignoring δLi , δH and δRi terms in the limit of small stringers of height H that are separated by the constant deformations, we obtain distance Ro − Ri, which defines the step width. 2 2 2 2 According to Fig. 1(c), the equilibrium length Li of Li + 2LiδLi = H + 2HδH + 16π (Ri + 2RiδRi) : (9) the inner stringer coil with two turns is related to its Subtracting Eq. (1) from Eq. (9), we obtain equilibrium radius Ri and its equilibrium height H by 2 2 2 2 2 2LiδLi = 2HδH + 16π 2RiδRi : (10) Li = H + (4π) Ri : (1) The equivalent relation applies, of course, to the outer With the assumption δRi = δRo = δR, we can rewrite Eq. (10) and its counterpart for the outer stringer as stringer of radius Ro. We first consider an incompressible stringer of constant 2 LiδLi = HδH + 16π RiδR ; length Li, which is stretched axially by δH, causing the 2 LoδLo = HδH + 16π RoδR : (11) radius Ri to change by δRi. Then, 2 2 2 2 Combining all terms containing δR on one side, we can Li = (H + δH) + 16π (Ri + δRi) : (2) eliminate δR by dividing the two equations. This leads Subtracting Eq. (1) from Eq. (2) and ignoring δH2 and to δR2 terms in the limit of small deformations, we obtain L δL − HδH R i i i = i (12) 2 LoδLo − HδH Ro 2HδH + 16π (2RiδRi) = 0 (3) and, by rearranging terms, to and consequently Li H Lo H H δL − δH = δL − δH = κ ; (13) R i R R o R δRi = − 2 δH : (4) i i o o 16π Ri where κ is a variable to be determined by minimizing the Considering the outer helical stringer to behave inde- strain energy U of the deformed stringers. U is given by pendently for the moment, we expect the counterpart 2 2 of Eq. (4) 1 δLi 1 δLo U = C Li + C Lo H 2 Li 2 Lo 2 2 δRo = − 2 δH (5) 1 δL δL 16π Ro = C i + o ; (14) 2 Li Lo to describe the outer stringer. The step width should then change by where C is the force constant describing the elastic re- sponse of the stringers to stretching. For a macroscopic HδH 1 1 stringer with the Young's modulus E and the cross- δ(Ro − Ri) = δRo − δRi = − 2 − : (6) 16π Ro Ri sectional area A, C = EA. From Eq. (13) we get The only way to keep the step width constant, corre- HδH + κRi δLi = (15) sponding to δ(Ro − Ri) = 0, is to have either zero step Li width Ri = Ro, reducing the double-helix to a single- helix, or to suppress any change in height with δH = 0. for the inner stringer. Similarly, we get Even though each individual stringer coil can change HδH + κRo its height H while keeping its length L constant, the δLo = (16) assumed rigid connection between the inner and outer Lo stringers makes the staircase completely rigid. for the outer stringer and can now rewrite the strain en- Next, we relax the constraint that each stringer should ergy as maintain its length when the staircase changes its height H. Nevertheless, we will still maintain the assumption 1 (HδH + κR )2 (HδH + κR )2 U = C i + o : (17) of a fixed step width 3 3 2 Li Lo Ro − Ri = const: (7) The optimum value of κ is obtained from requiring @U=@κ = 0. This leads to that translates to δRi = δRo = δR. The compressible inner stringer helix will still be characterized by its equi- (HδH + κR )R (HδH + κR )R i i + o o = 0 ; (18) librium length Li and equilibrium radius Ri. Its defor- 3 3 L Lo mation caused by changes of its axial height by δH is i then described by which can be rewritten as 2 2 2 2 3 3 (Li + δLi) = (H + δH) + (4π) (Ri + δRi) : (8) (HδH + κRi)RiLo + (HδH + κRo)RoLi = 0 : (19) 3 We can regroup the terms to get For the sake of a fair comparison to a straight staircase with a slope given by tan('), as seen in Fig. 1(c), we do 3 3 2 3 2 3 HδH RiLo + RoLi + κ Ri Lo + RoLi = 0 ; (20) not use the pitch, but rather the local slope tan('i) = H=(4πRi) of the inner stringer to characterize how steep which yields expressions for the optimum values of κ, δLi the staircase is. The Loretto staircase is rather steep near and δLo. We get the inner stringer with tan('i) = 1:9, corresponding to ◦ 3 3 'i≈61 . RiLo + RoLi The stringers of the Loretto staircase have a rectan- κ = −HδH 2 3 2 3 ; (21) Ri Lo + RoLi gular cross-section of 6:4 cm×19:0 cm, and so we have for the cross-section area A = 121 cm2. Considering the elastic modulus E≈1010 N/m2 for wood along the HδH R L3 + R L3 i o o i grains, we obtain C = E·A = 1:2×108 J/m. Thus, the δLi = 1 − Ri 2 3 2 3 Li Ri Lo + RoLi spring constant of the double-helix structure describing 2 6 Li Ro the staircase could be as high as k = 4:8×10 N/m.

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