Persistence of a pinch in a pipe

The response of low-dimensional solid objects combines geometry and physics in unusual ways, exemplified in structures of great utility such as a thin-walled tube that is ubiquitous in nature and technology. Here we provide a consequence of this confluence of geometry and physics in tubular structures: our analysis shows that the persistence of a localized pinch in an elastic pipe whose effect decays as an oscillatory exponential with a persistence length that diverges as the thickness of the tube vanishes, which we confirm using simulations and simple experiments. The result is more a consequence of geometry than material properties, and is thus equally applicable to carbon nanotubes as it is to oil pipelines. Copyright c © EPLA, 2007 Introduction and scaling. – Thin sheets and membranes are ubiquitous in nature and technology over a range of length scales. Their mechanical behaviour is strongly determined by the geometrical separation of scales that allows out-of-plane bending deformations to be much softer than in-plane shearing or stretching deformations. This leads to all manner of interesting phenomena such as the wrinkling, draping and crumpling of thin sheets [1–5]. In all these situations, the primary focus has been on understanding how small scales arise from larger ones as energy and stress are quite literally focused strongly in response to imposed far field boundary conditions, and some progress has been made in understanding fine scales and singularities arising in these systems. A different set of questions that one can ask of these same thin structures is the effective geometric stiffening on scales much larger than the thickness induced purely by geometry. For example, it is well known [6] that a complete sphere is perfectly rigid with respect to isometric deformations, i.e. it cannot be deformed without stretching which would lead to changes in the metric. Similarly, a cylindrical sheet is much stiffer than its flat counterpart with the same dimensions and explains the relatively large effective stiffness of a carpenter’s rule. Indeed the separation of geometric scales inherent in these structures gives them their large specific stiffness (stiffness per unit weight) and leads to strongly geometrical modes of deformation that (a)E-mail: [email protected] involve global buckling on the one hand and localized modes on the other when subject to pressure, compression, twisting and bending. Here we focus on a peculiar global mode of deformation in cylindrical structures which are ubiquitous in nature and technology over a range of length scales, from carbon nanotubes and cytoskeletal microtubules to oil pipelines and grain silos. Our starting point is the simple observation that when a paper or plastic drinking straw of length L, radius R and thickness t (L R t) is pinched at an end it becomes elliptical locally, as shown in fig. 1 (try it yourself); however the deformation of the straw persists over a length that is much larger than the radius of the straw. This raises a natural question: what is the persistence length of a pinch? Indeed even when a naturally flat relatively narrow elastic plate of width D and length L( D) (equivalently, we could consider a periodically pinched sheet such as a drape) is pinched at an end to make it slightly narrower so that the amplitude of the pinch is a, casual observations show that the persistence of the pinch is much larger than the width of the strip. To understand this persistence, we note that the dominant component of the curvature is transverse to the length of the sheet and scales as as a/D. Then the bending energy in the sheet [7] scales as Ub ∼Et3(a/D2)2Dlp, where E is Young’s modulus of the material of the plate, and lp is the unknown persistence length of a pinch. Over the scale of this persistence length, the sheet is weakly curved in two orthogonal directions so that it must be stretched (since the Gauss curvature of a