Session H52: Focus Session: Extreme Mechanics - Rods

Dynamic curling of a naturally curved Elastica

We consider the motion of a naturally curved Elastica that has been flattened onto a hard surface. When it is released from one end, the Elastica lifts off the surface and curls dynamically into a moving spiral. The motion is governed by inertia, bending and geometric nonlinearity. At long times, the dynamics follows a self-similar regime: the size of the spiral grows like the cubic root of time, while the velocity of the front reaches a constant value. The asymptotic velocity is derived analytically, and compared to numerical simulations and to experiments.   

The Mechanics of Curly Hair

We explore the oft-neglected role of intrinsic natural curvature on the mechanics of elastic rods. Our testbed, a hanging hair, is a deceivingly simple system that exhibits complex mechanics and geometrically nonlinear behavior. Through a combination of precision desktop-scale experiments, numerical simulations, and theoretical analysis, we seek physical insight into the nontrivial configurations adopted by a naturally curved elastic rod that is suspended under its own weight. In particular, we aim to gain predictive understanding of the transition from planar to non-planar solutions as well as the localization of torsion in the non-planar configurations. The experimentally observed behavior of our custom-fabricated naturally curved rods is captured well by simulations and is rationalized through scaling arguments.   

The Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles

From Leonardo to the Brothers Grimm our fascination with hair has endured in art and science. Yet, a quantitative understanding of the shapes of a hair bundles has been lacking. Here we combine experiment and theory to propose an answer to the most basic question: What is the shape of a ponytail? A model for the shape of hair bundles is developed from the perspective of statistical physics, treating individual fibers as elastic filaments with random intrinsic curvatures. The combined effects of bending elasticity, gravity, and bundle compressibility are recast as a differential equation for the envelope of a bundle, in which the compressibility enters through an ``equation of state.'' From this, we identify the balance of forces in various regions of the ponytail, extract the equation of state from analysis of ponytail shapes, and relate the observed pressure to the measured random curvatures of individual hairs.   

Following the equilibria of slender elastic rods

We present a novel continuation method to characterize and quantify the equilibria of elastic rods under large geometrically nonlinear displacements and rotations. To describe the kinematics we exploit the synthetic power and computational efficiency of quaternions. The energetics of bending, stretching and torsion are all taken into account to derive the equilibrium equations which we solve using an asymptotic numerical continuation method. This provides access to the full set of analytical equilibrium branches (stable and unstable), a.k.a bifurcation diagrams. This is in contrast with the individual solution points attained by classic energy minimization or predictor-corrector techniques. We challenge our numerics for the specific problem of an extremely twisted naturally curved rod and perform a detailed comparison against a precision desktop-scale experiments. The quantification of the underlying 3D buckling instabilities and the characterization of the resulting complex configurations are in excellent agreement between numerics and experiments.   

The elasticity of magnetic chains: From self-buckling to self-assembly

Spherical neodymium magnets have become a popular toy in recent years. In this talk, we present the results of some experimental and theoretical investigations into the peculiar elastic-like behaviour exhibited by chains of these magnetic spheres. We show how the dipole-dipole interactions between spheres penalise deformation, and we find that the form of this penalty is different for a long chain compared to a short chain. Finally, we investigate the dynamic self-assembly of these chains into cylindrical structures.   

Spontaneous and Deterministic Three-dimensional Curling of Pre-strained Elastomeric Strips: From Hemi-helix to Helix

A variety of three dimensional curls are produced by a simple generic process consisting of pre-straining one elastomeric strip, joining it to another and then releasing the bi-strip. The hemi-helix, one kind of three dimensional curls, consists of multiple, alternating helical sections of half wavelength in opposite chiralities and separated by perversions. The hemi-helix wavelength and the number of perversions are determined by the strip cross-section, the constitutive behavior of the elastomer and the value of the pre-strain. Topologically, the perversions also separate regions of the helix deforming principally by bending from those where twisting dominates. Changing the prestrain and the ratio between the thickness and the width induce a phase separation of hemi-helical structure, helical structure and hybrid structure which have similarities to coiled polymer molecules and plant tendrils.   

Snakes Out of the Plane

We develop a new computational model of elastic rods, taking into account shear and full rotational dynamics, as well as friction, adhesion, and collision. This model is used to study the movement of snakes in different environments. By applying different muscular activation patterns to the snake, we observe many different patterns of motion, from planar undulation to sudden strikes. Many of the most interesting behaviors involve the snake rising out of the horizontal plane in the vertical direction. Such behaviors include a sand snake sidewinding over the hot desert sand and a cobra rearing up into a defensive striking position. Experimental videos of live snakes are analyzed and compared with computational results. We identify and explain a new form of movement previously unobserved: ``collateral locomotion.''   

Analysis of the fluid mechanical sewing machine

A thin thread of viscous fluid falling onto a moving belt generates a surprising variety of patterns, similar to the stitch patterns produced by a traditional sewing machine. By simulating the dynamics of the viscous thread numerically, we can reproduce these patterns and their bifurcations. The results lead us to propose a new classification of the stitch patterns within a unified framework, based on the Fourier spectra of the motion of the point of contact of the thread with the belt. The frequencies of the longitudinal and transverse components of the contact point motion are locked in most cases to simple ratios of the frequency $\Omega_c$ of steady coiling on a surface at rest (i.e., the limit of zero belt speed). In particular, the ``alternating loops'' pattern involves the first five multiples of $\Omega_c/3$. The dynamics of the patterns can be described by matching the upper (linear) and the lower (non-linear) portions of the thread. Following this path we propose a toy model that successfully reproduces the observed transitions from the steady dragged configuration to sinusoidal meanders, alternating loops, and the translated coiling pattern as the belt speed is varied.   

Microfabrication of a spider-silk analogue through the liquid rope coiling instability

Spider capture silk outperforms most synthetic materials in terms of specific toughness. We developed a technique to fabricate tough microstructured fibers inspired by the molecular structure of the spider silk protein. To fabricate microfibers (with diameter $\sim 30\mu m$) with various mechanical properties, we yield the control of their exact geometry to the liquid rope coiling instability. This instability causes a thread of honey to wiggle as it buckles when hitting a surface. Similarly, we flow a filament of viscous polymer solution towards a substrate moving perpendicularly at a slower velocity than the filament flows. The filament buckles repetitively giving rise to periodic meanders and stitch patterns. As the solvent evaporates, the filament solidifies into a fiber with a geometry bestowed by the instability. Microtraction tests performed on fibers show interesting links between the mechanical properties and the instability patterns. Some coiling patterns give rise to high toughness due to the sacrificial bonds created when the viscous filament loops over itself and fuse. The sacrificial bonds in the microstructured fiber play an analogous role to that of the hydrogen bonds present in the molecular structure of the silk protein which give its toughness to spider silk.   

Buckling Instability and Stress Propagation in Rods with Elastic Support

The cytoskeleton of living cells is a composite material consisting of a network of biopolymers including f-actin and microtubules (MTs). MTs are able to bear significant compressive loads in cells as a result of reinforced short wavelength buckling, due to the surrounding actin network. However, the length scale of compressive force propagation, even for macroscopic rods, remains poorly understood. Here we propose a minimal theory that incorporates elastic restoring forces from the surrounding network, elucidating the compressive force-dependence of the buckled rod shape. We identify a threshold length as the effective distance stresses can propagate in such network, and show that the decay length is tunable by modifying the longitudinal mechanical coupling coefficients. We test these predictions with experiments in macroscopic rods, and show that the degree of mechanical coupling directly controls the penetration depth of buckling, in agreement with theoretical and numerical predictions. Our results suggest that the length scale over which mechanical signals are transduced in cells may be actively controlled, and could provide design principles for novel types of fiber composite materials based on biomimetic control of the longitudinal coupling coefficients.   

Undulatory buckling of a rod constrained by an elastic matrix

Elastic instabilities of rods constrained by an elastic matrix and subjected to axial compression have long been recognized as essential for structural applications in the context of failure mitigation and, more recently, towards exploitation of functionality. Relevant fields for this class of problems include drilling, biomedical instrumentation and root growth in plants. We explore the two possible scenarios observed when, above a threshold load, compression is applied to a rod constrained by a matrix: i) the rod can develop a planar oscillatory solution (sinusoidal buckling) or ii) it can take the configuration of a helix (helical buckling). We identify the principal parameters of this system, perform a systematic parametric study and rationalize the phase diagram through a hybrid of theoretical and numerical analyses. Particular attention is devoted to the effect of the mechanical properties of the constraining matrix which is found to have a critical influence on this buckling scenario.   

Slack, stress, and noisy structures in inertial strings

Strings and chains are inextensible filaments with negligible bending and twist resistance. Local arc length conservation is enforced by the stress, a Lagrange multiplier field screened by curvature. Uniform stress fields are generated by a wide class of inertial motions that includes travelling waves of curvature and torsion, while gradients in stress result in more complicated dynamics. We will discuss a theoretical example inspired by experimental and numerical observations of the growth of an arch in a straightening chain, involving the amplification, rectification, and advection of slack.   

Sinusoidal to helical buckling of an elastic rod under a cylindrical constraint

We investigate the buckling and post-buckling behavior of an elastic rod loaded under cylindrical constraint. Our precision desktop-scale experiments comprise of axially compressing a hyper-elastic rod inside a transparent acrylic pipe. These experiments are also modeled using a discrete elastic rod simulation that includes frictional effects. Under imposed displacement, the initially straight rod first buckles into a sinusoidal mode and eventually undergoes a secondary instability into a helical buckling regime. The buckling and post-buckling behavior is found to be highly dependent on the systems' geometry, in particularly the aspect ratio of the rod to pipe diameter. We quantify the wavelength and pitch of the periodic patterns through direct digital imaging and record the reaction forces at both ends of the pipe. The observed behavior is rationalized through scaling arguments and captured by numerical simulations.