Session L40: Geometry and Dynamics of Slender Structures

Hamiltonian formulations in the computation of extremely deformed nano-scale hyper-elastic rods.

There has been a recent resurgence of interest in models exploiting elastic filaments and ribbons, motivated in large part by nano-scale applications, including DNA. Such models are frequently nearly inextensible or nearly unshearable. I will describe how such systems can be modelled as a smooth limit within a hierarchy of perturbed Hamiltonian formulations of the governing equations. Examples include a sequence-dependent double-strand birod model of DNA, where a more familiar rod model can be obtained as a~smooth limit in which the intra-strand degrees of freedom are frozen..   

A conserved quantity in thin body dynamics

We use an example from textile processing to illustrate the utility of a conserved quantity associated with metric symmetry in a thin body. This quantity, when combined with the usual linear and angular momentum currents, allows us to construct a four-parameter family of curves representing the equilibria of a rotating, flowing string. To achieve this, we introduce a non-material action of mixed Lagrangian-Eulerian type, applicable to fixed windows of axially-moving systems. We will point out intriguing similarities with Bernoulli's equation, discuss the effects of axial flow on rotating conservative systems, and make connections with 19th- and 20th-century results on the dynamics of cables.   

Tearing of thin spherical shells adhered to equally curved rigid substrates

Lasik (Laser-Assisted in Situ Keratomileusis) eye surgery involves the tearing of the corneal epithelium to remodel the corneal stroma for corrections such as myopia, hyperopia and astigmatism. One issue with this procedure is that during the tearing of the corneal epithelium, if the two propagating cracks coalesce, a flap detaches which could cause significant complications in the recovery of the patient. We seek to gain a predictive physical understanding of this process by performing precision desktop experiments on an analogue model system. First, thin spherical shells of nearly uniform thickness are fabricated by the coating of hemispherical molds with a polymer solution, which upon curing yields an elastic and brittle structure. We then create two notches near the equator of the shell and tear a flap by pulling tangentially to the spherical substrate, towards its pole. The resulting fracture paths are characterized by high-resolution 3D digital scanning. Our primary focus is on establishing how the positive Gaussian curvature of the system affects the path of the crack tip. Our results are directly contrasted against previous studies on systems with zero Gaussian curvature, where films were torn from planar and cylindrical substrates.   

Cracks in Sheets Draped on Curved Surfaces

Conforming materials to surfaces with Gaussian curvature has proven a versatile tool to guide the behavior of mechanical defects such as folds, blisters, scars, and pleats.~In this talk, we show how curvature can likewise be used to control material failure. In our experiments, thin elastic sheets are confined on curved geometries that stimulate or suppress the growth of cracks, and steer or arrest their propagation. By redistributing stresses in a sheet, curvature provides a geometric tool for protecting certain regions and guiding crack patterns. A simple model captures crack behavior at the onset of propagation, while a 2D phase-field model successfully captures the crack's full phenomenology.   

Pick-up, impact, and peeling.

We consider a class of problems involving a one-dimensional, inextensible body with a propagating discontinuity (shock) associated with partial contact with a rigid obstacle providing steric, frictional, or adhesive forces. This class includes the pick-up and impact of an axially flowing string or cable, and the peeling of an adhesive tape. The dynamics are derived by applying an action principle to a non-material volume. The resulting boundary conditions provide momentum and energy jump conditions at the shock. These are combined with kinematic conditions on velocities and accelerations to obtain families of steady-state solutions parameterized by the shock velocity and momentum and energy sources. We find relationships between the jump in stress, injection of momentum, and dissipation of energy, which we apply to specific cases, and compare with other results in the literature on chain fountains, falling folded chains, and impulsively loaded cables. Time permitting, we will briefly discuss the possibility of using kinematic conditions and information about accelerating or otherwise unsteady forms of the adjoining bulk solutions to construct an equation of motion of the shock.   

Effect of boundary conditions on the buckling instabilities of a ribbon under twist

We investigate the buckling instabilities of a thin flat sheet in the shape of a ribbon which is held at its ends and twisted under tension. Recently it was shown that such a system with clamped boundary conditions exhibited a rich variety of buckled shapes with longitudinal and transverse wrinkles as a function of applied twist and tension for a given ribbon aspect ratio and elastic modulus [1], which could be described by a far from threshold analysis of the covariant form of the F\”oppl-von K’arm’an equations [2]. Here, we focus on the effect of the boundary conditions on the observed buckling patterns by constraining the ends only at the midpoint towards imposing free boundary conditions normal to the ribbon. In particular, we compare and contrast the observed phase diagram and the shape of the longitudinal and transverse buckled modes as a function of applied constraints. [1] J. Chopin and A. Kudrolli, Phys. Rev. Lett. 111, 174302 (2013). [2] J. Chopin, V. D’emery, and B. Davidovitch J. Elast. 119, 137 (2015).   

Multi-stability and bifurcations of a thin band.

Thin band- or strip-like structures are common motifs in flexible and deployable systems, serving as integrated connectors, hinges, and umbilicals. The morphing systems impose variable constraints on these components, inducing complex responses. We experimentally investigate a simple configuration representing the above type, a thin elastic band with end constraints on position and orientation. These constraints correspond to a combination of compression and shear with respect to a flat rectangular rest configuration. We vary the aspect ratio of the band, and the position and clamping angle at its ends. The buckled structure remains developable up to limiting deformations that approach one of two states, each dominated by two singularities. At intermediate deformations, the structure may adopt many distinct stable states. Transitions between these states can be smooth or violent, and depend strongly on constraints such as the clamping angle. Time permitting, we will relate our results to the behavior of anisotropic rods, and of strips subjected to twisting and extension.   

Finite and infinite wavelength elastocapillary instabilities with cylindrical geometry

In an elastic cylinder with shear modulus $\mu$, radius $R_0$ and surface tension $\gamma$ we can define an emergent elastocapillary length $l=\gamma/\mu$. When this length becomes comparable to $R_0$ the cylinder becomes undergoes a Rayleigh-Plateaux type instability, but surprisingly, with infinite wavelength $\lambda$ rather than with wavelength $\lambda\sim R_0\sim l$. Here we take advantage of this infinite wavelength behaviour to construct a simple 1-D model of the elastocapillary instability in a cylindrical gel which permits a high-amplitude fully non-linear treatment. In particular, we show that the instability is sub-critical and entirely dependent on the elastic cylinder being subject to tension. We also discuss elastocapillary instabilities in a range of other cylindrical geometries, such a cylindrical cavities through a bulk elastic solid, or a solid cylinder embedded in a bulk elastic solid, and show that in these cases instability has finite wavelength. Thus infinite wavelength behaviour is a curiosity of elastic cylinders rather than the generic behaviour or elasto-capiliarity.   

Sequential buckling of an elastic wall

A beam under quasistatic compression classically buckles beyond a critical threshold. In the case of a free beam, the lowest buckling mode is selected. We investigate the case of a long “wall” grounded of a compliant base and compressed in the axial compression. In the case of a wall of slender rectangular cross section, the selected buckling mode adopts a nearly fixed wavelength proportional to the height of the wall. Higher compressive loads only increase the amplitude of the buckle. However if the cross section has a sharp shape (such as an Eiffel tower profile), we observe successive buckling modes of increasing wavelength. We interpret this unusual evolution in terms of scaling arguments. At small scales, this variable periodicity might be used to develop tunable optical devices.   

Mechanics of a Knitted Fabric

A simple knitted fabric can be seen as a topologically constrained slender rod following a periodic path. The non-linear properties of the fabric, such as large reversible deformation and characteristic shape under stress, arise from topological features known as stitches and are distinct from the constitutive yarn properties. Through experiments we studied a model stockinette fabric made of a single elastic thread, where the mechanical properties and local stitch displacements were measured. Then, we derived a model based on the yarn bending energy at the stitch level resulting in an evaluation of the displacement fields of the repetitive units which describe the fabric shape. The comparison between the predicted and the measured shape gives very good agreement and the right order of magnitude for the mechanical response is captured. This work aims at providing a fundamental framework for the understanding of knitted systems, paving the way to thread based smart materials.   

Crumpling of an Elastic Ring in Two Dimensions

We use molecular dynamics simulations to study the crumpling of an elastic ring (i.e., a circular elastic line) in two dimensions. The crumpling is triggered by reducing the radius of a circular repulsive wall that is used to confine the ring. The ring is modeled as a bead-spring chain. A harmonic potential describing the bonds between neighboring beads is parameterized to reproduce the Young's modulus of the elastic line in the continuum limit. A modified harmonic angle interaction is used to capture the bending of the elastic line including situations where the line is locally stretched or compressed. We have confirmed that the bead-spring model has the correct continuum limit by comparing results on rings made of different numbers of beads but with parameters derived from the same elastic line. With the computational model, we study the morphological transition of the ring and the local distribution of the bond and bending energies as the ring is compressed at various rates, forced to crumple, and finally confined into a dense-packed structure. We find that the crumpling transition signals a sharp energy transfer from the compression to the bending mode. We further explore the possibility of defining an effective temperature for such crumpled systems.   

Natural Curvature as Effective Confinement in Elastic Sheets

The wrinkling and folding transitions of thin elastic sheets have been extensively studied in the last decade. The exchange of energy from stretching to bending acts as a paradigm for a wide range of elastic instabilities, including the wrinkling of the gut, and the crinkling of leaves. In two dimensions this type of problem is typically considered by the model of an Euler-\textit{elastica} in compressive confinement. We show that, even without any external forces, an elastic surface supported by a fluid can bend and wrinkle when it acquires a non-\textbf{z}ero natural curvature. Locally, we will demonstrate how a preferential curvature can be related to an effective compression, and hence a confining force that can vary spatially. This suggests a simple experimental setup, where we have characterised a variety of wrinkle patterns that can be generated for different mechanical properties and natural curvatures.   

Anomalously soft non-Euclidean spring

In this work we study the mechanical properties of a frustrated elastic ribbon spring - the non-Euclidean minimal spring. This spring belongs to the family of non-Euclidean plates: it has no spontaneous curvature, but its lateral intrinsic geometry is described by a non-Euclidean reference metric. The reference metric of the minimal spring is hyperbolic, and can be embedded as a minimal surface. We argue that the existence of a continuous set of such isometric minimal surfaces with different extensions leads to a complete degeneracy of the bulk elastic energy of the minimal spring under elongation. This degeneracy is removed only by boundary layer effects. As a result, the mechanical properties of the minimal spring are unusual: the spring is ultra-soft with rigidity that depends on the thickness, $t$ , as $t^{\raise0.7ex\hbox{$7$} \!\mathord{\left/ {\vphantom {7 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}$, and does not explicitly depend on the ribbon's width. These predictions are confirmed by a numerical study of a constrained spring. This work is the first to address the unusual mechanical properties of constrained non-Euclidean elastic objects. We also present a novel experimental system that is capable of constructing such objects, along with many other non-Euclidean plates.