Session G44: Focus Session: Extreme Mechanics: Contortion of Filaments, Ribbons and Bundles

Spontaneous formation of singularities in twisted ribbons

We present experimental results on the spontaneous formation of a triangular tessellation of a thin elastic ribbon which is twisted with a prescribed longitudinal tension. We find that triangular patterns arise out of a period doubling of a primary longitudinal instability as the twist is increased in contrast with theoretical development assuming infinitely thin, inextensible sheet. Using x-ray tomography, we are able to reconstruct the 3D shape of the ribbon which can then be precisely characterized by measuring locally the mean and Gaussian curvature. We discuss quantitatively the structures of singularities (d-cones and ridges) as a function of nondimensional parameters characterizing the twist, the tension, and the geometry of the ribbon. Because the observed singularities occur away from walls and boundaries, the twisted ribbon configuration provides a unique opportunity to address the spontaneous formation of localized structures with great experimental flexibility.   

Perversions driven spontaneous symmetry breaking in heterogeneous elastic ribbons

Perversion structures in an otherwise uniform helical structure are associated with several important concepts in fundamental physics and materials science, including the spontaneous symmetry breaking and the elastic buckling. They also have strong connections with biological motifs (e.g., bacteria shapes and plant tendrils) and have potential applications in micro-muscles and soft robotics. In this work, using a three-dimensional elastomeric bi-stripe model, we investigate the properties of perversions that are independent of the specific ribbon shapes. Several intrinsic features of perversions are revealed, including the spontaneous condensation of energy as well as the distinct energy transfer modes within the perversion region. These properties of perversions associated with the storage of elastic energies can be exploited in the design of actuator devices.   

Deswelling and buckling of a temperature-sensitive hydrogel toroid

Temperature-sensitive hydrogels lose volume with increasing temperature by expelling water from their polymer matrix, which becomes effectively hydrophobic above a certain critical temperature. Whilst the temperature response of a {\it spherically shaped\/} sample of hydrogel has been well studied, less is known about the response of a {\it toroidal\/} sample. We present a model for the behavior of a hydrogel toroid for two cases of heating protocol: (i)~the quasistatic limit, in which the sample loses volume but is found to maintain its toroidal shape; and (ii)~the rapid quench limit, after which the sample is found to have maintained its volume but may have undergone a macroscopic, qualitative change of shape to a buckled toroid. For the quench-limit case, we develop a criterion for the stability of the rotationally symmetric state of the toroid, by utilizing an effective elastic ring model. When this criterion is no longer met, a long-wavelength deformation leads to a buckling instability of the toroid in a manner analogous to the buckling of an Euler rod.   

Morphoelastic Rods and Birods: Theory and Applications

In many engineered or biological structures long thin elastic filaments are bundled together. Due to heating expansion, growth, or remodelling, the reference configuration of each filament can evolve independently and become incompatible with respect to its neighbours leading to internal stresses, deformations and, possibly, instabilities. A simple example of these structures is the bi-metallic strip first described by Timoshenko in 1925. To capture these phenomena in space and for large deformations, we have developed a general theory of growing elastic rods and birods. The theory provides a natural framework to consider the shape and dynamics of a single or multiple elastic rods with evolving reference configuration. In this talk, I will present the general theory and apply it to a number of interesting situations commonly found in engineering and biology. I will also describe new analytical methods to determine the shape and stability of growing birods. This work is done in collaboration with Thomas Lessinnes and Derek Moulton.   

Stretchability of freestanding and polymer-supported serpentine thin films

High-performance stretchable electronics integrate high-quality inorganic electronic materials such as metal, semiconductor and oxide with deformable polymer substrates. To minimize strains in inorganic materials under large deformation, metal and ceramic thin films can both be patterned into meandering serpentine ribbons which can rotate and twist to accommodate the applied strain. We have systematically investigated the effects of geometry and substrate stiffness on the stretchability of serpentines through both theoretically and experimental means. For freestanding serpentines, closed-form analytical results are obtained and validated by experiments. To investigate the effect of substrates, indium tin oxide (ITO) serpentines are patterned on both polyimide and elastomeric substrates with systematically changing geometries. While stiff substrates such as polyimide almost completely prevents the rotation or twist of the serpentines, soft substrates can provide serpentines with reasonable freedom of rotation and twisting, which yields stretchability of ITO ribbons beyond 100{\%}. But new failure mechanisms have been found on soft substrates.   

Coiling of elastic rods from a geometric perspective

We present results from a systematic numerical investigation of the pattern formation of coiling obtained when a slender elastic rod is deployed onto a moving substrate; a system known as the elastic sewing machine (ESM). The Discrete Elastic Rods method is employed to explore the parameter space, construct phase diagrams, identify their phase boundaries and characterize the morphology of the patterns. The nontrivial geometric nonlinearities are described in terms of the gravito-bending length and the deployment height. Our results are interpreted using a reduced geometric model for the evolution of the position of the contact point with the belt and the curvature of the rod in its neighborhood. This geometric model reproduces all of the coiling patterns of the ESM, which allows us to establish a universal link between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system referred to as the fluid mechanical sewing machine.   

The shape of strings to come: How topological defects twist, bend, and wrinkle filament bundles

Topological defects are crucial to the thermodynamics and structure of condensed matter systems. For instance, when incorporated into crystalline membranes like graphene, 5- and 7-fold disclinations produce conical- and saddle-like geometries respectively. A recently discovered mapping between the inter-filament spacing within a deformed bundle and the metric properties of curved surfaces, suggests previously unexplored parallels between the two, specifically in regards to how 2D patterning promotes 3D shape transitions. This discovery is poised to describe the structure of a host of filamentous materials--both biological and microfabricated--that exhibit distinctive shapes and packings. Motivated by the filamentous analogs to the conical and saddles shapes found in thin membranes, we investigate for the first time the interplay between defects in the cross section of a bundle and its global structure, using a combination of continuum elasticity theory and numerical simulation of cohesive bundles with a fixed packing topology. Focusing primarily on the instability response to disclinations, we predict a host of new equilibria structures, some of which are without direct parallel to the analogous membrane, including torsional wrinkling, radial kinking, and helical winding.   

Rotation of a Thin Elastic Rod Injected into a Cylindrical Constraint

We report the results from an experimental investigation of the buckling of a thin elastic rod injected into a horizontal cylindrical constraint, with an emphasis on comparing the two cases of rotating, or not, the rod at the injection site. We are particularly interested on the total length of rod that can be injected into the pipe prior to the onset of helical buckling. This instability arises due to the frictional rod-constraint contact that eventually leads to the buildup of axial stress on the rod, above a critical value. We explore the dependence of the buckling conditions on the physical and control parameters of the system (e.g. material and geometric parameters, injection speed and rotation frequency) and rationalize the underlying physical mechanism through a reduced model.   

Stress localisation in annular sheets

For very thin sheets stretching is much more costly in terms of energy than bending. The limiting behaviour of thin sheets is therefore governed by geometry only and thus applies to a wide range of materials at vastly different scales: it is equally valid for a microscopic graphene sheet and a macroscopic solar sail. We derive new geometrically-exact equations for the deformation of annular strips. We use a formulation in which the inextensibility constraint is used to reduce the problem to a suitably-chosen reference curve (here the circular centreline). The equations are therefore ODEs, which allow for a detailed bifurcation analysis. Closed conical solutions are found for centreline lengths $L$ less than $L_c=2\pi\kappa_g$, where $\kappa_g$ is the geodesic curvature of the strip. For such `short' strips we find in addition a second branch of stable solutions easily reproduced in a paper strip. For `long' strips ($L>L_c$) we find modes of undulating solutions. All non-conical solutions turn out to feature points of stress localisation on the edge of the annulus, the outer edge for short solutions and the inner edge of long solutions. Our theory may be used to investigate singularities of constrained or loaded sheets more general than conical ones.   

Large-deformation dynamics of an elastic filament at a fluid interface

We study the dynamics of a thin elastic filament at the interface of two fluids and observe the time evolution in its shape when released from an initial configuration with a large curvature. The unfolding of the filament is driven by a competition between bending energy and viscous dissipation. We experimentally study the overdamped regime of this system by varying fluid viscosity ($\eta$), length ($L$), diameter ($d$) and elastic modulus ($E$) of the filament with similar initial conditions and observe the kinematics of the filament straightening. The time-dependence for this process can be collapsed by scaling time by $\eta L^4/Ed^3$. However, the characteristic time is a very small fraction of this time-scale. We perform numerical computations parallel to the experiments to get access to the dynamics of the filament to resolve this puzzle. An understanding of the time-dependence will enable the use of this technique to measure interfacial properties.   

Elasto-capillary windlass: from spider web to synthetic actuators

Spiders' threads display a wide range of materials properties. The glue-covered araneid capture silk is unique among all silks because it is self tensing and remains taut even if compressed, allowing both thread and web to be in a constant state of tension. Here we demonstrate how this effect is achieved by unraveling the physics allowing the nanolitre glue droplets straddling the silk thread to induce buckling, coiling and spooling of the core filaments. Our model examines this windlass activation as a structural phase transition, which shows that fibre spooling results from the interplay between elasticity and capillarity. Fibre size is the key as such a capillary windlass requires micrometer-sized fibres in order to function. Our synthetic capillary windlasses point towards design principles for new bioinspired synthetic actuators.   

Variability of Fiber Elastic Moduli in Composite Random Fiber Networks Makes the Network Softer

Athermal fiber networks are assemblies of beams or trusses. They have been used to model mechanics of fibrous materials such as biopolymer gels and synthetic nonwovens. Elasticity of these networks has been studied in terms of various microstructural parameters such as the stiffness of their constituent fibers. In this work we investigate the elasticity of composite fiber networks made from fibers with moduli sampled from a distribution function. We use finite elements simulations to study networks made by 3D Voronoi and Delaunay tessellations. The resulting data collapse to power laws showing that variability in fiber stiffness makes fiber networks softer. We also support the findings by analytical arguments. Finally, we apply these results to a network with curved fibers to explain the dependence of the network's modulus on the variation of its structural parameters.