Session Q16: Extreme Mechanics: (more) Toys, Theory, and Fluid-Structure Interaction

A bicycle with compliant training wheels, half way between a bicycle and a tricycle, is uncontrollable

We have built and tested a vehicle that can balance and steer like a bicycle, a tricycle, or anything in between. A \textit{bricycle} is essentially a bicycle with springy training wheels. The stiffness of the training wheel suspension can be varied from infinite, when the bricycle is a tricycle, to zero, when it is a bicycle. One might expect a smooth transition from tricycle to bicycle as the stiffness is varied, in terms of handling, balance and feel. But the situation is more complicated. Rather, the controllability of a bicycle depends on gravity. Without gravity, lean and direction cannot be controlled independently. Springy training wheels effectively reduce or even negate gravity. Indeed, experiments with the bricycle show problems when the total effective gravity is about zero. People can then still balance easily but can no longer turn the brike. The theory and experiment show a qualitative difference between bicycles and tricycles. A difference that cannot be met halfway.   

A string falling onto a table

In the light of recent thought-provoking experiments on different types of chains, I consider the problem of an inextensible/incompressible string falling under gravity and undergoing a collision with a rigid table. The question of interest is whether the free upper end of the string experiences an acceleration greater than a free-falling string. I find that the answer is yes, subject to an assumption about the boundary condition at the colliding lower end. I propose some new experiments and simulations to assess the validity of the assumption and test the prediction.   

Computational Design of Animated Mechanical Characters

A factor key to the appeal of modern CG movies and video-games is that the virtual worlds they portray place no bounds on what can be imagined. Rapid manufacturing devices hold the promise of bringing this type of freedom to our own world, by enabling the fabrication of physical objects whose appearance, deformation behaviors and motions can be precisely specified. In order to unleash the full potential of this technology however, computational design methods that create digital content suitable for fabrication need to be developed. In recent work, we presented a computational design system that allows casual users to create animated mechanical characters. Given an articulated character as input, the user designs the animated character by sketching motion curves indicating how they should move. For each motion curve, our framework creates an optimized mechanism that reproduces it as closely as possible. The resulting mechanisms are attached to the character and then connected to each other using gear trains, which are created in a semi-automated fashion. The mechanical assemblies generated with our system can be driven with a single input driver, such as a hand-operated crank or an electric motor, and they can be fabricated using rapid prototyping devices.   

Slinky Mechanics: Static Shapes and Unstable States

The floppy nature of a tumbling Slinky has captivated children and adults alike for over half a century. Highly flexible, the spring will walk down stairs, turn over in your hands, and--much to the chagrin of children everywhere--become easily entangled. The Slinky is an educational tool for demonstrating standing waves, and a structural inspiration due to its ability to extend to many times beyond its initial length without imparting plastic strain on the material. In this work, we provide a mechanical model that captures the static equilibrium configurations of the Slinky in terms of its geometric and material properties. We present both continuous and discrete models to capture a Slinky's static equilibria and unstable transitions. We compare these with experimental results obtained for the Slinky's static equilibrium shapes. We emphasize the importance of modeling coil contact, and determine the critical criteria for the Slinky to topple over in terms of a tilt angle, and the vertical displacement of one bale of coils. Finally, we provide a general description of highly flexible helical springs by considering the nondimensional potential energy of the spring, which characterizes the ``Slinkiness'' of a spring.   

The Shape of a Developable M\"{o}bius Strip via Classical Elastic Cosserat Rod Theory

Recent efforts to find the equilibrium shape of an inextensible elastic M\"{o}bius strip have produced apparently conflicting approaches and results. While an earlier approach uses the traditional one dimensional Kirchhoff elastic rod, the latest effort claims that the strip must be modeled as an elastic two dimensional developable surface. This study explains the source of the discrepancy and demonstrates that a classical one dimensional Cosserat elastic rod can capture both types of behavior. Using numerical continuation methods, we show how to adapt traditional rod theory to approximate developable elastic strips and apply our method to the M\"{o}bius problem. We further analyze the stability of the equilibria obtained. The adapted rod theory holds promise for modeling the mechanics of other thin structures subject to curvature constraints.   

Euler-Lagrange Elasticity: elasticity without stress or strain

A Euler-Lagrange (E-L) approach to elasticity is proposed that produces differential equations of elasticity without the need to define stress or strain tensors. The positions of the points within the body are the independent parameters instead of strain. Force replaces stress. The advantage of this approach is that the E-L differential equations are the same for both infinitesimal and finite deformations. Material properties are expressed in terms of the energy of deformation. The energy is expressed as a function of the principal invariants of the deformation gradient tensor. This scalar invariant representation of the energy of deformation enters directly into the E-L differential equations so that there is no need to define fourth order tensor material properties. By experimentally measuring the force and displacement of materials the functional form of the energy of deformation can be determined. The E-L differential equations can be input directly into finite element, finite difference, or other numerical models. If desired, stress and stain can be calculated as dependent parameters.   

The ridge between two fracture tips

The shape of a fracturing thin sheet is governed by Griffith's criterion, wherein both the system's energy and the applied force are minimized. For a thin sheet adhered to a substrate, the important energies are those of adhesion and bending of the sheet. Without adhesion, the ridge connecting the crack tips need not be developable, and in-plane stretching energy may become important. A reasonable assumption is that this ridge take the shape of a minimal ridge. We present experimental and numerical results that show the shape of this configuration does resemble the minimal ridge. However, an anomalous energy scaling is observed. We also show that the ridge shape, and therefore energy balance, depends on the length of the flap being pulled, which suggests a mechanism for controlling crack shapes.   

Kinks in topological soft matter

Weakly connected mechanical systems near the isostatic threshold are fragile in the sense that they exhibit large deformations in response to tiny perturbations. Kane and Lubensky have recently defined a new topological invariant of isostatic mechanical lattices which leads within linear elasticity to zero energy modes at the boundary akin to the edge modes studied in topological quantum matter. What happens when such prototype topological soft materials are subject to an external mechanical perturbation? In our work, we demonstrate that the linear soft modes can often integrate to non-linear deformations described by topological solitons. These solitons that are moving kinks between distinct topological phases are the basic excitations of fragile mechanical systems. We illustrate the general soliton construction in the context of a 1D chain of rotors connected by springs that can be considered the archetype of a topological mechanical structure. In the continuum limit, this chain is described by a Lorentz invariant $\phi^4$ theory and the corresponding solitons exhibit a Lorentz contraction of the width, as their speed is raised.   

The sedimentation of flexible filaments

The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process.   

A flexible fiber in a turbulent flow: a macroscopic polymer?

We describe, for the first time, an experiment devoted to the study of the spatial conformation of a flexible fiber in a turbulent flow. We propose a model for the transition from rigid to flexible regimes as the intensity of turbulence is increased or the elastic energy of the fiber is decreased. This transition occurs for a fiber typical length which is observed experimentally and recovered by our analysis. We also demonstrate that the conformations of flexible fibers in a turbulent flow are analog to conformations of flexible polymers in a good solvent. This last result opens some new and creative ways to model flexible fiber distortions in turbulent flows while addressing fundamental problems in polymer dynamics.   

Propulsion at low Reynolds number via beam extrusion

We present experimental and theoretical results on the extrusion of a slender beam in a viscous fluid. We are particularly interested in the force necessary to extrude the beam as it buckles with large amplitude due to viscous friction. The problem is inspired by the propulsion of Paramecium via trichocyst extrusion. Self-propulsion in micro-organisms is mostly achieved through the beating of flagella or cilia. However, to avoid a severe aggression, unicellular Paramecium has been observed to extrude trichocysts in the direction of the aggression to burst away. These trichocysts are rod-like organelles which, upon activation, grow to about $40~\mathrm{\mu m}$ in length in 3 milliseconds before detaching from the animal. The drag force created by these extruding rods pushing against the viscous fluid generates thrust in the opposite direction. We developed an experimental setup to measure the force required to push a steel piano wire into an aquarium filled with corn syrup. This setup offers a near-zero Reynolds number, and allows studying deployments for a range of constant extrusion speeds. The experimental results are reproduced with a numerical model coupling a large amplitude Euler-Bernoulli beam theory with a fluid load model proportional to the local beam velocity.   

The propulsion of filaments with natural curls

We consider a macroscopic analogue model for the locomotion of prokaryotic bacteria with a single flagellum and study the dynamics of a flexible helical filament that is rotated in a viscous fluid, at low Reynolds numbers. The scaling from the original micron-scale onto the desktop-scale is made possible by the prominence of geometry in the deformation process. Our filaments are custom fabricated with different geometric and material properties (with an emphasis on varying their intrinsic curvature), clamped at one end and rotated in a bath of glycerin. Geometrically nonlinear configurations of the filament can result from the coupling of the elastic forces of the filament and the viscous drag. Using digital imaging, we reconstruct the 3D deformed configurations of the rotating filament and quantify its dynamics. Our precision model experiments are combined with numerical tools ported from the computer graphics community. We couple the results from the experiments and simulations to quantify the effect of the control parameters on the propulsive force exerted by the rotating filament and rationalize the underlying mechanical instabilities.   

Mechanics of fluid flow over compliant wrinkled polymeric surfaces

Skin friction coefficients (based on frontal area) of sharks and dolphins are lower than birds, fish and swimming beetles. By either exploiting flow-induced changes in their flexible skin or microscale textures, dolphins and sharks can change the structure of the fluid flow around them and thus reduce viscous drag forces on their bodies. Inspired by this ability, investigators have tried using compliant walls and riblet-like textures as drag reduction methods in aircraft and marine industries and have been able to achieve reductions up to 19{\%} [1]. Here we investigate flow-structure interaction and wrinkling of soft polymer surfaces that can emulate shark riblets and dolphin's flexible skin. Wrinkling arises spontaneously as the result of mismatched deformation of a thin stiff coating bound to a thick soft elastic substrate. Wrinkles can be fabricated by controlling the ratio of the stiffness of the coating and substrate, the applied displacement and the thickness of the coating. In this work we will examine the evolution in the kinematic structures associated with steady viscous flow over the polymer wrinkled surfaces and in particular compare the skin friction with corresponding results for flow over non-textured and rigid surfaces. 1. K-S Choi et al.: Proc. R. Soc. Lond. A. 1997   

Micro-mechanical lengthscales in soft elastic solids

We provide numerical evidence and supporting scaling arguments that the response of soft elastic solids to a local force dipole is characterized by a lengthscale $\ell_c$ that diverges as unjamming is approached as $\ell_c \sim (z - 2d)^{-1/2}$, where $z \ge 2d$ is the mean coordination, and $d$ is the spatial dimension, at odds with previous claims based on numerics. We also show how the magnitude of the lengthscale $\ell_c$ is amplified by the presence of internal stresses in the disordered solid. Our data raise the possibility of a divergence of $\ell_c$ with proximity to a critical internal stress at which a buckling instability takes place.