Session M25: Fabric, Knits and Knots I

What Can Knitting Machines Make?

Industrial knitting machines are used to fabricate many complex 3D objects, including gloves, sweaters, and shaped fiber reinforcement for composites. But what are the limits of these machines? This talk will introduce two of the approaches my group has taken to answering this question.
The first -- top-down -- approach views a knitting machine as a device for shaping, splitting, and merging tubes. This leads to the observation that all manifolds with boundary whose Reeb graph under some smooth function has an upward-planar embedding are machine-knittable. This observation forms the basis of our work on knitting design tools.
The second -- bottom-up -- approach translates low-level machine instructions into a geometric description of intertwined yarns, allowing direct enumeration of structures. The important insight made in this work is that the way the yarn is routed between stitches is as important to model as the way the yarn behaves within stitches.
Taken together, these two views of knitting machine capabilities have enabled us to develop new design and simulation tools for machine knitting.   

Knitting Machine State Representation Using the Artin Braid Group

Industrial knitting machines are are incredibly powerful, flexible machines that are capable of creating a wide variety of shapes and structures using just a few basic operations. However, effectively using these operations to take full advantage of the machine’s capabilities is nontrivial; when creating a machine knitting pattern, the designer must decide how to move loops of yarn around the machine’s hundreds of needles, all while tracking interloop connections and tangles. Furthermore, they should do so in a way that minimizes construction time and error rate. To enable the computation of optimal machine knitting patterns, we developed an abstraction of the knitting machine state that uses the Artin braid group when describing interstitch relationships. We provide a formulation for each machine operation’s effect on this abstract representation. In addition, we leverage key properties of the braid group when considering the distance between any two machine states. These insights make searching the knitting machine state space for efficient patterns more computationally tractable, and we show several computationally planned patterns.   

Mechanics-Based Simulation of Multistable Knitted Fabrics

The pattern formation process of a one-dimensional yarn into a two-dimensional sheet of knitted fabric exhibits intricate deformation modes with complex contact description. Given this complexity, we are motivated to develop a mechanics-based predictive model that accounts for mechanical parameters in this mechanics-based process. We first characterize parameters that significantly dominate the pattern formation, such as pretension and boundary conditions on the knitted fabric. Then integrate this with development of a simulation scheme, where yarn-yarn interaction and mechanical forces are computed and cross validated against parametric study on the macroscopic mechanical response of knitted elastic fabrics. Our model is generalized to explore the rich landscape of knitted fabrics as each yarn can be parameterized with varying material properties, enabling the design space for functionality of the fabrics to be enormously enlarged. Projecting forward, we hope to extend this simulation scheme for coupled analysis on functionality and data augmentation for optimization of pattern design for multifunctional knitted fabrics.   

Constructing a constitutive model of knitted fabric

Knitting is an ancient technology wherein yarn is manipulated into an array of slipknots to form fabric. By patterning these stitches in different ways, one can create fabrics possessing a wide array of elasticities and geometries, without needing to change the type of yarn being used. Thus, knitting is a way of programming material properties via a code that describes stitch patterns. However, while there have been mechanical models of aspects of knitted fabric, there is currently no unifying description that models knitted fabric elasticity given a prescribed stitch pattern. To make progress towards this end, we seek a method of generating constitutive relations for a given stitch pattern. Using a geometric framework in which yarn degrees of freedom are modeled as elastic space curves, with stitches held together by contact interactions, we solve for energy-minimizing stitch structures numerically within the unit cell of a given pattern. We then extract the constitutive relations by monitoring the deformation of the stitch given external stresses on the unit cell. This method not only gives us predicted stress-versus-strain relations that compare favorably with experiments but it allows us to probe the role of yarn geometry in determining fabric mechanics.   

Stress-Strain Studies of Knitted Swatches

The properties of a knitted fabric are highly dependent on what stitches the fabric is made of and what patterns they are used in. Different stitches correspond to different linking topologies of the yarn that constrain the yarn configurations and therefore the overall fabric mechanics and shape. Here we examine the relationship between large scale properties of knitted swatches and the small scale yarn shape within each stitch via experiments that probe the elasticity of the swatches while imaging corresponding changes in the shape of the constituent yarn. Under stress, knitted fabric deforms not only as a bulk response, but the yarn-level deformations give rise to nonlinear elasticity. To quantify this shape change we are developing a method for tracking the 3D path of the yarn. By tracking the yarn within the fabric while we measure the stress-strain relationship, we can untangle the effects of curvature, compression, and friction on the emergent nonlinear elasticity of the textile.   

Emergent yarns and fabrics by twist origami

A first step to making functionalized fabrics is to spin yarns by twisting together a natural or synthetic soft material with a prescribed structure. We will discuss experiments that demonstrate large spontaneous shape transformations that couple with mechanical response starting with a hyperelastic rectangular sheet which is simply held under tension and twisted around its central long axis. For twists larger than half a turn, the sheet adopts an accordion shape with self-contacts and folds oriented at an angle with respect to the axis of rotation. We reconstruct the 3D shape of the folded sheet using x-ray Computed Tomography and calculate the Gaussian and mean curvature to characterize their morphology. Modeling the full soft elastic sheet undergoing such a large transformation is prohibitively difficult. Thus, we propose an origami starting with an inextensible sheet with a prescribed number of triangular folds corresponding those observed at the onset of transverse wrinkling of the elastic sheet as a starting point to analyze the emergence of the structure. Building on this insight, we then present a string model based on the origami kinematics to explain not only the emergence of the accordion folds but also its torsional response.   

A Study of Two-Periodic Knitted Fabrics using Tangles

We have previously created a framework to classify and characterize two-periodic stitch patterns of knitted fabrics that is based on knot and link theory. Using this theory, we are able to associate a unique link in three dimensional sphere S³, to every two-periodic stitch pattern of a knitted fabric. In every such link, the subset of loops representing the yarn satisfy ribbonness or forms a ribbon link. A link is said to be ribbon if each of its component loops bound a 2D disk that pierces through itself making a finite number of slit like intersections lying in the interior of the collection of disks. Based on the ribbonness, and different types of yarn-needle moves catalogued in the knitting literature, we are able to describe knittable stitch patterns composed of -- knit and purl with twists, yarn overs, knit or purl two together, knit front back, slip stitch, cable etc. The description uses the notion of a tangle -- a collection of properly embedded arcs inside a euclidean three ball. This set up lets us define operator invariants for two-periodic knitted stitch patterns because tangles can be described using tensor products and composition by contraction of indices.   

Untangling the mechanics of the clove hitch knot

Knots can impart unique mechanical function to filamentary structures, with examples ranging across length scales, including DNA, polymer-chains, shoelaces climbing ropes, tennis racket, and surgical sutures. Still, the predictive understanding of the mechanics of this class of structures is limited. The fundamental challenge arises from the complex interplay between topology, geometry, elasticity, and friction. Here, we focus on the clove hitch knot, which typically attaches a flexible rod/filament/rope to a rigid post. The clove hitch can sustain a remarkably large tension ratio between the two ends of the rod when compared to the simpler strategy of spooling the rod around the rigid post in a helical configuration. In our study, we combine experiments (mechanical testing and X-ray tomography), finite element and a theory based on the Kirchhoff equations. We find that the twist in the rod increases with the ratio of the tensions applied to the two ends. Furthermore, in contrast to helical spooling, the clove hitch loses a significant amount of its tension at the regions of self-contact, thereby enhancing its mechanical performance.   

To stop or not to stop: The stopper knot as a friction device

A stopper knot tied to the end of a climbing rope prevents it from retracing through a narrow passage in the climber’s belay device. Numerous other applications of stopper knots are found in stringing of tennis rackets, sailing and fishing. If a single-stranded stopper knot meets the requirement of preventing a rope from escaping the system, it is able to unfold its full potential by converting a high-traction force into an insignificant stress state. This sharp tension drop results from the complex interplay of the topology, the tightness of the configuration, and the nontrivial frictional interactions in regions of self-contact of the elastically deformed rod. Existing models in knot theory or Kirchhoff’s theory for elastic rods are insufficient to describe this functional behavior due to the importance of finite elastic deformations of the cross-section and frictional interactions. We tackle this problem by performing a combination of mechanical testing and X-ray tomography on a variety of stopper knots. Our experimental data, combined with finite element simulations, allows us to systematically explore the different ingredients at play that conspire to dictate the mechanical performance of stopper knots.   

Topological Mechanics of Knots and Tangles

Knotted structures play a fundamental role in the dynamics of biological and physical systems, from DNA and polymers to liquid crystals and turbulent plasmas, as well as in climbing, weaving and sailing. Despite having been empirically studied for centuries, the subtle interplay between topology and mechanics in knotted elastic materials remains poorly understood. Here, we combine optomechanical experiments with theory and simulations to analyze the behavior of knots in flexible fibers that change their color in response to mechanical deformations. Exploiting a previously unrecognized analogy with long-range ferromagnetic spin systems, we identify simple counting rules to predict the relative mechanical stability of knots and tangles, in agreement with numerical simulations and experimental measurements for commonly used climbing and sailing knots. The underlying topological principles provide a conceptual foundation for understanding the roles of twist and writhe in untangling processes, and are expected to find broad applications in the description and control of systems with complex entanglements.   

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