Physicists decipher the mathematical secrets of knitting to make tailor-made materials

Enlarge / Introducing topological defects into knitted patterns can shape (a) the out-of-plane elasticity of a plush rabbit (left) and (b) in-plane deformations of knitted textiles (right).

Elisabetta matsumoto

Knitted fabrics like a scarf or socks are very elastic, able to stretch up to twice their length, but the individual strands of yarn hardly stretch. It’s how these strands form a network of interlocking stitches that gives knitted fabrics their elasticity. Physicists are trying to unlock the “code” of knitting – the underlying mathematical rules that govern how different combinations of stitches give rise to different properties like elasticity – in hopes of creating new “tunable” materials including properties can be tailored for specific purposes.

“Knitting is this incredibly complex way of converting a one-dimensional yarn into an intricate fabric,” said Elisabetta Matsumoto, a physicist at the Georgia Institute of Technology. “So basically it’s a type of coding.” Determining how different types of stitches determine shape and strength could help create design materials for future technologies, from better materials for the aerospace industry to stretch materials to replace torn ligaments. The models his team is developing can also be useful in enhancing the realistic animation of clothes and hair in video game graphics. Matsumoto described his research at the March 2019 American Physical Society meeting held this week in Boston.

Knitted fabrics can technically be considered a type of metamaterial (engineered materials that derive their properties not from base materials but from their designed structures), according to Matsumoto, who cites the medieval embroidery technique known as ” smocks ”as a first example. From a physical standpoint, smocking uses knots to essentially convert local bending energy into bulk stretching energy.

“There is this enormous wealth of knowledge in the knitting community that has yet to be translated into a quantitative model.”

The elasticity (or stretchability) of knitted fabrics is an emerging property: the whole is more than the sum of its parts. How these components (strands of yarn) are arranged at an intermediate scale (the structure) determines the macro-scale properties of the resulting fabric. It’s analogous to a piece of gold, which is made up of millions of atoms on a microscopic scale and has macro-scale properties like hardness and its golden hue. But individual atoms themselves do not have these properties, just as individual strands of yarn do not stretch like a knitted scarf does.

An avid knitter since childhood, Matsumoto began to think about the underlying math when she dabbled in science and developed a new appreciation for all of the math and material physics behind her hobby. “There is this enormous wealth of knowledge in the knitting community that has yet to be translated into a quantitative model,” she said. “We’re trying to use that knowledge and bring it into the world of physics, where we can study it as materials and look at elasticity and other emerging properties.”

Essentially, knitted fabrics are made up of a series of interlocking sliding knots made up of a single thread hanging back and forth on itself. (Woven fabrics, on the other hand, are made up of several threads that cross each other.) To make a knitted stitch, you pull the slip knot through the front of the fabric; to purl a stitch, you pull it through the back of the fabric. Experienced knitters know how to combine these stitches in different ways, playing with topology and creating complex new shapes, including elaborate 3D shapes, like a stuffed bunny. And changing the topology will also change the emerging properties (like elasticity).

(a) Knitting is a periodic structure of slip knots.  (b) Textiles with intricate patterns are knitted by combining sliding knots in specific combinations.
Enlarge / (a) Knitting is a periodic structure of slip knots. (b) Textiles with intricate patterns are knitted by combining sliding knots in specific combinations.

“There are hundreds of books with thousands of stitch patterns, with seemingly limitless complexity,” Matsumoto said. “And each type of point has a different elasticity. By choosing a point, you choose not only the geometry, but also the elastic properties, which means you can incorporate the right mechanical properties for everything from aerospace engineering to materials. of fabric scaffolding. “

Matsumoto is not the only physicist intrigued by the remarkable complexity of this ancient machine. Last year, a team of French physicists developed a rudimentary mathematical model to describe the deformation of a common type of knitting. Their work was inspired when co-author Frédéric Lechenault watched his pregnant wife knit baby slippers and blankets, and he noted how the items returned to their original shape even after being stretched. With a few colleagues, he was able to summarize the mechanics in a few simple equations, adaptable to different point models.

It all comes down to three factors: the “curvature” of the thread, the length of the thread and the number of cross points in each stitch. The stretchability of the knitted fabric results from the loops created when the yarn in a row of stitches weaves through the rows above and below, as pulling or folding the fabric creates energy, which in turn distorts the loops. . The exact amount of stretch is limited by the number of times the yarn crosses with neighboring stitches, as well as the length of the yarn. They tested their model by stretching some knitted fishing line (which doesn’t generate as much friction as the yarn) to see if its behavior matched the model’s predictions, and it did.

Naturally, more research is needed to realize the full potential of knitting in what is known as “additive manufacturing” (that is, creating an object by building it one layer at a time). But there may soon come a day when the secrets of knitting will be fully revealed, allowing scientists to program topological defects, just as they introduce defects into crystal structures to achieve desired material properties. They will be able to customize knitted materials with very specific shapes and properties, in the same way that skilled knitters transform strands of yarn into complex three-dimensional shapes. They just have to crack the code.

About Tracy G. Larimore

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