American Scientist

If you’ve ever done a PhD in physics, you’ll know that you usually begin by plowing through lots of background reading, learning how to use your laboratory’s equipment, and maybe even carrying out some provisional experiments. My PhD was a bit different. I started off watching YouTube videos to improve my needlework.

The project I had accepted at the École Normale Supérieure in Paris was about the mechanics of knitted fabrics. The research was to have two sides: a theoretical one to determine what equations described the system, and an experimental one to mechanically test actual knits to guide and verify the theory. The trouble was, I barely knew what a knit was when I accepted the project.

I quickly learned that there are differences—both structural and mechanical—between a knit (such as a sweater, scarf, or hat) and a weave (such as a tablecloth, shirt, or pair of jeans). Indeed, those differences are easy to demonstrate. If you pull on your jeans, you should notice that the weave hardly deforms. Pull on a knitted sweater, in contrast, and it can be effortlessly elongated by up to two times its length. The stretchiness of a knit is also obvious if you wrap it around something: By locally stretching, a knit can fit complex shapes; a woven fabric, however, has to fold to conform to a complex shape.

The discrepancies between knits and weaves are true even if they are made of the same yarn. That’s because the structure of how the yarn is interlaced is what dictates the mechanical behavior, not the precise composition of the material. As I soon learned in my research, a woven fabric is made from two bundles of yarns that intertwine perpendicularly (see figure above). Pulling on a weave is therefore not very different to pulling on a single yarn.

Knitted fabrics, however, are generally made from a single yarn shaped into a network of loops called stitches. Pulling a knit is therefore equivalent to deforming loops and not directly pulling on the yarn itself. This property illustrates how different interlacing structures can affect the mechanics of a textile.

Knit Science

I should make it clear that my fellow researchers—Mokhtar Adda-Bedia and Frédéric Lechenault—and I were not the first people to wonder how a knit deforms. Anyone knitting for fun needs a profound empirical knowledge of knit structure and mechanics. A knowledge of knits is also important in industry—and not just for the clothing sector. Knitted materials can, for example, be used to reinforce composites in planes, cars, or trains. In addition, knitting has inspired work in other fields, such as mathematics. For instance, mathematician sarah-marie belcastro of the University of Massachusetts-Amherst illustrates and studies mathematical surfaces that have complex topology and geometry by knitting them (see Adventures in Mathematical Knitting). And knit properties are even of use in the emerging field of soft robotics, because a knit’s easily configurable structure and its mechanics can be harnessed to create deformable media whose properties can be spatially tuned. Combined with an activation process such as pneumatics or heat, textiles can be used to achieve complex, customized motions.

The first studies of knit mechanics began in the 1960s. Inspired by mechanical engineers, the research involved modeling the exact path of a yarn in a stitch to determine how this yarn deformed if slightly pulled. This work provided some beautiful equations, but they described just one stitch and not a whole fabric.

The structure of how the yarn is interlaced is what dictates the fabric’s mechanical behavior, not the precise composition of the material in the yarn.

• Facebook
• Twitter

More recent studies were encouraged by the computer graphics community, which might seem odd, but animated characters must be dressed properly, too. This work led to a fully numerical approach, published by Cem Yuksel at the University of Utah and his colleagues, which starts from the yarn and goes up to the whole knit. It provided very realistic mechanics, but did not give an analytical expression of the fabrics’ mechanical constants, or reveal the interplay between structural parameters (for example, the size of the stitch) and material parameters (such as the rigidity of the yarn).

My research therefore involved looking between these two extremes, where there’s a gap in our under-standing. I wanted to find out if we can get equations to directly describe the mechanics of a whole fabric while at the same time defining the role of the individual parameters.

Knitting Like a Physicist

Two major issues arose early on in my quest to get answers. First, despite the YouTube tutorials and some precious advice from my grandmother, my early knits were terrible and useless for proper testing. The second problem was that the number of parameters in commonly used knits is colossal, because even the most standard yarns are themselves incredibly complicated objects.

To tackle my lack of knitting skills, I contacted staff at a nearby art school, which luckily had a workshop containing manual knitting machines and looms. More important, the head of the workshop kindly agreed to teach me how to craft the perfect knit. With this knowledge—and a 40-year-old, secondhand retail knitting machine we bought for the laboratory—I was now able to make experiment-worthy knits.

Despite the YouTube tutorials and some precious advice from my grandmother, my early knits were terrible and useless for proper testing.

• Facebook
• Twitter

To deal with the second problem—the complexity of the knit—we did what all physicists love to do: We simplified the system as much as possible so that only the essential parameters remained. We knew that what defines a knit is the pattern of the crossing points, so we began by picking the simplest yarn we could find—nylon fishing lines. Then, we made a very loose knit so the yarn did not get too deformed. Though a bit daring to be a garment, the resulting knit (similar to that in the figure above) is a wonderful system for a physicist to play with. It means we only have to take into account a few factors (which may include several numerical parameters): the elasticity of the yarn, the structure imposed by the interlacing pattern, and yarn–yarn friction at the crossing points.

Elasticity and Tremors

We finally had a knit we could do experiments on. To assess its mechanical response, we measured the force needed to pull the knit, and took pictures to evaluate how it locally deformed. The results were not as simple as we had expected.

The mechanical response (as shown in the graph above) had two features—one elastic and the other noisy. The elastic element could be identified because of how it repeats with the stretching cycles, and it therefore was predictable. The noisy response—which warped the elastic one by small perturbations—was not identical over cycles and thus had to be considered from a statistical perspective. By simplifying things this way, we could easily spot the culprits for each response.

The elasticity of a fabric naturally derives from the yarn elasticity and the periodic looping stitches, so we need to define how these factors interplay. This definition means predicting how the elastic energy of the yarn varies as the knit is deformed.

Instead of basing the model on the yarn itself, as in standard mechanical studies, we looked at it as a network of subunits, or stitches. This approach makes the problem much simpler because one stitch is characterized only by the distance and orientation of its neighboring stitches, not by the full path of the yarn. The tricky bit is to express the energy of the yarn as a function of the stitch dimensions.

In our simplified knit, the yarn deforms when it bends because stretching is much more energetically costly. The loop geometry of a stitch means that the curvature of the yarn is tightly linked to loop dimensions. As a stitch gets smaller, the bending energy increases, providing a simple relationship between energy and the network parameters. However, a lack of stretching implies that the yarn is inextensible, a constraint we also have to express. Again, the link between the length of yarn in a stitch and the stitch dimensions is direct. If a loop expands in all directions, the yarn’s length must increase. So if the yarn is inextensible, loop expansion in one direction must be compensated for by a shrinking in a different direction.

In our simplified knit, the yarn deforms when it bends because stretching is much more energetically costly.

• Facebook
• Twitter

Using these ideas, we obtained a mathematical formula giving the mechanical response of one stitch, and therefore also for a knit where all the stitches are deformed identically. The model perfectly captures the observed elastic response of a knit being pulled (see figure above), even when the knit is stretched to twice its initial size. At further elongations, however, stitches cannot laterally shrink any more because of the finite yarn diameter. The yarn then stretches and compresses—factors that are not taken into account in the model, which therefore underestimates the pulling force. To predict more realistic cases where deformation is inhomogeneous, we can keep the same approach, but we have an additional constraint: Stitches must keep the same neighbors. The math then gets a bit trickier because stitch dimensions are no longer simply described by mere numbers but by functions that vary depending on where we are on the fabric. By assuming a small deformation of the yarn, the equations remain solvable and provide us with a prediction of the overall shape of the fabric. The figure at the top of the opposite page shows this prediction layered on top of a picture of a knit pulled in such a way that deformation is highly inhomogeneous, and the agreement is fairly good. This theoretical process, called homogenization, is used to describe a complex material with just a limited set of equations, and consequently, we are able to ignore its complex fine structure.

Crackling the Knits

Let’s now look at the noisy part of the response in the graph at the bottom of page 109. By zooming in on the force curve, we notice that the fluctuations follow a very specific shape: a slow linear increase interrupted by an abrupt drop. We know that this behavior did not arise from the limitations of our equipment, because it was up to 100 times larger than the precision of the experiment. Instead it can be explained by the fact that when one object is pushed along the surface of another, friction resists the pushing force. Below a critical force, friction dominates and the two objects stick together; above this force, the push overcomes the friction and the objects begin to slide across each other.

This phenomenon appears at each crossing point in our knit. As you pull the knit, the contacts will suddenly slide when the critical force is reached and the friction is overcome. That’s why you get the slowly increasing force, interrupted by the sliding-induced plunges.

The drops have many different sizes, meaning that the contacts don’t slide one by one but slip in groups. Indeed, contacts are not isolated from one another because they are linked by the elastic yarn. To spot those sliding, we took sequences of knit pictures separated by less than 0.1 percent of stretching— not enough to induce visible global deformation, but sufficient to trigger sliding of some contacts. It turns out that sliding occurs on large and elongated groups of stitches that also display varied sizes (see the figure above).

To characterize those sliding events we first needed to look at statistical quantities, such as the probability distribution of the drops’ amplitude. It turns out that there are lots of small drops but few big ones, following a power-law distribution. One feature of this law is scale invariance, which means that some event properties are independent of their size. To illustrate this effect in the distribution, we can simply zoom in on a small portion of the curve and see that the size of the corresponding events cannot be distinguished anymore: The decreased rate remains constant.

This property is characteristic of so-called crackling noise—an intermittent response displayed by lots of systems that exhibit sudden events when they are slowly loaded. The most widely studied example is the Earth’s crust. When two tectonic plates (such as the Pacific Plate and the North American Plate) rub against each other while traveling in opposite directions, they slowly build up energy while trying to overcome friction, but will then suddenly shift, resulting in an earthquake. The probability distribution of the earthquake’s size, known as the Gutenberg–Richter law, shows the same features as those measured in our knit.

Earthquakes, Robots, and Sweaters

From structural mechanics to earthquakelike statistics, the physics behind pulling a knit is very rich. Although understanding the elasticity of knits may help scientists find direct applications in composite reinforcement, soft robotics, or architecture, comprehending the statistical part may help fundamental physicists understand why such different systems show similar behavior.

By simplifying the knits, we have managed to isolate and understand different mechanisms that might otherwise be hiding behind other complex phenomena in standard knits. But we have to be careful to not oversimplify— for instance, if we had completely gotten rid of friction, as we initially planned to, we would have missed the crackling phenomenon.

The next stage in this research is to add complexity, step by step and in a controlled way, by changing the knit pattern or the yarn properties. Maybe by the time we unravel all the intricacies of knitting, I may even have learned how to knit a sweater for my grandmother.

This article has been expanded and adapted from one that appeared in Physics World.