Curly Hair Physics
February 13, 2014

Researchers Create First Detailed Model For The 3D Shape Of Curly Hair

April Flowers for - Your Universe Online

In animated films, the heroes and villains tend to be on opposite ends of the moral spectrum. Their hair, however, tends to be remarkably similar. It is usually rigid, or — if it moves at all — it is straight and swings to and fro, according to MIT researchers. An animated character with bouncy, curly hair is rare because computer animators don't have a simple mathematical means for describing it.

You may soon notice a change in animated features, however. A research team from MIT and the Université Pierre et Marie Curie in Paris has provided the first detailed model for the 3D shape of a strand of curly hair.

The applications of this work to the computer animation film industry are obvious, but the research team says it may also be used by engineers to predict the curve that long steel pipes, tubing, and cable develop after being coiled around a spool for transport. Like a stubborn garden hose whose intrinsic curves make it behave in unpredictable ways, these materials can be a challenge in the field. These types of items, including hair, are examples of what engineers would call a slender, flexible rod.

"Our work doesn't deal with the collisions of all the hairs on a head, which is a very important effect for animators to control a hairstyle," Pedro Reis, an assistant professor in MIT's Department of Civil and Environmental Engineering and Department of Mechanical Engineering, told MIT's Denise Brehm. "But it characterizes all the different degrees of curliness of a hair and describes mathematically how the properties of the curl change along the arc length of a hair."

Reis wasn't thinking of hair when he set out to study the natural curvature in flexible rods. As he studied several small flexible, curved segments of tubing suspended from a structure in his lab, however, he realized that they weren't so different from strands of curly hair hanging from a person's head. He then enlisted the help of Basile Audoly of the Université Pierre et Marie Curie, who had previously developed a theory to explain the 2D shape of human hair.

Reis and Audoly used lab experimentation, computer simulation, and theory — "the perfect triangle of science," Reis said — to identify the main parameters for curly hair. They simplified these parameters into two dimensional parameters for curvature (relating to the ratio of curvature and length) and weight (relating to the ratio of weight and stiffness). Their computer model will use curvature, length, weight and stiffness to predict the shape of a hair, steel pipe, or Internet cable suspended under its own weight.

The 2D hook of a curly hair grows larger as it curls up from the bottom, until it reaches a point where it becomes unstable under its own weight and falls out of plane to become a 3D helix. If only a portion of the strand is curled, the researchers call it a localized helix. If, on the other hand, the entire strand is curled, it is known as a global helix.

If its parameters change, a curl can change phase from 2D to 3D local helix to 3D global helix, and back again. Because gravity weights a strand of hair from the bottom, the top portion has more weight under it than the tip, which has none. This means that if the weight on a hair is too great for its innate curliness, the curl will fail and become either straight or helical, depending on the length and stiffness of the strand.

To understand the curvature, flexible, thin rods using molds as small as a bottle of Tabasco sauce and as large as the columns in MIT's Lobby 7 — approximately 3 feet in diameter — were created by former MIT graduate student James Miller, who is now a research associate at Schlumberger-Doll Research. Miller injected a rubber-like material inside hollow flexible tubing wrapped around these molds. The tubing was cut away once the rubber material cured, leaving flexible polyvinyl thin rods whose natural curvature was based on the size of the object around which they had been wrapped.

The research team used dimensionless numbers to describe innate curvature, meaning that their equation will hold true at all scales. The steel piping used by the oil industry, for example, is flexible enough to be spooled even with measurements in kilometers. "We think of steel pipes as being nice and straight but usually at some point they're getting wrapped around something," Miller told Brehm. "And at large dimensions, they're so flexible that it's like you and I dealing with a limp spaghetti noodle."

"The mathematician [Leonhard] Euler first derived the equation for a slender elastic body — like a hair strand — in 1744," Audoly says. "Even though the equations are well-known, they have no explicit solution and, as a result, it is challenging to connect these equations with real shapes."

"The fact that I am bald and worked on this problem for several years became a nice running joke in our lab," Reis added. "But joking aside, for me the importance of the work is being able to take the intrinsic natural curvature of rods into account for this class of problems, which can dramatically affect their mechanical behavior. Curvature can delay undesirable instability that happens at higher loads or torsion, and this is an effect that engineers need to be able to understand and predict."

The findings are described in a recent issue of Physical Review Letters.