Quite a while ago I wrote a post on materials that are called auxetic. This is where the Poisson ratio, the ratio of how much thinner a material gets when stretched a certain amount, is negative. This means that if you had a rod of auxetic material and you tried stretching it at the ends, it would get thicker in the middle. To the same degree compressing actually makes it get thinner, the opposite of most materials we encounter in every day life. The actual production of auxetic materials, a type of mechanical metamaterial, is actually quite easy with multiple methods being suggested such as moulding, 3D printing and the use of precise incisions into certain regular materials.
One of the structures that auxetic materials can be used to make is called the hierarchical rotating square system which consists of squares of metamaterials that rotate. These squares themselves can both make up, and be made up of, larger and smaller rotating squares. The diagram below hopefully visualises this:
Despite being reliably produced the dynamics and changing mechanical properties of the hierarchical systems has still not been studied. Recently researchers have created a set of dynamics equations to describe the action and rotation of the whole but also the sections of the squares. The model assumes the base squares do not distort as well as the fact that the hinges connecting any two squares are symmetrical in acting, both facts which may not be completely true in all of these systems but can easily be ensured if designed well.
Once other factors such as the hinge friction and the harmonic oscillations of the system are taken into account you’ve got a group of second order differential equations to solve. The numerical solutions reveal that the relative rotation of the constituent units is caused by a tensile force between the squares. The extent of deformation, and therefore the Poisson ratio, is caused by the force that is the resistance to motion. This resistance to motion is caused purely by the hinges and so this means that if the magnitude of resistance that the hinges exert could be changed, then the mechanical properties, like the Poisson ratio, would change with it. There are some suggestions about the possibility of this mechanical control, perhaps using magnetic interactions, but these are still ideas that require some work.