## Soft disks in contact (27th of May 2013)

A recently published paper >> Many-Body Contact Repulsion of Deformable Disks
deals with elasticity of soft disks in polygonal constraints (image above). The paper is a result of my collaboration with
>> Primož Ziherl, which
started in December of 2010, during Christmas Biophysics Workshop in Ptuj, when Primož and I first spoke about the problem. In the end it turned
out to be something very different from what we imagined in the beginning, but that is how it goes in research; research is like a box of chocolate -
you never know what you're gonna get.

Tha paper we wrote turned out to be fairly general - one could even say that it deals with the theory of elasticity. Many physicists think that
the theory of elasticity is a completely closed area of research, something which has been long ago researched in detail, something which was
last time taken seriously by Hertz. This is, of course, completely wrong. In fact, the "usual" theory of elasticity represents a fairly
inadequate description of the deformation of materials: it works only in the limit of linear response and for small displacements / deformations.

Here is one shocking prediction of a most standard theory of elasticity: **It takes a finite energy to compress any body to a point (!)**.
Here you are. I bet that many physicists still haven't seriously think about it, especially those dealing with the "most important"
questions of physics, e.g. whether God had any choice when he created the universe ...

The model that Primož and I used (which is, by itself, not original but is an extension of known models) does not have the usual deficiencies of
classical linear theories. In our model, it takes an infinite energy to compress a two-dimensional disk into a point. Furthermore, the model
consistently describes small and large deformations of disks, and it has a natural limit for small deformations, reducing to the theory of linear
response, i.e. it predicts the Young's modulus and the Poisson ratio which depends on the parameters of our "microscopic" model.

Our research, among other things, also shows that calling a body "soft", does not really mean much. According to our results, there are (at least)
two importantly different types of "softness" - the one in which the volume of the body changes with its deformation (the image above, top row;
shown is the deformation of the disk compressed in a square), and the one which allows the body to "easily" deform, but only in such a way that
its volume is preserved (isochoric, incompressible deformation; the image above, bottom row). For the disks of the later sort, Primož invented
a catchy name which we proposed in the paper: **semi-hard**.

It is interesting to note that the disks were made of about 40000 vertices (points), and that the elastic energy around each vertex was visualized by a colored polygon, as is shown in the image below (it is a zoomed-in part of the image above). The shape of each polygon depends of course on the neighbors of the vertex represented by a polygon.

Last updated on 27th of May 2013.