Expression for nano scale crack propogation

Question

The following image is of a nano-scale self replicating crack on a thin film/substrate pair. I want to construct a "position" function for the crack front's propagation. So far the only way I can think to achieve this is with a piece wise function, but I am not sure. Does anyone have an idea of how to define this shape as a continuous function?

here is the image

Answer 1

You need a parametric form, since the curve is multi-valued both along the horizontal axis and the vertical axis. I started from $$x=\sin(t)\\y=t$$ to get

Then I noted that as you go away from the center the points are moved upwards, so I added a term proportional to $x^2$ $$x=\sin(t)\\y=t+3x^2$$ and I get

You can now play with some parameters, or the exact functional representation for $y$