Plot a distribution around a circle in Mathematica?

Questions
1

I want to plot the distribution below around a circle and shade it. Currently, I have it in a polar plot. My expectation is to have something close to the image below. The goal is to show damage distribution around the circle. Thank you.

PolarPlot[
 1/(4 Pi) + (5/16 Pi) (1 - 3 Cos[2 t]) + 
  9/(256 Pi) (9 - 20 Cos[2 t] + 35 Cos[4 t]), {t, 0, 5 Pi}]

damagedistribution.PNG
damagedistribution.PNG286Ɨ173 35.9 KB

2

This may be a start:

PolarPlot[
  Abs[1/(4 Pi) + (5/16 Pi) (1 - 3 Cos[2 t]) + 
    9/(256 Pi) (9 - 20 Cos[2 t] + 35 Cos[4 t])], {t, 0, 2 Pi}] /. 
 Line -> Polygon
3

Crossposted here.