Consider an infinite medium with a circular stress$-$free hole of radius $r_0$ under a uniform farfield tension loading $F$ as shown:

Find the stress field in the medium. Show all work for full credit.

Hint $1$: Consider the general Michell solution to the biharmonic equation:

$=a_0+a_{1}logr+a_2{r}^2+a_3{r}^{2}logr$

$+(a_4+a_{5}logr+a_6{r}^2+a_7{r}^{2}logr)$

$+(a_{11}r+a_{12}rlogr+\frac{a_{13}}{r}+a_{14}{r}^3+a_{13}r+a_{16}rlogr)cos$

$+(b_{11}r+b_{12}rlogr+\frac{b_{13}}{r}+b_{14}{r}^3+b_{15}r+b_{16}rlogr)sin$

$+{\displaystyle \underset{n=2}{\overset{}{}}}(a_{n1}{r}^n+a_{n2}{r}^{2+}+a_{nj}{r}^{-n}+a_{n4}{r}^{2-n})cosn$

$+{\displaystyle \underset{n=2}{\overset{}{}}}(b_{n1}{r}^n+b_{n2}{r}^{2+x}+b_{n3}{r}^{-n}+b_{n4}{r}^{2-x})sinn$

Hint $2$: You may find the transformation equations helpful:

${}_{rr}={}_{}co{s}^2+{}_{yy}si{n}^2+{}_{xy}sin2$

${}_e={}_{xz}si{n}^2+{}_{yy}co{s}^2-{}_{xy}sin2$

${}_r=-\frac{1}{2}({}_{xz}-{}_{yy})sin2+{}_{xy}cos2$