$4\mathrm{.}1($a$)($Meyers $7\mathrm{.}1)$ In a polyvinyl chloride $($PVC$)$ plate, there is an elliptical, through$-$the$-$thickness cavity. The dimensions of the cavity are:

Major axis $\backslash (=1\backslash$mathrm$\{$~mm$\}\backslash )$

Minor axis $\backslash (=0\mathrm{.}1\backslash$mathrm$\{$~mm$\}\backslash )$

Compute the stress concentration factor $\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$t$\}\}\backslash )$ at the extremities of the cavity. $(2$ points$)$

$($b$)($Meyers $7\mathrm{.}2)$ Calculate the maximum tensile stress at the surfaces of a circular hole $($in the case of a thin sheet$)$ and a spherical hole $($in the case of a thick specimen$)$ subjected to a tensile stress of $200$ MPa $?$ The material is $\backslash (\backslash$mathrm$\{$Al$\}\_\{2\}\backslash$mathrm$\{$O$\}\_\{3\}\backslash )$ with $\backslash ($ v$=0\mathrm{.}2\backslash )(3$ points$)$

$($c$)$ Describe the design methodology based on stress concentration factor concept.

What's the limitation of this method? $(5$ point$)$