$2\mathrm{.}7.(20$ pts$)$ A sample of a particular ideal rubber has an average molecular mass between cross$-$links of $8500$ and a density of $\backslash (950\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ A block of this rubber, initially a cube with a side length of $120$ mm $,$ is tested at a temperature of $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Assume the rubber obeys Gaussian statistics and use the approximation $\backslash (\backslash$left$\backslash$langle r$^\{2\}\backslash$right$\backslash$rangle$\_\{$i$\}=\backslash$left$\backslash$langle r$^\{2\}\backslash$right$\backslash$rangle$\_\{0\}\backslash ).$

a$)$ Calculate its tensile modulus.

b$)$ Calculate its shear modulus.

c$)$ Axes $\backslash (\backslash$mathrm$\{$X$\},\backslash$mathrm$\{$Y$\}\backslash ),$ and Z are chosen parallel to the cube's edges. A compressive force $\backslash ($ F$\_\{$x$\}\backslash )$ is applied in the X direction to reduce the X$-$dimension from $120$ mm to $80$ mm $.$ Determine $\backslash ($ F$\_\{$x$\}\backslash ).$

d$)$ What do the Y and Z dimensions become?

e$)$ A further compressive force $\backslash ($ F$\_\{$y$\}\backslash )$ is applied in the Y$-$direction to reduce the Y$-$dimension to $80$ mm $,$ with the X $-$dimension remaining at $80$ mm $.$ Calculate the magnitudes of Fx and Fy $.$

f$)$ What is the Z$-$dimension now?

g$)$ Calculate the strain energy stored in the block after these deformations.