a$)(10$ pts$)$ A rubber band is stretched to $\backslash$lambda $=8$ at room temperature and hold there to reachQUESTION $1$: $(35$ pts$)$

$($a$)(10$ pts$)$ A rubber band is stretched to $=8$ at room temperature and hold there to reach

thermal equilibrium. It is then suddenly released. Would the temperature increase or

decrease? Explain using thermodynamics concepts and statistical nature of polymer

molecules. Estimate the temperature change assuming the specific heat capacity $c=2$

$k\frac{J}{k}g.K,$ density $=1\frac{g}{c}{m}^3$ and $M_x=10,000\frac{g}{m}ol.$

$($b$)(8$ pts$)$ Draw a nominal stress extension curve of a typical $($real$)$ polymer network. On

the same plot, show the prediction from ideal rubber theory. Identify where the curves

merge and deviate? Discuss the sources of deviations between the two curves. Assume

the polymer does not crystallize upon stretching.

$($c$)(10$ pts$)$ The statistical mechanical result of the entropy change of a polymer network

upon deformation is given by

$S=\frac{-k{v}_e}{2}({}_x^2+{}_y^2+{}_z^2-3)$

Where $k$ is the Boltzmann constant, $v_e$ is the number of elastically effective strands.

Obtain the relations between engineering stress$-$ extension ratio and engineering stress$-$

extension ratio for the case of equal biaxial stress $({}_x={}_y=)$ and extension

$$

$($d$)(7$ pts$)$ A thin bi$-$axially stretched polystyrene $($PS$)$ sheet is used to see$-$through envelope

that allows the addressee's name and address to be seen. The uncross$-$linked PS film is

stretched above its glass transition temperature $(T_g)$ with equal forces in two directions.

Assume that PS behaves as an ideal rubber at the stretching temperature. Estimate the

nominal stress to be imposed on each side of the film to double its original length.

thermal equilibrium. It is then suddenly released. Would the temperature increase or

decrease? Explain using thermodynamics concepts and statistical nature of polymer

molecules. Estimate the temperature change assuming the specific heat capacity c$=2$

kJ$/$kg$.$K$,$ density $\backslash$rho $=1$ g$/$cm$3$ and Mx$=10,000$ g$/$mol$.$

$($b$)(8$ pts$)$ Draw a nominal stress extension curve of a typical $($real$)$ polymer network. On

the same plot, show the prediction from ideal rubber theory. Identify where the curves

merge and deviate? Discuss the sources of deviations between the two curves. Assume

the polymer does not crystallize upon stretching.

$($c$)(10$ pts$)$ The statistical mechanical result of the entropy change of a polymer network

upon deformation is given by

$$

$=$ kv

S

e

$($

$$

$222++$

x

y

z

$3)$

$2$

Where k is the Boltzmann constant, ve is the number of elastically effective strands. Obtain the relations between engineering stress$-$ extension ratio and engineering stress$-$ extension ratio for the case of equal biaxial stress $(\backslash$sigma x $=\backslash$sigma y $=\backslash$sigma $)$ and extension $(\backslash$lambda x $=\backslash$lambda y $=\backslash$lambda $)$

$($d$)(7$ pts$)$ A thin bi$-$axially stretched polystyrene $($PS$)$ sheet is used to see$-$through envelope

that allows the addressee$$s name and address to be seen. The uncross$-$linked PS film is stretched above its glass transition temperature $($Tg$)$ with equal forces in two directions. Assume that PS behaves as an ideal rubber at the stretching temperature. Estimate the nominal stress to be imposed on each side of the film to double its original length.