a$)(10$ pts$)$ A rubber band is stretched to $\backslash$lambda $=8$ at room temperature and hold there to reachQUESTION $3$: $(25$ pts$)$ Rubber Elasticity

$($a$)$ The statistical mechanical result of the entropy change upon affine deformation is given

by

$S=\frac{-k{v}_e}{2}({}_1^2+{}_2^2+{}_3^2-3)$

to obtain the relation between stress and extension ratio for the case of biaxial stress $({}_x,$

$\{$:${}_y)$ with extension ratios $({}_x=2,{}_y=5)$

$($b$)$ In case of an ideal rubber of density $$ and specific heat capacity $c$ under nominal tensile

stress $,$ show that the temperature rises with extension ratio $$ during adiabatic

stretching according to

$(\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}}{)}_S=\frac{}{c}$

and find the temperature rise on rapid stretching of a cross$-$linked natural rubber from

the stress$-$free state to $200\%$ extensions $($i$.$e$.=3).$ Take the properties of the rubber to

be $=909k\frac{g}{m^3},c=1\mathrm{.}92k\frac{J}{k}g.K,$ and $G($shear modulus$)=500$kPa. Assume it behaves

as an ideal rubber and obeys Gaussian statistics.

$($c$)$ Following the above question, deduce the number of monomers between cross$-$links

per $m^3$ and hence the number average degree of polymerization between cross$-$links.

$($d$)$ Discuss the main limitations of ideal rubber theory?

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.