where t is expressed in seconds. Time$-$temperature superposition is obeyed. The shift factor for $60\backslash$deg C$,$ relative to $20\backslash$deg C$,$ is a$60=10-3.$ A section of PVC pipe is of length $2$ m$,$ outside diameter $40$ mm$,$ and wall thickness $2$ mm$.$ Calculate the relative rotation of the ends of the pipe when an axial torque of $8$ N$-$m is applied for $20$ hours under each of the following conditions: $4\mathrm{.}16$ A certain pipe$-$grade PVC deforms in shear with a creep compli$-$

ance function at $20C$ of the form

$J(t)=0\mathrm{.}75+0\mathrm{.}15lo{g}_{10}t+0\mathrm{.}018(lo{g}_{10}t{)}^{2}GP{a}^{-1}$

where $t$ is expressed in seconds. Time$-$temperature superposition is

obeyed. The shift factor for $60C,$ relative to $20C,$ is $a_{60}={10}^{-3}.$ A

section of PVC pipe is of length $2$ m $,$ outside diameter $40$ mm $,$ and

wall thickness $2$ mm $.$ Calculate the relative rotation of the ends of

the pipe when an axial torque of $8$ Nm is applied for $20$ hours

under each of the following conditions:

$(1)$ temperature constant at $20C$;

$(2)$ temperature constant at $60C$; and,

$(3)$ temperature constant at $20C$ for $19\frac{3}{4}$ hours, followed by temper$-$

ature constant at $60C$ for $\frac{1}{4}$ hour.

$($a$)$ temperature constant at $20\backslash$deg C;

$($b$)$ temperature constant at $60\backslash$deg C; and,

$($c$)$ temperature constant at $20\backslash$deg C for $19$ hours, followed by temperature constant at $60\backslash$deg C for $$ hour.