Temperature rise in simple deformation

A cylindrical specimen 25 mm in diameter and 25 mm high is being compressed by dropping a weight of 50 kg on it from a certain height. The material has the following properties: K = 104 MPa, n = 0.5, density = 2768 kg/ $m^3$ , and specific heat = 1255 J/kg.K. Assuming no heat loss and no friction, calculate the final height of the specimen if the temperature rise is 55 K

Question

Temperature rise in simple deformation

A cylindrical specimen 25 mm in diameter and 25 mm high is being compressed by dropping a weight of 50 kg on it from a certain height. The material has the following properties: K = 104 MPa, n = 0.5, density = 2768 kg/[latex]m^3[/latex], and specific heat = 1255 J/kg.K. Assuming no heat loss and no friction, calculate the final height of the specimen if the temperature rise is 55 K

Accepted Answer

The expression for heat is given by

[latex] ext { Heat }=\left(c_{p} ight)( ho)( ext { Volume })(\Delta T)[/latex]

or

[latex]\begin{aligned} ext { Heat } &=(1255)(2768)\left(\frac{\pi}{4} ight)\left(\frac{25}{1000} ight)^{2} imes\left(\frac{25}{1000} ight) imes 55 \mathrm{~J} \&=2345 \mathrm{~J}\end{aligned}[/latex]

Also we know specific energy

[latex]\begin{aligned}u=\frac{K \epsilon^{n+1}}{n+1} &=\frac{104 \epsilon^{0.5+1}}{0.5+1} \mathrm{MPa}\&=\frac{104 \epsilon^{1.5}}{1.5} imes 10^{6} \mathrm{~Pa}\end{aligned}[/latex]

Also, ideally

[latex]\begin{aligned} ext { Heat }= ext { Work }=(u)( ext { Volume }) &=\left[\frac{\pi}{4} imes\left(\frac{25}{1000} ight)^{2} imes \frac{25}{1000} ight] \frac{104 imes 10^{6}}{1.5}\epsilon^{1.5} \mathrm{~N}.\mathrm{m}\&=850.95 \epsilon^{1.5} \mathrm{~N} \cdot \mathrm{m}.\end{aligned}[/latex]

Now equating heat to work

[latex]2345=850.95 \epsilon^{1.5}[/latex]

hence,

[latex]\epsilon=1.965[/latex]

Using absolute values, we have

[latex]\ln \left(\frac{h_{o}}{h_{f}} ight)=\ln \left(\frac{1}{h_{f}} ight)=1.965[/latex]

and therefore, [latex]h_{f}=3.5 \mathrm{~mm}[/latex]

Question 2.8: Temperature rise in simple deformation A cylindrical specime...

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