First slide

Thermal expansion

Question

Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of 57°C is drunk. You can take body (tooth) temperature to be 37°C and $α_{Cu}$ = $1.7 \times 10^{- 5} /^{0} C$ bulk modulus for copper $B_{Cu}$ = 140 x $10^{9} N / m^{2}$

Moderate

A

$42 \times 10^{6} N / m^{2}$
B

$22 \times 10^{6} N / m^{2}$
C

$36 \times 10^{6} N / m^{2}$
D

$18 \times 10^{6} N / m^{2}$

Solution

Given, decrease in temperature
$(∆ t) = 57 - 37 = 20^{0} C$
Coefficient of cubical expansion of copper
$γ_{Cu} = 3 α_{Cu} = 5.1 \times 10^{- 5} /^{0} C$
Let initial volume of the cavity be V and its volume increases by $∆ V$ due to increase in temperature.
$∴ ∆ V = γV ∆ t$
$\Rightarrow \frac{∆ V}{V} = γ ∆ t$
Thermal stress produced = B x Volumetric strain
= $B \times \frac{∆ V}{V} = B \times γ ∆ t$
= 140 x $10^{9}$ x (1.5 x $10^{- 5}$ x 20) = 42 x $10^{6} N / m^{2}$
This is about $10^{3}$ times of atmospheric pressure.

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