Question

Thermal expansion seems like a small effect, but it can engender tremendous, often damaging, forces. For example, steel has a linear expansion coefficient of \(\alpha=1.2 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\) and a bulk modulus of \(B=160\) GPa. Calculate the pressure engendered in steel by a \(1.0^{\circ} \mathrm{C}\) temperature increase if no expansion is permitted.

Short answer

Answer: The pressure created in steel due to a temperature increase of 1.0°C if no expansion is allowed is 5.76 x 10⁵ Pa.

Step-by-step solution

Find the fractional change in length

Using the linear expansion coefficient, we can find the fractional change in length resulting from the temperature change by applying the formula: \(\delta L / L = \alpha \delta T\), where \(\delta L\) is the change in length, \(L\) is the original length, \(\alpha\) is the linear expansion coefficient and \(\delta T\) is the temperature change. We have \(\alpha = 1.2 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\) and \(\delta T = 1.0^{\circ} \mathrm{C}\). Let's plug those in: \(\delta L / L = 1.2 \cdot 10^{-5} \cdot 1.0 = 1.2 \cdot 10^{-5}\).

Calculate the fractional change in volume

Since steel is close to isotropic, the fractional change in volume will be three times the fractional change in length. Thus: \(\delta V / V = 3 \cdot (\delta L / L) = 3 \cdot (1.2 \cdot 10^{-5}) = 3.6 \cdot 10^{-5}\).

Find the pressure

Now, we can use the bulk modulus, \(B\), to calculate the pressure engendered in steel due to the volume change. The bulk modulus is defined as the ratio of pressure change to the fractional change in volume. Thus: \(P = B \cdot (\delta V / V)\). We have \(B = 160\) GPa \(= 160 \cdot 10^{9}\) Pa. Let's plug in the values of \(B\) and the fractional change in volume: \(P = 160 \cdot 10^9 \cdot (3.6 \cdot 10^{-5}) = 5.76 \cdot 10^5\) Pa. The pressure engendered in the steel by a \(1.0^{\circ} \mathrm{C}\) temperature increase if no expansion is permitted is \(5.76 \cdot 10^5\) Pa.