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.

Short answer

Answer: The approximate pressure engendered in steel by a 1.0°C temperature increase is 2.304 × 10^6 Pa.

Step-by-step solution

Calculate the change in length due to temperature increase

Using the given linear expansion coefficient \(\alpha\), we can find the change in length (\(\Delta L\)) due to the temperature change (\(\Delta T\)) using the linear expansion equation: $$ \Delta L = \alpha L_0 \Delta T $$ where \(L_0\) is the initial length and \(\Delta T\) is the temperature increase. Since we want to calculate pressure, we will keep the unknown quantities as variables and move on to the next step.

Calculate change in volume due to change in length

Now, let's find the change in volume (\(\Delta V\)) resulting from the change in length. Since the steel is undergoing a linear expansion, we can assume that each side of the steel expands by the same proportion. Then, the change in volume can be expressed as: $$ \Delta V = V_0 (\alpha \Delta T)^{3} $$ where \(V_0\) is the initial volume.

Calculate the pressure engendered using the bulk modulus

Finally, we will use the bulk modulus (\(B\)) to calculate the pressure engendered (\(P\)) due to the change in volume. The bulk modulus is defined as: $$ B = -V_0 \frac{\Delta P}{\Delta V} $$ We want to solve for \(\Delta P\), so we will rearrange the equation: $$ \Delta P = -\frac{B \Delta V}{V_0} $$ Now, we can substitute the expression for \(\Delta V\) found in Step 2: $$ \Delta P = -\frac{B V_0 (\alpha \Delta T)^{3}}{V_0} $$ Since \(V_0\) appears in both the numerator and denominator, we can cancel it out: $$ \Delta P = B (\alpha \Delta T)^{3} $$ Now we can plug in the given values for \(\alpha\), \(B\), and \(\Delta T\): $$ \Delta P = (160 \times 10^9{ }\mathrm{Pa})(1.2 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1})(1.0^{\circ} \mathrm{C})^{3} $$ Calculating the final result: $$ \Delta P \approx 2.304 \times 10^6{ }\mathrm{Pa} $$ Therefore, the pressure engendered in steel by a \(1.0^{\circ}\mathrm{C}\) temperature increase is approximately \(2.304 \times 10^6{ }\mathrm{Pa}\).

Key concepts

Linear Expansion Coefficient

When materials are heated, they usually expand in size—a phenomenon known as thermal expansion. The linear expansion coefficient, symbolized by \( \alpha \), quantifies this change for linear dimensions (length). It represents the fractional change in length per degree of temperature change. For instance, steel's linear expansion coefficient of \( \alpha=1.2 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1} \) means that steel expands by \(1.2\times10^{-5}\) of its original length for every Celsius degree increase in temperature.

Understanding this concept is critical in engineering and construction as it helps in designing structures that can accommodate the thermal expansion, reducing stress and potential damage. A bridge, for example, may have expansion joints to allow for this expansion and protect the structure's integrity.

Bulk Modulus

Whereas the linear expansion coefficient addresses one-dimensional changes, the bulk modulus encompasses three-dimensional volumetric responses to pressure. Represented by \(B\), the bulk modulus defines a material's resistance to uniform compression. It is related to the material's incompressibility—higher values indicate that the material is harder to compress. With a bulk modulus of \(B=160\) GPa, steel is relatively incompressible, indicating that it can withstand significant compressive forces without a large change in volume.

The relationship between pressure change \(\Delta P\) and volume change \(\Delta V\) is illustrated by the equation \(B=-V_0 \frac{\Delta P}{\Delta V}\), where \(V_0\) is the original volume. This equation shows that a large negative bulk modulus value corresponds to a small change in volume under pressure. The negative sign indicates that an increase in pressure results in a decrease in volume.

Engendered Pressure Calculation

When a material undergoes thermal expansion without being able to freely expand—imagine a steel beam in a rigid frame—a pressure is engendered due to the material's resistance to changing shape. This engendered pressure can be calculated by leveraging both the linear expansion coefficient and the bulk modulus of the material.

From the equation for the bulk modulus, rearranged as \(\Delta P=-\frac{B \Delta V}{V_0}\), we can substitute the change in volume due to thermal expansion and solve for the pressure engendered. For steel that is heated by \(1.0^{\circ}\mathrm{C}\), the engendered pressure is calculated to be \(2.304 \times 10^6{ }\mathrm{Pa}\) as shown in the exercise solution.

This concept is essential in preventing structural failures in engineering, such as thermal cracking or undue stress on components, by allowing for the appropriate design modifications, such as expansion gaps in materials that undergo significant thermal expansion.