When materials get hotter, their atoms and molecules jiggle more, pushing their neighbours further apart. This causes the material to expand. But not all materials expand by the same amount. To quantify this, we use a property called the coefficient of linear expansion, symbolised by the Greek letter alpha ($\\\alpha$).

This coefficient tells us the fractional change in length per degree of temperature change. For a solid object like a rod or a wire, the change in its length ($\\\\\\Delta L$) is directly proportional to its original length ($L_0$) and the change in temperature ($\\\\\\Delta T$).

For example, structural steel has an $\\\alpha$ of about $12 \\times 10^{-6}$ per degree Celsius. This means for every degree Celsius increase in temperature, a steel beam will expand by 12 millionths of its length. It sounds tiny, but for a 500-metre-long bridge span heating up by 30°C on a summer day, the expansion is 18 centimetres. Engineers must account for this by including expansion joints in their designs.

Expansion isn't just a one-dimensional affair. When a flat plate is heated, it expands in both length and width, resulting in an increase in its surface area. This is called superficial or area expansion. Similarly, a three-dimensional object will expand in all directions, leading to an increase in volume.

We can define coefficients for these as well. The coefficient of area expansion is beta ($\\\\\\beta$), and the coefficient of volume expansion is gamma ($\\\\\\gamma$). For isotropic materials, which expand uniformly in all directions, these coefficients are directly related to the linear coefficient.

What happens if a material tries to expand but is blocked? Imagine a steel rail with its ends fixed firmly in place. As it heats up, it can't get longer. The expansion that would have happened is converted into internal pressure, or stress. This phenomenon is known as thermal stress.

The material experiences a compressive strain, which is the fractional deformation it would have undergone if it were free to expand. The resulting stress ($\\\\\\sigma$) can be calculated using the material's Young's Modulus ($E$), a measure of its stiffness.

Not everything follows the simple rule of 'expand when heated'. Water exhibits a strange behaviour. While it expands when heated above 4°C, it actually becomes denser as it cools from 4°C down to 0°C. This is the anomalous expansion of water and has profound consequences for life on Earth.

We can also harness differing expansion rates for practical uses. A bimetallic strip is made by bonding two different metals, like steel and brass, together. Brass expands more than steel when heated. When the strip is heated, the brass side gets longer than the steel side, forcing the strip to bend.

This bending action is the principle behind simple mechanical thermostats used in ovens and old heating systems. When the temperature reaches a certain point, the strip bends enough to touch an electrical contact, switching a circuit on or off.

Understanding these principles allows us to predict how materials will behave in changing environments and to design robust structures and clever devices that work with, or in spite of, thermal expansion.