Thermal Expansion

Most materials expand when heated and contract when cooled. For small temperature changes, the fractional change in length (strain) is approximately proportional to the temperature change. This property is crucial in engineering, physics, and many practical applications where temperature fluctuations occur, and neglecting it can cause significant structural issues.

Linear Expansion Model

The constant-alpha small-strain length model is a fundamental way to describe this change in one dimension. It is given by the equation:
L = L0 * (1 + alpha * deltaT)
Where L0 is the initial length, alpha is the linear expansion coefficient specific to the material, and deltaT is the temperature change. The fractional change in length, or strain, is equal to alpha * deltaT.

Volume Expansion

For an isotropic material (a material that expands equally in all directions), the volume expansion can be derived and approximated from the linear expansion. A cube of side length L has a volume V = L^3.
Using the linear model for each side, the new exact volume (within this model) is:
V = L0^3 * (1 + alpha * deltaT)^3
Expanding this cubic term and keeping only the first-order term (which is a valid approximation for very small strains) gives the commonly used linear approximation:
Vlin = V0 * (1 + 3 * alpha * deltaT)

Worked Example Calculation

Let us calculate the expansion for an isotropic cube. This shows how a linear expansion in all dimensions translates to a significantly larger volume expansion when compared to the simple linear change. By walking through these calculation steps carefully, you will see how the components combine to give both linear and volumetric expansions.

Set Geometry to Isotropic cube
Set Initial length L0 to 2 m
Set Expansion coefficient alpha to 20 micro/K (which is 20 * 10^-6 = 0.00002 /K)
Set Temperature change deltaT to 50 K

The linear strain is calculated as 0.00002 * 50 = 0.001. The new length is therefore L = 2 * (1 + 0.001) = 2.002 m. This matches the known result of 2.002 m for the length.
The new volume is calculated as V = 2^3 * (1 + 0.001)^3 = 8 * 1.003003001 = 8.024024008 m³.
The linear approximated volume is calculated as Vlin = 8 * (1 + 3 * 0.001) = 8 * 1.003 = 8.024 m³.
The difference is the remainder: 0.000024008 m³. We can see that the new length L is 2.002.

Experiment Simulation

Use the Thermal Expansion module to observe these effects in action. Try setting alpha or deltaT to 0 to see that the length and volume remain completely unchanged, proving the dependence on both factors. Try a negative temperature change (cooling) to see the material contract rather than expand. Finally, compare the "Isotropic cube" expansion to the "Rod" expansion to visualize the difference between one-dimensional and three-dimensional scaling.

Set Geometry to Isotropic cube, Initial length L0 to 2, Expansion coefficient alpha to 20 and Temperature change deltaT to 50. Read strain 0.001, length 2.002 m, V = 8.024024008 m^3, Vlin = 8.024 m^3 and positive remainder 0.000024008 m^3.
Set Initial length L0 to 1, Expansion coefficient alpha to 10 and Temperature change deltaT to -100. Read strain -0.001, length 0.999 m, V = 0.997002999 m^3 and Vlin = 0.997 m^3.
Set Geometry to Rod, Initial length L0 to 5, Expansion coefficient alpha to 0 and Temperature change deltaT to 50. Length stays 5 m and change is 0; the corresponding cube comparison stays 125 m^3.
Set Temperature change deltaT to 0. All changes vanish while the initial length and volume remain.
Set Initial length L0 to 0. The positive-length error replaces all computations. Restore L0 = 2, choose Isotropic cube, set alpha = 30 and deltaT = 100. Read strain 0.003 and a nonzero volume remainder 0.000216216 m^3.
Set Initial length L0 = 1, Expansion coefficient alpha = 0.01 and Temperature change deltaT = 1. The small positive remainder is approximately 3.00000001e-16 m^3; it is not replaced with zero.
Pan and choose Reset view. Inputs remain unchanged. The upper body keeps actual relative dimensions; the separately labelled lower inset magnifies strain by 100.

Limitations and Constraints

This model assumes the linear expansion coefficient alpha is strictly constant, which is generally a good approximation only for relatively small temperature changes; in reality, it can vary with temperature. The cubed expression is mathematically exact within the chosen linear-length model, but it is not an exact nonlinear material law. The domain in this simulation is restricted to keep the absolute strain |strain| <= 0.003. The diagram auto-fits the body; a separately labelled strain-times-100 inset exaggerates changes for visibility and must not be read as actual geometry. Numerically, the remainder is evaluated as V0(3s² + s³), where s is strain, to avoid cancellation for small changes. Additionally, no thermal stress or phase changes (like melting or freezing) are modeled here.

External References

The physics of thermal expansion and related derivations can be explored further in comprehensive university textbooks. The detailed mathematical breakdown of the linear and volume expansion models, as well as tables for real-world material coefficients, are widely available in these academic resources. You can read more about it below:

OpenStax University Physics Volume 2, 1.3 Thermal Expansion

Thermal Expansion