Theory — Thermal Expansion in Composites
A composite rod made of different materials expands by different amounts in each section.
This tool treats the rod as axial sections in series: the same axial force \(F\) passes through every section,
while the thermal strain depends on each material’s coefficient of linear expansion \( \alpha \).
1) Free thermal expansion (no constraint)
For one uniform section:
For a composite rod with sections \(i=1,\dots,N\):
This is the default calculation: each section expands independently and the net elongation is the sum.
2) Effective coefficient for the whole composite
Define total length \(L_{\text{tot}}=\sum_i L_i\). The effective coefficient is:
This lets you replace the entire composite by a single “equivalent” uniform rod for free expansion only.
3) Constrained composites: thermal stress (series model)
If the rod is constrained, an axial force \(F\) develops. For a section with area \(A_i\) and Young’s modulus \(E_i\),
the mechanical strain is \(\varepsilon_{mech,i}=\sigma_i/E_i\) with \(\sigma_i=F/A_i\).
Total elongation is the sum of thermal + mechanical contributions:
Fully constrained means \(\Delta L=0\). More generally you may impose a target \(\Delta L_{\text{target}}\):
With equal areas \(A_i\), the stress is the same in all layers; if areas differ, the force is the same but \(\sigma_i=F/A_i\) differs.
4) Worked example (from the prompt)
Brass section + steel section, each \(0.5\,\text{m}\), heated by \(\Delta T=100^\circ\text{C}\):
5) Notes & limitations
- This calculator models a straight rod with axial sections in series (1D).
- It does not model bimetallic-strip bending (that needs curvature and geometry).
- Material properties can vary with temperature; here \( \alpha \) and \(E\) are treated as constants.
