Thermal Expansion Strain Calculator
Thermal expansion is the tendency of materials to change their dimensions in response to temperature changes. The linear thermal strain is simply the product of the coefficient of thermal expansion (CTE or alpha) and the temperature change (delta T). This calculator also computes the resulting change in length for a given initial length, and the thermal stress that would arise if the expansion were fully constrained. Enter the CTE (in 1/C or 10^-6/C), the temperature change, the original length, and optionally the Young's modulus to compute thermal stress.
Thermal expansion formula
epsilon = alpha * delta T
delta L = L0 * epsilon
sigma (constrained) = E * epsilon
Where alpha is the linear CTE (1/C), delta T is the temperature change (C), L0 is the original length, and E is Young's modulus. The strain epsilon is dimensionless; delta L has the same units as L0.
Practical applications
Thermal expansion analysis is critical in bridge expansion joints (steel bridges can expand by tens of millimeters over seasonal temperature swings), pipeline design (expansion loops accommodate thermal growth), electronic packaging (CTE mismatch between die and substrate causes solder fatigue), and precision metrology (temperature control to 0.1 C is required for micrometer-level measurements on steel parts).
Thermal expansion strain: frequently asked questions
What is thermal strain?
Thermal strain (epsilon) is the fractional change in length a material experiences due to a temperature change, given by epsilon = alpha * delta T. It is dimensionless (m/m). When constrained, thermal strain produces thermal stress: sigma = E * alpha * delta T.
What are typical CTE values for engineering materials?
Aluminum: 23.1 x 10^-6 /C. Carbon steel: 11.7 x 10^-6 /C. Copper: 17.0 x 10^-6 /C. Invar (Fe-Ni): 1.2 x 10^-6 /C. Glass (borosilicate): 3.3 x 10^-6 /C. PTFE: 135 x 10^-6 /C. Invar's near-zero CTE makes it valuable for precision instruments.
What is the difference between linear and volumetric CTE?
Linear CTE (alpha_L) gives the fractional change per unit length in one direction. Volumetric CTE (alpha_V) gives the fractional change in volume; for isotropic materials, alpha_V is approximately 3 * alpha_L. This calculator uses linear CTE.
How is thermal stress calculated from thermal strain?
If thermal expansion is fully constrained, thermal stress sigma = E * alpha * delta T, where E is Young's modulus. A steel beam (E = 200 GPa, alpha = 12e-6 /C) prevented from expanding over delta T = 100 C develops: sigma = 200e9 * 12e-6 * 100 = 240 MPa, approaching yield strength.
Why do composite materials have direction-dependent CTE?
In unidirectional fiber composites, stiff fibers constrain thermal expansion along their axis while the softer matrix dominates transverse expansion. This results in very low CTE along the fiber direction and higher CTE perpendicular to fibers. Carbon fiber composites can achieve near-zero axial CTE.
Official sources
- ASTM E228, "Standard Test Method for Linear Thermal Expansion of Solid Materials with a Push-Rod Dilatometer": astm.org.
- NIST, "NIST Materials Data Repository - Thermal Properties": materialsdata.nist.gov.
- ASM International, "ASM Handbook Vol. 2: Properties and Selection: Nonferrous Alloys": asminternational.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.