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For a small temperature change, a homogeneous bar’s free length change is approximately proportional to its initial length and temperature change.
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Formula
ΔL and L0 in m; α in K−1; ΔT in K
| Symbol | Meaning | Unit |
|---|---|---|
| Thermal length change | m | |
| Linear coefficient of thermal expansion | K−1 | |
| Initial length | m | |
| Temperature change | K |
\Delta L=\alpha L_0\Delta TThis relation integrates dL/L=αdT when α is approximately constant across the temperature range. A varying coefficient or temperature field requires integration. It predicts free expansion; a fully restrained bar instead develops thermal stress depending on constraints and material response. Temperature differences have equal numerical values in kelvin and Celsius degrees.
This relation integrates dL/L=αdT when α is approximately constant across the temperature range. A varying coefficient or temperature field requires integration. It predicts free expansion; a fully restrained bar instead develops thermal stress depending on constraints and material response. Temperature differences have equal numerical values in kelvin and Celsius degrees.
For a small temperature change, a homogeneous bar’s free length change is approximately proportional to its initial length and temperature change.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Linear thermal expansion” and check its unit and magnitude.
A 2.0 m rail with α=12×10−6 K−1 warms by 30 K: ΔL=0.72 mm.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: ΔL and L0 in m; α in K−1; ΔT in K.
Formula cluster
For a small temperature change, a homogeneous bar’s free length change is approximately proportional to its initial length and temperature change.
A 2.0 m rail with α=12×10−6 K−1 warms by 30 K: ΔL=0.72 mm.
Estimate expansion gaps, thermal fits and free component deformation.
Here is how to work through a typical Linear thermal expansion (ΔL = αL₀ΔT) task step by step:
Task
Work through the numerical example for “Linear thermal expansion” and check its unit and magnitude.
Solution path
A 2.0 m rail with α=12×10−6 K−1 warms by 30 K: ΔL=0.72 mm.
For a temperature difference, kelvin and Celsius have the same numerical value. A rise of 20 °C is ΔT=20 K. Do not confuse a difference with absolute temperature; radiation laws use absolute kelvin differently. The unit of α is K−1, making αΔT dimensionless. The relation describes free, uniform length expansion with an approximately constant thermal expansion coefficient. It predicts no stress while the part can expand freely. Restraint, temperature gradients and temperature-dependent properties require an extended model and boundary conditions. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
If the thermal expansion coefficient changes appreciably over the temperature range, integrate α(T) over temperature. With gradients, local strain depends on the temperature field. The equation also describes a homogeneous freely expanding bar; geometry, composites or constraints can add effects. The relation describes free, uniform length expansion with an approximately constant thermal expansion coefficient. It predicts no stress while the part can expand freely. Restraint, temperature gradients and temperature-dependent properties require an extended model and boundary conditions. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Yes. Fully restrained thermal expansion can create a mechanical reaction. In a simple linear-elastic bar with complete axial restraint, σ≈−EαΔT; the sign indicates compression on heating. Compliant supports, temperature gradients and plastic behavior can change this estimate substantially. The relation describes free, uniform length expansion with an approximately constant thermal expansion coefficient. It predicts no stress while the part can expand freely. Restraint, temperature gradients and temperature-dependent properties require an extended model and boundary conditions. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Write the quantity and unit from the variable list next to every value: ΔL and L0 in m; α in K−1; ΔT in K. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example A 2.0 m rail with α=12×10−6 K−1 warms by 30 K: ΔL=0.72 mm. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. The relation describes free, uniform length expansion with an approximately constant thermal expansion coefficient. It predicts no stress while the part can expand freely. Restraint, temperature gradients and temperature-dependent properties require an extended model and boundary conditions. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
The background states this starting condition: This relation integrates dL/L=αdT when α is approximately constant across the temperature range. A varying coefficient or temperature. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. The relation describes free, uniform length expansion with an approximately constant thermal expansion coefficient. It predicts no stress while the part can expand freely. Restraint, temperature gradients and temperature-dependent properties require an extended model and boundary conditions. Finally check units, assumptions, measurement conditions and reference quantities against the problem before using the result.
Quanta exam set
Curated exam set for "Linear thermal expansion":
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Which formula describes Linear thermal expansion?
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How do you rearrange Linear thermal expansion for Base equation for the target quantity?
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Which common mistake happens with Linear thermal expansion?
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