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MAT-31

Linear thermal expansion

ΔL = α L ΔT. Length change from a temperature step.

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ElasticityISO 11359

Governing equation

ΔL=αLΔT\Delta L=\alpha L\Delta T

ΔT∈[−200,400]\Delta T \in \left[-200, 400\right]

where

\alpha
CTE (10^{-6}/K)
L
Length (m)
\Delta T
Temperature rise (K)
\Delta L
Extension (mm)

The figure, part by part

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①

Governing law — red

ΔL=αLΔT\Delta L=\alpha L\Delta T

ΔT∈[−200,400]\Delta T \in \left[-200, 400\right]

ΔT=−200\Delta T = -200ΔL=−4.8\Delta L = -4.8

②

Domain of the variable — blue

ΔTmin⁡=−200,ΔTmax⁡=400\Delta T_{\min} = -200,\quad \Delta T_{\max} = 400

ΔT∈[−200,400]\Delta T \in \left[-200, 400\right]

ΔT=−200\Delta T = -200ΔT=−200\Delta T = -200

③

Response curve — green

ΔL=αLΔT\Delta L=\alpha L\Delta T

ΔT∈[−200,400]\Delta T \in \left[-200, 400\right]

ΔT=−200\Delta T = -200ΔL=−4.8\Delta L = -4.8

④

Rate of change — amber

dΔLdΔT\frac{\mathrm{d}\Delta L}{\mathrm{d}\Delta T}

ΔT∈[−200,400]\Delta T \in \left[-200, 400\right]

ΔT=−196.2\Delta T = -196.2ΔL′=0.024\Delta L^{\prime} = 0.024

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-31 — Linear thermal expansion) is the form associated with ISO 11359. Working symbols: α\alpha, LL, ΔT\Delta T →\rightarrow ΔL\Delta L. The linear CTE α is (1/L) dL/dT. Integrated at constant α it is ΔL/L = α ΔT, the origin of thermal stress E α ΔT.

Purpose

Purpose: compute ΔL\Delta L from α\alpha, LL, ΔT\Delta T in Materials via ΔL=αLΔT\Delta L=\alpha L\Delta T ΔL = α L ΔT. Length change from a temperature step. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given α=12.000 10−6/K\alpha = 12.000\,\mathrm{10^-6/K}, L=2.000 mL = 2.000\,\mathrm{m}, ΔT=80.000 K\Delta T = 80.000\,\mathrm{K}, the governing relation ΔL=αLΔT\Delta L=\alpha L\Delta T yields ΔL=1.920 mm\Delta L = 1.920\,\mathrm{mm}. A bar, a thermometer, a stretch. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Extension \Delta L1.920 mm
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Narration of this film

A bar, a thermometer, a stretch.

The linear CTE α is (1/L) dL/dT. Integrated at constant α it is ΔL/L = α ΔT, the origin of thermal stress E α ΔT.

Reading speed

Watch on YouTube