Material Selection Index Stiffness Thermal⚠ unverified
Mechanical / Materials · Compute the thermal distortion resistance selection index
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | Pa | 1.0 | Young's modulus of the material |
| alpha | α | 1/K | 1.0 | Coefficient of linear thermal expansion |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | E/α | Pa*K | Thermal distortion resistance index, in pascal kelvin (Pa*K) |
The science & history
Understanding the Parameters
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Young's modulus $E$ — the mechanical stiffness. A stiff frame resists deflection under load and holds alignment.
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Thermal expansion $\alpha$ — how much the material grows per kelvin. Low $\alpha$ means small thermal movements; putting it in the denominator rewards low-expansion materials.
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Index $M = E/\alpha$ — combines "stiff" and "dimensionally stable". Invar (very low $\alpha$) and ceramics/carbon composites score high; ordinary aluminium (high $\alpha$) scores low despite being light.
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Related thermal indices — for thermal-shock resistance the relevant group is different ($\sigma_f/(E\alpha)$, resisting cracking under thermal gradients); this $E/\alpha$ index targets distortion/precision, not fracture.
Derivation (Approaching a Proof)
Consider a precision structure that must remain both stiff (resist load deflection) and dimensionally stable (resist thermal drift). A temperature change $\Delta T$ produces a free thermal strain $\varepsilon_{th} = \alpha\,\Delta T$; the same structure under a working stress $\sigma$ has an elastic strain $\sigma/E$. For the thermal drift to be small relative to the (fixed, allowable) elastic response, the figure of merit is the ratio of mechanical to thermal sensitivity:
$$\frac{\text{stiffness}}{\text{thermal strain rate}} \sim \frac{E}{\alpha}.$$
Maximising $E/\alpha$ simultaneously favours small deflection (large $E$) and small thermal movement (small $\alpha$). It is the material group that survives when both requirements are combined into one objective.
Dimensional check. $M = \dfrac{E}{\alpha} = \dfrac{\text{Pa}}{1/\text{K}} = \text{Pa}\cdot\text{K}$ — as the registry label states.
History and Development
Thermal-distortion indices are part of Ashby's multi-objective materials selection for precision engineering: metrology frames, lithography stages, telescope structures, and length standards all need stiffness and dimensional stability. The development of ultra-low-expansion materials (Invar, 1896; Zerodur and ULE glass-ceramics) was driven precisely by maximising this class of index, trading raw stiffness or weight for near-zero $\alpha$.
Related Concepts: Specific Modulus, Thermal Stress Index, Thermal Stress, Ashby Chart Index, Ashby Charts, Material Selection Index Stiffness
Notes: Targets precision/dimensional stability (stiff + low expansion), not thermal-shock fracture (that uses $\sigma_f/(E\alpha)$). High for Invar, ceramics, carbon composites; low for aluminium. Complements Thermal Stress Index ($\alpha E$).