Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Thermal Expansion and the Bi-material Strip Summary
Thermal Expansion and the Bi-material Strip AimsBefore you startIntroductionExperiment: Measurement of Young's modulusResults: Measurement of Young's modulusSimulation: Measurement of Young's modulusOrigin of thermal expansionThe bi-material stripExperiment: Estimating the boiling temperature of nitrogenResults: Estimating the boiling temperature of nitrogenExperiment: Measuring the thermal expansivity of polycarbonateResults: Measuring the thermal expansivity of polycarbonateSummaryQuestionsGoing further
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Summary

In this TLP we have:

  1. Learned how the Young's modulus of a material may be determined using the relationship between the deflection of a cantilever beam δ and the load P applied to it, given by

    \[\delta = \frac{1}{3}\frac{{P{L^3}}}{{EI}}\]

    where L is the distance from support to point of load application, I is the second moment of area of the beam's cross section, and E is the Young's modulus of the beam's material.

  2. Used measurements of the deflection and load of a steel cantilever beam to determine the Young's modulus of steel as 210 GPa.

  3. Looked at how the asymmetry in the energy-separation graph for atoms in a solid gives rise to an increase in the average interatomic spacing of vibrations as the temperature is increased, the origin of thermal expansion.

  4. Derived a formula relating curvature κ to the length x and deflection δ of a bi-material strip:

    \[\kappa = \frac{{2\sin \left[ {{{\tan }^{ - 1}}\left( {\delta }/{x} \right)} \right]}}{{\sqrt {\left( {{x^2} + {\delta ^2}} \right)} }}\]

  5. Defined the misfit strain Δε for the two components of a bi-material strip as

    Δε  = (αA - αB) ΔT

    and quoted a formula relating the curvature κ to the misfit strain, the thickness of the strips h (of equal thickness) and the ratio of the Young's moduli E*

    \[\kappa = \frac{{12 \Delta \varepsilon }}{{{h_{}}\left( {{E_*} + 14 + 1 / {E_*}} \right)}}\]

  6. Used measurements of the change in shape of a steel-aluminium bi-material strip immersed in liquid nitrogen to estimate the boiling temperature of nitrogen as -193ºC.

  7. Used measurements of the change in shape of an aluminium-polycarbonate bi-material strip immersed in liquid nitrogen to estimate the thermal expansivity of the polycarbonate as 4.5 x 10-5 K-1.

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