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DoITPoMS Teaching & Learning Packages Thermal Expansion and the Bi-material Strip Definition of Young's Modulus
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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Definition of Young's Modulus

For solids that obey Hooke's Law:

Young's Modulus (E) is the ratio of tensile stress (s) to tensile strain (e) in a specimen subject to uniaxial tension.

\[E = \frac{\sigma }{\varepsilon }\]

This definition then begs for definitions of the terms within it.

For a force tending to elongate a specimen, the tensile stress is the ratio of the *force* (F) applied in a particular direction in the specimen to the cross-sectional area (A) of the specimen in a plane normal to the direction of the applied force.

\[\sigma = \frac{F}{A}\]

(* Strictly speaking, to achieve equilibrium this has to be a pair of equal and opposite forces acting along that direction.)

[If the direction of application of the force is reversed so that the force tends to shorten the sample, the stress is called "compressive".]

If an increment in the tensile stress applied to a specimen of length (l) produces an increment in length (dl) parallel to the direction of application of the stress, the increment in tensile strain (de) is given by

\[\delta \varepsilon = \frac{{\delta l}}{l}\]

and the "true" or "logarithmic" tensile strain (e) is given by

\[\varepsilon = \int_{{l_0}}^{{l_1}} {\frac{{dl}}{l}} = ln\frac{{{l_1}}}{{{l_0}}}\]

For small strains it is often sufficient to approximate the true strain by the "nominal" or "engineering" strain (e)

\[e = \frac{{{l_1} - {l_0}}}{{{l_0}}}\]

Uniaxial tension means that the only externally applied forces acting on the specimen are a pair of equal and opposite forces acting along a line.

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