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

# 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.