The Young's Modulus

I will introduce two new quantities:

Stress  =     Force / Area	= F / A  (N / m2  or Pa)
Strain  =     Extension / Original length	e = x / l (no units)

Stress is the force which acts over unit area. When a tensile force acts on a wire it produces a tensile stress, the smaller the c.s.a. of the wire the greater the stress. If the stress exceeds a certain value, the breaking stress (or ultimate tensile strength) the wire breaks.

This stress produces a certain strain, the ratio of the extension to the original length. As a ratio it has no units. It can be expressed as a percentage.

Consider this example;

A wire of original length 1.2 m and diameter 0.8 mm extends by 2.5 mm when it supports a load of 3kg. Take g = 9.8 N/kg.

Stress = F / A      = mg / ( r² )          =  3 x 9.8 / (   (0.4 x 10^-3)² )     = 29.4 / 5.03 x 10^-7     = 58.5 x 10⁶ N/m²

Strain = x / l          = 2.5 x 10-3 / 1.2    = 2.08 x 10^-3

Now just as a certain force produces a certain extension and the two are proportional up to a limit the same is true for stress and strain, i.e.   is proportional to e. 

The quantity / e is called the Young's Modulus E. As strain has no units it has the same units as stress, i.e. N/m². It is a material property, because we have taken the dimensions of the wire into account it applies to a wire with any length or area.

We find the Young's Modulus from the straight line section of a stress strain graph. Several other useful quantities are shown.

Beyond the yield point the material will no longer return to its original length. It is permanently deformed. For simplicity the limit of proportionality and the elastic limit are often assumed to be the same.

Here's another example.

The Young's modulus of one type of steel is 210 GPa and its yield stress is 700 MPa. What force is needed to permanently deform a wire of length 0.8 m and diameter 0.9 mm? How much will it have extended at this point?

y = F_max / A     so      F_max  = y  x A        =  700 x 10⁶  x  x ( 0.45 x 10^-3 )²     =  445 N

E = / e             so        e = / E       as e = x / l    then  x  =  l / E  =  ( 700 x 10⁶ x 0.8 )  / 210 x 10⁹  = 2.67 x 10^-3  = 2.67 mm