Modulus of Elasticity

Classification of different types of substance

a) Ductile:
The substances which elongate considerably up to plastic deformation until they break are known as ductile substances.

Example: copper, gold, iron, etc.

b) Brittle:
Those substances which break just after the elastic limit are called brittle substances. Example: glass

c) Elastomers:
Those substances that do not obey Hooke’s law within the elastic limit is called elastomer. Example: rubber

Types of Modulus of elasticity

a) Young’s Modulus of elasticity

It is defined as the ratio of normal stress to the longitudinal strain within the elastic limit.

$$ Y =\frac {\text {Normal stress}}{\text {Longitudional strain}} $$

Consider a wire of length ‘l’ and radius ‘r’ as shown in the figure. If a force ‘F’ is applied along the length of the wire so that its length increases by Δl. Then,

$$ \text {Normal stress} = \frac FA $$

Where A is the cross sectional area

$$ A = \pi r^2 $$

$$\text {Longitudinal strain} = \frac {\Delta L}{L} $$

$$ Y = \frac {\frac FA}{\frac {\Delta L}{L}} $$

$$=\frac {Fl}{A\Delta L} $$

If extension produced in the wire is due to the load of mass ‘m’, then the above formula becomes

$$ F = mg$$

$$A = \pi r^2$$

$$ Y = \frac {mg}{\pi r^2} $$

b) Bulk modulus of elasticity (k)

It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.

$$ Y =\frac {\text {Normal stress}}{\text {Volumetric strain}} $$

Consider a spherical object of volume ‘V’ and area ‘A’ as shown in the figure above. If a force ‘F’ is applied normally on the entire surface of the object so that its volume decreases by ΔV.

Then,

$$ \text {Normal stress} = \frac FA $$

$$ \text {Volumetric strain} = -\frac {\Delta V}{V} [-\text{ sign indicates that the volume decreases with the application of force}] $$

$$\text {Bulk modulus} = \frac {\frac FA}{\frac {\Delta V}{V}} = \frac {FV}{A\Delta V} $$

The reciprocal of bulk modulus of elasticity is called compressibility. It is denoted by ‘C’.

$$ C = \frac 1K $$

Unit of C = N^-1m²

c) Shear modulus or modulus of rigidity (η)

$$ \eta =\frac {\text {Tangential stress}}{\text {Shear strain}} $$

Consider a cubical object as shown in the figure. If a force ‘F’ is acting tangentially to the upper surface of the cube and lower surface is fixed, then,

$$ \text {Tangential stress} = \frac FA $$

$$\text {Shear strain} (\theta) = \frac XL $$

$$\eta = \frac {\frac FA}{\frac {X}{L}} = \frac {FL}{AX} $$