Modulus of Elasticity | kullabs.com

## Note on Modulus of Elasticity

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Classification of different type 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,

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

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

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

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

Those substances which break just after the elastic limit are called brittle substances.

Those substances that does not obey Hooke’s law within the elastic limit is called elastomer.

Young's modulus of elasticity is defined as the ratio of normal stress to the longitudinal strain within the elastic limit.

Bulk modulus of elasticity is defined as the ratio of normal stress to the volumetric strain within the elastic limit.

Shear modulus of elasticity is defined as the ratio of tangential stress to the shear strain within the elastic limit.

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