Applications of Elasticity

Mechanical properties like strength, stiffness (Rigidity), ductility, malleability and brittleness have to be carefully studied to select a material for a particular job.

The metallic parts of machines should not be subjected to stress beyond the elastic limit otherwise they will be deformed.

Beams are the simplest and most common parts of large structures. When beams are

subjected to stress, the different parts are strained in different way as shown in the above diagram. For this purpose, the beam's cross-section is I in shape, where there is advantage of lightness. The flanges are able to withstand the compression and tension force due to loading.

In an arched stone bridge, the stone is compressed and this makes the stone weak. Hence, steel arch is used instead, as steel arch is stronger than the stone arched bridge.

The thickness of the metallic rope needed to lift a given load is decided using the knowledge of elastic limit of the material of the rope and the factor of safety.

Electric poles are made hollow

A hollow shaft is found to be stronger than a solid one because the torque required to twist a hollow cylinder is greater than the torque required to twist a solid cylinder of same length & radius.

Maximum height of a mountain at any place along its length, can be estimated from the elastic behaviour of earth.

Its known that at the base of the mountain, the pressure is p = hrg where h is height of mountain, r is density of material of mountain (3 x 10³kg/m³) and g the acceleration due to gravity (~10m/s²). The pressure at the base should be less than the elastic limit of the Earth's supporting material (3 x 10⁸N/m²)

The stress vs strain for elastomers is a curve like this

While using a material, it is ensured that the working stress is always larger than its breaking stress. The difference is the safety factor.

Every material becomes less elastic under the action of repeated alternating deforming forces.

Some materials regain their original form long after the removal of the deforming force. This lag or delay is called elastic after-effect. Generally, suspensions are made of quartz or phosphor bronze, whose elastic after-effect is very small.

When wires are stretched, work is done against the internal restoring force and this appears as elastic P.E. in wire.

Elastic hysteresis: As the consequence of elastic after-effect, the strain persists even after the stress is removed. The lagging of strain is called elastic hysteresis. The area of the loop indicates the energy dissipated by the material in some form.

Sub Topics

Table of elastic coefficients

Table of elastic coefficients

(The values given below should be treated as approximate, as they vary with different specimens of the same material)

SI. No	Substance	Yong’s modulus (Y) 10¹¹ dyne cm^-2 or 10¹⁰ N m^-2	Bulk modulus (K) 10¹¹dyne cm^-2 or 10¹⁰ N m^-2	Rigidity modulus (Å) 10¹¹dyne cm^-2 or 10¹⁰ N m^-2
1.	Aluminium	7.0	7.5	2.5
2.	Brass	9.0 - 10.2	6.0	3.5
3.	Copper	11.0 - 13.0	13.0 - 14.0	3.4 - 4.6
4.	Glass (Flint)	5.0 - 6.0	3.7	2.3
5.	Glass (Crown)	6.0 - 7.8	4.0 - 5.0	2.6 - 3.2
6.	India (Rubber)	0.05	-	0.00015
7.	Iron (Cast)	10.0 - 13.0	9.5 - 9.7	3.5 - 5.3
8.	Iron (Wrought)	19.0 - 21.0	14.6 - 16.0	7.7 - 8
9.	Lead	1.6	4.5	0.6
10.	Quartz fibre	5.4	1.5	3.0
11.	Silver	7.8	10.5	2.2
12.	Steel (Cast)	19.0 - 21.0	16.5 - 17.5	7.4 - 7.6
13.	Steel (Mild)	22.0	16.0-19.0	8.0 - 8.9
14.	Tungsten	36.0	20.0	15.0
15.	Zinc	8.0 - 11.0	3.0 - 6.0	3.6 - 3.8

For a given material, there can be different moduli of elasticity depending on the type stress applied and strain produced.

Gases being most compressible, are least elastic.

It is easier to slide layers of atoms on another rather than pulling them apart or squeezing them, that is why shear modulus for a materials is less than Y.