# Elasticity

The property by virtue of which a body resists any change in its size, shape or both and tends to regain its configuration on withdrawal of the deforming forces is known as the elasticity of the body.

If different parts of a body do not change their positions with respect to each other under the action of a system of balanced forces or couples, however large they may be, the body is called a perfectly rigid body. Glass is a good approximation to that. A perfectly elastic body completely regains its original configuration once the deforming forces are withdrawn. A quartz fibre is almost perfectly elastic. A perfectly inelastic body or plastic body does not show any tendency to recover its original configuration even after removal of the deforming forces. Putty is a close example.

The average stress, σ, across any given cross section of a body is defined as the ratio of the resultant internal force acting on the section to the area of that section. Thus, if the resultant internal force is F and the area of the cross section is A then:

` σ = F / A `

Suppose we resolve F into a component vector Fn normal to the section and a component vector Ft tangent to the section. Normal stress is defined as:

``` σn = Fn / A ```

Tangential stress or shear stress is defined as:

``` σt = Ft / A ```

The dimensions of stress are [ML-1T-2], and its SI unit is newton per square metre (N/m2).

Suppose two equal and opposite forces, F and - F, are applied to the ends of a solid rod of uniform cross section. If the forces act normally outward, the rod is said to be under tension. The stress at every perpendicular section of the rod is a normal tensile stress. On the other hand, if the forces act normally inward, the rod is under compression and the corresponding stress is a normal compressive stress. Both types of stress are known as longitudinal stress, because they act along the length of the rod.

A solid rectangular block, which attains static equilibrium under two pairs of forces applied tangentially to opposite faces, experiences shear stress. The same block, taken to a depth under the sea, experiences hydrostatic pressure of water acting normally inward on each of its six faces. The corresponding stress is called volume stress.

The strain of a body is the fractional change in its length, volume or shape relative to that in its original configuration. The ratio of two similar quantities, strain is a dimensionless number without units. As stress is the cause and strain is its effect, it is possible to relate each type of stress to a corresponding type of strain. Thus, longitudinal stress gives rise to longitudinal strain – either tensile strain or compressive strain, shear stress produces shear strain, and volume stress causes volume strain.

The fundamental law of elasticity is the Hooke's law which states that: "Provided the strain is small, the stress is proportional to the strain". The law implies that, within proportional limit, the ratio of stress to strain is a constant known as the elastic modulus of the material. The dimensions of elastic modulus are [ML-1T-2], and its unit is N/m2.

In accordance with three types of stress and strain mentioned above, three types of elastic modulus can be defined. Young's modulus is the ratio of longitudinal stress to longitudinal strain. Shear modulus is the ratio of shear stress to shear strain. Bulk modulus is the ratio of volume stress to volume strain. While only a solid can have Young's modulus and shear modulus, bulk modulus is a characteristic of solids, liquids and gases.

Another elastic constant of importance is Poisson's ratio. A longitudinal strain produced in a rod is always accompanied by a lateral strain produced simultaneously. The negative of the ratio of lateral strain to longitudinal strain is known as Poisson's ratio for the material of the rod. The relations among elastic constants will be discussed in due course.

Although the relation between stress and strain follows Hooke's law for small strains, it becomes complex if the strain is large. Therefore, it is instructive to plot a stress versus strain graph for a typical ductile metal wire. There are certain terms of interest regarding this graph, such as proportional limit, elastic limit or yield point, breaking stress, necking, breaking point, crushing point etc. The lack of coincidence between the curves for increasing and decreasing stress is known as elastic hysteresis.

The elastic potential energy stored in a wire stretched within proportional limit is given by the formula:

`U = YA(∆l)2 / 2l `

where Y is the Young's modulus, A is the cross-sectional area, ∆l is the extension and l is the original length of the wire. Another useful formula for elastic potential energy stored per unit volume, or energy density, is:

`Energy density = 1/2 × stress × strain `

This last relationship is valid also if the body is under shear stress or volume stress.

#### Basic level videos

Internal Forces And Stress 58:07 Basic
300 5
Strain 33:25 Basic
200 3
Hooke's Law And Young's Modulus 58:42 Basic
300 5
More Problems On Young's Modulus 58:14 Basic
300 5
Shear Modulus 53:48 Basic
300 5
Bulk Modulus & Relations Among Elastic Constants 1:01:41 Basic
350 7.5
More Problems On Shear Modulus And Bulk Modulus 46:27 Basic
300 5
Stress Versus Strain Graph And Elastic Potential Energy Of A Deformed Body 1:09:29 Basic
350 7.5
More Problems On Elastic Potential Energy 56:19 Basic
300 5

#### Advanced level Videos Note: (CE) Stands for Problems from Competitive Examination Papers

Advanced-level Problems On Elasticity I 01:32:23
450
8
Advanced-level Problems On Elasticity II 58:27
250
4
Advanced-level Problems On Elasticity III 01:09:22
350
7.5