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 A **perfectly elastic body** completely regains its original configuration once the deforming forces are withdrawn.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 **F**_{n} normal to the section and a
component vector F_{t} tangent to the section. **Normal stress** is defined as:

```
σ
```

_{n} = F_{n} / A

**Tangential stress** or **shear stress** is defined as:

```
σ
```

_{t} = F_{t} / A

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

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^{-1}T^{-2}], and its unit is N/m^{2}.

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.

Internal Forces And Stress
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Strain
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Hooke's Law And Young's Modulus
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Bulk Modulus & Relations Among Elastic Constants
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More Problems On Elastic Potential Energy
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