# Relationship between three modulus of Elasticity

## STRESS, STRAIN AND THE BASIC DEFORMATIONS.

The study of elasticity is concerned with how bodies deform under the action of pairs of applied forces. In this study there are two basic concepts: stress and strain. The pairs of forces act in opposite directions along the same line. Thus there is no resulting acceleration (change of motion) but there is a resulting deformation or change in the size or shape of the body. This is described in terms of strain. The strain is the relative change in dimensions of a body resulting from the external forces. As a result of the deformation, internal forces are set up and these give rise to stresses. In many simple cases, these stresses are simply related to the external forces, because when these two are in balance the deformation will be maintained without further change. For these simple cases we make the following definition. The stress is the external force divided by the area over which this force is applied. There are three particular cases we will consider.

** Linear Extension**

The first type is the linear extension. An oppositely directed pair of forces along a line extend the body in along that line. Write the magnitude of these forces as F , the cross-sectional area at right angles to F as A, the original length as L and the extension as e.

**Uniform Compression**

If the forces are applied uniformly in all directions, we have a deformation typified by that produced by a uniform hydrostatic pressure. Write the pressure as p, the original volume as V and the change in volume as DV.

** HOOKE’S LAW and ELASTIC MODULI**

As we have seen, when a material is stressed there are basically two different regimes: the elastic and the non-elastic.

The latter is difficult to describe in a way which is easily applicable but in the former the stress is proportional to the strain. This proportionality between stress and strain is known as Hooke’s law; it applies to all of the three basic deformations.

Hence the ratio stress/strain is a constant; this constant is known as the elastic modulus.

There are three elastic moduli, one for each of the three basic deformations.

Linear Extension Stress Strain = **F/A **/** e/L** = Y named Young’s modulus

Uniform Compression Stress Strain = **-P **/** ΔV/V** = **– pV **/** ΔV** = k named bulk modulus

Shear Stress Strain = **F/A **/ **θ = F **/** Aθ** = n named shear modulus or modulus of rigidity