There are three types of elastic modulus. These are

1. Young’s Modulus(Y):

This is defined as the ratio of longitudinal stress to the corresponding strain within the elastic limit. READ MORE

2. Bulk Modulus(K):

This is defined as the ratio of the volume stress to the volume strain within the elastic limit. READ MORE

3.Modulus of Rigidity(\( \eta \)):

This is defined as the ratio of tangential stress to shear strain within the elastic limit. READ MORE

Realation Among Young’s Modulus(Y), Bulk Modulus(K), Modulus of Rigidity(\( \eta \)) and Poisson’s Ratio(\( \sigma \)):

Let us consider, a tensile force \( P \), acting normally outward be applied to each surface of a cube of unit side, as shown in the adjoining Fig.1. Now the force \( P \) acts along the direction of \( x\ axis \), as a result there is an extension \( e \) along that direction. Sine the cube is a unit cube, so \( P \) and \( e \) respectively indicate the tensile stress and tensile strain along that axis. If \( Y \) be the Young’s Modulus, then \( Y=\frac{P}{e} \), or, \( e=\frac{P}{Y} \).

Again \( \sigma \) is the Poisson’s Ratio, then the contraction in other two directions along \( y\ axis \) and \( z\ axis \) are \( \sigma\cdot{e}=\displaystyle\frac{\sigma\cdot{P}}{Y} \).

Thus the force \( P \) acting parallel to \( x\ axis \) produces extensions \( \displaystyle\frac{P}{Y} \), \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \) (contraction), and \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \) (also contraction) along \( x\ axis \), \( y\ axis \) and \( z\ axis \) respectively.

Similarly, the tensile force \( P \) acting along \( y\ axis \) produces the extensions \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \), \( \displaystyle\frac{P}{Y} \), and \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \) along \( x\ axis \), \( y\ axis \) and \( z\ axis \) respectively.

Again the tensile force \( P \) acting parallel to the \( z\ axis \) produces the extensions \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \), \( \displaystyle-\frac{\sigma\cdot{P}}{Y} \) and \( \displaystyle\frac{P}{Y} \) along \( x\ axis \), \( y\ axis \) and \( z\ axis \) respectively.

So the total extension along each of the three axes is \( \displaystyle{\frac{P}{Y}-\frac{2\sigma\cdot{P}}{Y}=\frac{P}{Y}(1-2\sigma)} \)

All the tensile forces acting together produce volume stress of magnitude \( P \). This stress produces a volume strain of magnitude \( \frac{P}{K} \), this \( K \) is known as Bulk Modulus. Now, this volume strain is equivalently equal to three times the longitudinal strain along each direction.

\displaystyle{\frac{P}{K}=3\times\frac{P}{Y}(1-2\sigma)}

or, \displaystyle{Y=3K(1-2\sigma)}

Now, let us consider the cube is subjected to a tensile stress along \( x\ axis \) and an equal compressional stress along \( y\ axis \). Due to the tensile stress along \( x\ axis \) there is a linear stain along that axis is \( \frac{P}{Y} \). Again Due to compression stress along \( y\ axis \) there is a linear strain along \( x\ axis \) is \( \frac{\sigma{P}}{Y} \). So the resultant strain along \( x\ axis \) is \( \displaystyle{\frac{P}{Y}+\frac{\sigma{P}}{Y}=\frac{P}{Y}(1+\sigma)} \)

Similarly the resulatnt linear strain along \( y\ axis \) is \( \displaystyle{-\frac{P}{Y}-\frac{\sigma{P}}{Y}=-\frac{P}{Y}(1+\sigma)} \) .

The resultant linear strain Long \( z\ axis \) is \( \displaystyle{-\frac{\sigma{P}}{Y}+\frac{\sigma{P}}{Y}=0} \) .

The simultaneous extensional and compressive strain each equal to \( \displaystyle{\frac{P}{Y}(1+\sigma)} \) and at the right angle to each other rise to a strain \( \theta \) in the \( xy\ plane \) which is given by

\( \displaystyle{\theta=2\frac{P}{Y}(1+\sigma)} \).

Hence, Modulus of Rigidity \( \displaystyle{\eta=\frac{P}{\theta}=\frac{PY}{2P(1+\sigma)}=\frac{Y}{2(1+\sigma)}} \)

therefore, \( \displaystyle{Y=2\eta(1+\sigma)} \)

Now we get,

\( \displaystyle{3K(1-2\sigma) = 2\eta(1+\sigma) \\ or, 3K-6K\sigma=2\eta+2\eta\sigma \\ or, (2\eta+6K)\sigma=3K-2\eta \\ or,\sigma=\frac{3K-2\eta}{6K+2\eta}} \).

Limiting value of Poisson’s Ratio \( (\sigma) \):

We get,

\( \displaystyle{3K(1-2\sigma)=2\eta(1+\sigma)} \),

(i) if \( \sigma \) is positive quatuty, the right hand expression and also the left hand expression must be positive. This is possble only if \( (1-2\sigma)>0 \ or, 2\sigma<1 \ or, \sigma<\frac{1}{2} \).

(ii) if \( \sigma \) is a negative quantity, then the left-hand expression is positive, the right-hand expression must be positive. This is possible only if \( (1+\sigma)>0 \ or, \sigma>-1 \)

Therefore, \( \displaystyle{-1<\sigma<\frac{1}{2}} \)

Practical Limit of \( (\sigma) \):

The negative value of \( \sigma \) means that on being extended, a bodylognitudinally, will also expand laterally keeping its volume constant, which is impossible in practical cases. So \( \sigma \) lies between 0 and \( \frac{1}{2} \). i.e,