Question

The ratio of modulus of the rigidity to bulk modulus for a Poisson's ratio of 0.25 would be
$\begin{array}{rl}& A.{\displaystyle \frac{2}{3}}\\ & B.{\displaystyle \frac{2}{5}}\\ & C.{\displaystyle \frac{3}{5}}\\ & D.1.0\end{array}$

Answer 1

Hint: Poisson's ratio is defined as a measure of the Poisson effect that explains the expansion or contraction of a material in the directions perpendicular to the direction of loading. The value of Poisson's ratio is given as the negative of the ratio of transverse strain to the axial strain.

Complete step by step answer:
Poisson's ratio is a needed constant in the engineering analysis for detecting the stress and deflection properties of materials such as plastics, metals, and so on. Basically it is a constant for measuring the stress and deflection properties of structures such as beams, plates, shells, and rotating discs. Young's modulus is referred to as the ability of a material to withstand variation in length when under lengthwise tension or compression. Often it is known as the modulus of elasticity also, Young's modulus is equivalent to the longitudinal stress divided by the strain.
Now let us check the relation between young modulus,$Y$, bulk modulus,$K$, rigidity modulus,$\eta$ and the poisson's ratio$\sigma$
$\begin{array}{rl}& Y=2\eta (1+\sigma )\\ & Y=3K(1-2\sigma )\end{array}$
Rearranging the above mentioned equation will give,
${\displaystyle \frac{Y}{2\eta }}=(1+\sigma )$
Now substitute $Y$in the second equation to the first one will give,
${\displaystyle \frac{3K(1-2\sigma )}{2\eta }}=(1+\sigma )$
 As we know, the poisson's ratio in the question is given as,
$\sigma =0.25$
Substituting this in equation,
${\displaystyle \frac{3K(1-2\times 0.25)}{2\eta }}=(1+0.25)$
From this we can derive the ratio ${\displaystyle \frac{\eta }{K}}$
Rearranging the terms will give as,
$2.50\eta =1.50K$
That is,
${\displaystyle \frac{\eta }{K}}={\displaystyle \frac{1.5}{2.5}}={\displaystyle \frac{3}{5}}$
Hence the correct answer for the question is option C.

Note:
The bulk modulus of a material is defined as the measure of how resistant to compression that material is. It is described as the ratio of the small pressure increase to the resulting relative decrease of the volume.