Question

The ratio of modulus of rigidity to modulus of elasticity for a Poisson's ratio of 0.25 would be
$\left( a \right)0.5$
$\left( b \right)0.4$
$\left( c \right)0.3$
$\left( d \right)1.0$

Answer 1

Hint: In this question use the relation between modulus of elasticity, modulus of rigidity and Poisson's ratio which is given as $\lambda = 2G\left( {1 + \sigma } \right)$ so use this concept to reach the solution of the question.
Formula used – $\lambda = 2G\left( {1 + \sigma } \right)$

Complete Step-by-Step solution:
 As we know there is direct relation between modulus of elasticity, modulus of rigidity and Poisson's ratio which is
$ \Rightarrow \lambda = 2G\left( {1 + \sigma } \right)$, where $\lambda = $ modulus of elasticity
                                                  G = modulus of rigidity
                                                  $\sigma $ = Poisson's ratio
Now it is given that Poisson's ratio $\sigma = 0.25$
So substitute this value in above equation we have,
$ \Rightarrow \lambda = 2G\left( {1 + 0.25} \right)$
$ \Rightarrow \lambda = 2G\left( {1.25} \right) = 2.5G$
Now we have to find out the ratio of modulus of rigidity (G) to modulus of elasticity ($\lambda $) so we have,
$ \Rightarrow \dfrac{G}{\lambda } = \dfrac{1}{{2.5}} = 0.4$
So this is the required answer.
Hence option (B) is the correct answer.

Note – Whenever we face such types of questions always recall the relation between modulus of elasticity, modulus of rigidity and Poisson's ratio which is stated above. Shear modulus also known as modulus of rigidity is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain. Often denoted by G. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region. Often denoted by $\lambda $. Poisson's ratio is a measure of the Poisson effect that describes the expansion or contraction of a material in directions perpendicular to the direction of loading. Often denoted by $\sigma $.