Question

The modulus of rigidity and the bulk modulus of a material are found as 70 GPa and 150 GPa respectively. Then

1. elasticity modulus is 200 GPa

2. Poisson’s ratio is 0.22

3. elasticity modulus is 182 GPa

4. Poisson’s ratio is 0.3

Which of the above statements are correct?

1. 1 and 2 
2. 1 and 4
3. 2 and 3 
4. 3 and 4 

Answer 1

Correct Answer - Option 4 : 3 and 4 

Concept:

Elastic Modulus (E)

When the body is loaded within its elastic limit, the ratio of stress and strain is constant. This constant is known as Elastic modulus or Young's Modulus.

\({\rm{E}} = \frac{{{\rm{Stress}}}}{{{\rm{Strain}}}} = \frac{{\rm{\sigma }}}{\epsilon}\)

Rigidity modulus (G)

When a body is loaded within its elastic limit, the ratio of shear stress and shear strain is constant, this constant is known as the shear modulus.

\({\rm{G}} = \frac{{{\rm{Shear\;stress\;}}}}{{{\rm{Shear\;strain}}}} = \frac{{\rm{\tau }}}{\phi }\)

Bulk modulus (K)

When a body is subjected to three mutually perpendicular like stresses of same intensity then the ratio of direct stress and the volumetric strain of the body is known as bulk modulus

\({\rm{K}} = \frac{{{\rm{Direct\;stress}}}}{{{\rm{Volumetric\;strain}}}} = \frac{{\rm{\sigma }}}{{\frac{{{\rm{\delta V}}}}{{\rm{V}}}}}\)

Relationship between various elastic constants is:

E = 2G(1 + μ)      - (1)  

E = 3K (1 – 2μ)      - (2) 

where μ is Poisson's ratio.

Calculation:

Given G = 70 GPa, K = 150 GPa;

Dividing equation 1 and 2,

2G (1 + μ) = 3K (1 - 2μ) ⇒ 140 (1 + μ) = 450 (1 - 2μ)

⇒ μ = 0.298 ≈ 0.3

Now E = 140 (1 + 0.3) = 182 GPa