If E is the modulus of elasticity, N the modulus of rigidity and K the bulk modulus, which relation is correct?
Try it first: pick an option, then read the explanation.
Correct answer: A. E = 9NK/(N + 3K)
Asked in SSC JE 23/03/2021 (Evening)
Detailed Explanation
Eliminating μ between E = 2N(1 + μ) and E = 3K(1 − 2μ) gives a relation among E, N and K alone.
From the two relations: \( \mu = \dfrac{E}{2N} - 1 \) and \( \mu = \dfrac{1}{2} - \dfrac{E}{6K} \).
Equating: \( \dfrac{E}{2N} + \dfrac{E}{6K} = \dfrac{3}{2} \Rightarrow E\left(\dfrac{3K + N}{6NK}\right) = \dfrac{3}{2} \Rightarrow E = \dfrac{9NK}{N + 3K} \)
Why “E = 9NK/(N + 3K)” Is Correct
The derived relation is E = 9NK/(N + 3K) — option (a).
| Relation | Formula |
|---|---|
| Young's modulus and rigidity | E = 2G(1 + μ) |
| Young's modulus and bulk modulus | E = 3K(1 − 2μ) |
| All three moduli | E = 9KG/(3K + G) |
| Poisson's ratio | μ = (3K − 2G)/(6K + 2G) |
Only two elastic constants are independent for a homogeneous, isotropic, linearly elastic material.
Why the Other Options Are Wrong
- B. E = 9NK/(3N + 3K): The denominator should be N + 3K, not 3N + 3K.
- C. E = 6NK/(N + 3K): The numerator should be 9NK.
- D. E = 9NK/(N + K): The denominator should be N + 3K.
Key Point
E = 9KG/(3K + G).
Key Concepts for Students
- Check with steel: K ≈ 167, G ≈ 77 GPa → E ≈ 200 GPa.
- Any two constants give the other two.
- Some books write C or N for the modulus of rigidity.
Practice more: this is question 8 in Stress and Strain MCQs – Part 1. Try the whole set and check every answer instantly.