The relation between bulk modulus K, Young's modulus E and Poisson's ratio 1/m is:
Try it first: pick an option, then read the explanation.
Correct answer: D. K = mE/[3(m − 2)]
Asked in SSC JE 23/09/2019 (Morning)
Detailed Explanation
Here Poisson's ratio is written as 1/m (m is called Poisson's number).
\[ E = 3K\left(1 - \frac{2}{m}\right) = 3K\,\frac{m-2}{m} \;\Rightarrow\; K = \frac{mE}{3(m-2)} \]
Why “K = mE/[3(m − 2)]” Is Correct
Rearranging gives K = mE/[3(m − 2)] — option (d).
| 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
- A. K = mE/(3m): mE/3m simplifies to E/3, which ignores Poisson's ratio.
- B. K = mE/3: mE/3 is missing the (m − 2) term.
- C. K = E/[3(m − 2)]: The factor m is missing from the numerator.
Key Point
With μ = 1/m: K = mE/[3(m − 2)], G = mE/[2(m + 1)].
Key Concepts for Students
- m = 4 (μ = 0.25) gives K = 2E/3.
- m = 3 (μ = 1/3) gives K = E.
- m must exceed 2 for K to be positive.
Practice more: this is question 13 in Stress and Strain MCQs – Part 1. Try the whole set and check every answer instantly.