A material has a modulus of elasticity equal to 3 times its modulus of rigidity. Which of the following statements may be INCORRECT?
Try it first: pick an option, then read the explanation.
Asked in SSC JE 05/06/2024 (9 am–11 am)
Detailed Explanation
Use the relations between elastic constants. Knowing E/G fixes Poisson's ratio; Poisson's ratio then fixes the bulk modulus and the volumetric strain.
\[ E = 2G(1+\mu) \Rightarrow 3G = 2G(1+\mu) \Rightarrow \mu = 0.5 \]
\[ K = \frac{E}{3(1-2\mu)} = \frac{E}{3(0)} \to \infty \]
\[ \varepsilon_v = \frac{\sigma}{E}(1-2\mu) = 0 \]
Why “There is a change in the volume of the material.” Is Correct
With μ = 0.5 the material is incompressible: the bulk modulus is infinite and the volumetric strain is zero, so its volume does not change. The statement “there is a change in the volume” is therefore the incorrect one — option (c).
| Relation | Formula | What it tells you |
|---|---|---|
| E and G | E = 2G(1 + μ) | G = E/2 when μ = 0; G = E/3 when μ = 0.5 |
| E and K | E = 3K(1 − 2μ) | K → ∞ when μ = 0.5 (incompressible) |
| E, G and K | E = 9KG/(3K + G) | Links all three moduli without μ |
| Poisson's ratio | 0 ≤ μ ≤ 0.5 (isotropic) | Steel ≈ 0.3, concrete ≈ 0.15–0.2, rubber ≈ 0.5 |
Why the Other Options Are Wrong
- A. The bulk modulus is infinite.: Correct statement: K = E/3(1 − 2μ) becomes infinite at μ = 0.5.
- B. Volumetric strain is equal to zero.: Correct statement: (1 − 2μ) = 0, so volumetric strain is zero.
- D. Poisson's ratio is equal to 0.5.: Correct statement: E = 3G gives μ = 0.5 directly.
Key Point
E = 3G ⇔ μ = 0.5 ⇔ K = ∞ ⇔ no volume change.
Key Concepts for Students
- Rubber is close to incompressible (μ ≈ 0.5).
- Volumetric strain of a bar under axial stress = ε(1 − 2μ).
- The upper limit μ = 0.5 comes from requiring a positive bulk modulus.
Practice more: this is question 4 in Bending Moment and Shear Force MCQs – Part 1. Try the whole set and check every answer instantly.