A material has linear strains of 0.05, 0.10 and 0.10 along the X, Y and Z directions. The hydrostatic stress at the point is 25 N/mm². Calculate the bulk modulus.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 07/06/2024 (Morning)
Detailed Explanation
Bulk modulus K compares a uniform (hydrostatic) pressure with the volume change it causes. The volumetric strain is the sum of the three linear strains (for small strains).
\[ \varepsilon_v = \varepsilon_x + \varepsilon_y + \varepsilon_z = 0.05 + 0.10 + 0.10 = 0.25 \]
\[ K = \frac{p}{\varepsilon_v} = \frac{25}{0.25} = 100\ \text{N/mm}^2 \]
Why “100 N/mm²” Is Correct
Dividing the hydrostatic stress by the volumetric strain gives K = 100 N/mm².
| 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. 125 N/mm²: 125 would need εv = 0.2.
- C. 500 N/mm²: 500 uses only one strain (25/0.05).
- D. 250 N/mm²: 250 uses 25/0.10 — a single strain, not the sum.
Key Point
Volumetric strain = εx + εy + εz; K = p/εv.
Key Concepts for Students
- Bulk modulus has the units of stress.
- For water, K ≈ 2.1 GPa.
- Hydrostatic stress has no shear component.
Practice more: this is question 1 in Stress and Strain MCQs – Part 1. Try the whole set and check every answer instantly.