Linear strains along X, Y and Z in a homogeneous object are 0.05, 0.10 and 0.10. The object measured 20 cm × 10 cm × 10 cm before deformation. Find the volumetric strain and the change in volume.
Try it first: pick an option, then read the explanation.
Correct answer: A. Volumetric strain = 0.25, change in volume = 5 × 10⁵ mm³
Asked in SSC JE 06/06/2024 (Evening)
Detailed Explanation
Volumetric strain is the change in volume divided by the original volume, and for small strains it equals the sum of the three linear strains.
\[ \varepsilon_v = 0.05 + 0.10 + 0.10 = 0.25 \]
\( V = 20 \times 10 \times 10 = 2000\ \text{cm}^3 \)
\[ \Delta V = \varepsilon_v V = 0.25 \times 2000 = 500\ \text{cm}^3 = 500 \times 10^3 = 5 \times 10^5\ \text{mm}^3 \]
Why “Volumetric strain = 0.25, change in volume = 5 × 10⁵ mm³” Is Correct
Both values match option (a). Watch the unit conversion: 1 cm³ = 1000 mm³.
Why the Other Options Are Wrong
- B. Volumetric strain = 0.7, change in volume = 1400 cm³: 0.7 is not the sum of the strains.
- C. Volumetric strain = 0.3, change in volume = 60 × 10⁴ mm³: 0.3 is not the sum of the strains.
- D. Volumetric strain = 0.25, change in volume = 500 mm³: 500 is in cm³, not mm³ — the unit is wrong by a factor of 1000.
Key Point
ΔV = (εx + εy + εz) × V; 1 cm³ = 10³ mm³.
Key Concepts for Students
- Under pure shear, volumetric strain is zero.
- For a bar in uniaxial tension, εv = ε(1 − 2μ).
- Volume can only increase in tension if μ < 0.5.
Practice more: this is question 2 in Stress and Strain MCQs – Part 1. Try the whole set and check every answer instantly.