Linear strains along X, Y and Z directions in an object with homogenous material are given as 0.05, 0.10 and 0.10, respectively. The size of the object before deformation was 20 cm × 10 cm × 10 cm. Determine the volumetric strain and change in volume of the material after deformation.
- AVolumetric strain = 0.25, change in volume = 500 mm³
- BVolumetric strain = 0.3, change in volume = 60 × 10⁴ mm³
- CVolumetric strain = 0.25, change in volume = 5 × 10⁵ mm³Correct
- DVolumetric strain = 0.7, change in volume = 1400 cm³
Solution
Concept
For small strains, volumetric strain is the sum of the three linear strains.
Formula
ev = ex + ey + ez ; ΔV = ev × V
Step-by-step solution
1. Linear strains: ex = 0.05, ey = 0.10, ez = 0.10. 2. Volumetric strain ev = 0.05 + 0.10 + 0.10 = 0.25. 3. Original volume V = 20 × 10 × 10 = 2000 cm³ = 2 × 106 mm³. 4. Change in volume ΔV = 0.25 × 2 × 106 = 5 × 105 mm³ (= 500 cm³). 5. So the answer is option C.
Explanation
Volumetric strain is the sum of the strains in the three directions, which gives 0.25. Original volume is 2000 cm³, or 2 × 106 mm³. So the change in volume is 0.25 × 2 × 106 = 5 × 105 mm³.
Why the other options are wrong
A: strain is correct but 500 is in cm³, not mm³. B: 0.3 is wrong strain and 60 × 104 mm³ is wrong volume change. D: 0.7 is wrong strain (strains are added, not combined like this), so 1400 cm³ is also wrong.
Shortcut
Add the strains (0.25), then take 25% of 2000 cm³ = 500 cm³ = 5 × 10
Correct answer: C) Volumetric strain = 0.25, change in volume = 5 × 10⁵ mm³
