In a plane strain condition, εx = 400×10–6, εy = –100×10–6 and γxy = 300×10–6. What is the volumetric strain?
- A300 × 10⁻⁶Correct
- B100 × 10⁻⁶
- C0
- D–200 × 10⁻⁶
Solution
Concept
Volumetric strain determination in plane strain states
Formula
εv = εx + εy + εz (with εz = 0)
Step-by-step solution
1. Volumetric strain is the first invariant of strain: εv = εx + εy + εz. 2. Under plane strain conditions, out-of-plane normal strain is zero by definition: εz = 0. 3. Therefore, εv = εx + εy. 4. Given εx = 400x10⁻⁶ and εy = -100x10⁻⁶: εv = (400 - 100)x10-6 = 300x10-6. Answer: A
Explanation
Volumetric strain (εv) is the sum of normal strains along the principal coordinate axes: εv = εx + εy + εz. Under plane strain conditions, out-of-plane normal strain is constrained to zero (εz = 0). Therefore, εv =εx + εy = 400x10⁻⁶ + (-100x10⁻⁶) = 300x10⁻⁶.
Why the other options are wrong
B) 100x10⁻⁶ results from improperly subtracting shear strain. C) Zero volumetric strain corresponds to incompressible materials. D) -200x10⁻⁶ is an arithmetic sign error.
Shortcut
Plane strain: εz = 0 => Volumetric strain = εx + εy = 400 - 100 = 300x10⁻⁶
Common mistake
Adding the engineering shear strain γxy to the volumetric strain calculation.
Correct answer: A) 300 × 10⁻⁶