Consider the following statements regarding the difference between true strains and engineering strains : 1. True strains for equivalent amounts of deformation in tension and compression are equal including for sign. 2. True strains are additive. For a deformation consisting of several steps the overall strain is the sum of the strains in each step. 3. The volume change is related to the sum of the three normal strains. For constant volume, εx + εy + εz = 0. Which of the above statements are correct ?
- A1 and 2 only
- B1 and 3 only
- C2 and 3 onlyCorrect
- D1, 2 and 3
Solution
Concept
True (natural) strain is ln(L/L0). It is additive, equal and opposite for equivalent tension and compression, and its normal strains add to zero at constant volume.
Formula
εtrue = ln(L/L₀)
Step-by-step solution
1. True strain ε = ln(L/L0}). Engineering strain e = (L - L0)/L0. 2. Statement 1: Stretch a bar to twice its length: ε = ln 2 = +0.693. Compress it to half its length: ε = ln(1/2) = -0.693. The values are equal in size but opposite in sign. Engineering strain gives +1.0 and -0.5, which are not equal. So true strains are equal EXCEPT for sign. Statement 1 says 'including for sign', so it is wrong. 3. Statement 2: ln(L1/L0) + ln(L2/L1) = ln(L2/L0). True strains add up over the steps. Statement 2 is correct. 4. Statement 3: Volume V = Lx·Ly·Lz, so ln(V/V0) = εx + εy + εz. For constant volume, ln(1) = 0, so the sum is 0. Statement 3 is correct. 5. Only 2 and 3 are correct, so the answer is (C).1. Doubling the length gives ln 2 and halving it gives ln(1/2) = −ln 2: equal in magnitude but opposite in sign (so "equal including for sign" is false). 2. True strains add over successive steps (true). 3. For constant volume the sum of the true normal strains is zero (true).
Correct answer: C) 2 and 3 only