Consider the following statements regarding the effective stress and effective strain : 1. σ̄ and ε̄ will reduce to σx and εx in an x-direction tension test. 2. The incremental work per volume done in deforming a material plastically is dω = ε̄ dσ̄. 3. It is usually assumed that the σ̄ vs ε̄ curve describes the strain hardening for loading under a constant stress ratio α, regardless of α*. Which of the above statements is/are correct ?
- A2 and 3
- B3 only
- C1 and 3Correct
- D2 only
Solution
Concept
Equivalent (effective) stress and strain in plasticity
Formula
dW = σ̄ dε̄
Step-by-step solution
1. In a uniaxial tension test σ̄ = σx and ε̄ = εx (true). 2. The plastic work increment is σ̄ dε̄, not ε̄ dσ̄ (false). 3. The effective stress–strain curve is assumed to be independent of the stress ratio (true).
Explanation
Effective quantities are defined so that a multiaxial state is reduced to an equivalent uniaxial test.
Why the other options are wrong
Statement 2 interchanges the variables of the work expression.
Shortcut
dW = σ̄ dε̄.
Common mistake
Accepting the swapped product.
Correct answer: C) 1 and 3