True stress; stress-strain curve of steel

Definition of true stress and true strain

True stress is the load divided by the instantaneous (actual) cross-sectional area, $\sigma_t = P/A_i$. True strain is the sum of all small strains, each based on the instantaneous length:

$$\varepsilon_t = \int_{L_0}^{L}\frac{dL}{L} = \ln\frac{L}{L_0}$$

Up to the start of necking: $\sigma_t = \sigma(1+\varepsilon)$ and $\varepsilon_t = \ln(1+\varepsilon)$, where $\sigma$ and $\varepsilon$ are the engineering stress and strain.

Stress-strain curve of structural steel

 stress
   |            C
   |        B ,-'''-. D        true curve rises
   |   A  ,.'/        \ - - - - -> T
   | ,-'  /             \  E
   |/    /               x  engineering curve
   +------------------------ strain
 A elastic limit, B yield, C ultimate, D necking, E fracture

Significant points

1. A, proportional/elastic limit: linear region where $\sigma = E\varepsilon$ and deformation is recoverable.
2. B, yield point: large plastic strain at almost constant stress. This stress is the yield strength (for example, about 250 MPa for Fe 250 mild/structural steel). High-strength bars such as Fe 415 show no clear yield point, so their 415 MPa is a 0.2% proof stress.
3. Strain hardening: the stress rises again with strain.
4. C, ultimate tensile strength: the highest engineering stress.
5. D, necking: local reduction of area; the engineering curve falls because the load decreases.
6. E, fracture: the specimen breaks. The true stress curve keeps rising up to fracture because the actual area keeps decreasing.