Stress-strain curve of ductile material

A ductile material such as mild steel is tested in a universal testing machine and the stress $\sigma = P/A_0$ is plotted against the strain $\varepsilon = \Delta L/L_0$.

 stress
   |            C
   |        B ,-'''-. D
   |   A  ,.'/        \
   | ,-'  / yield      \ E (fracture)
   |/ P  /              x
   +--------------------- strain
   O
 P proportional limit, A elastic limit,
 B upper yield, C ultimate, D necking, E fracture

Significant points

1. Proportional limit (P): up to this point stress is proportional to strain (Hooke's law). The slope is Young's modulus $E$.
2. Elastic limit (A): the maximum stress up to which the material returns to its original length on unloading. It is very close to the proportional limit.
3. Upper and lower yield points (B): the stress drops slightly and the material elongates at nearly constant stress. This is the start of plastic flow. The yield stress is used as the design strength.
4. Strain hardening region: after yielding, the stress rises again with strain because of work hardening.
5. Ultimate stress (C): the maximum stress on the curve, equal to the maximum load divided by original area.
6. Necking (D): the cross-section reduces locally, so the engineering stress falls although the true stress keeps rising.
7. Fracture point (E): the specimen breaks. The area under the curve to this point is the toughness.

Percentage elongation and reduction in area measure the ductility.