Tensile test and stress-strain diagram

Tensile test: a standard specimen (gauge length $L_0$, area $A_0$) is gripped in a universal testing machine and pulled axially at a slow, constant rate until it fractures. The load $P$ and extension $\Delta L$ are recorded. Then

$$\sigma = \frac{P}{A_0}, \qquad e = \frac{\Delta L}{L_0}$$

give the engineering stress and engineering strain. The test gives the modulus, yield strength, tensile strength and ductility.

Stress-strain diagram of mild steel

 stress
   |            D (UTS)
   |          ,-'-.
   |    B C ,'     `.
   |   /`--'         `. E
   |  /                `x  (fracture)
   | / A
   |/
   +------------------------> strain
   O

Salient points

• O-A: proportional limit. Stress is proportional to strain (Hooke's law), $\sigma = E e$. The slope is Young's modulus (about 200 GPa for steel).
• A-B: elastic limit. Up to this point the specimen returns to its original length on unloading.
• B: upper yield point; C: lower yield point. Dislocations break away from their atmospheres of interstitial (carbon, nitrogen) atoms, so the stress drops and the specimen extends at nearly constant load (Lüders bands). Permanent deformation starts here.
• C-D: strain hardening. Stress rises with strain as the dislocation density increases, and elongation is uniform along the gauge length.
• D: ultimate tensile strength (UTS). The maximum engineering stress, $P_{max}/A_0$. A neck starts to form here.
• D-E: necking. Deformation concentrates in the neck and its area falls rapidly, so the engineering stress falls although the true stress still rises.
• E: fracture (cup-and-cone type for ductile steel).

Properties obtained

Yield strength, UTS, percentage elongation $=\dfrac{L_f - L_0}{L_0}\times100$, percentage reduction in area $=\dfrac{A_0 - A_f}{A_0}\times 100$, resilience and toughness (areas under the curve).