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Strength of Materials · Stress and Strain

The stress-strain diagram

Read a tensile-test curve: elastic region, yield, strain hardening, ultimate strength, necking and fracture.

40 minYear 1-2Prerequisites: Normal and shear stress, strain

By the end of this topic you will be able to

  • Identify the elastic limit, yield point, ultimate strength and fracture point on a stress–strain curve.
  • Distinguish ductile from brittle behaviour.
  • Estimate the permanent strain left after loading beyond yield and unloading.

The tensile test

To measure a material's strength, a standard specimen is pulled in a testing machine while the force and extension are recorded. Dividing force by the original area gives stress, and extension by the original gauge length gives strain. Plotting one against the other gives the stress–strain curve, a fingerprint of the material.

Reading the curve (mild steel)

  1. Elastic region. The graph is a straight line of slope EE. Remove the load and the specimen returns to its original length. Hooke's law applies.
  2. Yield. At the yield stress σy\sigma_y the curve flattens: the material stretches with little extra stress, and the deformation is no longer recoverable.
  3. Strain hardening. The material becomes stronger as it deforms, and the stress rises again.
  4. Ultimate strength. The peak of the curve is the ultimate tensile strength σu\sigma_u.
  5. Necking and fracture. The specimen thins locally (necks); the force falls, and it finally fractures. (On a graph that uses the original area the curve falls; with the true, reduced area it would still rise.)

Ductile and brittle

A ductile material (mild steel, aluminium, copper) stretches a lot before breaking, giving plenty of warning. A brittle material (cast iron, glass, concrete in tension) fractures with little strain and no clear yield point.

Aluminium has no flat yield plateau, so its yield stress is defined by the 0.2% offset method: draw a line parallel to the elastic slope starting at 0.2% strain; where it meets the curve is the yield strength.

Unloading beyond yield

If you stretch a ductile specimen past yield and then release the load, it follows a line parallel to the original elastic slope, not back along the loading curve. The strain left behind is the permanent (plastic) strain:

εp=ε−σE\varepsilon_p = \varepsilon - \frac{\sigma}{E}
Interactive modelChoose a material and drag the extension. The dashed line shows what is left when you unload.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Practice problems

  1. A steel stressed to 300 MPa in the elastic range has E=200E = 200 GPa. Find the strain.
    Show answer
    ε=300/200000=1.5×10−3\varepsilon = 300/200\,000 = 1.5\times10^{-3}.
  2. Why does cast iron give no warning before it fails?
    Show answer
    It is brittle: it fractures at a small strain without a yield plateau or visible necking, so the member deforms very little before breaking.
  3. A specimen is loaded to a stress of 350 MPa, beyond yield, and the strain is 0.5%. Unloading follows the elastic slope with E=200E = 200 GPa. What permanent strain remains?
    Show answer
    εp=0.005−350/200000=0.005−0.00175=0.00325\varepsilon_p = 0.005 - 350/200\,000 = 0.005 - 0.00175 = 0.00325 (0.325%).
  4. In the model, which material has the greatest elongation at fracture, and which has the highest ultimate strength?
    Show answer
    Mild steel has the greatest elongation (about 25%) and the highest ultimate strength (400 MPa) of the three modelled.

Key formulae

Elastic modulus (slope of the straight part)
E=σεE = \frac{\sigma}{\varepsilon}
Percent elongation
%elongation=Lf−L0L0×100\%\,\text{elongation} = \frac{L_f - L_0}{L_0}\times 100
Permanent strain after unloading
εp=ε−σE\varepsilon_p = \varepsilon - \frac{\sigma}{E}
Lecturer notes: suggested plan, activity and assessment

Suggested plan (40 min). 10 min: video or demonstration of a tensile test if available; 15 min: reading the curve stage by stage with the model; 10 min: brittle vs ductile and the offset method; 5 min: summary.

Activity. Ask half the room to sketch steel and the other half aluminium from memory, then compare with the model.

Discussion. Why do designers prefer ductile materials for structures?

Assessment idea. Provide force–extension data and ask for E, yield and elongation.