Stress-Strain Curve of Mild Steel: Regions, Points and Tensile Test
The tensile test is the most common mechanical test in any materials lab. A standard specimen is pulled in a universal testing machine (UTM) until it breaks, and the load and extension are recorded. Plotting stress against strain gives the stress-strain curve, and from it you read the properties that every design calculation depends on: yield strength, ultimate tensile strength, ductility and stiffness. Mild steel is the classic example because its curve shows a clear yield point, which many other metals do not.
In this article you will learn what each region of the curve means and then process a complete set of tensile test readings into the values you would write in a lab record.
Points and Regions on the Curve
Moving from zero load to fracture, the engineering stress-strain curve of mild steel passes through these stages.
| Point or region | What happens | Key fact |
|---|---|---|
| Proportional limit (A) | Stress is directly proportional to strain | Hooke's law holds; slope is Young's modulus |
| Elastic limit (B) | Load removal still returns specimen to original length | Very close to A for steel |
| Upper yield point (C) | Load drops suddenly as dislocations break free of carbon and nitrogen atoms | Sensitive to test speed and specimen alignment |
| Lower yield point (D) | Strain increases at nearly constant stress (Luders bands form) | Used as the yield strength in design |
| Strain hardening (D to E) | Material becomes stronger as it deforms plastically | Basis of cold working |
| Ultimate stress (E) | Maximum load the specimen can carry | Necking starts here |
| Necking and fracture (E to F) | Local cross-section shrinks rapidly; engineering stress falls | Cup-and-cone fracture surface |
The drop in the curve after point E does not mean the material is getting weaker. It happens because engineering stress is calculated on the original area, while the real area at the neck is shrinking fast. The true stress at the neck keeps rising until fracture.
Formulas Used in Tensile Test Calculations
Original area A0 = pi/4 * d0^2 Yield strength sigma_y = P_y / A0 Ultimate strength sigma_u = P_max / A0 Percentage elongation = (Lf - L0) / L0 * 100 Reduction in area = (A0 - Af) / A0 * 100 Proportional gauge L0 = 5.65 * sqrt(A0) True stress sigma_t = sigma * (1 + eps) (valid up to necking) True strain eps_t = ln(1 + eps) Modulus of resilience U_r = sigma_y^2 / (2E)
The gauge length rule L0 = 5.65 times the square root of A0 is used so that results from specimens of different diameters can be compared. For a 10 mm diameter bar it gives almost exactly 50 mm, which is why 10 mm by 50 mm specimens are so common in college labs.
Worked Example: Processing Tensile Test Readings
A mild steel specimen has an initial diameter of 10 mm and a gauge length of 50 mm. The UTM records a lower yield load of 20 kN and a maximum load of 33 kN. After fracture the two halves are fitted together; the final gauge length is 62 mm and the neck diameter is 6.5 mm. Take E = 200 GPa.
A0 = pi/4 * 10^2 = 78.54 mm^2 Af = pi/4 * 6.5^2 = 33.18 mm^2 Yield strength = 20,000 / 78.54 = 254.6 MPa Ultimate strength = 33,000 / 78.54 = 420.2 MPa Elongation = (62 - 50) / 50 = 24 % Reduction in area = (78.54 - 33.18) / 78.54 = 57.8 % Yield ratio = 254.6 / 420.2 = 0.61 Resilience = 254.6^2 / (2 * 200,000) = 0.162 N-mm/mm^3
These values are consistent with a structural steel such as IS 2062 grade E250, whose specified minimum values are 250 MPa yield, 410 MPa tensile strength and 23 percent elongation. A reduction in area above 50 percent indicates a ductile material that will give visible warning before failure.
True stress at maximum load
Suppose the engineering strain at maximum load was 0.18. Then the true stress at that point is 420.2 x 1.18 = 495.8 MPa and the true strain is ln(1.18) = 0.166. The difference of almost 18 percent shows why engineering values understate what the material is really carrying once deformation is large. Forming and crash simulations always use true stress-strain data.
Doing the Test Correctly
- Measure the diameter at three places along the gauge length with a micrometer and use the smallest value.
- Mark the gauge length with a punch or scriber so it can be measured after fracture.
- Grip the specimen in the UTM wedge jaws, keeping it aligned with the loading axis.
- Attach an extensometer if you need an accurate modulus; crosshead movement includes machine and grip deflection.
- Load at a steady rate and note the load where the pointer pauses or drops (yield).
- Record the maximum load and observe neck formation.
- After fracture, fit the pieces together and measure final gauge length and neck diameter.
Common Mistakes and Points to Check
- Calculating Young's modulus from crosshead travel. The result is usually far below 200 GPa because the machine frame also stretches.
- Using the final area for ultimate strength. Engineering UTS always uses the original area.
- Measuring elongation when fracture occurs outside the middle third of the gauge length; the value is then not representative.
- Confusing elastic limit with yield strength. In design, the lower yield point or 0.2 percent proof stress is used.
- Reporting percentage elongation without the gauge length. Elongation depends on L0, so always quote it.
Summary
The stress-strain curve of mild steel has a linear elastic part, a distinct yield plateau, a strain hardening region up to the ultimate stress, and a necking region ending in fracture. From a single test you can extract yield strength, tensile strength, elongation, reduction in area, resilience and, with an extensometer, Young's modulus. These numbers feed directly into factor of safety calculations, selection of steel grades and decisions about cold forming.