The stress-strain curve is one of the most important graphs in engineering because it shows how a material responds when it is pulled, compressed, or otherwise loaded. It connects the applied stress, σ, to the resulting strain, ε, so engineers can compare stiffness, strength, and ductility. For a ductile material such as mild steel, the curve reveals a sequence of behavior from elastic stretching to permanent deformation and finally fracture.
Understanding this curve helps engineers choose safe materials for bridges, vehicles, machines, and structures.
At small strain, stress is proportional to strain, and the slope of the graph is Young's modulus. After the proportional limit and yield point, the material begins to deform plastically, so it will not fully return to its original shape when the load is removed. As stretching continues, strain hardening raises the stress needed for further deformation until the ultimate tensile strength is reached.
Beyond that point, necking concentrates deformation in a smaller region, causing the engineering stress to drop until fracture occurs.
Key Facts
- Engineering stress is σ = F / A0, where F is the applied force and A0 is the original cross-sectional area.
- Engineering strain is ε = ΔL / L0, where ΔL is the change in length and L0 is the original length.
- In the linear elastic region, Hooke's law applies: σ = Eε.
- Young's modulus is the slope of the elastic region: E = Δσ / Δε.
- The yield strength marks the start of significant plastic deformation.
- Ultimate tensile strength is the maximum engineering stress on the curve: UTS = Fmax / A0.
Vocabulary
- Stress
- Stress is the internal force per unit area in a material caused by an external load.
- Strain
- Strain is the fractional change in length or shape of a material compared with its original size.
- Elastic region
- The elastic region is the part of the stress-strain curve where a material returns to its original shape after unloading.
- Yield point
- The yield point is the point where a material begins to undergo permanent plastic deformation.
- Necking
- Necking is the localized thinning of a material after it reaches ultimate tensile strength.
Common Mistakes to Avoid
- Confusing stress with force is wrong because stress depends on both the applied force and the cross-sectional area.
- Treating strain as a length is wrong because strain is a ratio, ΔL / L0, and has no units.
- Using the full curve to calculate Young's modulus is wrong because Young's modulus only comes from the initial linear elastic region.
- Assuming ultimate tensile strength is the fracture point is wrong because ductile materials usually neck and continue deforming after the maximum engineering stress.
Practice Questions
- 1 A metal rod has an original cross-sectional area of 2.0 x 10^-4 m^2 and is pulled with a force of 12,000 N. Calculate the engineering stress.
- 2 A 0.50 m wire stretches by 1.0 mm while it remains in the elastic region. Calculate the engineering strain. If the stress is 160 MPa, calculate Young's modulus.
- 3 Compare a ductile stress-strain curve with a brittle stress-strain curve. Explain which material gives more warning before fracture and why.