Stress–Strain Curve

When you stretch a rubber band, it snaps back. When you bend a paperclip far enough, it stays bent. When you pull a piece of chalk hard enough, it breaks. All three outcomes — elastic recovery, permanent deformation, and fracture — are captured in a single, powerful graph called the stress–strain curve. This curve is the fingerprint of a material's mechanical personality, and engineers use it every day to decide whether a steel beam is safe, whether a medical implant will hold, or whether a car's crumple zone will absorb the right amount of energy in a crash. Stress is the internal force per unit area that a material develops in response to an applied load, measured in pascals (Pa). Strain is the dimensionless fractional change in length that results from that stress. Plotting stress on the vertical axis against strain on the horizontal axis as you slowly stretch a sample to destruction reveals several distinct regions, each with its own physics. The shape of the curve differs dramatically between materials — a glass rod and a copper rod of identical dimensions will produce very different curves — but the same vocabulary of elastic limit, yield point, ultimate tensile strength, and fracture point applies to all of them. Understanding the stress–strain curve is essential not just for engineers but for anyone studying the physics of matter. It bridges atomic-level bonding (why do atoms resist being pulled apart?) with macroscopic behaviour (why does a bridge not collapse under traffic?). In this article, we will walk through each region of the curve systematically, define every key point precisely, and show you how to extract real numbers — like Young's modulus — directly from the graph.

Key takeaways
  • Proportional (Hookean) Region — OA
  • Elastic Region — up to point B (Elastic Limit)
  • Yield Region — B to C (Yield Point)
  • Strain-Hardening Region — C to D (Ultimate Tensile Strength)
  • Necking and Fracture — D to E

Proportional (Hookean) Region — OA

From the origin O to point A (the proportional limit), stress and strain are directly proportional: σ = Eε. The graph is a perfectly straight line, and its slope is Young's modulus E. At the atomic level, you are simply stretching the interatomic bonds like tiny springs without permanently displacing atoms from their equilibrium positions. Remove the load anywhere in this region and the material returns exactly to its original length. For steel, this linear region extends to roughly 250 MPa of stress.

Elastic Region — up to point B (Elastic Limit)

Just beyond A, the curve may begin to deviate slightly from a straight line, but the material is still fully elastic: remove the load and it still returns to its original dimensions. Point B — the elastic limit — is the last point at which this complete recovery is possible. For many metals the proportional limit and elastic limit are so close together that they are treated as the same point in introductory courses, but they are conceptually distinct. Beyond B, deformation becomes permanent.

Yield Region — B to C (Yield Point)

Past the elastic limit, the material enters plastic deformation. In ductile metals like mild steel, there is often a dramatic upper yield point C followed by a lower yield point C′, where the stress actually drops slightly while strain continues to increase rapidly. This happens because dislocations (line defects in the crystal lattice) suddenly break free from pinning atoms and glide through the material. Think of it like a traffic jam that suddenly clears — once the first car moves, all the others rush forward. Remove the load here and the sample is permanently longer than it started.

Strain-Hardening Region — C to D (Ultimate Tensile Strength)

After yielding, continued deformation actually makes the material stronger — a phenomenon called strain hardening or work hardening. The dislocations that were gliding freely now start colliding with each other and becoming tangled, requiring ever-greater stress to keep deforming the material. The curve rises again to point D, the ultimate tensile strength (UTS), which is the maximum engineering stress the material can withstand. For structural steel, UTS ≈ 400–500 MPa.

Necking and Fracture — D to E

Beyond the UTS, the weakest cross-section of the sample begins to thin rapidly in a localised region — a process called necking. Because the cross-sectional area is shrinking faster than the material is hardening, the engineering stress (force divided by original area) actually decreases, even though the true stress in the neck is still rising. The curve falls until fracture occurs at point E. Ductile materials show a significant necking region; brittle materials like glass or concrete fracture with little or no plastic deformation, so their curves drop sharply right after the elastic region.

The Key Equation

E=σε=F/A0ΔL/L0E = \frac{\sigma}{\varepsilon} = \frac{F / A_0}{\Delta L / L_0}
  • E — Young's Modulus (Pa (N m⁻²)): The slope of the linear (proportional) region of the stress–strain curve; a measure of stiffness.
  • \sigma — Engineering Stress (Pa (N m⁻²)): Force F divided by the original cross-sectional area A₀ of the sample.
  • \varepsilon — Engineering Strain (dimensionless): Change in length ΔL divided by the original length L₀; often expressed as a percentage.
  • A_0 — Original Cross-Sectional Area (m²): The cross-sectional area of the sample before any load is applied.
  • L_0 — Original Length (m): The gauge length of the sample before any load is applied.

Worked Example

Problem. A cylindrical steel rod of original length L₀ = 2.00 m and diameter d = 10 mm is subjected to a tensile force of 40 kN. Young's modulus for steel is E = 200 GPa. (a) Calculate the engineering stress. (b) Calculate the engineering strain. (c) Find the extension ΔL. (d) If the proportional limit is 250 MPa, is the rod still in the elastic region?

σ=509 MPa,ε=2.55×10−3,ΔL≈5.1 mm\sigma = 509\text{ MPa},\quad \varepsilon = 2.55\times10^{-3},\quad \Delta L \approx 5.1\text{ mm}

Answer. Stress = 509 MPa; Strain = 2.55 × 10⁻³; Extension ≈ 5.1 mm; the rod has exceeded its elastic limit.

Figure
Diagram
Stress–Strain Curve (Ductile Metal)Strain ε →Stress σ (MPa) →AProportionalLimitBElastic LimitCUpper YieldLower Yield C′DUTSEFractureOElasticRegionPlastic / YieldStrainHardeningNeckingslope = E
Origins · 1 minRobert Hooke and how this was discovered
RH
Robert Hooke
1635–1703 · English

Hooke formulated his Law of Elasticity in 1676 (published as an anagram, then decoded in 1678): 'Ut tensio, sic vis' — 'As the extension, so the force.' This is the foundational principle behind the linear (Hookean) region of the stress–strain curve. Without Hooke's insight, the entire framework of linear elasticity — including Young's modulus — would have no theoretical starting point. Hooke originally published his law as the Latin anagram 'ceiiinosssttuv' to secure intellectual priority without revealing the result — a common trick among 17th-century scientists to date their discoveries.

  • Hooke's Law — Hooke's Law describes the linear elastic region (O→A) of the stress–strain curve, where stress is directly proportional to strain.
  • Young's Modulus — Young's modulus is the slope of the stress–strain curve in the proportional region and quantifies a material's stiffness.
  • Elastic and Plastic Deformation — The stress–strain curve explicitly maps out the boundary between recoverable elastic deformation and permanent plastic deformation.
  • Ultimate Tensile Strength — The UTS is the peak point D on the stress–strain curve, representing the maximum load-bearing capacity before necking.