The Stress-Strain Curve

The stress-strain diagram is the most informative single picture in solid mechanics. It compactly tells us how a material responds from gentle loading all the way to fracture.

Concept

A typical tensile stress-strain curve for a ductile metal (mild steel) has the following landmarks, traversed as load increases:

1. OA — Proportional region. Stress and strain are strictly linear. The slope is Young's modulus $Y$. Hooke's law holds.
2. A — Proportional limit. End of strict linearity.
3. AB — Elastic but nonlinear. Strain is still recovered on unloading, but stress grows nonlinearly with strain.
4. B — Elastic limit (yield point). Beyond this stress, removing the load leaves a permanent (plastic) set. For mild steel, B is sometimes split into an upper and lower yield point.
5. BC — Plastic flow. Strain increases rapidly with small increases in stress. The material flows.
6. C — Ultimate tensile strength. Maximum stress the material can bear.
7. CD — Necking and fracture. Cross-section reduces locally, true stress rises further while engineering stress falls, and the sample breaks at D.

The area under the curve equals the energy per unit volume absorbed by the material — a measure of toughness.

Derivation

We do not derive a single equation here — the curve is empirical. But we can derive two key quantities from it.

Step 1 — Young's modulus from the linear portion:

$Y = \left.\frac{\sigma}{\varepsilon}\right|_{OA}$

Step 2 — Strain energy per unit volume up to a strain $\varepsilon$:

$u = \int_0^{\varepsilon} \sigma(\varepsilon')\,d\varepsilon'$

For the linear region $\sigma = Y\varepsilon$, the integral evaluates to:

$u = \int_0^{\varepsilon} Y\varepsilon'\,d\varepsilon' = \frac{1}{2} Y\,\varepsilon^2 = \frac{1}{2}\,\sigma\,\varepsilon$

Step 3 — Toughness:

$U_T = \int_0^{\varepsilon_f} \sigma\,d\varepsilon$

where $\varepsilon_f$ is the fracture strain.

Step 4 — Modulus of resilience: same as step 2 but evaluated up to the elastic limit. This is the elastic energy a unit volume can absorb without permanent damage.

Worked Example

A mild-steel rod has the following points on its stress-strain curve:

• Proportional limit at $\sigma_A = 2.5\times 10^8\,\text{Pa}$ and $\varepsilon_A = 1.25\times 10^{-3}$
• Ultimate stress $\sigma_C = 4.0\times 10^8\,\text{Pa}$
• Fracture at $\varepsilon_f = 0.20$

(a) Young's modulus: $Y = \sigma_A/\varepsilon_A = 2.5\times 10^8 / 1.25\times 10^{-3} = 2.0\times 10^{11}\,\text{Pa}$.

(b) Modulus of resilience: $u_R = \tfrac{1}{2}\sigma_A \varepsilon_A = \tfrac{1}{2}(2.5\times 10^8)(1.25\times 10^{-3}) \approx 1.6\times 10^5\,\text{J/m}^3$.

(c) Approximate toughness: take an average stress of $\sim 3.5\times 10^8\,\text{Pa}$ over the strain range; $U_T \approx 3.5\times 10^8 \times 0.20 = 7\times 10^7\,\text{J/m}^3$. Steel is enormously tough relative to its resilience.

Common Confusions

• Ductile vs brittle. A brittle material (like cast iron, glass) breaks just beyond the elastic limit, with little or no plastic region — its curve has no plateau.
• Engineering vs true stress. Beyond ultimate stress, engineering stress (using initial area) decreases while true stress (using actual area) continues to rise.
• Strength vs stiffness. High Young's modulus means high stiffness, not necessarily high strength.
• Toughness vs strength. Toughness is the total energy absorbed; strength is the maximum stress.

Key Takeaways

• The stress-strain curve marks five characteristic points: proportional limit, elastic limit, yield, ultimate, and fracture.
• Slope of the linear part gives Young's modulus.
• Area under the curve gives strain energy density (toughness).
• Ductile materials have an extended plastic region; brittle materials do not.