Typical specimen: gage length $50$ mm, diameter $12.5$ mm. Strain is controlled and the load (stress) measured. Gives $Y$ and UTS strength, $E$, ductility, toughness, strain-hardening.

Elastic vs plastic:

• Elastic — Hooke’s law, no permanent set: $\sigma = E e$
• Plastic — permanent deformation remains.

Engineering stress & strain:

$\sigma = \frac{P}{A_0} = \frac{\text{Load}}{\text{original area}}, \qquad e = \frac{l - l_0}{l_0}$

$E$ = slope of the elastic region. Past UTS a ductile bar necks. Unloading parallels the elastic slope — the part is work-hardened (↑yield, ↓ductility).

Ductility

Percentage elongation $= \dfrac{l_f - l_0}{l_0}\times 100$

Percentage reduction of area $= \dfrac{A_0 - A_f}{A_0}\times 100$

True Stress & True Strain

$\sigma = \frac{P}{A}\quad(\text{instantaneous area}), \qquad \varepsilon = \ln\frac{l}{l_0}$

Engineering strains are not additive; true strains are.

Plastic Region & Necking

$\sigma = K \varepsilon^{n}$

True strain at the onset of necking equals the strain-hardening exponent:

$\varepsilon_{\text{neck}} = n$

Volume is conserved, $A_0 l_0 = A_{text{neck}} l_{text{neck}}$, so

$\varepsilon_{\text{neck}} = \ln\frac{l_{\text{neck}}}{l_0} = \ln\frac{A_0}{A_{\text{neck}}} = n \;\Rightarrow\; A_{\text{neck}} = A_0 e^{-n}$

True stress at necking, and the engineering UTS:

$\sigma_{\text{neck}} = K n^{n}, \qquad \text{UTS} = K n^{n} e^{-n}$

Temperature & Strain-Rate Effects

• Higher temperature: $Y$, UTS, $E$ ↓; ductility & toughness ↑.
• Strain rate $dot{varepsilon} = dvarepsilon/dt;[text{s}^{-1}]$. UTS rises with rate: $\sigma = C \dot{\varepsilon}^{m}$
• Superplasticity: huge uniform elongation (hundreds–2000%) before necking — fine-grain (10–15 µm) Ti and Zn–Al alloys, glass, thermoplastics.

Compression

• For ductile metals, tension and compression true curves coincide.
• Brittle materials are stronger / more ductile in compression.
• Platen friction causes barreling (bulging toward the center).

Disk test (brittle ceramics/glass): diametral compression → uniform tensile stress on the centerline; the disk splits:

$\sigma = \frac{2P}{\pi d t}$

Torsion (shear strength)

Thin tube, average radius $r$, wall $t = r_o - r_i$:

$\tau = \frac{Tc}{J} = \frac{T r_o}{\pi(r_o^4 - r_i^4)/2} \approx \frac{T}{2\pi r^2 t}, \qquad \gamma \approx \frac{r\phi}{l}$

Elastic: $tau = Ggamma$, with $G = dfrac{E}{2(1+nu)}$.

Bending (Flexure)

Used for brittle materials. Three-point bending:

$\sigma_{\text{bending}} = \frac{Mc}{I} = \frac{(PL/4)(h/2)}{bh^3/12} = \frac{3PL}{2bh^2}$

This fracture stress = modulus of rupture. Four-point bending gives a lower rupture strength than three-point.

Hardness

Resistance to indentation/scratching; hardness ≈ load / indentation area.

• Knoop — light loads (25 g–5 kg), microhardness.
• Brinell — heavy loads (500 / 1500 / 3000 kg); tungsten-carbide ball.
• Shore (durometer) — rubbers/plastics; inverse to penetration depth.
• Hot hardness matters for tools and dies.
• Indent ≥ 2 indenter-diameters from an edge; thickness ≥ 10× indentation depth.

Fatigue

Cyclic loads fail parts below the static strength. S–N curve: stress amplitude $S$ vs cycles $N$.

• Endurance (fatigue) limit: stress sustainable for unlimited cycles.
• No clear limit → quote fatigue strength at e.g. $N = 10^7$.
• A fine surface finish improves fatigue life.

Creep

Permanent elongation under static load over time (primary / secondary / tertiary stages). Important at high $T$; resistance rises with melting temperature.

• Stress relaxation: stress drops over time at fixed dimensions (bolts, rivets, tensioned wire).

Impact

Notched specimen broken by a pendulum — Charpy & Izod. Finds the ductile–brittle transition temperature; absorbed energy = impact toughness.

Fracture

• Ductile: plastic deformation + necking; fibrous, dimpled surface; voids nucleate at inclusions. Hard inclusions seed voids; soft ones are usually OK.
• Brittle: little/no plasticity; bright granular surface. Low temperature + high strain rate promote it.

Residual Stresses

Non-uniform plastic deformation leaves residual stresses after unloading (recovery is elastic). They can distort a part after cutting or slitting.