Mechanics of Materials — Stress & Strain

Strain & Hooke's law — stiffness is a material property

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: strain ε = ΔL / L is the fractional stretch — pure ratio, no units. For most metals, stress and strain rise together in lockstep: σ = E·ε. The constant E (Young's modulus) is the material's stiffness — steel's is ~200 GPa, and it's the same whether the bar is huge or tiny.

E_{steel} := 200 GPa

= 200 GPa Young's modulus for steel

strain_{axial} = \frac{\sigma_{axial}}{E_{steel}}

0.0008

fractional stretch ≈ 0.0008

L_{rod} := 2 m

= 2 m original length

\delta _{rod} := strain_{axial} * L_{rod} in mm

= 1.5915 mm how much it stretches ≈ 1.6 mm

✓ pass

abs(\delta _{rod} - F_{axial} * L_{rod} / (A_{cross} * E_{steel})) < 1e-6 mm

matches the direct formula δ = F·L / (A·E)

200000 * x

stress (MPa) vs strain x in the elastic region — a straight line whose slope is E

Real-world hook: Hooke's law is why a bridge deck flexes and springs back, how a bathroom scale turns your weight into a reading, and why guitar strings hold a pitch.