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Hooke's Law states that within the proportional limit, stress is directly proportional to strain: sigma = E * epsilon, where E is the modulus of elasticity. This linear relationship is analogous to F = kx for springs and forms the foundation of linear elasticity theory.
The stress-strain curve for a ductile metal reveals six key regions. The proportional limit (A) marks the end of linearity — Hooke's Law is valid only up to this point. The elastic limit (B) is the maximum stress for complete recovery — slightly beyond A. The yield point (C-D) is where plastic deformation begins; mild steel shows distinct upper and lower yield points. Strain hardening (D to E) occurs as dislocations pile up, requiring increasing stress for further deformation. The ultimate tensile strength (E) is the maximum stress — necking begins here. Finally, fracture (F) is where the material breaks.
Different materials show characteristically different curves. Brittle materials (glass, cast iron) fracture near the elastic limit with minimal plastic deformation. Ductile materials (copper, steel) show extensive plastic regions. Elastomers (rubber) can sustain huge elastic strains (up to 800%) but with non-linear stress-strain behavior.
The slope of the linear region equals Young's modulus. The area under the curve to the elastic limit is resilience (elastic energy per unit volume). The total area to fracture is toughness (total energy absorbed per unit volume).