Stress and Strain
The two fundamental measures of how materials respond to loads: force per unit area and fractional deformation.
Stress and strain describe how materials respond to applied loads. Stress σ is force per unit area, σ = F/A, measured in pascals; strain ε is the fractional change in dimension, ε = ΔL/L, and is dimensionless. Engineering stress uses the original cross-section, while true stress uses the instantaneous one — a distinction that matters once deformation is large.
In the elastic region, stress is proportional to strain: Hooke's law σ = E·ε, where E is Young's modulus, a material property that measures stiffness. Steel has E ≈ 200 GPa, aluminum ≈ 70 GPa, and wood varies widely with grain direction. Beyond the elastic limit, materials yield, deform permanently, strain-harden, and eventually fracture. Ductile materials such as steel and aluminum deform noticeably before breaking; brittle materials such as cast iron, glass, and concrete in tension fail with little warning.
Three basic stress states — tension, compression, and shear — combine in real structures. A beam in bending puts one side in tension and the other in compression, which is why structural loads and beam bending is analyzed with the same stress–strain framework. For combined loading, engineers use the von Mises yield criterion to predict when yielding begins.
Design applies a safety factor: allowable stress equals material strength divided by the safety factor, so components operate well below failure. Fatigue — failure under repeated loading at stresses below the static strength — is a major practical concern in rotating machinery, aircraft, and bridges, and it explains why many failures occur suddenly after years of service. Reading and interpreting the stress–strain curve is the foundation of mechanical design and materials selection.
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elasticity engineering materials science mechanics stress
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