Hooke's law
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Robert Hooke noticed in 1660 that a spring pulls back in proportion to how far you stretch it: pull twice as far and it resists twice as hard. Hooke's law is that simple straight-line rule, and it is the backbone of all elastic behaviour — double the stress and you get double the strain, right up until the material starts to yield.
For a material under simple tension it is written sigma = E x epsilon: stress equals the elastic (Young's) modulus E times strain. E is just the slope of the straight part of the stress-strain curve, the constant of proportionality. Rearranged, strain = stress / E, so a stiffer material (bigger E) strains less under the same stress. Example: apply 200 MPa to steel with E = 200 GPa (200000 MPa) and the elastic strain is 200 / 200000 = 0.001, one tenth of one percent. The classic spring form, F = k x (extension), is the same law wearing everyday clothes.
Hooke's law is a superb approximation, but only within the elastic range. It fails once you pass the elastic limit into plastic deformation, where strain grows out of proportion to stress. It also assumes small strains and simple loading; rubber and other polymers curve away from a straight line even at modest strain, so Hooke's law is a linear starting model, not a universal truth.
Hang a 10 kg mass on a wire that stretches 1 mm; hang 20 kg and it stretches 2 mm; hang 30 kg and it stretches 3 mm — strain tracks stress in a straight line, exactly as Hooke's law says, until the wire yields.
Stress = modulus times strain: the straight, proportional part of the curve.
Hooke's law holds only in the elastic (straight-line) region. Beyond yield, strain grows faster than stress and the proportionality breaks down; polymers can deviate even earlier.