The Stress-Strain Curve

Hooke's law describes only the initial, small-stress part of a material's elastic response. To see the complete picture -- how a material behaves as the stress on it is increased steadily, all the way from zero up to the point where it finally breaks -- physicists and engineers plot the material's stress-strain curve: a graph of the stress developed in a test specimen (typically a wire or a rod, held in a testing machine) against the strain produced in it, as the load is increased gradually and continuously.

For a typical ductile metal, such as mild steel, the curve has a characteristic shape with several distinct, physically meaningful landmarks, read off in the order they occur as stress increases from zero:

• The straight-line (Hooke's-law) region, from the origin up to the proportional limit ($P$): here stress is directly proportional to strain, exactly as Hooke's law predicts, and the slope of this straight portion of the graph is, by definition, equal to the material's Young's modulus $Y$.
• The elastic limit ($E$), just beyond $P$: up to this point (which, for many metals, lies extremely close to $P$), the material still returns completely to its original length if the load is removed, even though the stress-strain relation is no longer perfectly linear between $P$ and $E$.
• The yield point ($Y$): beyond the elastic limit, the material reaches a point where it suddenly begins to stretch by a comparatively large amount for only a very small further increase in stress. Physically, this marks the onset of plastic deformation -- if the load were removed anywhere beyond this point, the specimen would be left permanently longer than its original length. The nearly flat stretch of the curve just after the yield point is called the plastic region.
• Ultimate tensile strength ($U$): as loading continues into the plastic region, the stress needed to keep stretching the material eventually rises to a maximum value, called the ultimate tensile strength -- the greatest stress the material can withstand before its cross-section starts to visibly thin out locally (a phenomenon called "necking").
• The fracture point ($F$): once necking sets in, the actual stress needed to continue deforming the (now much thinner) neck region actually falls even as the true local stress in the thinned region keeps rising, until the specimen finally snaps at the fracture point. …