Stress-Strain Curve

To map out how a material behaves under increasing stress, all the way up to the point where it finally breaks, a metal wire is suspended vertically from a rigid support and progressively loaded at its lower end, in small increments, until it snaps. At each step, the elongation is measured, and the corresponding stress and strain are calculated and plotted, with strain along the x-axis and stress along the y-axis — this plot is the stress-strain curve (Fig. 6.7).

The initial part of this curve, from the origin $O$ up to a point $A$, is a straight line: this is exactly the region in which Hooke's law (section 6.4) is obeyed, with stress directly proportional to strain. The stress value at $A$ — where the straight-line behaviour ends — is called the proportional limit. Beyond $A$, as loading continues to a point $B$, the curve bends: stress and strain are no longer proportional, so Hooke's law no longer strictly holds, but the material is still elastic in the sense that matters most — if the load is removed at any point between $O$ and $B$, the curve is exactly retraced and the wire fully regains its original length. This entire region $OB$ is therefore called the elastic region, point $B$ is called the yield point, and the stress at $B$ is the elastic limit.

Beyond $B$, the material's behaviour changes fundamentally: strain keeps increasing even for very little further increase in stress, and — crucially — if the load is now removed at some point beyond $B$ (say at point $C$), the wire does not return to its original length. Instead, unloading follows a different path, the line $CE$, ending at a nonzero residual strain at point $E$ on the strain axis — the wire is now permanently longer than it started. This irreversible deformation is called a permanent set, and the region from $B$ onward, where it appears, is the region of plastic flow or plastic deformation. Loading continues to increase strain rapidly for only a small further rise in stress until point $D$ is reached, at which the wire finally fractures.

Not every material shows this full sequence in equal proportion. Many metallic wires — copper, aluminium, silver — do trace out something close to this complete curve, but most everyday materials show only part of it:

• Brittle materials, such as glass and ceramics, fracture while still within the elastic limit — they break at (or very near) point $B$, without ever entering a substantial plastic region.
• Ductile materials, such as copper, aluminium and wrought iron, have a large plastic range: they lengthen considerably and undergo substantial plastic deformation before finally breaking.
• Malleable materials, such as gold and silver, can be hammered out into thin sheets — a related but distinct property from ductility (which refers to being drawn into wires).
• Rubber has an unusually large elastic region: it can be stretched to many times its original length and still fully recover once released, but its stress-strain curve within that elastic region is markedly non-linear (curved), not the straight Hooke's-law line of a metal. A material capable of being elastically stretched to such large strain values is called an elastomer. …