Mechanical Properties of Solids

• Longitudinal (tensile or compressive) stress and strain: a force applied along the length of a rod or wire, stretching or compressing it. Strain here is $\Delta L/L$, the fractional change in length.
• Shearing stress and strain: a tangential force applied parallel to a face, while the opposite face is held fixed, so the body’s shape distorts (a cube leans into a slanted box) without any real change in volume. Strain here is the angle of shear $\theta \approx \Delta x/L$, the ratio of the sideways displacement to the perpendicular height.
• Volume (hydraulic) stress and strain: a uniform force per unit area pressing inward from every direction at once, as a fluid does on a submerged body. The stress here is just the pressure change $\Delta P$, and the strain is the fractional change in volume, $\Delta V/V$.

Load a wire gradually and plot the stress you apply against the strain it develops. From the unloaded state O up to a point A, the graph is a straight line through the origin, exactly as Hooke’s law predicts, and the slope of this line is the material’s Young’s modulus (§8.3). Point A is called the elastic limit (sometimes the yield point): load the wire to anywhere on OA and release it, and the stress-strain point retraces the same straight line exactly back down to O. Nothing is left behind, the wire returns to precisely its original length. This full reversibility, not just the straight line, is what the word “elastic” means here.

Push past A and the curve bends over, strain now grows much faster than stress for the same extra load, a behaviour called plastic flow. The curve climbs, with decreasing slope, to a peak at point B, the ultimate tensile strength, the largest stress the material can bear. Beyond B, something counterintuitive happens: the wire keeps stretching, often thinning visibly at one point (necking), while the stress it can actually sustain drops, until it snaps at the fracture point, E. A material that stretches a long way between A and E (a large plastic region) is called ductile (most metals); one that fractures soon after, or even at, the elastic limit, with almost no plastic region, is brittle (glass, cast iron, ceramics).

Here is the detail that trips almost everyone up the first time: release the load from a point on OA and the wire returns to zero strain. Release it from a point past A, say somewhere between A and B, and it does not retrace the loading curve back down. It follows a new straight line instead, parallel to the original OA (same slope, same Young’s modulus, the material hasn’t changed, only its starting point has), down to zero stress at some strain that is greater than zero.

That leftover strain is the permanent set: with absolutely no load left on it, the wire is now longer than it started, for good. Crucially, this depends on the highest strain the wire has ever reached, not on where you currently have it, pulling it back down part-way and re-stretching it later will not erase that history; the wire “remembers” the worst stretch it has survived. This is precisely the behaviour the interactive stress-strain simulation lets you feel directly: drag the strain slider past the elastic limit, release it, and watch the unloading line peel away from the loading curve instead of folding back onto it. The common misconception it is built to break is the assumption that a stretched wire always “springs back” to its original length, true only while you stay inside OA.

Young’s modulus spans an enormous range. Steel is around $2\times10^{11}\text{ Pa}$, one of the stiffest common engineering materials, which is exactly why it is the default choice for anything that must not visibly sag or stretch, bridge cables, rails, structural beams. Rubber, by contrast, sits around $10^{5}\text{–}10^{6}\text{ Pa}$, five to six orders of magnitude smaller, which is precisely why it is chosen when large, easily reversible deformation is the entire point, tyres, gaskets, elastic bands. A large $Y$ means a material is rigid (resists stretching); it says nothing by itself about how strong or how brittle the material is, those are governed by the elastic limit and ultimate strength from §8.2, not by the slope OA.

A crane cable or a bridge girder is never operated anywhere near its ultimate tensile strength (point B in §8.2), it is kept well inside the elastic region, usually at a stress that is some small fraction of the elastic limit itself, a factor of safety. The reason is exactly the permanent-set behaviour from §8.2.3: a cable loaded past its elastic limit even once comes away permanently longer and weaker, and repeating that even a little will eventually cause fatigue failure, so safe engineering design is bounded by the elastic limit, not by how strong the material is at the point of breaking.

It is also why metal, not wood, is the default choice for load-bearing beams in bridges and buildings: for a given cross-section, metal combines a far larger Young’s modulus (so it sags less under load) with a far higher elastic limit (so it can be loaded much harder before any permanent bending sets in), and modern beams are shaped (as a hollow box or an I-section rather than a solid bar) to concentrate material away from the centre, where bending stresses are smallest, and toward the top and bottom edges, where they are largest, giving much greater resistance to bending for very little extra weight.