The graph below is a detail of the stress – separation diagram for a typical metal. The curve crosses the x-axis (zero stress) at r/r0 =1, i.e. the standard separation of the atoms/molecules. If an attempt is made to separate the particles then a negative (attractive) force results that attempts to restore them to their original position. The theoretical maximum stress occurs at 25 % strain.
The theoretical maximum stress is achieved at point D. The line BD is approximately half that BC. Using the triangle ABC show that the theoretical maximum stress is equal to E/8 where E is the Young’s Modulus for the metal.
Piano wire, also called music wire, is one of the strongest steels readily available. It is a high carbon steel that has been extensively work hardened. Its Young’s Modulus is around 84 GPa and has an ultimate tensile strength of approximately 2400 MPa. Compare this to the theoretical maximum stress. Repeat your comparison for structural steel (Young’s Modulus 200 GPa, ultimate tensile strength 400 MPa.
The chart is in effect a standard stress – strain graph with the strain axis displaced by 1.0 (10 % strain is re-plotted as 1.10 and 20 % as 1.20) and the stress axis is inverted by multiplying by -1. This however does not affect the significance of the slope at zero strain r/r0 = 1.0) it is the Young’s Modulus.
Young’s Modulus is given by the slope of the graph at zero stress (r/r0 = 1.0).
For piano wire:
This is actually surprisingly good for a real material. For structural steel:
Even this is quite good for a real material. Figures are usually much closer to 1 %.