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Consider a cylindrical rope of rubbery elastomer with a radius (in inches) equal to the square root of the inverse of $\pi \rightarrow \sqrt \frac{1}{\pi} $ so that the cross sectional area is $1 \, \mathrm{inch^2}$ in its relaxed state of no tension.

Now, toss the rope over a tree limb and hang a horse-thief from it. As it stretches, it gets skinnier constantly while stretching within its elastic limits. Consider the rope is supporting a $175 \, \mathrm{lb}$ horse-thief (including boots/spurs). Assume that the rope is not stretched into the non-elastic region. Supporting the weight of the horse thief, the rope diameter is smaller and its cross sectional area is no longer one square inch.

What is acceptable scientific practice? Can one simply say the stress in the rope (above the horse thief's neck) is $175 \, \mathrm{\frac{lb}{inch^2}}$? Or does one measure and use the smaller diameter to compute a value of say $195 \, \mathrm{\frac{lb}{inch^2}}$?

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The so-called nominal or engineering stress is force over original area. The true stress is force over current area. Both are useful: the former is easier to calculate, the latter is better related to how the material fails. The only essential is that you tell your reader/listener/whatever which one you're using.

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