Physicopoeia

φυσικοποιΐα
THINGS I WISH I'D KNOWN WHEN I STARTED TEACHING PHYSICS

How do you get the Young Modulus from a Stress-Strain graph?

Very often, you will see either textbooks or exam board mark schemes showing or stating that you can find the Young's modulus of a substance by taking the gradient of the initial straight-line region of its stress-strain graph. But Young's Modulus is not defined as a rate of change of anything; E = Stress / Strain, and not d(stress) / d(strain):

So there's no need for any gradient calculation. In fact, all you need to do is pick any point on the linear-region of the graph, read off the values of stress and strain and divide one by the other. Of course, numerically, this is identical to a gradient calculation, which is why it's often described that way, but it's very likely to lead to misunderstandings. This is a very similar situation to that with finding values of resistance from a V-I graph.

For example, when you have a stress-strain graph for a non-Hookean substance like rubber, you can pick a point and divide stress by strain, but the resulting values would vary over the range of the graph so wouldn't be very useful. There are various things you can do instead, such as calculating the secant modulus (the gradient of a straight line drawn from the origin to the point you're interested in) or the tangent modulus (the gradient of a tangent to the curve at the point you're interested in). However, neither of these is the Young's modulus.