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For an elastomer, what could be the modulus of elasticity. My thought process is that the modulus of elasticity is the tangent of a point of the stress-strain graph. And therefore looking at the graph it makes sense that the value is high. But then I realised that value was low at small stress and then got progressively higher as we went up...

I realised that since the modulus of elasticity is not a constant like moduli of other materials, is it even a correct question to ask? This was a question we discussed in class as it had appeared on a worksheet and I argued that the value should be high, but then later (after the academic year to be precise) it struck me that the modulus of elasticity of aorta is not a single value, and so it should not be existing. Am I close to being correct?

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Rubber (vulcanised, aka cross-linked) elasticity actually takes on an $S$-shape, like:

Rubber elasticity

Source.

You're correct in thinking there's no elastic modulus in the Hookean sense of the word.

Instead, rubber technologists usually define a 'modulus' at various point of strain, e.g. $E_{100}$ (at $100\text{ percent}$ strain), $E_{200}$ (at $200\text{ percent}$ strain) and $E_{300}$ (at $300\text{ percent}$ strain).

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Here is why I think it is a correct question to ask.

The stress-strain curve we get when pulling a macroscopic sample of any engineering material on an Instron machine is a macroscopic reflection of phenomena taking place on a nanoscale, inside the test coupon.

In this connection, a "Hookean" response is an idealized model of a complicated system, and deviations from it provides evidence of physical processes that were not contained in our model.

To improve our understanding of Stress-Strain graphs, we then study the test results and the microstructure of the sample and revise our model to include the physics we left out of the first iteration.

Now because elastomers respond so differently to applied stresses than most engineering metals, the test techniques developed to characterize metals can not be expected to furnish us with results that are nearly as simple to explain when testing nonmetals- but those results are still very useful in understanding what's going on.

That is why an Instron test of a strip of rubber is always a good thing to do, especially in front of a lab full of undergraduate materials science students:

"OK, pals'n'gals. We didn't get a straight line there, did we? Why is that? And if you are selling rubber compounds, how will you characterize their different stiffnesses to your customers in a useful way?"

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