Recently I have been studying solid structural mechanics, and one of the points I find really confusing is how elasticity and flexibility are closely intertwined.
Consider an Euler-Bernoulli beam, for instance. The flexural rigidity of the beam is given by $EI$, where $E$ is the Young's Modulus of a beam, while $I$ is the second moment of area (basically moment of inertia, but with area instead of mass). This suggests that an object that is easy to bend (with a small flexural rigidity) must also be easy to stretch (and possess a small Young's Modulus).
I find it hard to wrap my head around this idea because there are so many objects in our everyday lives that are difficult to stretch but very easy to bend. Consider a thick string, for instance. Although it is effectively inextensible, it can be bent around objects. Paper, too, is very hard to stretch, but it can bend very easily.
Is there anything that I misunderstood from this concept? What other understanding should I need to make sense of this apparent contradiction?