Question

What does the area under a straight stress -strain graph tell you?

Answer 1

A stress-strain graph represents a dependence between axial normal stress $\sigma$ and axial normal strain $\varepsilon$ of materials. In the linear region (where the stress obeys Hook's law):

$$\sigma = E\varepsilon$$

By definition, the area under this graph is:

$$A = \int_0^\varepsilon \sigma d\varepsilon = E\int_0^\varepsilon \varepsilon d\varepsilon = \dfrac{E\varepsilon^2}{2}$$

which is the energy stored in the deformed material.

Answer. Absorbed energy.