Define strain energy and complementary

Strain energy is the energy stored in an elastic body when it is deformed by external loads. It equals the work done by the loads and is recovered fully on unloading. It is the area under the load-displacement curve:

$$U = \int_0^\Delta P\,d\Delta \quad(= \tfrac12 P\Delta \text{ for a linear body})$$

Complementary strain energy is the area between the load-displacement curve and the load axis:

$$U^* = \int_0^P \Delta\,dP \quad(= \tfrac12 P\Delta \text{ for a linear body})$$

The sum $U + U^* = P\Delta$. For linearly elastic materials $U^* = U$; they differ only for non-linear materials.