For completeness, we’ll derive the volume change and area change:
Derivation starts from the triple scalar product. Cross products and determinants (written in tensor form) is a math topic, so we’ll skip the derivation of volumetric deformation. A quick derivation of volumetric deformation without the use of indices can be found in [Bonet].
Now we know that , but what about area?
If is the initial area and is the final area:
For small volumes, we can say that and .
It was previously shown that
Substituting, we get
We also know that
where and are the normal vectors to the surface in the respective initial and final configurations.
Eq. 2 is called Nanson’s Equation and will be useful later when we look at “true” stress versus “nominal” stress, for example.