For completeness, we’ll derive the volume change and area change:

(1)

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].

note: “incompressible”

Nanson’s Equation:

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

and

Substituting, we get

We also know that

(2)

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.