Consider an infinitesimal volume element in the reference configuration. If a right-handed orthonormal basis in the reference configuration is then the vectors representing the edges of the element are
The volume of the element is given by
Upon deformation, these edges go to where
Therefore, the deformed volume is given by
Recall that from conservation of mass we have
Therefore, an alternative form of the conservation of mass is
Distortional component of the deformation gradient
For many materials it is convenient to decompose the deformation gradient in a volumetric part and a distortional part. This is particularly useful when there is no volume change in the material when it deforms - for example in muscles, rubber tires, metal plasticity, etc.
Let us assume that the deformation gradient can be decomposed into a volumetric part and a distortional part, i.e,
Since there is no volume change due to the pure distortion, we have
If we have
Therefore the distortional component of the deformation gradient is given by
We can use this result to find the distortional components of various strain and deformation tensors. For example, the left Cauchy-Green deformation tensor is given by
Nanson's formula is an important relation that can be used to go from areas in the current configuration to areas in the reference configuration and vice versa.
This formula states that
where is an area of a region in the current configuration, is the same area in the reference configuration, and is the outward normal to the area element in the current configuration while is the outward normal in the reference configuration.
To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:
The reference and current volumes of an element are