Proof of the Piola Transform Proof of the Piola Transform. As I understand it, the relationship between the second order tensor $\bf T$ over a reference configuration and the same tensor in a deformed configuration $\bf T^\prime$ is given defined as follows:
$
{\bf T} := {\bf T^\prime} \, \textrm{cof} \,  {\bf F}
$
Where $\bf F$  is the deformation gradient. The above relation is presented as a definition without proof in every text that I've come across, but it looks like a relationship that ought to have a proof, or some intuition ought to be given for the relation.
 A: The Piola transform appears to be based on Nanson's formula for which there is a proof.
A: The Piola stress tensor ${\bf P}$ delivers the traction per unit reference area acting at a material point: ${\bf p} = {\bf P} {\bf N}$, where ${\bf p}$ is the Piola traction, and ${\bf N}$ is a unit normal in the reference configuration.
The Cauchy stress tensor ${\bf T}$ delivers the traction per unit current area acting at a material point: ${\bf t} = {\bf T} {\bf n}$, where ${\bf t}$ is the Cauchy traction, and ${\bf n}$ is a unit normal in the current configuration.
These tractions deliver the same small amount of force $d{\bf f}$ transmitted over the small area of interest: $d {\bf f} = {\bf p} dA = {\bf t} da$, where $dA$ is an infinitesimal area in the reference configuration and $da$ is that same material area in the current configuration.
So we have ${\bf P} {\bf N} dA = {\bf T} {\bf n} da$. From here, you can use Nanson's formula to relate ${\bf N} dA$ to ${\bf n} da$ and you will recover the desired relation.
