  We consider the volumetric-isochoric split in planar isotropic
hyperelasticity and give a precise analysis of rank-one convexity criteria for
this case, showing that the Legendre-Hadamard ellipticity condition separates
and simplifies in a suitable sense. Starting from the classical two-dimensional
criterion by Knowles and Sternberg, we can reduce the conditions for rank-one
convexity to a family of one-dimensional coupled differential inequalities. In
particular, this allows us to derive a simple rank-one convexity classification
for generalized Hadamard energies of the type $W(F)=\frac{\mu}{2}\frac{\lVert
F\rVert^2}{\det F}+f(\det F)$; such an energy is rank-one convex if and only if
the function $f$ is convex.
