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 \(W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)\); such an energy is rank-one convex if and only if the function \(f\) is convex.

我们考虑了平面各向同性超弹性中的体积-等速分裂，并针对这种情况给出了秩一凸度准则的精确分析，表明Legendre-Hadamard椭圆率条件在适当的意义上进行了分离和简化。从Knowles和Sternberg的经典二维准则开始，我们可以将秩为一的凸的条件简化为一维耦合微分不等式的族。特别是，这使我们能够为\（W（F）= \ frac {\ mu} {2} \ hspace {0.07em} \ frac {\ lVert F类型的广义Hadamard能量得出简单的秩一凸分类。\ rVert ^ {2}} {\ det F} + f（\ det F）\） ; 当且仅当函数\（f \）是凸的时，这种能量才是一阶凸的。