A non-ellipticity result, or the impossible taming of the logarithmic strain measure

Similarly, we consider the related isochoric strain measure ‖devnlogU‖2, where devnlogU is the deviatoric part of logU. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n=2, we show that for n≥3, no strictly monotone function Ψ:[0,∞)→R exists such that F↦Ψ(‖devnlogU‖2) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F↦Ψ(‖devnlogU‖2)+Wvol(detF) cannot be rank-one convex for any function Wvol:(0,∞)→R if Ψ is strictly monotone.