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

Abstract

Constitutive laws in terms of the logarithmic strain tensor logU, i.e. the principal matrix logarithm of the stretch tensor U=FTF corresponding to the deformation gradient F, have been a subject of interest in nonlinear elasticity theory for a long time. In particular, there have been multiple attempts to derive a viable constitutive law of nonlinear elasticity from an elastic energy potential which depends solely on the logarithmic strain measure ‖logU‖2, i.e. an energy function W:GL+(n)→R of the form (1)W(F)=Ψ(‖logU‖2)with a suitable function Ψ:[0,∞)→R, where ‖.‖ denotes the Frobenius matrix norm and GL+(n) is the group of invertible matrices with positive determinant.However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form (1) is Legendre–Hadamard elliptic.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.