An ellipticity domain for the distortional Hencky logarithmic strain energy

Abstract

We describe ellipticity domains for the isochoric elastic energy F ↦ ∥ dev n log ⁡ U ∥ 2 = ∥ log ⁡ F T F ( det F ) 1 / n ∥ 2 = 1 4 ∥ log ⁡ C ( det C ) 1 / n ∥ 2 for n =2,3, where C = F T F for F ∈ GL + ( n ). Here, dev n log U = log U − ( 1 / n )   tr ( log U ) ⋅ 1 is the deviatoric part of the logarithmic strain tensor log ⁡ U . For n =2, we identify the maximal ellipticity domain, whereas for n =3, we show that the energy is Legendre–Hadamard (LH) elliptic in the set E 3 ( W   H iso , LH , U , 2 3 ) := { U ∈ PSym ( 3 ) | ∥ dev 3 log ⁡ U ∥ 2 ≤ 2 3 } , which is similar to the von Mises–Huber–Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy W   H ( F ) = μ ∥ dev 3 log ⁡ U ∥ 2 + ( κ / 2 ) [ tr ( log ⁡ U ) ] 2 , U = F T F with μ >0 and κ > 2 3 μ , previously obtained by Bruhns et al.