An ellipticity domain for the distortional Hencky logarithmic strain energy

Abstract

We describe ellipticity domains for the isochoric elastic energy

A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes

“…The three-dimensional exponentiated Hencky energy, on the other hand, is not overall rank-one convex, although it is Legendre-Hadamard elliptic in a large neighbourhood of the identity tensor 1. Moreover, in couplings with multiplicative elasto-plasticity, the computation of the elastic trial step always leads to a rank-one convex problem provided the computation is carried out in a (large) neighbourhood of the yield surface [10,33]. This is true since the elastic domain in that model is always included in its rank-one convexity domain, at any given plastic deformation.…”

The exponentiated Hencky energy: anisotropic extension and case studies

“…While the elasticity model induced by the Hencky energy is in very good agreement with experimental observations for up to moderate strains for a large number of materials Neff et al [46], Anand [1], there are some major shortcomings of this model. For example, the qualitative behavior of materials under very large deformations is not modeled accurately, and since the energy function is neither polyconvex nor quasiconvex or rank-one convex (Neff [41], Ghiba et al [16]), no known methods are available to ensure the existence of energy minimizers for general boundary value problems. Moreover, the pressurecompression relation is not monotone.…”

A polyconvex extension of the logarithmic Hencky strain energy

“…This shortcoming raises some concern regarding the suitability of the Hencky model in finite element methods, although Bruhns et al [4] have explicitly determined a (rather large) ellipticity domain of the Hencky energy (cf. [7]).…”

“…The volumetric part ψ of the polyconvex extension and the original volumetric term ln 2 (t) of the classical Hencky energy. Bruhns et al[4] have shown that for Λ ≥ 0, the quadratic Hencky strain energy is elliptic on the set of all F ∈ GL + (3) with all singular values in the interval [α,3 √ e], where α ≈ 0.21 < e −2/3 , cf [7]…”