The exponentiated Hencky-logarithmic strain energy: part III—coupling with idealized multiplicative isotropic finite strain plasticity

Abstract

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energieswhere ∂ χ is the subdifferential of the indicator function χ of the convex elastic domain Ee(Wiso, Σe, 1 3 σ 2 y ) in the mixedvariant Σe-stress space and Σe = F T e DF e Wiso(Fe). While W eH may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of W eH on the one hand, and the elastic domain in Σe-stress space on the other hand, are closely related. Thus the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true-stress-true-strain relation remains monotone, as observed in experiments.

An ellipticity domain for the distortional Hencky logarithmic strain energy

“…In order to visualize the established domain of ellipticity for n = 3, we go back to the initial substitution and find that the unbounded ellipticity domain given by Proposition 4.1 is the set enclosed by the cone presented in Figures 4 and 5, which is completely defined by since on the one hand t → µ e t is convex and monotone increasing, and therefore the composition with this mapping preserves ellipticity, and on the other hand the function F → e k (log det F ) 2 is rank-one convex on GL + (3) for k ≥ 1 8 (see [37] for more details). However, numerical tests suggest that the ellipticity domain of the exponentiated Hencky energy is far bigger than all ellipticity domains which are known for various energies of quadratic Hencky energy type, see also [37,36]. This remark might be useful in the study of large deformations which do not belong to the known ellipticity domains of the energies of the quadratic Hencky energy type.…”

Homogeneous Cauchy stress induced by non‐homogeneous deformations

“…where V = √ F F T is the left stretch tensor. Then σ eH is invertible, while W eH is not rank-one convex [3][4][5].…”

The exponentiated Hencky energy: anisotropic extension and case studies

“…In order to alleviate some of these shortcomings, Neff et al introduced the exponentiated Hencky energy W eH in a series of articles Neff et al [46; 47], Neff and Ghiba [42], Ghiba et al [17]. This energy function closely approximates the classical quadratic Hencky energy for small deformations, but aims to provide a more accurate model for large deformations as well as an improvement in terms of common constitutive requirements; for example, W eH is polyconvex in the two-dimensional case Neff et al [47], and in the three-dimensional case the rank-one convexity domain contains the extremely large set {F ∈ GL + (3)| dev 3 log U ≤ 6}.…”

“…In this article we consider a novel Hencky-type hyperelasticity model, the exponential Hencky-logarithmic strain energy proposed by Neff et al [46], Neff et al [47] and Neff and Ghiba [42]. Here, we focus on an extension to anisotropy in a coordinate invariant setting.…”