Related papers: The exponentiated Hencky energy: Anisotropic exten…
In this paper we improve the result about the polyconvexity of the energies from the family of isotropic volumetric-isochoric decoupled strain exponentiated Hencky energies defined in the first part of this series, i.e. $$ W_{_{\rm eH}}(F)=…
The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are…
We consider a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{\rm eH}(F):=\widehat{W}_{\rm eH}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log…
We investigate a finite element formulation of the exponentiated Hencky-logarithmic model whose strain energy function is given by \[ W_\mathrm{eH}(\boldsymbol{F}) = \dfrac{\mu}{k}\, e^{\displaystyle k \left\lVert\mbox{dev}_n…
We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energies \begin{align*} F\mapsto W_{_{\rm eH}}(F):=\hat{W}_{_{\rm…
Adapting a method introduced by Ball, Muite, Schryvers and Tirry, we construct a polyconvex isotropic energy function $W\colon\operatorname{GL^+}(n)\to\mathbb{R}$ which is equal to the classical Hencky strain energy \[ W_{\mathrm{H}}(F) =…
We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$ F\mapsto W_{_{\rm eH}}(F):=\widehat{W}_{_{\rm eH}}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{2\,…
We describe ellipticity domains for the isochoric elastic energy $ F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2 $ for $n=2,3$,…
The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n) of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity…
In this paper we present a novel formulation for phenomenological anisotropic finite visco-hyperelasticity. The formulation is based on a multiplicative decomposition of the equilibrated deformation gradient into nonequilibrated elastic and…
This work presents a hyper-viscoelastic model based on the Hencky-logarithmic strain tensor to model the response of a Tire Derived Material (TDM) undergoing moderately large deformations. TDM is a composite made by cold forging a mix of…
We consider the two logarithmic strain measures\[\omega_{\rm iso}=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log \sqrt{F^TF}\|\quad\text{ and }\quad \omega_{\rm vol}=|\mathrm{tr}(\log U)|=|\mathrm{tr}(\log\sqrt{F^TF})|\,,\]which are…
In this paper we consider the additive logarithmic finite strain plasticity formulation from the view point of loss of ellipticity in elastic unloading. We prove that even if an elastic energy $F\mapsto W(F)=\hat{W}(\log U)$ defined in…
A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for…
We discuss a completely forgotten work of the geologist G.F. Becker on the ideal isotropic nonlinear stress-strain function. In doing this we provide the original paper from 1893 newly typeset in LaTeX and with corrections of typographical…
Anisotropy of a space naturally leads to direction dependent electromagnetic tensors and electromagnetic potentials. Starting from this idea and using variational approaches and exterior derivative formalism, we extend some of the classical…
A general expression for the elastic potential energy of a helical spring is derived using the basic concepts of elasticity theory and geometry. Both the translational and the rotational displacement of the spring's moving ends are…
We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a…
A general expression for the strain energy of a homogeneous, isotropic, plane extensible elastica with an arbitrary undeformed configuration is derived. This energy constitutes the correct expression for one-dimensional models of polymers…
We give conditions on the strain-energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the…