$\newcommand{\Dev}{\operatorname{Dev}}$ $\newcommand{\dev}{\operatorname{dev}}$ $\newcommand{\grad}{\operatorname{grad}}$ $\newcommand{\Grad}{\operatorname{Grad}}$ $\newcommand{\divg}{\operatorname{div}}$ $\newcommand{\Ei}{\operatorname{Ei}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\Divg}{\operatorname{Div}}$ $\newcommand{\cay}{\operatorname{cay}}$ $\newcommand{\rot}{\operatorname{rot}}$

This is a compressible isotropic neo-Hookean material that uses the natural (Hencky) strain tensor invariants to formulate its strain energy density. These invariants are reviewed in [28]. The left Hencky strain is evaluated from $\boldsymbol{\eta}=\ln\mathbf{V}$ where $\mathbf{V}$ is the left stretch tensor in the polar decomposition of the deformation gradient $\mathbf{F}=\mathbf{V}\cdot\mathbf{R}$ . To evaluate $\boldsymbol{\eta}$ we first evaluate the left Cauchy-Green tensor $\mathbf{b}=\mathbf{V}^{2}$ from $\mathbf{F}$ as in eq.(2.3.2-2) and get its eigenvalues $\lambda_{i}^{2}$ and eigenvectors $\mathbf{n}_{i}$ . Then $$\begin{equation} \boldsymbol{\eta}=\sum_{i=1}^{3}\left(\ln\lambda_{i}\right)\mathbf{n}_{i}\otimes\mathbf{n}_{i}\,.\label{eq:left-Hencky} \end{equation}$$ The invariants $K_{i}$ of the natural strain tensor are $$\begin{equation} \begin{aligned}K_{1} & =\tr\boldsymbol{\eta}=\ln J & \text{amount of dilatation}\\ K_{2} & =\left|\dev\boldsymbol{\eta}\right|=\sqrt{\dev\boldsymbol{\eta}:\dev\boldsymbol{\eta}} & \text{amount of distortion}\\ K_{3} & =3\sqrt{6}\det\boldsymbol{\Phi} & \text{mode of distortion} \end{aligned} \label{eq:Hencky-isotropic-invariants} \end{equation}$$ where $J=\det\mathbf{F}$ as usual, and $$\begin{equation} \boldsymbol{\Phi}=\frac{1}{K_{2}}\dev\boldsymbol{\eta}\,.\label{eq:Hencky-3rd-basis} \end{equation}$$ It can be shown that $$\begin{equation} \boldsymbol{\eta}=\frac{1}{3}K_{1}\mathbf{I}+K_{2}\boldsymbol{\Phi}\,.\label{eq:left-Hencky-redux} \end{equation}$$ Note that $K_{2}\boldsymbol{\Phi}\to\mathbf{0}$ as $K_{2}\to0$ . It also follows that $\boldsymbol{\eta}:\boldsymbol{\eta}=\frac{1}{3}K_{1}^{2}+K_{2}^{2}$ . As explained in [28], $K_{1}\in\left(-\infty,\infty\right)$ with positive $K_{1}$ implying expansion and negative $K_{1}$ implying contraction. Similarly, $K_{2}\in[0,\infty)$ , with $K_{2}=0$ implying no distortion. Finally, $K_{3}\in\left[-1,1\right]$ with $K_{3}=1$ representing uniaxial extension, $K_{3}=-1$ representing uniaxial contraction and $K_{3}=0$ representing pure shear.