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

The traction $\mathbf{t}$ on a plane bisecting the body is given by, $$\begin{equation} \mathbf{t}=\boldsymbol{\sigma}\cdot\mathbf{n},\label{eq48} \end{equation}$$ where $\boldsymbol{\sigma}$ is the Cauchy stress tensor and $\mathbf{n}$ is the outward unit normal vector to the plane. It can be shown that by the conservation of angular momentum that this tensor is symmetric ($\sigma_{ij}=\sigma_{ji})$ [73]. The Cauchy stress tensor, a spatial tensor, is the actual physical stress, that is, the force per unit deformed area. To simplify the equations of continuum mechanics, especially when working in the material configuration, several other stress measures are often used. The Kirchhoff stress tensor is defined as $$\begin{equation} \boldsymbol{\tau}=J\boldsymbol{\sigma}\,.\label{eq49} \end{equation}$$ The first Piola-Kirchhoff stress tensor is given as $$\begin{equation} \mathbf{P}=J\boldsymbol{\sigma}\cdot\mathbf{F}^{-T}\,.\label{eq50} \end{equation}$$ Note that $\mathbf{P}$ , like $\mathbf{F}$ , is not symmetric. Also, like $\mathbf{F}$ , $\mathbf{P}$ is known as a two-point tensor, meaning it is neither a material nor a spatial tensor. Since we have two strain tensors, one spatial and one material tensor, it would be useful to have similar stress measures. The Cauchy stress is a spatial tensor and the second Piola-Kirchhoff (2$^{nd}$ PK) stress tensor, defined as $$\begin{equation} \mathbf{S}=J\mathbf{F}^{-1}\cdot\boldsymbol{\sigma}\cdot\mathbf{F}^{-T}\,,\label{eq51} \end{equation}$$ is a material tensor. The inverse relations are: $$\begin{equation} \boldsymbol{\mathbf{\sigma}}=\frac{1}{J}\boldsymbol{\tau}\,,\quad\boldsymbol{\sigma}=\frac{1}{J}\mathbf{P}\cdot\mathbf{F}^{T}\,,\quad\boldsymbol{\sigma}=\frac{1}{J}\mathbf{F}\cdot\mathbf{S}\cdot\mathbf{F}^{T}\,.\label{eq52} \end{equation}$$ In many practical applications it is physically relevant to separate the hydrostatic stress and the deviatoric stress $\tilde{\boldsymbol{\sigma}}$ of the Cauchy stress tensor: $$\begin{equation} \boldsymbol{\sigma}=\tilde{\boldsymbol{\sigma}}+p\mathbf{I}\,.\label{eq53} \end{equation}$$ Here, the pressure is defined as $p=\frac{1}{3}\tr\boldsymbol{\sigma}$ . Note that the deviatoric Cauchy stress tensor satisfies $\tr\tilde{\boldsymbol{\sigma}}=0$ .