Continuum mechanics cauchy stress I have a question regarding the stress tensors. In a publication I found the following definition:

I havnt seen this equation before, and I am wondering how to get there. T is the Cauchy stress, Je the elastic volume ratio, μ the lamé constant, λe the elastic principal stretch. The ' means the deviatoric part. κ is the bulk modulus. The formula can be found in (4) in the paper: https://doi.org/10.1016/j.jpowsour.2010.03.047
Probably the first part of the formula is the expression for the hydrostatic stress and the second the deviatoric. Wikipedia (https://en.wikipedia.org/wiki/Cauchy_stress_tensor), continuummechanics.org (http://www.continuummechanics.org/energeticconjugates.html) as well as Kontinuumsnechanik (J. Betten, 2001, p.88) did not help.
Can somebody help please?
 A: (My aim is to explain the likely origin of the expression $T_{Ii}=\frac{1}{J^e}[2\mu\ln(\lambda_{Ii}^e)^\prime+\kappa\ln J^e]$, not to defend its use in any particular context.)
First, some simplification: The index $I$ refers to a particular model in the paper, so let's strip that along with the label $e$ for elastic (we'll consider only elastic deformation) and multiply by $J$ to give
$$JT_{i}=2\mu\ln(\lambda_{i})^\prime+\kappa\ln J.$$
We seek the origin of this equation.
The equation generalizes Hooke's law (applicable for small strains of isotropic elastic materials), or $$\boldsymbol{\sigma}=2\mu\boldsymbol{\varepsilon}+\Lambda \mathrm{tr}(\boldsymbol{\varepsilon})\boldsymbol{I},$$ where $\boldsymbol{\sigma}$ is the stress tensor, $\mu$ and $\Lambda$ are the Lamé parameters, $\boldsymbol{\varepsilon}$ is the infinitesimal strain tensor, and $\boldsymbol{I}$ is the identity matrix. (We could write this in index form as $\sigma_i=2\mu \varepsilon_i+\Lambda \mathrm{tr}(\boldsymbol{\varepsilon}).$)
The generalization aims to address finite strains of isotropic elastic materials, or hyperelasticity, using the so-called Hencky or logarithmic measure of strain.
To suitably modify Hooke's Law for large displacements, we return to the differential definition of strain, namely, $d\varepsilon=\frac{dL}{L}$, and—rather than assuming nearly constant $L$ to integrate as $\frac{1}{L}\int dL$ and thus obtain the engineering strain $\frac{\Delta L}{L}$—we instead integrate directly as $\int\frac{dL}{L}$ to obtain the Hencky*, true,  or logarithmic strain $\varepsilon=\ln\left(\frac{L_\mathrm{final}}{L_\mathrm{original}}\right)=\ln(\lambda)$, where $\lambda$ is termed the stretch. For small stretches ($\lambda\approx 1$), the true strain is well approximated by the engineering strain, as one can show with a Taylor series expansion of the logarithmic function for arguments close to 1.
On to the next point. The generalization uses** the Kirchhoff stress $\boldsymbol{\tau}=J\boldsymbol{T}$, where $J$ is called the elastic volume ratio (it's also the Jacobian or determinant of the deformation matrix, unity for incompressible materials and in shear) and $\boldsymbol{T}$ is the Cauchy stress, so the modified Hooke's Law has become
$$J\boldsymbol{T}=2\mu\ln\boldsymbol{\lambda}+\Lambda \mathrm{tr}(\ln\boldsymbol{\lambda})\boldsymbol{I},$$
where $\ln\boldsymbol{\lambda}$ refers to the logarithm taken elementwise. Again, for small strains, we recover Hooke's Law, as $J\approx 1+\mathrm{tr}(\boldsymbol{\varepsilon})\approx 1$.
I see a prime ($^\prime$) symbol in the original equation. Let's define the deviatoric strain
$$\ln\boldsymbol{\lambda}^\prime =\ln\boldsymbol{\lambda}-\frac{1}{3}\mathrm{tr}(\ln\boldsymbol{\lambda}).$$
Since $\Lambda\equiv\kappa-\frac{2\mu}{3}$, where $\kappa$ is the bulk modulus, working through the algebra, we now have
$$J\boldsymbol{T}=2\mu\ln\boldsymbol{\lambda}^\prime+\kappa \mathrm{tr}(\ln\boldsymbol{\lambda})\boldsymbol{I}.$$
Let's write this in index form:
$$JT_i=2\mu\ln(\lambda_i)^\prime+\kappa \mathrm{tr}(\ln\boldsymbol{\lambda}).$$
But the trace of a matrix of logarithmic stretches is simply the logarithm of the determinant of the principal stretch matrix $\lambda_1\lambda_2\lambda_3$, or $\ln J$, so we finally obtain
$$JT_i=2\mu\ln(\lambda_i)^\prime+\kappa \ln J,$$
which matches the expression in the paper.
*Other measures of finite strain can be defined; a broader expression (called the Seth–Hill or Doyle–Erickson strain measures) is
$$\boldsymbol{E^{(m)}}=\frac{\boldsymbol{C}^{m/2}-\boldsymbol{I}}{m},$$
where $\boldsymbol{E^{(m)}}$ is the finite strain, $\boldsymbol{C}$ is the right Cauchy–Green tensor, and $m$ is any real number. For the case of $m=0$, a limiting procedure ($m\to 0$) is used to obtain the Hencky strain measure $\boldsymbol{E^{(0)}}=\sum_{k=1}^3(\ln\lambda_k)\boldsymbol{C_k}\otimes \boldsymbol{C_k}$, where $\lambda_k^2$ and $\boldsymbol{C_k}$ are the eigenvalues and eigenvectors of $\boldsymbol{C}$. This is another way of obtaining the characteristic logarithmic form of the Hencky strain.
**Why the Kirchhoff stress tensor instead of the Cauchy stress tensor? Although perhaps outside the scope of this answer, the reason—involving the need to weight the stress tensor to accommodate volumetric changes and maintain path independence upon repeated work—is discussed in Neff et al.'s "The axiomatic deduction of the
quadratic Hencky strain energy by
Heinrich Hencky." Put another way, if you incorporate the volume ratio $J$ into the stress term, you don't have to linearize it. See also Hajhashemkhani et al.'s "Identification of material
parameters of a hyper-elastic
body with unknown boundary conditions", Xiao and Chen's "Hencky's logarithmic strain and dual stress–strain and
strain–stress relations in isotropic finite hyperelasticity" and Neff et al.'s "A brief history of logarithmic strain measures in nonlinear elasticity."
(This answer was guided strongly by Xiao et al.'s "Hill’s class of compressible elastic materials and finite bending problems:
Exact solutions in unified form" and the references within, especially Hill's "On constitutive inequalities for simple materials—I". It would have been useful if Silberstein had provided a citation upon stating the equation to one or more of these and Anand's "On H. Hencky’s Approximate Strain-Energy Function for Moderate Deformations", say, to avoid reader confusion. This thought may not have arisen because the framework is so familiar to those in the Boyce group.)
