Hyperelastic material models - intepretation of distortional and deviatoric part of Cauchy-Green tensor I have a problem with understanding a meaning behind deviatoric of distortional part of Cauchy-Green tensor $ dev[\textbf{B}^{*}] $ used in stress equations for hyperelastic material models (like in Mechanics of Solid Polymers by J. Bergstrom, equations 5.92 or 5.104): 
$$ \text{Yeoh model: } \boldsymbol{\sigma} = \frac{2}{J}[C_{10}+2C_{20}(I_1^{*}-3)+3C_{30}(I_1^{*}-3)^2] \cdot dev[\textbf{B}^{*}] + \kappa \cdot (J-1)\cdot \textbf{I}  $$
$$ \text{Eight-chain model: } \boldsymbol{\sigma} = \frac{\mu}{J \bar{\lambda^*}} \frac{\mathcal{L}^{-1}(\frac{\bar{\lambda^*}}{\lambda^{lock}}) }{\mathcal{L}^{-1}(\frac{1}{\lambda^{lock}} )} dev[\textbf{B}^{*}]+\kappa \cdot (J-1)\cdot \textbf{I} $$
Deformation gradient tensor can be decomposed into dilatational (volumetric) $\textbf{F}^{vol}$ and distortional (isochoric) part $\textbf{F}^{*}$:
$$ \textbf{F}=\textbf{F}^{vol}\cdot \textbf{F}^{*}  $$
where
$$ \textbf{F}^{vol}= J^{1/3}\cdot \textbf{I}   $$
$$ \textbf{F}^{*} = J^{-1/3}\cdot \textbf{F} $$
$$ J = det(\textbf{F})  $$
As I understand, if we deal with incompressible material model, then we are interested only in deformations under constant volume, so we have to extract from tensor F specific part which is $ \textbf{F}^{*} $. Consequently, a Cauchy-Green tensor is:
$$ \textbf{B}^{*} = J^{-2/3}\cdot \textbf{F} \cdot \textbf{F}^T = J^{-2/3}\cdot \textbf{B}$$
Now comes the main problem - what is the physical interpretation of
$ dev[\textbf{B}^{*}] $ which is used in stress equations?
As I rembember from small strains theory, deviatoric strains represents shape change at constant volume, so I would think of something similar in this case but:


*

*what is the reason to use it for large deformations (specific for hyperelastic materials) when deviator loses its physical meaning?

*what is the interpretation of deviatoric part of distortional Cauchy-Green tensor?


I feel I'm missing something and I would like ask for advice.
 A: Context
I think there is no direct geometric interpretation of the spheric and deviatoric parts of the left Cauchy-Green (LCG) tensor $\mathbf B$.
We do know for a fact that a strictly isochoric deformation gradient does not correspond to a strictly deviatoric LCG, which makes me think the answer below is relevant.
Indeed, there is an meaningful explanation for the form of the constitutive models you mentioned.
The additional notions I'm assuming known are those of configuration, work conjugate and continuum thermodynamics applied to constitutive modeling. No computation detail is given.
Recall
By definition, hyperelasticity refers to a family of constitutive models that derive from an energy-like function. Let us consider here the free energy density $\rho\psi$. The work conjugate of $\mathbf F$ is the first Piola-Kirchoff (PK1) stress tensor $\mathbf\Pi$, and we deduce from thermodynamics considerations:
$$ \mathbf\Pi = \frac{\partial\rho\psi}{\mathbf F} .$$
Constitutive modeling
Let's get in the shoes of Yeoh and others and build a hyperelastic constitutive model that only depends on isochoric changes. To do so in large deformations, $\mathbf F^*$ is the best (the only?) strain measure candidate. We choose the simplest quadratic form:
$$\rho\psi = \frac\mu2(\mathbf F^*:\mathbf F^*-3)=\frac\mu2(J^{-2/3}\mathbf F:\mathbf F-3)$$
so that $\rho\psi$ is null when $\mathbf F=\mathbf I$ and variates otherwise. Computing the associated stress measure:
$$ \mathbf\Pi = \frac{\partial\rho\psi}{\mathbf F} =\mu J^{-2/3}(\mathbf F - \frac13 (\mathbf F:\mathbf F)\mathbf F^{-\top}),$$
we end up with a constitutive relation that involves quantities of the initial (or material) configuration. Since the goal is to express it in terms of the Cauchy stress measure $\sigma$, we can apply a push-forward to the current configuration: by definition,
$$\sigma = J^{-1}\mathbf\Pi\mathbf F^{\top}=\mu J^{-1} (J^{-2/3}\mathbf F \mathbf F^{\top}- \frac13 (J^{-2/3}\mathbf F:\mathbf F)\mathbf I).$$
Since $\mathbf F:\mathbf F=\mathrm{trace}\,(\mathbf F \mathbf F^{\top})$, we plug in $\mathbf B$ and eventually identify:
$$\sigma =\mu J^{-1} \mathrm{dev}\,\mathbf B^*.$$
The models you mentioned are fancier but exhibit the same linearity with respect to $1/J$ and $\mathrm{dev}\,\mathbf B^*$. I think they share identical conceptual goals.
Conclusion
The occurrence of $\mathrm{dev}\,\mathbf B^*$ is simply the consequence of:

*

*a constitutive model formulated in terms of $\mathbf F^*$;

*a push-forward to the current configuration to find an expression of $\mathbf\sigma$.

Therefore, we can still interpret the deviatoric part of the left Cauchy-Green tensor in the context of hyperelastic materials as the strain measure that intervenes in an isochoric work.
