1 Conclusion and results

1.1 Different stress tensors

Stress	Stress measure	Generally Symmetrical ?
\(\mathbf {\tilde {\tau }}\) Cauchy	\(\mathbf {df}=\left ( da\ \mathbf {n}\right ) \cdot \mathbf {\tilde {\tau }}\)	Yes
\(\mathbf {\tilde {t}}\) First Piola-Kirchhoff	\(\mathbf {\tilde {t}}=J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\)	No
\(\mathbf {\tilde {s}}_{1}\)Second Piola-Kirchhoff	\(\mathbf {\tilde {s}}_{1}=J\ \mathbf {\tilde {F}}^{-1}\cdot \mathbf {\tilde {\tau }}\cdot \mathbf {\tilde {F}}^{-T}\)	Yes
\(\mathbf {\tilde {\sigma }}\) Kirchhoff	\(\mathbf {\tilde {\sigma }}=J\ \mathbf {\tilde {\tau }}\)	Yes
\(\boldsymbol {\tilde {\Gamma }}\)	\(\boldsymbol {\tilde {\Gamma }=\tilde {R}}^{T}\cdot \mathbf {\tilde {\tau }\cdot \tilde {R}}\)	Yes
\(\mathbf {\tilde {r}}^{\ast }\) Biot-Lure	\(\mathbf {\tilde {r}}^{\ast }=J\ \mathbf {\tilde {F}}^{-1}\mathbf {\cdot \tilde {\tau }}\cdot \mathbf {\tilde {R}}\)	No
\(\mathbf {\tilde {r}}\) Jaumann	\(\mathbf {\tilde {r}=}\frac {\left ( \mathbf {\tilde {r}}^{\ast }+\mathbf {\tilde {r}}^{\ast T}\right ) }{2}\)	Yes
\(\mathbf {\tilde {T}}^{\ast }\)	\(J\ \mathbf {\tilde {V}}^{-1}\mathbf {\cdot \ \tilde {\tau }}\)	No
\(\mathbf {\tilde {T}}\)	\(\mathbf {\tilde {T}=}\frac {\left ( \mathbf {\tilde {T}}^{\ast }+\mathbf {\tilde {T}}^{\ast T}\right ) }{2}\)	Yes

1.2 Deformation gradient tensor under rigid body transformation \(\tilde {Q}\)

Tensor	\(\mathbf {\tilde {Q}}\) based transformation
\(\mathbf {\tilde {F}}\) The deformation gradient	\(\mathbf {\tilde {F}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {F}}\)
\(\mathbf {\tilde {U}~}\)Stretch before rotation \(\mathbf {\tilde {R}}\)	\(\mathbf {\tilde {U}}_{q}=\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {V}~}\)Stretch after rotation \(\mathbf {\tilde {R}}\)	\(\mathbf {\tilde {V}}_{q}=\mathbf {\tilde {Q}\cdot \tilde {V}\cdot \tilde {Q}}^{T}\)

1.3 Stress tensors under rigid body transformation \(\tilde {Q}\)

Stress	\(\mathbf {\tilde {Q}}\) based transformation	Transforms Similar to
\(\mathbf {\tilde {\tau }}\) Cauchy	\(\mathbf {\tilde {\tau }}_{q}=\mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\)	\(\mathbf {\tilde {V}}\)
\(\mathbf {\tilde {t}}\) First Piola-Kirchhoff	\(\mathbf {\tilde {t}}_{q}=\mathbf {\tilde {Q}}\cdot \mathbf {\tilde {t}}\)	\(\mathbf {\tilde {F}}\)
\(\mathbf {\tilde {s}}_{1}\)Second Piola-Kirchhoff	\(\mathbf {\tilde {s}}_{1_{q}}=\mathbf {\tilde {s}}_{1}\)	\(\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {\sigma }}\) Kirchhoff	\(\mathbf {\tilde {\sigma }}=J\ \left ( \mathbf {\tilde {Q}\cdot \tilde {\tau }\cdot \tilde {Q}}^{T}\right ) \)	\(\mathbf {\tilde {V}}\)
\(\boldsymbol {\tilde {\Gamma }}\)	\(\boldsymbol {\tilde {\Gamma }}_{q}=\boldsymbol {\tilde {\Gamma }}\)	\(\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {r}}^{\ast }\) Biot-Lure	\(\mathbf {\tilde {r}}_{q}^{\ast }=\mathbf {\tilde {r}}^{\ast }\)	\(\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {r}}\) Jaumann	\(\mathbf {\tilde {r}}_{q}\mathbf {=\tilde {r}}\)	\(\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {T}}^{\ast }\)	\(\mathbf {\tilde {T}}_{q}^{\ast }=\mathbf {\tilde {T}}^{\ast }\)	\(\mathbf {\tilde {U}}\)
\(\mathbf {\tilde {T}}\)	\(\mathbf {\tilde {T}}_{q}=\mathbf {\tilde {T}}\)	\(\mathbf {\tilde {U}}\)

1.4 Conjugate pairs (Stress tensor/Strain tensor)

Let \(W\) be the current amount of energy stored in a unit volume as a result of the body undergoing deformation, then the time rate at which this energy changes will equal the stress tensor \(\mathbf {\tilde {B}}\) multiplied by the strain rate \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}\). Therefore

\[ \boxed { \dot {W} = \mathbf {\tilde {B}} \frac {\partial \mathbf {\tilde {A}}}{\partial t} } \]

The following table gives the stress tensor \(\mathbf {\tilde {B}}\), the strain rate \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}\) and the strain \(\mathbf {\tilde {A}}\)

Stress tensor \(\mathbf {\tilde {B}}\)	Strain tensor rate \(\frac {\partial \mathbf {\tilde {A}}}{\partial t}\)	Strain tensor \(\mathbf {\tilde {A}}\)
\(\mathbf {\tilde {\tau }}\) Cauchy	\(\frac {1}{J}\frac {1}{2}\left ( \mathbf {\dot {F}\cdot \tilde {F}}^{-1}+\left ( \mathbf {\dot {F}\cdot \tilde {F}}^{-1}\right ) ^{T}\right ) \)	Almansi strain tensor \(\mathbf {\tilde {\mu }}=\frac {1}{J} \frac {1}{2}\left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}} ^{-1}-\mathbf {\tilde {I}}\right ) \)
\(\mathbf {\tilde {\sigma }}\) Kirchhoff	\(\frac {1}{2}\left ( \mathbf {\dot {F} \cdot \tilde {F}}^{-1}+\left ( \mathbf {\dot {F}\cdot \tilde {F}}^{-1}\right ) ^{T}\right ) \)	\(\frac {1}{2}\left ( \mathbf {\tilde {F}}^{-T}\cdot \mathbf {\tilde {F}}^{-1}-\mathbf {\tilde {I}}\right ) \)
\(\mathbf {\tilde {t}\ }1^{st}\) Piola-Kirchhoff	\(\frac {1}{J}\mathbf {\dot {F} }^{T}\)	\(\frac {1}{J}\mathbf {\tilde {F}}^{T}\)
\(\mathbf {\tilde {s}}_{1}\) \(2^{nd}\) Piola-Kirchhoff	\(\frac {1}{J} \mathbf {\dot {\gamma }}\)	Green-Lagrange strain tensor \(\frac {1}{J} \boldsymbol {\tilde {\Gamma }=}\frac {1}{2J}\left ( \mathbf {\mathbf {\tilde {F}} ^{T}\cdot \tilde {F}-\tilde {I}}\right ) \)
\(\mathbf {\tilde {r}}^{\ast }\) Biot-Lure	\(\frac {1}{J}\mathbf {\dot {U}}\)	\(\frac {1}{J}\mathbf {\mathbf {U}}\)
\(\mathbf {\tilde {r}}\) Jaumann	\(\mathbf {\dot {U}}\)	\(\mathbf {\tilde {U}} \)
\(\boldsymbol {\tilde {\Gamma }}\)	\(\frac {1}{2}\left ( \mathbf {\tilde {U}}^{-1} \cdot \mathbf {\dot {U}+\dot {U}\cdot \tilde {U}}^{-1}\right ) \)	\(\ln \left ( \mathbf {\tilde {U}}\right ) \)   (For isotropic material only)
\(\mathbf {\tilde {T}}\)	\(\mathbf {\dot {V}}\) (for isotropic only)	\(\mathbf {\tilde {V}}\)   (For isotropic material only)