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\begin{document}
\textbf{\centering \huge{St. Venant Kirchhoff UMAT}}\\\\
This routine assumes a constant stiffness in the material configuration
\begin{equation}
\Psi=\frac{1}{2}\undertilde{E}:\undertilde{\undertilde{C}}:\undertilde{E}
\end{equation}
with the stiffness tensor
\begin{equation}
C_{IJKL}=\lambda\delta_{ij}\delta_{kl}+\mu\left[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right]
\end{equation}
and the Green Lagrange strain
\begin{equation}
	\undertilde{E}=\frac{1}{2}[\undertilde{F}^\text{T}\undertilde{F}-\undertilde{I}]
\end{equation}
The 2. Piola Kirchhoff stress then reads
\begin{equation}\label{PK2}
	\undertilde{S}=\dfrac{\partial\Psi}{\partial\undertilde{E}}=\undertilde{\undertilde{C}}:\undertilde{E}
\end{equation}
It is transformed to the Kirchhoff stress by
\begin{equation}\label{tau_to_PK2}
	\undertilde{\tau}=\undertilde{F}\undertilde{S}\undertilde{F}^\text{T}
\end{equation}
The cauchy stress $\undertilde{\sigma}$ is in relation to the Kirchhoff stress $\undertilde{\tau}$ via
\begin{equation}
	\undertilde{\tau}=\frac{1}{J}\undertilde{\sigma}
\end{equation}
For the UMAT needs the consistent tangent in the spatial configuration a incremental form is adopted
\begin{equation}\label{rate_Green_lagrange}
	\undertilde{\dot{E}}=\frac{1}{2}[\undertilde{F}^\text{T}\undertilde{\dot{F}}+\undertilde{F}^\text{T}\undertilde{\dot{F}}]=\undertilde{F}^\text{T}\undertilde{d}\undertilde{F}
\end{equation}
Thereby $\undertilde{d}$ is
\begin{equation}
	\undertilde{d}=\text{sym}(\undertilde{l}),\ \undertilde{w}=\text{skw}(\undertilde{l}),\ l=\undertilde{\dot{F}}\undertilde{F}^{-1}
\end{equation}
At first equation \ref{PK2} is differentiated after time
\begin{equation}
	\undertilde{\dot{S}}=\undertilde{\undertilde{C}}:\undertilde{\dot{E}}
\end{equation}
Now equation \ref{tau_to_PK2} is differentiated after time
\begin{equation}
	\undertilde{\dot{\tau}}=\undertilde{\dot{F}}\undertilde{S}\undertilde{F}^\text{T}+\undertilde{F}\undertilde{S}\undertilde{\dot{F}}^\text{T}+\undertilde{F}\undertilde{\dot{S}}\undertilde{F}^\text{T}=\undertilde{l}\,\undertilde{\tau}+\undertilde{\tau}\,\undertilde{l}^\text{T}+\undertilde{F}[\undertilde{\undertilde{C}}:\undertilde{\dot{E}}]\undertilde{F}^\text{T}
\end{equation}
The rate of Green Lagrange strain is substitued with \ref{rate_Green_lagrange}
\begin{equation}\label{Lie_deriv}
\undertilde{\dot{\tau}}=\undertilde{l}\,\undertilde{\tau}+\undertilde{\tau}\,\undertilde{l}^\text{T}+\undertilde{F}\undertilde{\undertilde{C}}:[\undertilde{F}^\text{T}\undertilde{d}\undertilde{F}]\undertilde{F}^\text{T}=[\undertilde{d}+\undertilde{w}]\undertilde{\tau}+\undertilde{\tau}[\undertilde{d}-\undertilde{w}]+\undertilde{F}\undertilde{\undertilde{C}}:[\undertilde{F}^\text{T}\undertilde{d}\undertilde{F}]\undertilde{F}^\text{T}
\end{equation}
The Jaumann rate of the Kirchhoff stress reads
\begin{equation}\label{Jaumann}
	\stackrel{\nabla}{\undertilde{\tau}}=\undertilde{\dot{\tau}}+\undertilde{\tau}\undertilde{w}-\undertilde{w}\undertilde{\tau}
\end{equation}
Now equation \eqref{Lie_deriv} is inserted into \eqref{Jaumann} 
\begin{equation}
\stackrel{\nabla}{\undertilde{\tau}}=\undertilde{\tau}\,\undertilde{d}+\undertilde{d}\,\undertilde{\tau}+\undertilde{F}\undertilde{\undertilde{C}}:[\undertilde{F}^\text{T}\undertilde{d}\undertilde{F}]\undertilde{F}^\text{T}
\end{equation}
The for the hypoelastic/update Lagrange scheme required tangent reads
\begin{equation}
	\text{DDSDDE}_{ijkl}=\frac{1}{J}\dfrac{\mathrm{d}[J\sigma_{ij}^{m+1}]}{\mathrm{d}\Delta d_{kl}}=\frac{1}{J}\dfrac{\mathrm{d}\tau_{ij}^{m+1}}{\mathrm{d}\Delta d_{kl}}=\frac{1}{J}\dfrac{\mathrm{d}\Delta\!\stackrel{\nabla}{\tau}_{ij}}{\mathrm{d}\Delta d_{kl}}
\end{equation}
\begin{equation}
	\dfrac{\mathrm{d}\Delta\!\stackrel{\nabla}{\tau}_{ij}}{\mathrm{d}\Delta d_{kl}}=\frac{1}{2}[\tau_{jl}\delta_{ik}+\tau_{il}\delta_{jk}+\tau_{ik}\delta_{jl}+\tau_{jk}\delta_{il}]+C_{IJKL}F_{iI}F_{jJ}F_{kK}F_{lL}
\end{equation}
\end{document}