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\begin{document}
\textbf{\huge{UMAT Neo-Hookean Hyperelasticity}}\\\\
\section{Theory}
This routine is based on the Neo-Hookean UMAT from the Abaqus Manual \cite{Abaqus}.\\\\
The strain energy function is defined as
\begin{equation}
	U=U(I_1,I_2,J)=C_{10}\cdot[I_1-3]+\frac{1}{D_1}\cdot[J-1]^2.
\end{equation}
The appearing invariants are those of the left Cauchy-Green tensor $\undertilde{B}$:
\begin{equation}
	I_1=\text{tr}(\undertilde{B}),\ I_2=\frac{1}{2}[I_1^2-\text{tr}(\undertilde{B}^2)],\ \undertilde{B}=\undertilde{F}\cdot\undertilde{F}^\text{T},\ J=\text{det}(\undertilde{F})
\end{equation}
The cauchy stress is yield with the deviatoric left Cauchy-Green tensor $\undertilde{\bar{B}}=\undertilde{B}/J^{2/3}$ as
\begin{equation}
	\sigma_{ij}=\frac{2}{J}C_{10}\left[\bar{B}_{ij}-\frac{1}{3}\delta_{ij}\bar{B}_{kk}\right]+\frac{2}{D_1}[J-1]\delta_{ij}
\end{equation}
The consistent stiffness is expected as
\begin{equation}
	DDSDDE_{ijkl}=\frac{1}{J}\dfrac{\text{d}(J\sigma_{ij})}{\text{d}(\Delta d_{kl})}
\end{equation}
The actual DDSDDE is the voigt presentation of the Stiffness $C$, which reads
\begin{alignat}{1}
	C_{ijkl}=&\frac{2}{J}C_{10}\left[\frac{1}{2}[\delta_{ik}\bar{B}_{jl}+\delta_{jl}\bar{B}_{ik}+\delta_{il}\bar{B}_{jk}+\delta_{ik}\bar{B}_{il}]-\frac{2}{3}\delta_{ij}\bar{B}_{kl}-\frac{2}{3}\delta_{kl}\bar{B}_{ij}-\frac{2}{9}\delta_{ij}\delta_{kl}\bar{B}_{mm}\right]\\ \notag
	&+\frac{2}{D_1}[2J-1]\delta_{ij}\delta_{kl}
\end{alignat}
\section{Usage}
The UMAT can be used within Abaqus directly, uel-large-deformation and MonolithFE\textsuperscript{2}. In the latter two it needs to be compiled into with setting the MATERIAL variable to 'NeoHooke'.\\\\
This UMAT needs no internal state variables and expects exactly two material parameters: $E$ (Young's Modulus) and $\nu$ (Poissons ratio).
%%=========================================================================
\begin{thebibliography}{[10]}
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	\bibitem{Abaqus}
	ABAQUS/Standard User's Manual, Version 6.9, Michael Smith, Dassault Syst{\`e}mes Simulia Corp, 2009 
\end{thebibliography}

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