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Hyper elasticity

Hyper elasticity is used to model rubbers. The stresses follow from a strain function (with $C_{ij}$ components of the matrix $C$, and where $F$ is the deformation tensor and $U$ is the stretch tensor following from the polar decomposition of the deformation tensor)


\begin{displaymath}
2 \frac{\partial W}{\partial {C_{ij}}}
\end{displaymath}


\begin{displaymath}
C = F^T F = U^T U
\end{displaymath}

Strictly speaking $W$ is not a strain energy function, because the Cauchy stresses $\sigma_{ij}$ are not conjugate to the strain matrix $C_{ij}$; the approach obeys the restriction of objectivity however. The stress rates follow from the time derivative of this law. Typically, this law is chosen such that it gives only a deviatoric stress contribution. The hydrostatic stress is obtained by including group_materi_elasti_compressibility. To obtain a purely deviatoric function, the following strain measures are used (with $I_1$, $I_2$ and $I_3$ the first, second and third invariant of the elastic strain matrix respectively)


\begin{displaymath}
J_1 = I_1 {I_3}^{\frac{-1}{3}} \; \; \; \;
J_2 = I_2 {I_3}^{\frac{-2}{3}}
\end{displaymath}

The group_materi_hyper_besseling function reads ( with $K_1$, $K_2$ and $\alpha$ user defined constants)


\begin{displaymath}
W = K_1 ( J_1 - 3 ) ^ \alpha + K_2 ( J_2 - 3 )
\end{displaymath}

The group_materi_hyper_mooney_rivlin function reads (with $K_1$ and $K_2$ user defined constants)


\begin{displaymath}
W = K_1 ( J_1 - 3 ) + K_2 ( J_2 - 3 )
\end{displaymath}


next up previous contents
Next: Viscoelasticity Up: Material deformation and flow Previous: Thermal stresses   Contents
root
1999-04-23