next up previous contents
Next: Plasticity Up: Material deformation and flow Previous: Memory   Contents

Elasticity

The elastic stress rate is


\begin{displaymath}
C_{ijkl} \dot{{\epsilon_{kl}}}^{{\rm elas}}
\end{displaymath}

where Cijkl is the elastic modulus tensor (which is a doubly symmetric tensor: Cijkl=Cjikl, Cijkl=Cijlk and Cijkl=Cjilk), and $\dot{{\epsilon_{kl}}}^{{\rm elas}}$ is the elastic strain rate. See the plasticity section for a definition of the elastic strain rate.

For an isotropic material


\begin{displaymath}
C_{0000} = C_{1111} = C_{2222} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)}
\end{displaymath}


\begin{displaymath}
C_{0011} = C_{0022} = C_{1122} = \frac{E\nu}{(1+\nu)(1-2\nu}
\end{displaymath}


\begin{displaymath}
C_{0101} = C_{0202} = C_{1212} = \frac{E}{1+\nu}
\end{displaymath}

with E group_materi_elasti_young modulus and $\nu$ group_materi_elasti_poisson ratio (the remaining non-zero moduli follow from the double symmetry conditions).

For a transverse isotropic material the material has one unique direction (think of an material with fibers in one direction). Here we take 'a' as the unique direction; 'b' and 'c' are the transverse directions. The material is fully defined by Caaaa, Cbbbb, Caabb, Cabab and Cbcbc and the unique direction in space (see group_materi_elasti_transverse_isotropy). The other non-zero moduli follow from Ccccc=Cbbbb, Cacac=Cabab, Cbbcc=Cbbbb-2Cbcbc and from the double symmetry conditions.



root
1998-11-16