 |
|
The implementation can be found in Marmot::Materials::LinearElastic
The constitutive law is given in total form as
\[ \displaystyle \boldsymbol{ \sigma } = \mathbb{ C } : \boldsymbol{ \varepsilon }, \]
relating the nominal stress tensor \( \boldsymbol{ \sigma } \) to the linearized strain tensor \( \boldsymbol{ \varepsilon } \) with the fourth order stiffness tensor \( \mathbb{ C } \). The latter is implemented in MarmotElasticity.h and can be specified for isotropic, transversely isotropic or orthotropic material behavior as follows:
Number of independent material parameters: 2
\[ \displaystyle \mathbb{ C }^{-1} = \begin{bmatrix} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G} \end{bmatrix} \]
with
\[ \displaystyle G = \frac{E}{2\,(1 + \nu)} \]
In case of transversely isotropic behavior, the user defined normal vector specifies the \( x_1 \) - axis of a local coordinate system, representing the principal material directions in which the material stiffness tensor is formulated. The isotropic plane is implementend with respect to the local \( x_2 \) and \( x_3 \) axes.
Number of independent material parameters: 5
\[ \displaystyle \mathbb{ C }^{-1} = \begin{bmatrix} \frac{1}{E_1} & \frac{-\nu_{12}}{E_2} & \frac{-\nu_{12}}{E_2} & 0 & 0 & 0 \\ \frac{-\nu_{12}}{E_2} & \frac{1}{E_2} & \frac{-\nu_{23}}{E_2} & 0 & 0 & 0 \\ \frac{-\nu_{12}}{E_2} & \frac{-\nu_{12}}{E_2} & \frac{1}{E_2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{12}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{23}} \end{bmatrix} \]
with
\[ \displaystyle G_{23} = \frac{E_2}{2\,(1 + \nu_{23})} \]
In case of orthotropic behavior, the user defined normal vector defines the \( x_1 \) - axis of a local coordinate system, representing the principal material directions in which the material stiffness tensor is formulated
Number of independent material parameters: 9
\[ \displaystyle \mathbb{ C }^{-1} = \begin{bmatrix} \frac{1}{E_1} & \frac{-\nu_{12}}{E_2} & \frac{-\nu_{13}}{E_3} & 0 & 0 & 0 \\ \frac{-\nu_{12}}{E_2} & \frac{1}{E_2} & \frac{-\nu_{23}}{E_3} & 0 & 0 & 0 \\ \frac{-\nu_{13}}{E_3} & \frac{-\nu_{23}}{E_3} & \frac{1}{E_3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G_{12}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G_{13}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G_{23}} \end{bmatrix} \]