von-Mises-3d¶
Module: solid
Category: materialprop
Type string: "von-Mises-3d"
Parameters¶
| Name | Description | Default | Units |
|---|---|---|---|
b |
concentration | 0 | [] |
Description¶
The fiber density distribution type von-Mises-3d models a transversely isotropic 3D distribution 1. It corresponds to
\[
R\left(\mathbf{n}\right)=\frac{1}{\pi}\sqrt{\frac{b}{2\pi}}\frac{\exp\left(2bn_{1}^{2}\right)}{\mbox{erfi}\left(\sqrt{2b}\right)}\,,
\]
where \(\left(n_{1},n_{2},n_{3}\right)\) are the components of \(\mathbf{n}\) in the local Cartesian basis \(\left\{ \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\right\}\) of each element of the global basis by default; and b is the concentration parameter (\(b>0\)).
Figure 1. Illustration of the von-Mises-3d distribution function.
Example:
<distribution type="von-Mises-3d">
<b>0.5</b>
</distribution>
-
Gasser, T Christian, Ogden, Ray W, and Holzapfel, Gerhard A, "Hyperelastic modelling of arterial layers with distributed collagen fibre orientations", J R Soc Interface 3, 6 (2006), pp. 15-35. ↩