elliptical

Module: solid

Category: materialprop

Type string: "elliptical"

Parameters

Name	Description	Default	Units
spa1	spa1	1	[]
spa2	spa2	1	[]

Description

The fiber density distribution type elliptical models an orthotropic 2D distribution. This distribution corresponds to

\[ R\left(\mathbf{n}\right)=C^{-1}\left[\left(\frac{n_{1}}{a}\right)^{2}+\left(\frac{n_{2}}{b}\right)^{2}\right]^{-1/2} \]

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 or the global basis by default, implying that the elliptical distribution lies in the \(\left\{ \mathbf{e}_{1},\mathbf{e}_{2}\right\}\) plane; and \(\left(a,b\right)\) are the semi-principal axes of the ellipse. Here, \(C=4bK\left(e\right)\) where \(K\) is the complete elliptic integral of the first kind and

\[ e=\sqrt{1-\frac{b^{2}}{a^{2}}} \]

is the ellipse eccentricity.

Figure 1. Illustration of the elliptical fiber distribution function.

Example:

<distribution type="elliptical">
  <spa1>8</spa1>
  <spa2>1</spa2>
</distribution>