CDF gamma

Module: solid

Category: materialprop

Type string: "CDF gamma"

Parameters

Name	Description	Default	Units
Dmax	Dmax	1	[]
alpha	alpha	2	[]
mu	mu	4	[]

Description

The material type for a gamma c.d.f. is CDF gamma.

For this material the c.d.f. is given by

\[ F\left(\Xi\right)=\frac{\gamma\left(\alpha,\Xi/\mu\right)}{\Gamma\left(\alpha,0\right)}\,, \]

where \(\gamma\left(\alpha,z\right)\) and \(\Gamma\left(\alpha,z\right)\) are the lower and upper incomplete gamma functions, respectively. The normalization by \(\Gamma\left(\alpha,0\right)\) ensures that this function is bounded by \(0\le F\left(\Xi\right)\le1\). The figure below uses \(\mu=0.15\).

Figure 1. Illustration of the CDF gamma cumulative distribution function.

Example:

<damage type="CDF gamma">
  <alpha>3</alpha>
  <mu>0.15</mu>
</damage>