CDF quintic

Module: solid

Category: materialprop

Type string: "CDF quintic"

Parameters

Name	Description	Default	Units
Dmax	Dmax	1	[]
mumin	mumin	0	[]
mumax	mumax	1	[]

Description

The material type for a quintic polynomial c.d.f. is CDF quintic.

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

\[ F\left(\Xi\right)=\begin{cases} 0 & \Xi\leqslant\mu_{\min}\\ x^{3}\left(10-15x+6x^{2}\right) & \mu_{\min}<\Xi\leqslant\mu_{\max}\\ 1 & \mu_{\max}<\Xi \end{cases}\,,\quad x=\frac{\Xi-\mu_{\min}}{\mu_{\max}-\mu_{\min}}\,. \]

Note that

\[ F\left(\frac{1}{2}\left(\mu_{\min}+\mu_{\max}\right)\right)=\frac{1}{2}\,, \]

which shows that \(\left(\mu_{\min}+\mu_{\max}\right)/2\) is the value of \(\Xi\) at which half of the bonds break. The range \(\mu_{\max}-\mu_{\min}\) regulates the rate at which damage increases with increasing \(\Xi\), with a narrower range producing a more rapid increase.

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

Example:

<damage type="CDF quintic">
  <mumin>0.3</mumin>
  <mumax>1.7</mumax>
  <Dmax>1</Dmax>
</damage>