CDF log-normal

Module: solid

Category: materialprop

Type string: "CDF log-normal"

Parameters

Name	Description	Default	Units
Dmax	Dmax	1	[]
mu	mu	1	[]
sigma	sigma	1	[]

Description

The material type for a log-normal c.d.f. is CDF log-normal.

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

\[ F\left(\Xi\right)=\frac{1}{2}\text{erfc}\left[-\frac{\ln\left(\Xi/\mu\right)}{\sigma\sqrt{2}}\right]\,. \]

Note that

\[ F\left(\mu\right)=1/2, \]

which shows that \(\mu\) is the value of \(\Xi\) at which half of the bonds break. Here, \(\sigma\) regulates the rate at which damage increases with increasing \(\Xi\), with smaller values of \(\sigma\) producing a more rapid increase.

Figure 1. Illustration of the CDF log-normal cumulative distribution function.

Example:

<damage type="CDF log-normal">
  <mu>1</mu>
  <sigma>0.5</sigma>
  <Dmax>1</Dmax>
</damage>