**process weibull

Description

This post computation provides the means for doing a Weibull analysis on a structure. Two modes of operation are available:

• eigenstress

  \[\sigma'= \left( \frac{\sigma_1^M-\sigma_0}{\sigma_u} \right)^m\]

• independent

  \[\sigma'= \left( \frac{\sigma_1^M-\sigma_0}{\sigma_u} \right)^m + \left( \frac{\sigma_2^M-\sigma_0}{\sigma_u} \right)^m + \left( \frac{\sigma_3^M-\sigma_0}{\sigma_u} \right)^m\]

The Weibull stress is then defined as:

\[\sigma_W =\left( \frac{\sigma_u}{V_0}\int_{V}^{}{\sigma'{}^m} \,dV\right){}^{1/m}\]

where \(\sigma_1^M\), \(\sigma_2^M\) and \(\sigma_3^M\) are the post-computation results from a local fmax applied successively to the three principal stresses \(\sigma_1\), \(\sigma_2\) and \(\sigma_3\), in descending order. These sub-posts are run automatically by the weibull processor.

\(V_0\), \(\sigma_u\), \(\sigma_0\) and \(m\) are material parameters.

The probability of failure \(P_r\) is is given by:

\[P_r = 1 -\exp\left(-\left(\frac{\sigma_W}{\sigma_u}\right)^m\right)\]

In addition to the output of \(\sigma_W\) and \(P_r\). The values of \(\sigma'\) are stored at each Gauss point under a name constructed by adding _wb to the variable name.

Syntax

**process weibull \(~\,\) *var name \(~\,\) *mode eigenstress | independant \(~\,\) *coefmin value1 \(~\,\) *coefmax value2 \(~\,\) *file namef

*var name

name of stress variable.

If the user has only specified a single map (with output_number), the history is reconstructed from the values read after the options *coefmin and *coefmax. \(\sigma_1^M\) varies linearly over 100 maps of value1 * \(\sigma_1\) to value2* \(\sigma_1\) (where \(\sigma_1\) is calculated at the specified map).

With the option *file, the history is read in the file namef, of the form:

0.01     10
0.02     15
0.03     20
0.04     40
0.05     60
...

where the first column represents the time and the second the value of \(\sigma_1\).

These two methods presented for the mode eigenstress extend to the mode independent as well.

Example

****post_processing
 ***global_post_processing
  **output_number 10
  **process weibull
   *var sig
   *mode independent
   *coefmin 0.5
   *coefmax 2.5
****return

   % material file
  ***post_processing_data
   **process weibull
     V0      10.
     m       20.
     sigma_u 1200.
     sigma_0 0.
  ***return