Crack propagation

In CalculiX a rather simple model to calculate cyclic crack propagation is implemented. In order to perform a crack propagation calculation the following procedure is to be followed:

In CalculiX, the following crack propagation law has been implemented:

$\displaystyle \frac{da}{dN} = \left ( \frac{da}{dN} \right ) _{ref} \left ( \frac{\Delta K}{\Delta K_{ref}} \right )^m \frac{f_{th}f_R}{f_c},$ (578)


$\displaystyle f_{th}$ $\displaystyle = 1-\exp \left[ \epsilon (1 - \frac{\Delta K}{\Delta K_{th}} ) \right], \;\;\; \Delta K > \Delta K_{th}$    
$\displaystyle f_{th}$ $\displaystyle = 0, \;\;\; \Delta K \le \Delta K_{th}$ (579)

accounts for the threshold range,

$\displaystyle f_c$ $\displaystyle =1-\exp \left[ \delta \left( \frac{K_{max}}{K_c} -1 \right) \right], \;\;\; K_{max} < K_c$    
$\displaystyle f_c$ $\displaystyle =0 \;\;\; K_{max} \ge K_c$ (580)

for the critical cut-off and

$\displaystyle f_R = \left[ \frac{1}{(1-R)^{1-w}} \right]^m$ (581)

for the $ R:=K_{min}/K_{max}$ influence. The material constants have to be entered by using a *USER MATERIAL card with the following 8 constants per temperature data point (in that order): $ \left ( \frac{da}{dN} \right ) _{ref} [L/cycle]$, $ \Delta
K_{ref}[F/L^{3/2}] $, $ m [-]$, $ \epsilon [-]$, $ \Delta K_{th} [F/L^{3/2}]$, $ \delta [-]$, $ K_c
[F/L^{3/2}]$ and $ w$[-], were $ [F]$ is the unit of force and $ [L]$ of length. Notice that the first part of the law corresponds to the Paris law. Indeed the classical Paris constant C can be obtained from:

$\displaystyle \left ( \frac{da}{dN} \right ) _{ref} \left ( \frac{1}{\Delta K_{ref}} \right )^m = C.$ (582)

Vice versa, $ \Delta K_{ref}$ can be obtained from $ C$ using the above equation once $ (da/dN)_{ref}$ has been chosen. Notice that $ (da/dN)_{ref}$ is the rate for which $ \Delta K=\Delta K_{ref}$ (just considering the Paris range). For a user material, a maximum of 8 constants can be defined per line (cf. *USER MATERIAL). Therefore, after entering the 8 crack propagation constants, the corresponding temperature has to be entered on a new line.

The crack propagation calculation consists of a number of increments during which the crack propagates a certain amount. For each increment in a LCF calculation the following steps are performed:

For a combined LCF-HCF calculation, triggered by the *HCF keyword in the *CRACK PROPAGATION procedure the picture is slightly more complicated. On the *HCF card the user defines a scaling factor and a step from the static calculation on which the HCF loading is to be applied. This is usually the static loading at which the modal excitation occurs. At this step a HCF cycle is considered consisting of the LCF+HCF and the LCF-HCF loading. The effect is as follows:

Right now, the output of a *CRACK PROPAGATION step cannot be influenced by the user. By default a data set is created in the frd-file consisting of the following information (most of this information can be changed in user subroutine crackrate.f):