Material properties
"Optional title"
did, erode, noic
$W_c$, $G_I$, $\sigma_s$, $t_s$, $\alpha_s$, $\beta_s$
Parameter definition
did Unique damage identification number
erode Element erosion flag
0 $\rightarrow$ failed element is not eroded
1 $\rightarrow$ failed element is eroded
2 $\rightarrow$ node splitting at failure (crack plane orthogonal to max principal strain)
3 $\rightarrow$ node splitting at failure (crack plane orthogonal max principal stress)
noic Flag to turn off cracking along interface between different materials
0 $\rightarrow$ material interface cracks are allowed
1 $\rightarrow$ material interface cracks are not allowed
$W_c$ Damage parameter
options: constant
$G_I$ Fracture energy parameter (only used with node splitting)
default: not used
$\sigma_s$ Spall strength (threshold stress)
default: not used
$t_s$ Time to develop spall fracture at threshold stress
default: not used
$\alpha_s$ Exponent controlling time to develop spall fracture
default: not used
$\beta_s$ Parameter controlling the pressure dependency
default: not used

This is the Cockcroft-Latham failure criterion. It has been complemented with a tensile fracture/spalling criterion. Ductile damage is defined as:

$\displaystyle{D_d = \frac{1}{W_c} \int_0^{\varepsilon_{eff}^p} \mathrm{max}(0,\sigma_1) \mathrm{d}\varepsilon_{eff}^p}$

where $\sigma_1$ is the maximum principal stress. The complementing tensile damage is defined as:

$\displaystyle{ D_t = \frac{1}{t_s} \int_0^{t} (\bar{\sigma}_1 / \sigma_s )^{\alpha_s} \mathrm{d}t}$

$\bar{\sigma}_1$ is defined as:

$\bar{\sigma}_1 = \sigma_1^{dev} - (1-\beta_s) \cdot p$

where $\sigma_1^{dev}$ is the maximum deviatoric principal stress and $p$ is the pressure. With $\beta_s = 0$, $\bar{\sigma}_1$ equals the maximum principal stress. Note that the tensile fracture/spalling term only contributes to the damage growth if $\bar{\sigma}_1 \geq \sigma_s$. The material is assumed to fail when one of the damage parameters reaches 1.

Defining damage parameter

The following input defines a damage parameter that depends on the distance from the material surface.

Wc0 = 200.0e6 # on the surface
Wc1 = 400.0e6 # in the interior
dist_ref = 0.001
%Wc0 + (%Wc1 - %Wc0)*(1 - exp(-dist_surf/%dist_ref))