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### PROP_DAMAGE_CL

###### Material properties
*PROP_DAMAGE_CL
"Optional title"
did, erode, noic, $\alpha_{irr}$, $\beta_{irr}$
$W_c$, $G_I$, $\sigma_s$, $t_s$, $\alpha_s$, $\beta_s$

#### Parameter definition

VariableDescription
did Unique damage identification number
erode Element erosion flag
options:
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
options:
0 $\rightarrow$ material interface cracks are allowed
1 $\rightarrow$ material interface cracks are not allowed
$\alpha_{irr}$ Irregularization factor
default: no irregularization
$\beta_{irr}$ Irregularization cap
default: $\beta_{irr}=1$
$W_c$ Damage parameter. If referring to a function (see FUNCTION_STATIC) the damage parameter can be defined to depend on the location
options: constant, fcn
$G_I$ 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

#### Description

This is the Cockcroft-Latham failure criterion. It has been complemented with a tensile fracture/spalling criterion. The material will lose its shear strength once the damage parameter, $D$, has evolved from 0 to 1. The damage is defined as:

$D = \displaystyle{\frac{1}{W_c} \int_0^{\epsilon_{eff}^p}} \mathrm{max}(0,\sigma_1) \mathrm{d}\epsilon_{eff}^p + \frac{1}{t_s} \displaystyle{\int_0^{t} (\bar{\sigma}_1 / \sigma_s )^{\alpha_s} } \mathrm{d}t$

where $\sigma_1$ is the maximum principal stress. $\bar{\sigma}_1$ 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 optional irregularization parameters $(\alpha_{irr}, \beta_{irr})$ are used to amplify the damage growth in regions where the Finite Element mesh is too coarse to accurately resolve the local variations of the strain field. The purpose is to significantly reduce the mesh dependency. Note that this irregularization procedure currently only is implemented for 64-node cubic hexahedra. It has no effect on other element types. The amplified rate of damage growth $\dot D_{amp}$ is defined as:

$\dot D_{amp} = \displaystyle{ (1 + \mathrm{min} (\alpha_{irr} \cdot \frac{\|\mathbf{\varepsilon}_a - \mathbf{\varepsilon}_b \|} {\|\mathbf{\varepsilon}_a + \mathbf{\varepsilon}_b \|}, \beta_{irr})) \cdot \dot D}$

where $(\mathbf{\varepsilon}_a, \mathbf{\varepsilon}_b)$ are the average strain tensors at the eight element center IP's and eight corner IP's, respectively. Hence, $\|\mathbf{\varepsilon}_a - \mathbf{\varepsilon}_b \|$ is a measure of the curvature of the strain field (in parametric space).

#### Example

Defining damage parameter

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

*PARAMETER
Wc0 = 200.0e6 # on the surface
Wc1 = 400.0e6 # in the interior
dist_ref = 0.001
*PROP_DAMAGE_CL
1
[fcn(23)]
*FUNCTION_STATIC
23
%Wc0 + (%Wc1 - %Wc0)*(1 - exp(-dist_surf/%dist_ref))