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### PROP_DAMAGE_HC

###### Material properties
*PROP_DAMAGE_HC
"Optional title"
did, erode, noic
$a$, $b$, $c$, $n$, $T_a$, $T_b$, $\sigma_s$, $t_s$
$\alpha_s$, $c_r$, $\dot\varepsilon_r$

#### 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
$a$, $b$, $c$, $n$ Hosford-Coulomb damage parameters
$T_a$ Threshold temperature for thermal sensitivity
default: no thermal sensitivity
$T_b$ Thermal sensitivity parameter
$\sigma_s$ Threshold spall stress
$t_s$ Time parameter for the development of fracture
$\alpha_s$ Spall fracture stress sensitivity parameter
$c_r$ Rate sensitivity exponent
default: no rate sensitivity
$\dot\varepsilon_r$ Reference strain rate

#### Description

This is a slightly modified version of Hosford-Coulomb ductile failure criterion. The original model has been extended with two parameters $(T_a, T_b)$, allowing for the definition of a temperature dependent ductility.

The material will fail once the damage parameter, $D$, has evolved from 0 to 1. The damage growth rate is defined as:

$\displaystyle{ \dot D = \frac{\dot\epsilon_{eff}^p}{\epsilon_f}}$

where:

$\displaystyle{ \epsilon_f = b \left( \frac{1+c}{g(\eta,\theta)} \right)^{1/n} \cdot \left( 1 + \frac{\dot\varepsilon_{eff}^p}{\dot\varepsilon_r} \right)^{c_r} \cdot \mathrm{max} \left( 1, e^{\frac{T-T_a}{T_b}} \right) }$

$g(\eta,\theta)$ is a function of stress triaxiality $\eta$ and Lode angle parameter $\theta$.

$\displaystyle{g(\eta,\theta) = \left( \frac{1}{2}\vert f_I-f_{II} \vert^a + \frac{1}{2}\vert f_{II}-f_{III} \vert^a + \frac{1}{2}\vert f_{III}-f_I \vert^a \right)^{1/a} + c \left( 2\eta + f_I + f_{III} \right)}$
$\displaystyle{ f_I(\theta) = \frac{2}{3} \mathrm{cos} \left( \frac{\pi}{6}(1-\theta) \right) }$
$\displaystyle{ f_{II}(\theta) = \frac{2}{3} \mathrm{cos} \left( \frac{\pi}{6}(3+\theta) \right) }$
$\displaystyle{ f_{III}(\theta) = -\frac{2}{3} \mathrm{cos} \left( \frac{\pi}{6}(1+\theta) \right) }$

The ductile failure criterion is complemented with an optional spall criterion. Spall fracture occurs when a spall damage parameter $D_s$ evolves from from 0 to 1.

$\dot{D}_s = \left\{ \begin{array}{ccc} \frac{1}{t_s}\left(\frac{\sigma_1}{\sigma_s}-D_s\right) & : & \sigma_1 \leq \sigma_s \\ \frac{1}{t_s} \left(1 + \alpha_s \left(1 - \mathrm{e}^{-\frac{\sigma_1 - \sigma_s}{\alpha_s \sigma_s}} \right) - D_s\right) & : & \sigma_1 \gt \sigma_s \\ \end{array} \right.$

Note that the spall damage parameter $D_s$ is not necessarily monotonically growing. It grows if the first princial stress $\sigma_1$ is large and drops is $\sigma_1$ is small. The main purpose of this unconventional formulation is to filter numerical noise (such as stress oscillations from element erosion in contact interfaces).

The figure below exemplifies how the spall damage parameter grows at different stress levels (for two different $\alpha_s$).