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### PROP_DAMAGE_IMP

###### Material properties
*PROP_DAMAGE_IMP
"Optional title"
did, erode, noic, $\alpha_{irr}$, $\beta_{irr}$
$W_c$, $n$

#### 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
$n$ Damage growth exponent

#### Description

IMPETUS failure criterion is similar to the Cockcroft-Latham failure criterion. However, it has been equipped with one extra parameter $n$ that allows for an anisotropic damage growth. The model is based on the assumption that defects deform with the material. It is further assumed that compressed defects exposed to tensile loading are more harmful than elongated defects. The damage growth is assumed proportional to the maximum eigenvalue $\hat\sigma_1$ of a distorted stress tensor $\hat{\mathbf\sigma}$.

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

The distorted stress tensor $\hat{\mathbf\sigma}$ is formed as:

$\hat{\mathbf\sigma} = \mathbf{A \sigma A}$

where $\mathbf{\sigma}$ is the current stress tensor and $\mathbf{A}$ is a symmetric tensor describing the defect compression. $\mathbf{A}$ is a function of the principal stretches $(\lambda_1, \lambda_2, \lambda_3)$ and of their corresponding eigenvectors $(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)$.

$\mathbf{A} = \displaystyle{\sum_{i=1}^3} \left( \frac{\lambda_1}{\lambda_i} \right)^n \mathbf{v}_i \otimes \mathbf{v}_i$

Note that $\lambda_1$ is the maximum principal stretch. The formulation ensures that the damage growth is equivialent to Cockcroft-Latham in proportional loading where $\lambda_1$ coincides with $\sigma_1$.

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).