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This is mesh dependent version of the Johnson-Cook failure criterion. The damage evolution is scaled (accelerated) if the mesh is considered too coarse for an accurate description of the local post necking behaviour.

The material will lose its shear strength pressure once the damage parameter, $D$, has evolved from 0 to 1. The damage growth rate is defined as:

$\displaystyle{\dot D = sf \cdot \frac{\dot\varepsilon_{eff}^p}{\mathrm{max}(\varepsilon_{min}, \varepsilon_f)}}$

where:

$\displaystyle{ \varepsilon_f = (d_1 + d_2 \cdot \mathrm{e}^{\frac{\vert d_3 \vert \, p}{\sigma_{eff}}}) \cdot (1 + d_4 \cdot \mathrm{ln}(\frac{\dot\varepsilon_{eff}^p}{\dot\varepsilon_0})) \cdot (1 + d_5 \cdot (\frac{\mathrm{T}-\mathrm{T}_0}{\mathrm{T}_m - \mathrm{T}_0}))}$

and:

$p = -(\sigma_{xx} + \sigma_{yy} + \sigma_{zz})/3$

The scale factor $sf$ accounts for the mesh dependency. $sf>1$ if the pressure $p \lt 0$, if damage $D \gt D_0$ and if the element size to component wall thickness ratio $R \gt R_0$.

$\displaystyle{ sf = \left\{ \begin{array}{ccc} 1 & : & D \leq D_0 \; \mathrm{or} \; R \leq R_0 \\ 1 + g(p/\sigma_{eff}) \left[ \left( \frac{R}{R_0} \right)^c - 1 \right] & : & D > D_0 \; \mathrm{and} \; R > R_0 \end{array} \right. }$

where:

$\displaystyle{ g(p/\sigma_{eff}) = \left\{ \begin{array}{ccc} 1 & : & p/\sigma_{eff} \leq -1/3 \\ -3 p/\sigma_{eff} & : & -1/3 \lt p/\sigma_{eff} \lt 0 \\ 0 & : & p/\sigma_{eff} \geq 0 \\ \end{array} \right. }$

The element size definition depends on element type. Hexahedra elements have three size values, one for each parametric direction. Sizes and directions are stored in a tensor $\mathbf Q$. The element size to wall thickness ratio $R$ is defined by projecting $\mathbf Q$ in the direction of the maximum principal stress $\boldsymbol{\lambda}_1$.

$\displaystyle{ R = \frac{\vert\vert \mathbf{Q} \cdot \boldsymbol{\lambda}_1 \vert\vert}{t_c} }$

Here $t_c$ is the local component wall thickness. It is computed automatically by the solver. Tetrahedra elements are assigned a scalar value for the element size. This scalar value is the average element height in the four different parametric directions.

Model parameters can be defined from a simple tensile specimen test. $W_c$ is a ductility parameter for an element grid that is so fine that the results have converged. $R_0$ is the ratio below which the results have converged. $D_0$ is typically the damage at onset of diffuse necking. That is, the damage level in uni-axial tension where $\sigma(\varepsilon_{eff}^p) = \mathrm{d}\sigma / \mathrm{d}\varepsilon_{eff}^p$. $c$ is a dimensionless tuning parameter. Typical values for both $R_0$ and $c$ are $\approx 0.5$.