# Appendix

## Basic Relations

$E,\nu $ | $E,G$ | $E,B$ | $G,\nu $ | $G,B$ | $B,\nu $ | $\lambda ,\mu $ | |
---|---|---|---|---|---|---|---|

$E$ | $E$ | $E$ | $E$ | $2\left(1+\nu \right)G$ | $\frac{9BG}{3B+G}$ | $3\left(1-2\nu \right)B$ | $\frac{\left(3\lambda +2\mu \right)\mu}{\lambda +\mu}$ |

$G=\mu $ | $\frac{E}{2\left(1+\nu \right)}$ | $G$ | $\frac{3EB}{9B-E}$ | $G$ | $G$ | $\frac{3\left(1-2\nu \right)B}{2\left(1+\nu \right)}$ | $\mu $ |

$B=K$ | $\frac{E}{3\left(1-2\nu \right)}$ | $\frac{EG}{9G-3E}$ | $B$ | $\frac{2\left(1+\nu \right)G}{3\left(1-2\nu \right)}$ | $B$ | $B$ | $\frac{3\lambda +2\mu}{3}$ |

$\nu $ | $\nu $ | $\frac{E-2G}{2G}$ | $\frac{3B-E}{6B}$ | $\nu $ | $\frac{3B-2G}{6B+2G}$ | $\nu $ | $\frac{\lambda}{2\left(\lambda +\mu \right)}$ |

${D}_{11}$ | $\frac{E\left(1-\nu \right)}{\left(1+\nu \right)\left(1-2\nu \right)}$ | $\frac{\left(4G-E\right)G}{3G-E}$ | $\frac{3B\left(3B+E\right)}{9B-E}$ | $\frac{2G\left(1-\nu \right)}{1-2\nu}$ | $\frac{3B+4G}{3}$ | $\frac{3B\left(1-\nu \right)}{1+\nu}$ | $\lambda +2\mu $ |

${D}_{12}=\lambda $ | $\frac{E\nu}{\left(1+\nu \right)\left(1-2\nu \right)}$ | $\frac{\left(E-2G\right)G}{3G-E}$ | $\frac{3B\left(3B-E\right)}{9B-E}$ | $\frac{2G\nu}{1-2\nu}$ | $\frac{3B-2G}{3}$ | $\frac{3B\nu}{1+\nu}$ | $\lambda $ |

${C}_{11}$ | $\frac{E}{1+{\nu}^{2}}$ | $\frac{4GG}{4G-E}$ | $\frac{36BE}{36-{\left(3-\raisebox{1ex}{$E$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)}^{2}}$ | $\frac{2G}{1-\nu}$ | $\frac{4G\left(3B+G\right)}{3B+4G}$ | $\frac{3B\left(1-2\nu \right)}{1-{\nu}^{2}}$ | |

${C}_{12}$ | $\frac{E\nu}{1+{\nu}^{2}}$ | $\frac{\left(E-2G\right)2G}{4G-E}$ | $\frac{6E\left(3-\raisebox{1ex}{$E$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)}{36-{\left(3-\raisebox{1ex}{$E$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\right)}^{2}}$ | $\frac{2G\nu}{1-\nu}$ | $\frac{2G\left(3B-2G\right)}{3B+4G}$ | $\frac{3B\left(1-2\nu \right)}{1-{\nu}^{2}}$ |

### Hook Law 3D (principal stress and strain)

$\sigma =D\epsilon $

${\sigma}_{1}={D}_{11}{\epsilon}_{1}+{D}_{12}{\epsilon}_{2}+{D}_{13}{\epsilon}_{3}$

${\sigma}_{1}=\left(\lambda +2\mu \right){\epsilon}_{1}+\lambda \left({\epsilon}_{2}+{\epsilon}_{3}\right)$

${\sigma}_{1}=\lambda \left({\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}\right)+2\mu {\epsilon}_{1}$

${\sigma}_{1}=K{\epsilon}_{kk}+2\mu {e}_{1}$ with ${\epsilon}_{kk}={\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}$ and ${e}_{1}={\epsilon}_{1}-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\left({\epsilon}_{1}+{\epsilon}_{2}+{\epsilon}_{3}\right)$

### Hook Law 2D (plane stress)

$\sigma =C\epsilon $

${\sigma}_{1}={C}_{11}{\epsilon}_{1}+{C}_{12}{\epsilon}_{2}$

## Unit Systems

Length | Time | Mass | Force | Pressure | Velocity | $\rho $ | Energy | G |
---|---|---|---|---|---|---|---|---|

m | s | Kg | Kg m/s2 | N/m2 | m/s | Kg/m3 | Kmg2/s2 | 9.81 |

m | s | Kg | N | Pa | m/s | m Kg/l | J | 9.81 |

m | s | g | mN | mPa | m/s |
$\mu $
Kg/l |
mJ | 9.81 |

m | s | Mg (ton) | KN | KPa | m/s | Kg/l | KJ | 9.81 |

m | ms | Kg | MN | MPa | Km/s | m Kg/l | MJ | 9.81e-6 |

m | ms | g | KN | KPa | Km/s |
$\mu $
Kg/l |
KJ | 9.81e-6 |

m | ms | Mg (ton) |
GN | GPa | Km/s | Kg/l | GJ | 9.81e-6 |

mm | s | Kg | mN | KPa | mm/s | M Kg/l | mJ | 9.81e+3 |

mm | s | g | mN | Pa | mm/s | K Kg/l | nJ | 9.81e+3 |

mm | s | Mg (ton) |
N | MPa | mm/s | G Kg/l | mJ | 9.81e+3 |

mm | ms | Kg | KN | GPa | m/s | M Kg/l | J | 9.81e-3 |

mm | ms | g | N | MPa | m/s | K Kg/l | mJ | 9.81e-3 |

mm | ms | Mg (ton) |
MN | TPa | m/s | G Kg/l | KJ | 9.81e-3 |

cm | ms | g | daN | 10^{5}Pabar |
dam/s | Kg/l | dJ | 9.81e-4 |

cm | ms | Kg | 10^{4}
N(KdaN) |
10^{8}Pa(Kbar) |
dam/s | K Kg/l | hJ | 9.81e-4 |

cm | ms | Mg (ton) |
10 (MdaN) |
10 (Mbar) |
dam/s | M Kg/l | 10^{5} J |
9.81e-10 |

cm | ${\mu}_{s}$ | g | 10^{7}
N(MdaN) |
10^{11}
Pa(Mbar) |
10^{4} m/s |
Kg/l | 10^{5} J |
9.81e-10 |

## Filtering

Often it is useful to filter results in a material or failure law to remove numerical noise. The most common filter is an exponential moving average filter. This is especially important for material models that include strain rate effects.

In most materials, the flag `F`_{smooth} = 1 must be defined to enable the filtering and the cutoff
frequency entered using `F`_{cut}. For the case of filtering strain rates, use:

- ${\dot{\epsilon}}_{filtered}(t)$
- Filtered strain rate.
- $\dot{\epsilon}(t)$
- Strain rate at the current timestep before filtering.
- $\alpha $
- Degree of weighting decrease, a constant smoothing factor between 0 and 1. A higher value discounts previous values faster which results in less filtering.
- $\text{}dt$
- Timestep of the simulation.
- ${\dot{\epsilon}}_{filtered}\left(t-dt\right)$
- Filtered strain rate at the previous time step.

For materials laws where `F`_{cut} can be entered.

Where, `F`_{cut} is the cutoff frequency.

Thus,

The cutoff frequency is a function of the model timestep. Experience shows that the speed of the deformation is important also. For slower speeds, like a car crash, 1 – 10 kHz (1000 – 10,000 Hz) is a good value, but for high-speed events, like ballistic, less filtering should be used - so 1 – 10 GHz is appropriate. Good engineering judgment should be used to determine a reasonable value for each simulation. Refer to RD-E: 1102 Strain Rate Effect for an example of strain rate filtering.