# 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.

`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.

`F`

_{cut}can be entered.

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

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.