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Material law describing a non viscous fluid.

Stresses are computed with

$$ \sigma_{ij} = s_{ij} + p\delta_{ij} $$

with $ s_{ij} = 0 $ in a non viscous fluid.

The equation which associates pressure and volume is

$$ dp = K \frac{dV}{V} $$

where $K$ is the bulk modulus.

Convergence when using this law may be difficult to achieve since there is no stiffness when shearing the material. If you encounter this kind of issue, you may use **NortonHoffHypoMaterial** discussed below with a low viscosity parameter and NORTON_M = 1 to model a Newtonian fluid.

Name | Metafor Code | Dependency |
---|---|---|

Density | `MASS_DENSITY` | |

Bulk Modulus | `BULK_MODULUS` |

Norton-Hoff law describing a viscous fluid.

Stresses are computed with $$ \sigma_{ij} = s_{ij} + p \delta_{ij} $$

where $p$ is the hydrostatic pressure and $ s_{ij} $ the stress deviator tensor.

The stress deviator tensor ($ s_{ij} $) is determined upon the strain rate deviator tensor ($ D_{ij} $) through the following equation

$$ s_{ij} = 2 \mu D_{ij} \left( \sqrt{3} \ \sqrt{\frac{2}{3} D_{wz}.D_{wz}} \right)^{m-1}$$

For a newtonian fluid : $ m=1 \rightarrow s_{ij} = 2 \mu D_{ij} $

Hydrostatic pressure is computed upon volume variation through the following eqation

$$ dp = K \frac{dV}{V} $$

where $K$ is the bulk modulus and $V$ the volume.

Name | Metafor Code | Dependency |
---|---|---|

Viscosity parameter | `NORTON_MU` | TM |

Bulk modulus | `BULK_MODULUS` | TM |

Parameter m | `NORTON_M` | TM |

Density | `MASS_DENSITY` | TM |

To model Newtonian fluid with low viscosity (0.3 Pa.s for instance) it is important to use SRI elements instead of SRIPR. Indeed, SRIPR elements might induce oscillations in pressure and velocity fields. This seems to come from the different pressure values that can be stored at the deviatoric Gauss points of SRIPR elements. For SRI elements, the hydrostatic pressure value is computed and stored at centre of each element. In that case, the pressure value on each element is unique.

Boundary conditions might also have a huge impact on the convergence and the validity of the results. It is therefore crucial to check the way the elements deform along the boundaries and the corners. From these observations, you will be able to adapt the boundary conditions.

A good compromise must be reached between the elements size and the time step value. If elements are too small, low viscous fluid elements may reach configurations where the Jacobian of some element can become negative during the Newton-Raphson procedure. The more elements you have for low viscous fluid, the more difficult is the convergence and therefore the time step is decreased automatically by Metafor. It is therefore better to avoid refining mesh elements too much where it is not needed.

The more stable the time step is, the better for the low viscous fluid behaviour.

This law is identical to NortonHoffHypoMaterial but it can account for the variation of the bulk modulus and the viscosity parameter with the hydrostatic pressure $p$.

The bulk modulus variation must be linear which is consistent with measurements made on oil used in cold rolling.

Both bulk modulus and the viscosity parameter are constant during a time step and are updated at the end of each time step. It is therefore important to assess the influence of the time step size: a time step too large might have an effect on the solution.

Name | Metafor Code | Dependency |
---|---|---|

Viscosity parameter | `NORTON_MU` | TM/IF_P |

Bulk modulus | `BULK_MODULUS` | TM/IF_P |

Parameter m | `NORTON_M` | TM |

Density | `MASS_DENSITY` | TM |

As mentionned above, the bulk modulus dependance with pressure must be linear.

$$ K(p) = K_0 + \frac{dK}{dp} p $$

Therefore two parameters must be provided by the user: the bulk modulus at atmospheric pressure and the derivative $\frac{dK}{dp}$

With Metafor, the pressure is negative in compression. Therefore, the value of $\frac{dK}{dp}$ has to be negative (otherwise the bulk modulus would decrease in compression which has no physical meaning).

The function $K(p)$ is provided by the user through a **PythonDirectorOneParameterFunction**:

class MyBulkFunction(PythonDirectorOneParameterFunction): def __init__(self,_p): PythonDirectorOneParameterFunction.__init__(self) self.debugRefs() self.p = _p def __del__(self): print "MyBulkFunction : __del__ begin\n" print "callToDestructor of MyBulkFunction not allowed." print "Add MyBulkFunction.__disown__()" exit(1) def evaluate(self, pres): if pres>=0.0: # pression positive en traction ! return self.p['k0'] else : # pression négative en compression ! return self.p['k0']+self.p['dkdp']*pres def computeDerivation(self, pres): if pres>=0.0: return 0.0 else : return self.p['dkdp']

The user has more freedom for the viscosity evolution with pressure. The viscosity variation and its derivative have to be provided trough a class similar to the one below

class MyViscoPFunction(PythonDirectorOneParameterFunction): def __init__(self,_p): print "MyViscoPFunction : __init__" PythonDirectorOneParameterFunction.__init__(self) self.debugRefs() self.p = _p print "MyViscoPFunction : __init__ finished" def __del__(self): print "MyViscoPFunction : __del__ \n" print "callToDestructor of MyViscoPFunction not allowed. Add MyViscoPFunction.__disown__()" exit(1) def evaluate(self,pres): if pres>0.0: # pression positive en traction ! return self.p['mu0'] else : # pression négative en compression ! return self.p['mu0']*math.exp(-self.p['alpha']*pres) def computeDerivation(self,pres): if pres>0.0: return 0.0 else : return -self.p['mu0']*self.p['alpha']*math.exp(-self.p['alpha']*pres)

The instances of MyBulkFunction and MyViscoPFunction are passed to the Norton-Hoff material law through the commands below

bulkLaw = MyBulkFunction(p) # <-- p stands viscoLaw = MyViscoPFunction(p) # <- " " # for parameters dictionnary and not pressure materset = domain.getMaterialSet() materset.define(1,NortonHoffPHypoMaterial) materset(1).put (BULK_MODULUS , 1.0) materset(1).depend(BULK_MODULUS , bulkLaw , Field(IF_P)) materset(1).put (NORTON_MU , 1.0) materset(1).depend(NORTON_MU , viscoLaw , Field(IF_P)) materset(1).put (NORTON_M , 1.0 ) materset(1).put (MASS_DENSITY , p['rho'] )

Norton-Hoff law including thermal aspects.

Name | Metafor Code | Dependency |
---|---|---|

Density | `MASS_DENSITY` | |

Bulk modulus | `BULK_MODULUS` | |

Parameter m | `NORTON_M` | |

Viscosity parameter | `NORTON_MU` | |

Thermal expansion | `THERM_EXPANSION` | `TO/TM` |

Conductivity | `CONDUCTIVITY` | `TO/TM` |

Heat capacity | `HEAT_CAPACITY` | `TO/TM` |

Dissipated thermoelastic power fraction | `DISSIP_TE` | - |

Dissipated (visco)plastic power fraction (Taylor-Quinney factor) | `DISSIP_TQ` | - |

doc/user/elements/volumes/fluid_iso_hypo_materials.txt · Last modified: 2017/12/01 12:11 by boemer