## Modeling of HIP

Modeling of the HIP process has tried to define mathematically consolidation, and its objective has been to either predict the properties or the shape of the finished product. This work has been going on for more than 20 years, and the starting points for most models have been either microscopic, in which constitutive equations are based on the interaction of particles under high stress, or macroscopic, in which yield criteria were developed for the powder, treating it as a porous material continuum. A third technique, confined to shape prediction, results in an empirically derived model based on an analysis of actual production experience over an extended period of time. This section reviews all three of these methods and shows the current status and capabilities of all models.

Table 1 lists the nomenclature used in this section on HIP modeling. However, readers should be aware that often different symbols are used in literature for some properties or parameters. Another case is the material property parameters in constitutive equations developed by different authors. In some cases, the terms are identical, but have different mathematical definitions. In some cases, they are unique with their own mathematical description.

Symbol |
Definition |

function of relative density, used in yield function with J! | |

j function of relative density, used in yield function with J2 | |

F |
strain |

strain rate | |

"ö |
material property in power-law creep equation |

Ww ea |
equivalent matrix strain increment |

0 |
temperature |

SX |
nonnegative constant |

Cf |
stress |

principal stresses | |

■^av |
average stress |

■^m |
hydrostatic stress |

'^ea |
equivalent stress |

■^ö |
material property parameter in creep equation |

yield stress of material | |

shear stress | |

: i function of relative density, used with hydrostatic stress in stress-strain equations | |

b |
material constant used in various yield functions. May have different specific meanings |

f, fl, f2, f |
function used in various yield functions. May have different specific meanings |

k |
thermal conductivity of fully dense product |

kD |
thermal conductivity as an exponential function of relative density |

li characteristic initial dimension of compact | |

If |
characteristic final dimension of compact |

m |
material constant in various yield functions |

n |
power-law creep exponent |

A |
power-law creep coefficient |

D |
relative density of the powder at any time |

Do |
initial relative density of the powder |

D |
densification rate |

F |
yield function |

J |
£7 [ + (T2 + (T3j first invariant of the stress tensor |

J 2 |
[((T1 - iT2)2 + (&2 - i?"3)2 + (&3 - & i)2l, second invariant of the deviatoric stress sensor |

P |
pressure |

Peff |
effective (contact) pressure on each powder particle |

Plim |
external pressure that will cause yielding |

Q |
activation energy for creep |

R |
radius |

Ro |
universal gas constant |

T |
temperature |

Vi initial volume of powder container | |

Vf |
final volume of powder container after HIP |

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