The values of A and n are calculated using fully dense samples, while f and b are determined with porous samples with relative densities ranging from 0.77 to 1.0. Experimental work with T316LN powder was done to determine these functions (Ref 47, 48).

These investigators also have proposed a function in which thermal conductivity is an exponential function of relative density:

where k is thermal conductivity. The results of this work have been incorporated into a multi-materials finite element modeling code, PRECAD/M, that allows the simulation of HIP of axisymmetric and three-dimensional parts. The finite element software was taken from a commercial package and modified by CEA/CEREM specifically to handle the problem of dimensional change during HIP. Data were presented to show the comparison between initial and deformed geometry in a part made with Ti-6Al-4V powder with a solid core (Ref 46).

Validation tests of the different models by different collaborators have also been run for actual HIP cycles on production type parts, where beginning and final dimensions are reported so that the precision of the model can be evaluated (Ref 49, 50, 51). This work was done with 316L stainless steel powder from a single source so that the variable of powder type and size is kept constant. Simulation results were reported by three investigators in a 1997 workshop with the following conclusions:

• Jeon and Kim (Ref 49) using the models of Abouaf and McMeeking concluded that it is not feasible to produce near-net shape parts under HIP when the part has an asymmetric shape.

• Eisen, Bouaziz, and Dellis (Ref 50) compared the prediction of the Crucible empirical model and the CEA/CEREM two-dimensional model using two different HIP cycles (one with a very rapid heat up and pressurization, the other with a slower ramp up for temperature and pressure) with the actual results. They found that the predictions of both models were qualitatively but not quantitatively accurate and that either model could be used for near-net-shape prediction. Errors (ratio of simulated final value minus measured final value to initial value minus measured final value) on the three major dimensions ranged from 4 to 32% for the Crucible model and from 7 to 42% for the CEA/CEREM model. They also showed that the isotropic predictions using Eq 5 gave an accuracy very close to actual results. On this basis, they hypothesized that in small parts and even in large parts that are heated slowly enough to minimize temperature gradients, isotropic shrinkage can be used as a first approximation.

• Dellis, Bouaziz, Baccino, and Moret (Ref 51) used a three-dimensional simulation on this same part and compared it to the actual shrinkage. Their error on the three main dimensions ranged between 1 and 10%, which is a significant improvement over the two-dimensional and empirical model. This three-dimensional model required a fine-mesh grid and took 72 h of central processing unit time compared with 4 h for the two-dimensional simulation.

It is also possible, however, that current models may not be accurate because the mathematics do not describe what is actually happening during the consolidation process. Work of DeLo and Piehler (Ref 52) using thin-walled containers holding Ti-6Al-4V powder that was hot isostatically pressed consolidated concluded that the vast majority of densification and shape change occurs early in the HIP cycle before steady state-temperature and pressure are achieved. They attributed particle motion as the reason for an underestimation of densification. It has also been suggested that because temperature and pressure are increased together in most HIP cycles, power-law creep may be the predominant mechanism during consolidation and that the plastic yielding contribution could be ignored. A model using power-law creep as the sole densification mechanism might work as well as any viscoplastic model.

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