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1.3 MODELING THE PROPERTIES OF METALLIC ALLOYS
The discussion about the types of metallic alloys, their strengthening mechanisms and processing was related to a simple phase diagram, but these are only available for two-component systems and most commercial alloys contain more than two elements. A large number of binary alloy phase diagrams and the results of extensive phase measurements have built up sizeable thermodynamic and diffusion databases for most alloy systems.
These allow the stability of different mixes of phases of varying composition to be calculated and total free energy minimized to predict the phase mix and composition for an overall alloy composition as a function of temperature. This can be readily carried out using commercial packages (Thermo-Calc, MTDATA, and ChemSage), as for the example of 4340 steel in Fig. 1.3. This indicates that heating the steel to 800°C will dissolve the carbides, releasing Cr, Mo, and C into the solution, but a temperature of 1100°C is needed to dissolve AlN and release Al and N into solution as well.
The energies and compositions of the different phases give the driving
forces and composition gradients for nucleation, growth, and dissolution
of phases so that the rates of precipitate dissolution and precipitation can
be calculated fromEqs. (1.8) and (1.9). A simple use of these rates is in an “Avrami”plot [based on separate analyses by Johnson, Mehl, Avrami, and Kolmogorov (Avrami, 1939; Johnson & Mehl, 1939; Kolmogorov,
1937)], which plots the characteristic sigmoidal shape to overall amount of transformation (e.g., precipitation) as a function of time at constant temperature, The Avrami parametersk and n depend on the precise balance of nucleation and growth which, in turn, will depend on the exact type of system being modeled and its prior processing. Example values of Avrami parameters are tabulated inChristian (1975). By combining the Avrami plots at different temperatures, the transformationtimetemperature (TTT) behavior can be described. Many TTT diagrams were determined experimentally, but they can now be modeled (Lee & Bhadeshia, 1993) to give the overall extent of transformation, and they can be used to determine the schedule needed to achieve the desired properties from that alloy. The isothermal relationships can be used to deal with continuous heating and cooling situations by replacing the temperature variation by a series of small isothermal steps and determining the proportion of precipitation/dissolution that would have occurred during that time step at that temperature. This can be used to define the heating and cooling rates which determine whether or not transformations can occur and would be used, for example, with FE thermal fields, to establish the maximum section size or appropriate quenching medium to achieve the desired properties without excessive distortion or cracking.
This modeling method will give the overall volume fraction, but does
not give the size and spacing of individual precipitates and, therefore, the accuracy of the property prediction could be improved by applying the nucleation and growth equations to individual regions in the alloy. This requires a refinement of the mesh from millimeters to microns and, thus, the computing time would increase dramatically (days cf. minutes typically) and this would only be carried out at critical positions, for example, around welds or where a small, representative area was fully modeled (local) and was then used as the repeat unit over the whole of the structure (global). Within the local region the approaches adopted would usually be:
1. Monte Carlo: Atoms are assigned a probability of transforming, which
allows grain boundaries to be distinguished from grain bulks and favors
the growth of preexisting particles over the formation of new ones.
For each time step, the untransformed atoms are assessed in turn to
determine whether they transform during that step or not (a simple
“yes”or“no”decision). If transformation occurs, then the probability
of transformation of the surrounding atoms for the next time step is
modified (along with any other modification, e.g., temperature
change). This approach is particularly suited for grain growth and grain
boundary precipitation.
2. Phase field: The yes/no decision of the Monte Carlo approach is very
good for sharp interfaces, but many interfaces between transformed
and nontransformed regions are not sharp but rather diffuse, for example, the“mushy”solid and liquid region in solidification. The phase
field method separates the transformed/nontransformed regions by a
third phase, whose nature varies gradually from 0 (nontransformed) to
1 (transformed). This was developed for solidification with the solid
growing at different rates along its length resulting in re-production of
the dendritic structure and prediction of the eutectic phase regions.
Phase field modeling should also be suitable for modeling grain boundary precipitation.
3. Cellular automata: in this approach the structure, for example, grain or
groups of grains, is represented by a number of cells which are then
assigned probabilities of transformation. As for the Monte Carlo
approach, each cell is assessed for transformation during each time step.
The use of appropriate cells allows more efficient modeling of threedimensional structures than a Monte Carlo approach.
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