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Franck Tancret
Laboratoire Génie des Matériaux,
Polytech' Nantes,
La Chantrerie, rue Christian Pauc,
BP50609,
44306 Nantes Cedex 3,
France
(1)E-mail: franck.tancret@polytech.univ-nantes.fr
Added to MAP: April 2003.
This MATLAB program caclulates the volume fraction of gamma-prime and the hardness or yield stress of nickel-base superalloys during precipitation ageing.
| Language: | MATLAB |
| Product form: | Source code (".m" file) |
Complete program.
The program calculates the isothermal diffusion-controlled growth of gamma-prime precipitates from a supersaturated
gamma solid solution containing an average (Al + Ti) atomic concentration, Cav.
First, the material is divided into elementary volumes, each containing a single growing precipitate.
The elementary volume is then divided into 50 slices. It is assumed that local equilibrium exists at the particle/matrix
interface. At each computing time step, the atom flux is computed between two consecutive slices using Fick's diffusion
law. This allows to calculate the number of (Al + Ti) atoms absorbed by the precipitate, hence the gamma-prime volume
fraction, as a function of ageing time.
Hardness or yield stress is then calculated using a simple Friedel-type strengthening model, provided that the mechanical
property before ageing, Mo, (i.e. after solutionising) and the hardening coefficient, k, are known in the expression
M = Mo + k.Vf1/2, where M is the hardness or yield stress and Vf the volume fraction of precipitates.
To run the program, you need first to create an "inputs.txt" text file in the same directory as the program. This file must contain, either in lines, or in space- or tab- separated columns, all the inputs for the program: 1 - The average atomic fraction of (Al + Ti) 2 - The equilibrium atomic fraction in (Al + Ti) in the gamma phase, which can be obtained from phase diagram calculations 3 - The elementary volume size (typically half the average interparticle spacing), in µm 4 - The computing time step, in s 5 - The effective diffusion coefficient of (Al + Ti), in nm2/s 6 - The number of computing steps (e.g. 50 000). 7 - The hardness or yield stress of the material before ageing (after solutionising), in MPa. If not known, set to 0, and the mechanical property will not be calculated. 8 - The hardening coefficient, k, in M = Mo + k.Vf1/2, in MPa. If not known, set to 0, and the mechanical property will not be calculated.
There is a compromise to find between a small time step and a large number of computing steps, for a given diffusion coefficient. A small time step gives better accuracy but will require long computing times to reach the "equilibrium" volume fraction. Conversely, a large time step may require shorter computing times, but reduces accuracy, and in some cases the program produces huge oscillations (not desired!).
Then, open MATLAB and run MAP_NICKEL_GROWTH.mDuring computing, the program displays the number of computed steps evey 1000 steps, for the user to check if the program is still running, for how long, etc.
The program produces a graphic window, with the plot of gamma-prime precipitates volume fraction vs. ageing time, and, if asked, the plot of the mechanical property (hardness or yield stress) vs. ageing time, both in a semi-log scale. The final MATLAB workspace contains a variable Vf with volume fraction values for all computed time steps, and, if asked, a variable M with hardness or yield stress values for all computed time steps.
None. However, if the time step is too large, the program may produce "oscillations" in the result, which is obviously not correct. In this case the time step must be reduced till a smooth result is obtained.
In the binary Ni / Ti system, for an alloy with 15 at.% Ti, let's take, at 800°C, D = 0.20 nm2/s. Inputs are: - average atomic fraction in (Al + Ti): 0.15 - equilibrium atomic fraction in (Al + Ti): 0.046 (obtained from phase diagram) - elementary volume size, in µm: 100 - time step, in seconds: 5 - effective diffusion coefficient, in nm2/s: 0.2 - number of computing time steps: 50000 - Vickers hardness before ageing, in MPa: 1000 - hardening coefficient k, in MPa: 3000
The "inputs.txt" file may thus look lik this: 0.15 0.046 100 5 0.2 50000 1000 3000
The program produces a graphic window with the evolution of hardness and gamma-prime volume fraction as a function of ageing time at 800°C. Saturation (~ near thermodynamical equilibrium) is reached in about 40 hours, with 40% gamma-prime in volume, and a maximum hardness of 2900 MPa.
diffusion-controlled growth, supersaturated solid solution, mechanical properties, ageing kinetics, precipitation, superalloys, gamma-prime
MAP originated from a joint project of the National Physical Laboratory and the University of Cambridge.
