Modeling strain-induced precipitation in solid-state alloys using a particle size density function

Dennis den Ouden

Supervisor: Fred Vermolen
Site of the project:
Delft University of Technology

start of the project: January 2009

In April 2009 the Interim Thesis has been appeared and a presentation has been given.

The Master project has been finished in December 2009 by the completion of the Masters Thesis and a final presentation has been given. For working address etc. we refer to our alumnipage.

Summary of the master project:
Solid-state alloys, such as aluminum or steel alloys, contain a wide variety of alloying elements. These alloying elements are often deliberately added in order to improve ductile and corrosive properties of the metal products. Since the alloys are manufactured in a liquid phase and subsequently cooled down, the decrease of solubility during the solification process results into precipitation (that is the formation of particles) of several chemical compounds. At the very early stages, in which the particle size ranges within the order of several Angstroms, the statistical model due to Krafman and Wagner provides a good approximation for the particle size distribution. Myhr and Grong were the first to apply the Krafman and Wagner formalism successfully to model precipitation in aluminum alloys. For the particle growth rate, a simplified equation due to Heckel was used. In this approach, the diffusion field around a precipitate is assumed to satisfy the equilibrium state, that is the time derivative vanishes.

Since metallurgical observations reveal that the stress pattern in an alloy possibly enhances or inhibits precipitation, we try to model this relation. The stress and strain tensors are related via constitutive laws, such as the simple linear Hooke's Law if the strains are small. The local strain influences both the solubility and the diffusion coefficient. The reason that this is important, is that during a mechanical treatment, especially at an elevated temperature (such as hot extrusion), (unwanted) particles appear. We would like to able to predict the particle density and particle size density at different locations in the alloy, so that we know how to set up a mechanical treatment with an optimal result in terms of the aforementioned parameters. In other words, the quality of the alloy is optimized.

To obtain this prediction, we solve the evolutional equation of the particle size density in combination with a force balance partial differential equation. For the stress-strain relation, we use an appropriate constitutive Law, so that the partial differential equation for the local displacements can be solved. In the present approach, Hooke's law no longer applies since the alloy is deformed significantly. The relation between stress and strain contains nonlinearities due to plasticity. Further, the coefficients depend on the amount of dissolved alloying elements and on the precipitate density. Hence, to deal with the coupled problem between the statistical model for the particle size evolution and the force balance equations faces us with a nice challenge!

Left a photo of molten steel and right a photo of an aluminum alloy

Contact information: Kees Vuik

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