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:
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
and a final presentation has been given.
For working address etc. we refer to our
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
Left a photo of molten steel and right a photo of an aluminum alloy
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