

Fluid model of crystal plasticity  mathematical properties and computer simulations


Piotr Gwiazda (MIM University of Warsaw) and Josef Malek (Faculty of Mathematics and Physics Charles University in Prague)


The project is carried out by Piotr Minakowski at University of Warsaw


Sufficient to describe ubiquitous fluids such as air or water, the popular and widely investigated
NavierStokes model fails to capture behaviour of materials that exhibit such phenomena as stress
relaxation, nonlinear creep, normal stress differences, shear thinning/shear thickening etc. Such
materials are being used, ever increasingly, but most of the models describing them are very recent
and still are beyond rigorous mathematical understanding.
The aim of this project is to investigate model of a plastic flow of a highly viscous,
incompressible fluid approximating ultrafine structure formation induced by severe
plastic deformation presented by Kratochvil (Faculty of Civil Engineering, Czech
Technical University in Prague) et al. in [2] and [4].
Mechanical description of this flow is possible due to the introduction of anisotropy,
higher gradients and dynamical (timedependent) internal variables into the governing
system of equations and is based on physical experiments presented in [1]. From mechanical
point of view this is the main extension of prior models of flows (NavierStokes model for instance).
It raises a lot of mathematical difficulties, in particular one obtains model of a flow with dynamical,
pressure and shearrate dependent viscosity.
The project will be divided into two main stages. The first stage will consist
mainly of numerical simulations. This stage aims in calibrating the model and
comparing it with other models of anisotropic fluids [3]. Such results are of high
interest and are very desirable  particularly among the community of scientists
interested in modelling. During this stage student is to stay in Prague. He or she will
cooperate with Prof. Malek's group, which has a lot of experience in carrying out numerical
simulation for nonNewtonian fluids and testing various models (see [5] for instance). It will also
enable a direct contact with Prof. Kratochv\'il and his group which is needed to identify model
parameters, to analyse, interpret and verify the results of computational experiments.
The collaboration with Prof. Maria Lukacova (Hamburg University of Technology) is also planned in the first stage of the project.
Mathematical analysis of the investigated system of equations will be subject of
research during the second stage of the project. Since the model involves implicit
constitutive relations (due to the dynamical viscosity) one cannot rely on the theory
for the NavierStokes equations. Mathematical analysis of the model will be carried out
in collaboration with Prof. Malek's group in Prague and dr hab. Gwiazda's group in Warsaw.
Both scientists have already published number of papers concerning fluids described by implicit
relations (see [6] and [7]). The Ph.D. student is to generalize their results to fluid model of
crystal plasticity or some of its geometric or constitutive simplifications.
[1] T. Hebesberger, H.P. Stuwe, A. Vorhauer, F. Wetscher and R. Pippan, Acta Materiala, 53: 393402.
[2] J. Kratochvil, M. Kruzik and R. Sedlacek, (2009) Acta Materialia, 57: 739748.
[3] K.R. Rajagopal, A.R. Srinivasa, (2001) J. NonNewtonian Fluid Mech., 99: 109124.
[4] J. Kratochvil et al. ''Viscoplastic model of a ultrafine structure formation induced by HPT'' in preparation.
[5] J. Hron, J. Malek and K.R. Rajagopal, (2001) Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 457: 16031622.
[6] J. Malek, (2008) Electronic Transactions on Numerical Analysis, 31: 110125.
[7] P. Gwiazda, J. Malek and A. Swierczewska, (2007) Computers and Mathematics with Applications, 53: 531546.




