research partners    
Ph.D. students    
consortium members    

Piotr Gwiazda
Institute of Applied Mathematics and Mechanics
University of Warsaw
Banacha 2,
02-097 Warsaw
phone: +48 22 5544551
fax: +48 22 5544700

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 Navier-Stokes model fails to capture behaviour of materials that exhibit such phenomena as stress relaxation, non-linear 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 (time-dependent) 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 (Navier-Stokes model for instance). It raises a lot of mathematical difficulties, in particular one obtains model of a flow with dynamical, pressure and shear-rate 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 non-Newtonian 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 Navier-Stokes 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: 393-402.
[2] J. Kratochvil, M. Kruzik and R. Sedlacek, (2009) Acta Materialia, 57: 739--748.
[3] K.R. Rajagopal, A.R. Srinivasa, (2001) J. Non-Newtonian Fluid Mech., 99: 109-124.
[4] J. Kratochvil et al. ''Visco-plastic 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: 1603-1622.
[6] J. Malek, (2008) Electronic Transactions on Numerical Analysis, 31: 110-125.
[7] P. Gwiazda, J. Malek and A. Swierczewska, (2007) Computers and Mathematics with Applications, 53: 531-546.