dr Karel Tůma


Doktorat
2014-04-04Identification of rate type fluids suitable for modeling geomaterials  (ChU P)
promotor -- prof. Josef Malek, ChU P
1234 
Ostatnie publikacje
1.Průša V., Řehoř M., Tůma K., Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots, ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK ZAMP, ISSN: 0044-2275, DOI: 10.1007/s00033-017-0768-x, Vol.68, No.24, pp.1-13, 2017

Streszczenie:

The response of mechanical systems composed of springs and dashpots to a step input is of eminent interest in the applications. If the system is formed by linear elements, then its response is governed by a system of linear ordinary differential equations. In the linear case, the mathematical method of choice for the analysis of the response is the classical theory of distributions. However, if the system contains nonlinear elements, then the classical theory of distributions is of no use, since it is strictly limited to the linear setting. Consequently, a question arises whether it is even possible or reasonable to study the response of nonlinear systems to step inputs. The answer is positive. A mathematical theory that can handle the challenge is the so-called Colombeau algebra. Building on the abstract result by Průša and Rajagopal (Int J Non-Linear Mech 81:207–221, 2016), we show how to use the theory in the analysis of response of nonlinear spring–dashpot and spring–dashpot–mass systems.

Słowa kluczowe:

Mechanical systems, Nonlinear ordinary differential equations, Jump discontinuities, Colombeau algebra

Afiliacje autorów:

Průša V.-Charles University in Prague (CZ)
Řehoř M.-Charles University in Prague (CZ)
Tůma K.-IPPT PAN
35p.
2.Hron J., Miloš V., Průša V., Souček O., Tůma K., On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients, INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, ISSN: 0020-7462, DOI: 10.1016/j.ijnonlinmec.2017.06.011, Vol.95, pp.193-208, 2017

Streszczenie:

We derive a class of thermodynamically consistent variants of Maxwell/Oldroyd-B type models for incompressible viscoelastic fluids. In particular, we study the models that allow one to consider temperature dependent material coefficients. This naturally calls for the formulation of a temperature evolution equation that would accompany the evolution equations for the mechanical quantities. The evolution equation for the temperature is explicitly formulated, and it is shown to be consistent with the laws of thermodynamics and the evolution equations for the mechanical quantities. The temperature evolution equation contains terms that are ignored or even not thought of in most of the practically oriented (computational) works dealing with this class of fluids. The impact of the additional terms in the temperature evolution equation on the flow dynamics is documented by the solution of simple initial/boundary value problems.

Słowa kluczowe:

Maxwell fluid; Oldroyd-B fluid; Temperature dependent material coefficients; Thermodynamics; Cylindrical Couette flow; Biaxial extension; Numerical simulations

Afiliacje autorów:

Hron J.-Charles University in Prague (CZ)
Miloš V.-Charles University (CZ)
Průša V.-Charles University in Prague (CZ)
Souček O.-Charles University (CZ)
Tůma K.-IPPT PAN
3.Tůma K., Stupkiewicz S., Petryk H., Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 0022-5096, DOI: 10.1016/j.jmps.2016.04.013, Vol.95, pp.284-307, 2016

Streszczenie:

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.

Słowa kluczowe:

Phase-field method, Microstructure, Martensite, Size effects, Shape memory alloys

Afiliacje autorów:

Tůma K.-IPPT PAN
Stupkiewicz S.-IPPT PAN
Petryk H.-IPPT PAN
40p.
4.Tůma K., Stupkiewicz S., Phase-field study of size-dependent morphology of austenite–twinned martensite interface in CuAlNi, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 0020-7683, DOI: 10.1016/j.ijsolstr.2016.07.040, Vol.97-98, pp.89-100, 2016

Streszczenie:

Size-dependent microstructure of the interface layer between austenite and twinned martensite is studied using a recently developed finite-strain phase-field model. The microstructure is assumed periodic and two-dimensional, however, non-zero out-of-plane displacements are allowed so that the basic microstructural features, specifically the nominal orientation of the twinning and habit planes and the twin fraction, are consistent with the crystallographic theory of martensite. The phase-field computations are carried out for the CuAlNi shape memory alloy undergoing the cubic-to-orthorhombic transformation, and the corresponding four crystallographically distinct microstructures of the austenite–twinned martensite interface are studied in detail. The focus is on size-dependent morphology of the interface layer and on size-dependent interfacial and elastic micro-strain energy contributions. Two mechanisms of reducing the elastic micro-strain energy are revealed: formation of a non-planar zigzag-like interface and twin branching.

Słowa kluczowe:

Microstructure, Phase transformation, Martensite, Phase-field method, Size effects

Afiliacje autorów:

Tůma K.-IPPT PAN
Stupkiewicz S.-IPPT PAN
35p.
5.Řehoř M., Průša V., Tůma K., On the response of nonlinear viscoelastic materials in creep and stress relaxation experiments in the lubricated squeeze flow setting, PHYSICS OF FLUIDS, ISSN: 1070-6631, DOI: 10.1063/1.4964662, Vol.28, No.10, pp.103102-1-25, 2016

Streszczenie:

Rigorous analysis of the response of nonlinear materials to step inputs requires one to simultaneously handle the discontinuity, differentiation, and nonlinearity. This task is however beyond the reach of the standard theories such as the classical theory of distributions and presents a considerable mathematical difficulty. New advanced mathematical tools are necessary to handle the challenge. An elegant and relatively easy-to-use framework capable of accomplishing the task is provided by the Colombeau algebra, which is a generalisation of the classical theory of distributions to the nonlinear setting. We use the Colombeau algebra formalism and derive explicit formulae describing the response of incompressible Maxwell viscoelastic fluid subject to step load/deformation in the lubricated squeeze flow setting.

Słowa kluczowe:

Lubricating, viscoelastic fluid, Maxwell

Afiliacje autorów:

Řehoř M.-Charles University in Prague (CZ)
Průša V.-Charles University in Prague (CZ)
Tůma K.-IPPT PAN
35p.
6.Malek J., Rajagopal K.R., Tůma K., A thermodynamically compatible model for describing asphalt binders: solutions of problems, International Journal of Pavement Engineering, ISSN: 1029-8436, DOI: 10.1080/10298436.2015.1007575, Vol.17, No.6, pp.550-564, 2016

Streszczenie:

In this sequel to the first paper (Málek et al., 2014. International Journal of Pavement Engineering), in which we identified a generalisation of the model due to Burgers which was corroborated against two sets of experiments, including a challenging one showing distinctly different relaxation times for shear and normal stresses, we solve several time-dependent boundary value problems wherein the boundary of the material is deforming, that have relevance to applications involving asphalt. Problems wherein the boundary is subject to time-varying compressive loads such as those due to moving automobiles and the attendant rutting, and the compaction due to rollers are considered in additions to other problems.

Słowa kluczowe:

rate type fluid, asphalt, finite element method, monolithic ALE method

Afiliacje autorów:

Malek J.-Charles University (CZ)
Rajagopal K.R.-Texas A&M University (US)
Tůma K.-IPPT PAN
25p.
7.Malek J., Rajagopal K.R., Tůma K., On a variant of the Maxwell and Oldroyd-B models within the context of a thermodynamic basis, INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS, ISSN: 0020-7462, DOI: 10.1016/j.ijnonlinmec.2015.03.009, Vol.76, pp.42-47, 2015

Streszczenie:

In this paper we develop models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and Oldroyd-B models but differ from them in one important aspect, the manner in which they unload instantaneously from the deformed configuration. As long as the response is not instantaneous, the models that are derived cannot be differentiated from the Maxwell and Oldroyd-B models, respectively. The models can be viewed within the context of materials whose natural configuration evolves, the evolution being determined by the maximization of the rate of entropy production of the material. However, the underpinnings to develop the model are quite different from an earlier development by Rajagopal and Srinivasa [8] in that while the total response of the viscoelastic fluid satisfies the constraint of an incompressible material, the energy storage mechanism associated with the elastic response is allowed to be that for a compressible elastic solid and the dissipative mechanism associated with the viscous response allowed to be that for a compressible fluid, the total deformation however being isochoric. The analysis calls for a careful evaluation of firmly held customs in viscoelasticity wherein it is assumed that it is possible to subject a material to a purely instantaneous elastic response without any dissipation whatsoever. Finally, while the model developed by Rajagopal and Srinivasa [8] arises from the linearization of the non-linear elastic response that they chose and leads to a model wherein the instantaneous elastic response is isochoric, here we develop the model within the context of a different non-linear elastic response that need not be linearized but the instantaneous elastic response not necessarily being isochoric.

Słowa kluczowe:

Rate type fluid, Maxwell, Oldroyd-B, Compressible neo-Hookean, Thermodynamical compatibility

Afiliacje autorów:

Malek J.-Charles University (CZ)
Rajagopal K.R.-Texas A&M University (US)
Tůma K.-IPPT PAN
30p.
8.Malek J., Rajagopal K.R., Tůma K., A thermodynamically compatible model for describing the response of asphalt binders, International Journal of Pavement Engineering, ISSN: 1029-8436, DOI: 10.1080/10298436.2014.942860, Vol.16, No.4, pp.297-314, 2015

Streszczenie:

In this paper, we develop a model from a thermodynamic standpoint that seems capable of describing the nonlinear response of asphalt binders. We test the efficacy of the model by comparing its predictions against two different sets of torsion experiments on asphalt binders. The first set of experiments that we use for corroborating the model was carried out by Narayan et al. [2012. Mechanics Research Communications, 43, 66–74] wherein for the first time it was found that the relaxation times associated with the torque and the normal forces, in a torsion experiment, are markedly different, and the second set of experiments that we use to corroborate the model documents the overshoot of torque in a torsion experiment [Krishnan and Narayan, 2007. Steady shear experiments on ashpalt. Chennai: IIT, Madras]. The model that is developed in this paper fits both sets of experiments well, and it seems to be a good candidate for describing the response of asphalt binders in general. As the deformation is nonlinear, it would be inappropriate to use the linearised viscoelastic model which is based on the linearised strain, and the models due to Maxwell, and the Oldroyd-B model are unable to capture the marked difference in the relaxation times, while the Burgers model is unable to describe the torque overshoot.

Słowa kluczowe:

stress relaxation, normal forces, torque, rate type fluid, asphalt, experiment fitting

Afiliacje autorów:

Malek J.-Charles University (CZ)
Rajagopal K.R.-Texas A&M University (US)
Tůma K.-other affiliation
25p.
9.Hron J., Rajagopal K.R., Tůma K., Flow of a Burgers fluid due to time varying loads on deforming boundaries, Journal of Non-Newtonian Fluid Mechanics, ISSN: 0377-0257, DOI: 10.1016/j.jnnfm.2014.05.005, Vol.210, pp.66-77, 2014

Streszczenie:

In this paper we study three boundary-initial value problems within the context of four rate type viscoelastic constitutive models, the Maxwell model, the Oldroyd-B model, Burgers model and the generalized Burgers model. We consider challenging problems wherein the boundary is deforming with time. The flows lead to a complex system of partial differential equations that require the development of a robust numerical procedure based on the arbitrary Lagrangian–Eulerian method.

Słowa kluczowe:

Rate type fluid, Maxwell, Oldroyd-B, Burgers, Finite element method, Monolithic ALE method

Afiliacje autorów:

Hron J.-Charles University in Prague (CZ)
Rajagopal K.R.-Texas A&M University (US)
Tůma K.-other affiliation
30p.
10.Hron J., Kratochvil J., Malek J., Rajagopal K.R., Tůma K., A thermodynamically compatible rate type fluid to describe the response of asphalt, Mathematics and Computers in Simulation, ISSN: 0378-4754, DOI: 10.1016/j.matcom.2011.03.010, Vol.82, No.10, pp.1853-1873, 2012

Streszczenie:

In this paper, we consider two models that have been recently developed from a thermodynamic standpoint and that are capable of describing the response of nonlinear viscoelastic fluids. We test the efficacy of both models by comparing their predictions against torsion experiments conducted for asphalt, a material that is notoriously difficult to model. Both the models seem to describe the response adequately, though neither is really very accurate. This should not be surprising as asphalt is a heterogenous material comprising of many components which is being homogenized and modeled as a single constituent viscoelastic fluid.

Słowa kluczowe:

Rate type fluid, Large deformation, Numerical simulation

Afiliacje autorów:

Hron J.-Charles University in Prague (CZ)
Kratochvil J.-other affiliation
Malek J.-Charles University (CZ)
Rajagopal K.R.-Texas A&M University (US)
Tůma K.-other affiliation
20p.
11.Pirkl L., Bodnar T., Tůma K., Viscoelastic Fluid Flows at Moderate Weissenberg Numbers Using Oldroyd Type Model, AIP Conference Proceedings, ISSN: 0094-243X, DOI: 10.1063/1.3636680, Vol.1389, pp.102-105, 2011

Streszczenie:

This paper presents the preliminary results of our numerical simulations designed and performed to address the high Weissenberg number problem that is the major challenge in the simulation of viscoelastic flows. The mathematical model used to explore this problem is based on Oldroyd type model. A new simple computational test case is proposed and solved to demonstrate the nature of the high Weissenberg number problem. Various finite‐volume as well as finite‐element methods are introduced to be tested for this test case. Some of our very first results are presented and discussed at the end.

Słowa kluczowe:

Viscoelasticity, Fluid flows, Numerical modeling

Afiliacje autorów:

Pirkl L.-other affiliation
Bodnar T.-other affiliation
Tůma K.-other affiliation

Prace konferencyjne
1.Tůma K., On Capability of a Class of Incompressible Rate-type Fluid Models to Fit Experimental Data for Asphalt, WDS 2009, 18th Annual Conference of Doctoral Students, 2009-06-02/06-05, Praga (CZ), pp.224-230, 2009

Streszczenie:

We consider a class of viscoelastic rate type models that in particular includes: (i) Oldroyd-B fluid model with three parameters, (ii) nonlinear fluid model derived in Rajagopal K. R., Srinivasa A. R. [2000] with three parameters, and (iii) nonlinear model with five parameters. We are interested in observing how well are these models capable to capture the experimental data for asphalt performed by J. Murali Krishnan, Indian Institute of Technology, Madras using dynamic shear rheometer. We find out that the model (i) is not able to capture
the experimentally observed overshoot for the torque, while we obtain overshoots for the models (ii) and (iii). Also, there are very small significant differences in the results established for models (ii) and (iii).

Afiliacje autorów:

Tůma K.-other affiliation

Abstrakty konferencyjne
1.Tůma K., Stupkiewicz S., Petryk H., The effect of twin spacing on the morphology of austenite-twinned martensite interface, SolMech 2016, 40th Solid Mechanics Conference, 2016-08-29/09-02, Warszawa (PL), No.P069, pp.1, 2016
2.Tůma K., Stupkiewicz S., Petryk H., Phase-field modelling of twinning and martensitic transformation at finite strain, PCM-CMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 2015-09-08/09-11, Gdańsk (PL), pp.815-816, 2015

Streszczenie:

We develop a micromechanical phase-field model that describes the transformation between the austenite and twinned martensites. The new model constrains the volume fractions of both parent and internally twinned phases such that the physically motivated bounds are not violated. As an application, we studied the twinned martensite and austenite-martensite interfaces in the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and estimated the elastic part of the interfacial energy.

Afiliacje autorów:

Tůma K.-IPPT PAN
Stupkiewicz S.-IPPT PAN
Petryk H.-IPPT PAN