Karel Tůma, PhD 
Doctoral thesis
20140404  Identification of rate type fluids suitable for modeling geomaterials (ChU P)
 1234 
Recent publications
1.  Rezaee Hajidehi M., Tůma K.^{♦}, Stupkiewicz S., Gradientenhanced thermomechanical 3D model for simulation of transformation patterns in pseudoelastic shape memory alloys, International Journal of Plasticity, ISSN: 07496419, DOI: 10.1016/j.ijplas.2019.08.014, Vol.128, pp.102589129, 2020 Abstract: Stressinduced martensitic transformation in polycrystalline NiTi under tension often proceeds through formation and propagation of macroscopic phase transformation fronts, i.e., diffuse interfaces that separate the transformed and untransformed domains. A gradientenhanced 3D finitestrain model of pseudoelasticity is developed in this work with the aim to describe the related phenomena. The underlying softening response is regularized by enhancing the Helmholtz free energy of a nongradient model with a gradient term expressed in terms of the martensite volume fraction. To facilitate finiteelement implementation, a micromorphictype regularization is then introduced following the approach developed recently in the 1D smallstrain context. The complete evolution problem is formulated within the incremental energy minimization framework, and the resulting nonsmooth minimization problem is solved by employing the augmented Lagrangian technique. In order to account for the thermomechanical coupling effects, a general thermomechanical framework, which is consistent with the second law of thermodynamics and considers all related couplings, is also developed. Finiteelement simulations of representative 3D problems show that the model is capable of representing the loadingrate effects in a NiTi dogbone specimen and complex transformation patterns in a NiTi tube under tension. A parametric study is also carried out to investigate the effect of various parameters on the characteristics of the macroscopic transformation front. Keywords:phase transformation, softening, strain localization, micromorphic regularization, finiteelement method Affiliations:
 
2.  RezaeeHajidehi M., Tůma K.^{♦}, Stupkiewicz S., A note on Padé approximants of tensor logarithm with application to Henckytype hyperelasticity, COMPUTATIONAL MECHANICS, ISSN: 01787675, DOI: 10.1007/s00466020019150, pp.114, 2020 Abstract: We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor series. As an application, Henckytype hyperelasticity models are considered, in which the elastic strain energy is expressed in terms of the Hencky strain, and of our particular interest is the anisotropic energy quadratic in the Hencky strain. Finiteelement computations are carried out to examine performance of the Padé approximants of tensor logarithm in Henckytype hyperelasticity problems. A discussion is also provided on computation of the stress tensor conjugate to the Hencky strain in a general anisotropic case. Keywords:logarithmic strain, Padé approximation method, hyperelasticity, anisotropy, finiteelement method Affiliations:
 
3.  Tůma K., Stupkiewicz S., Petryk H., Rateindependent dissipation in phasefield modelling of displacive transformations, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2018.02.007, Vol.114, pp.117142, 2018 Abstract: In this paper, rateindependent dissipation is introduced into the phasefield framework for modelling of displacive transformations, such as martensitic phase transformation and twinning. The finitestrain phasefield model developed recently by the present authors is here extended beyond the limitations of purely viscous dissipation. The variational formulation, in which the evolution problem is formulated as a constrained minimization problem for a global ratepotential, is enhanced by including a mixedtype dissipation potential that combines viscous and rateindependent contributions. Effective computational treatment of the resulting incremental problem of nonsmooth optimization is developed by employing the augmented Lagrangian method. It is demonstrated that a single Lagrange multiplier field suffices to handle the dissipation potential vertex and simultaneously to enforce physical constraints on the order parameter. In this way, the initially nonsmooth problem of evolution is converted into a smooth stationarity problem. The model is implemented in a finiteelement code and applied to solve two and threedimensional boundary value problems representative for shape memory alloys Keywords:Phasefield method, Microstructure, Martensite, Twinning, Nonsmooth optimization Affiliations:
 
4.  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: 00442275, DOI: 10.1007/s000330170768x, Vol.68, No.24, pp.113, 2017 Abstract: 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 socalled Colombeau algebra. Building on the abstract result by Průša and Rajagopal (Int J NonLinear 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. Keywords:mechanical systems, nonlinear ordinary differential equations, jump discontinuities, colombeau algebra Affiliations:
 
5.  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 NONLINEAR MECHANICS, ISSN: 00207462, DOI: 10.1016/j.ijnonlinmec.2017.06.011, Vol.95, pp.193208, 2017 Abstract: We derive a class of thermodynamically consistent variants of Maxwell/OldroydB 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. Keywords:Maxwell fluid, OldroydB fluid, Temperature dependent material coefficients, Thermodynamics, Cylindrical Couette flow, Biaxial extension, Numerical simulations Affiliations:
 
6.  Tůma K., Stupkiewicz S., Petryk H., Size effects in martensitic microstructures: Finitestrain phase field model versus sharpinterface approach, JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS, ISSN: 00225096, DOI: 10.1016/j.jmps.2016.04.013, Vol.95, pp.284307, 2016 Abstract: A finitestrain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharpinterface 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 phasefield 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 orderparameters 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, sizedependent microstructures with diffuse interfaces are calculated for the cubictoorthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharpinterface microstructures with interfacial energy effects. Keywords:Phasefield method, Microstructure, Martensite, Size effects, Shape memory alloys Affiliations:
 
7.  Tůma K., Stupkiewicz S., Phasefield study of sizedependent morphology of austenite–twinned martensite interface in CuAlNi, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, ISSN: 00207683, DOI: 10.1016/j.ijsolstr.2016.07.040, Vol.9798, pp.89100, 2016 Abstract: Sizedependent microstructure of the interface layer between austenite and twinned martensite is studied using a recently developed finitestrain phasefield model. The microstructure is assumed periodic and twodimensional, however, nonzero outofplane 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 phasefield computations are carried out for the CuAlNi shape memory alloy undergoing the cubictoorthorhombic transformation, and the corresponding four crystallographically distinct microstructures of the austenite–twinned martensite interface are studied in detail. The focus is on sizedependent morphology of the interface layer and on sizedependent interfacial and elastic microstrain energy contributions. Two mechanisms of reducing the elastic microstrain energy are revealed: formation of a nonplanar zigzaglike interface and twin branching. Keywords:Microstructure, Phase transformation, Martensite, Phasefield method, Size effects Affiliations:
 
8.  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: 10706631, DOI: 10.1063/1.4964662, Vol.28, No.10, pp.103102125, 2016 Abstract: 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 easytouse 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. Keywords:Lubricating, viscoelastic fluid, Maxwell Affiliations:
 
9.  Malek J.^{♦}, Rajagopal K.R.^{♦}, Tůma K., A thermodynamically compatible model for describing asphalt binders: solutions of problems, International Journal of Pavement Engineering, ISSN: 10298436, DOI: 10.1080/10298436.2015.1007575, Vol.17, No.6, pp.550564, 2016 Abstract: 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 timedependent 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 timevarying 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. Keywords:rate type fluid, asphalt, finite element method, monolithic ALE method Affiliations:
 
10.  Malek J.^{♦}, Rajagopal K.R.^{♦}, Tůma K., On a variant of the Maxwell and OldroydB models within the context of a thermodynamic basis, INTERNATIONAL JOURNAL OF NONLINEAR MECHANICS, ISSN: 00207462, DOI: 10.1016/j.ijnonlinmec.2015.03.009, Vol.76, pp.4247, 2015 Abstract: In this paper we develop models within a thermodynamic standpoint that are very similar in form to the classical Maxwell and OldroydB 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 OldroydB 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 nonlinear 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 nonlinear elastic response that need not be linearized but the instantaneous elastic response not necessarily being isochoric. Keywords:Rate type fluid, Maxwell, OldroydB, Compressible neoHookean, Thermodynamical compatibility Affiliations:
 
11.  Malek J.^{♦}, Rajagopal K.R.^{♦}, Tůma K.^{♦}, A thermodynamically compatible model for describing the response of asphalt binders, International Journal of Pavement Engineering, ISSN: 10298436, DOI: 10.1080/10298436.2014.942860, Vol.16, No.4, pp.297314, 2015 Abstract: 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 OldroydB model are unable to capture the marked difference in the relaxation times, while the Burgers model is unable to describe the torque overshoot. Keywords:stress relaxation, normal forces, torque, rate type fluid, asphalt, experiment fitting Affiliations:
 
12.  Hron J.^{♦}, Rajagopal K.R.^{♦}, Tůma K.^{♦}, Flow of a Burgers fluid due to time varying loads on deforming boundaries, Journal of NonNewtonian Fluid Mechanics, ISSN: 03770257, DOI: 10.1016/j.jnnfm.2014.05.005, Vol.210, pp.6677, 2014 Abstract: In this paper we study three boundaryinitial value problems within the context of four rate type viscoelastic constitutive models, the Maxwell model, the OldroydB 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. Keywords:Rate type fluid, Maxwell, OldroydB, Burgers, Finite element method, Monolithic ALE method Affiliations:
 
13.  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: 03784754, DOI: 10.1016/j.matcom.2011.03.010, Vol.82, No.10, pp.18531873, 2012 Abstract: 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. Keywords:Rate type fluid, Large deformation, Numerical simulation Affiliations:

Conference papers
1.  Pirkl L.^{♦}, Bodnar T.^{♦}, Tůma K.^{♦}, Viscoelastic Fluid Flows at Moderate Weissenberg Numbers Using Oldroyd Type Model, ICNAAM 2011, 9th International Conference of Numerical Analysis and Applied Mathematics, 20110919/0925, Halkidiki (GR), DOI: 10.1063/1.3636680, pp.102105, 2011 Abstract: 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. Keywords:Viscoelasticity, Fluid flows, Numerical modeling Affiliations:
 
2.  Tůma K.^{♦}, On Capability of a Class of Incompressible Ratetype Fluid Models to Fit Experimental Data for Asphalt, WDS 2009, 18th Annual Conference of Doctoral Students, 20090602/0605, Praga (CZ), pp.224230, 2009 Abstract: We consider a class of viscoelastic rate type models that in particular includes: (i) OldroydB 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

Conference abstracts
1.  Tůma K., Stupkiewicz S., Petryk H., The effect of twin spacing on the morphology of austenitetwinned martensite interface, SolMech 2016, 40th Solid Mechanics Conference, 20160829/0902, Warszawa (PL), No.P069, pp.1, 2016  
2.  Tůma K., Stupkiewicz S., Petryk H., Phasefield modelling of twinning and martensitic transformation at finite strain, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.815816, 2015 Abstract: We develop a micromechanical phasefield 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 austenitemartensite interfaces in the cubictoorthorhombic transformation in a CuAlNi shape memory alloy and estimated the elastic part of the interfacial energy. Affiliations:
