
Maticové rozklady v teorii konstitutivních vztahů pro spojité prostředí
Vejvoda, Martin ; Průša, Vít (advisor) ; Kružík, Martin (referee)
We study the application of the QR decomposition in the theory of Green elastic solids with emphasis on transversely isotropic materials such as fiberreinforced materials. We provide a methodology, how to use the QR decomposition to describe materials with gen eral fiber orientation including curved fibers. We then focus on the socalled conjugate stress / strain basis model, and we show that for isotropic materials the model is equiva lent to the standard model of Green elastic solid. We also provide a methodology, how to describe transversely isotropic materials using the QR decomposition. Next, we consider the popular standard reinforcing material model with spatiallyvarying fiber directions and fiber stiffnesses, and we perform numerical experiments in various geometries. To our best knowledge, our implementation is the first implementation of numerical solvers for QR based models with spatiallyvarying fiber directions. Finally, we compare the results for the conjugate stress / strain model with the results for the standard model of Green elasticity and linear elasticity. 1

 

Matematické modelování magnetosriktních látek
Vermach, Lukáš ; Kružík, Martin (advisor) ; Zeman, Jan (referee)
4 Title: Mathematical modeling of magnetostrictive materials Author: Lukáš Vermach Department: Mathematical Institute of Charles University Advisor: Priv.Doz. Dr. habil. RNDr. Martin Kružík Ph.D., Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic Advisor's email address: kruzik@utia.cas.cz Abstract In the present work we introduce an isotermic mathematical model of ferromagnetic shape memory alloys (FSMAs). FSMAs are a special class of magnetostrictive materials, i.e. materials which deform their shape on account of external magnetic field or which change magnetization as a consequence of strain. This property originates from phase transformations that occur within the material when being exposed to external loading. First, the stationary model of FSMA is formulated. The thermodynamical potential is composed (Helmholz free energy) and its nonquasiconvexity is discussed. The quasicon vexification is performed via the relaxation theory, i.e. quasiconvex envelope construction. For such a model the existence theory is built. Then, taking advantage of the stationary case the evolutionary model is developed. The attention is drawn to hysteresis, which arises from energy dissipation. The time discretization leads to a sequence of hysteresismodified stationary problems (the...


Elastic properties of blood veins with a scaffold
Frost, Miroslav ; Maršík, František (advisor) ; Kružík, Martin (referee)
Presented master's thesis deals with modeling of a NiTiNOL wire under thermal and uniaxial mechanical loading. NiTiNOL can undergo reversible martensitic phase transformation and thus belongs among shape memory alloys. In the form of a thin wire it is used in many applications (e.g. as a reinforcement for veins). MT is studied with respect to the extended nonequilibrium thermomechanics of mixtures and the ClusiusClapeyron equation is derived for it. A new phenomenological model iRLOOP, developed at AS CR, simulating thermomechanical behavior of a NiTiNOL wire is mathematically formulated. Restrictions on tting functions in proposed hysteresis mechanism are derived from the second law of thermodynamics. The existence and uniqueness of the solution of an initial problem are proven for the superelasticity model. Experiments are compared with results modeled by numerical implementation of iRLOOP.


Variational Methods in Thermomechanics of Solids
Pelech, Petr ; Kružík, Martin (advisor) ; Dondl, Patrick (referee) ; Zeman, Jan (referee)
The thesis is devoted to study of continuum mechanics and thermodynamics and the related mathematical analysis. It consists of four selfcontained chapters dealing with different aspects. The first chapter focuses on peridynamics, a nonlocal theory of continuum mechanics, and its relation to conventional local theory of CauchyGreen elasticity. Similar compar isons has been used for proving consistency and for determining some of the material coefficients in peridynamics, provided the material parameters in the local theory are known. In this chapter the formula for the nonlocal forceflux is computed in terms of the peridynamic interaction, relating the fundamental concepts of these two theories and establishing hence a new connection, not present in the previous works. The second and third chapters are both devoted to RateIndependent Systems (RIS) and their applications to continuum mechanics. RIS represents a suitable approximation when the internal, viscous, and thermal effects can be neglected. RIS has been proven to be useful in modeling hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in both small and large strain regimes. In the second chapter the existence of solutions to an evolutionary rateindepend ent model of Shape Memory Alloys (SMAs) is proven. The model...


Relaxace v mechanice kontinua tuhé fáze
Pathó, Gabriel ; Kružík, Martin (advisor) ; Zeman, Jan (referee)
This work deals with the modelling of shapememory alloys, in particular with the steadystate model of martensitic thin films. After the introductory motivation the crystallographic structure of the materials is described followed by the introduction of the link between the lattice and continuum model. The next parts of the work focus on the possible solutions of the given 3D variational problem (quasiconvexification, Young measures) and on derivation of thin film theories with the aid of different tools (regularization,convergence). The last part takes over an approximation of an obtained model and sketches numerical experiments on a NiMnGa alloy.


Mathematical modelling of thin films of martensitic materials
Pathó, Gabriel ; Kružík, Martin (advisor) ; Kalamajska, Agnieszka (referee) ; Šilhavý, Miroslav (referee)
The aim of the thesis is the mathematical and computer modelling of thin films of martensitic materials. We derive a thermodynamic thinfilm model on the mesoscale that is capable of capturing the evolutionary process of the shapememory effect through a twostep procedure. First, we apply dimension reduction techniques in a microscopic bulk model, then enlarge gauge by neglecting microscopic interfacial effects. Computer modelling of thin films is conducted for the static case that accounts for a modified Hadamard jump condition which allows for austenitemartensite interfaces that do not exist in the bulk. Further, we characterize $L^p$Young measures generated by invertible matrices, that have possibly positive determinant as well. The gradient case is covered for mappings the gradients and inverted gradients of which belong to $L^\infty$, a nontrivial problem is the manipulation with boundary conditions on generating sequences, as standard cutoff methods are inapplicable due to the determinant constraint. Lastly, we present new results concerning weak lower semicontinuity of integral functionals along (asymptotically) $\mathcal{A}$free sequences that are possibly negative and noncoercive. Powered by TCPDF (www.tcpdf.org)


Variational Methods in Thermomechanics of Solids
Pelech, Petr ; Kružík, Martin (advisor) ; Dondl, Patrick (referee) ; Zeman, Jan (referee)
The thesis is devoted to study of continuum mechanics and thermodynamics and the related mathematical analysis. It consists of four selfcontained chapters dealing with different aspects. The first chapter focuses on peridynamics, a nonlocal theory of continuum mechanics, and its relation to conventional local theory of CauchyGreen elasticity. Similar compar isons has been used for proving consistency and for determining some of the material coefficients in peridynamics, provided the material parameters in the local theory are known. In this chapter the formula for the nonlocal forceflux is computed in terms of the peridynamic interaction, relating the fundamental concepts of these two theories and establishing hence a new connection, not present in the previous works. The second and third chapters are both devoted to RateIndependent Systems (RIS) and their applications to continuum mechanics. RIS represents a suitable approximation when the internal, viscous, and thermal effects can be neglected. RIS has been proven to be useful in modeling hysteresis, phase transitions in solids, elastoplasticity, damage, or fracture in both small and large strain regimes. In the second chapter the existence of solutions to an evolutionary rateindepend ent model of Shape Memory Alloys (SMAs) is proven. The model...


Application of the spectral method to the simulation of the phasefield model for martensitic transformation
Sejková, Klára ; Tůma, Karel (advisor) ; Kružík, Martin (referee)
For some alloys martensitic transformation is responsible for the socalled shape memory effect and pseudoelasticity. These properties are used in a wide range of industry applications. Each of these materials is transformed to the shape it was manufactured in when heated to its critical temperature (austenite phase) no matter how seriously it was deformed at lower temperatures (martensite phase). Looking at the microstructure, one can observe significant change of crystalographic lattice depending on temperature and deformation. This the sis focuses on modelling the evolution of microstructure during deformation for materials in the martensite phase. In this case, the creation of multiple variants of martensite is observed, divided by interfaces where a part of energy is stored. This behaviour can be described by the phasefield model. The numerical im plementation of this model using the standard finite element method requires large computational costs. The aim of this thesis is to implement this model in MATLAB using a spectral method based on the fast Fourier transform, which is suitable for solving problems on a periodic domain. It is interesting to com pare the computation using spectral method on a conventional PC with the computation written in FEniCS computed on a cluster. However, the...

 