Ústav matematiky a deskriptivní geometrie

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Damage behaviour of quasi-brittle composites: mathematical and computational aspects
(MDPI, 2025-04-11) Vala, Jiří; Tomáš, Jiří
In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.
Some peculiarities of using the extended finite element method in modelling the damage behaviour of fibre-reinforced composites
(MDPI, 2025-04-14) Kozák, Vladislav; Vala, Jiří
The present study utilises the extended finite element method (XFEM) to model fibre-reinforced composites, with a focus on crack initiation and propagation. Silicon nitride-based ceramics were selected as a model material; they represent a broad class of short fibre ceramics and have received a lot of attention in recent decades. Some peculiarities when using the XFEM, including its selected modifications, are discussed in response to applied external stresses, mainly in the viscoelastic range. Promising approaches are recommended, which lead to a more accurate description of these materials under operating conditions, focusing on the correct calculation of the macroscopic stress ahead of the propagating crack front. The authors draw on years of experience with the material and investigate the XFEM.
Large time behavior of nonautonomous linear differential equations with Kirchhoff coefficients
(Elsevier, 2024-03-31) Diblík, Josef; Pituk, Mihaly; Szederkényi, Gábor
Nonautonomous linear ordinary differential equations with Kirchhoff coefficients are considered. Under appropriate assumptions on the topology of the directed graphs of the coefficients, it is shown that if the Perron vectors of the coefficients are slowly varying at infinity, then every solution is asymptotic to a constant multiple of the Perron vectors at infinity. Our results improve and generalize some recent convergence theorems.
Use of cohesive approaches for modelling critical states in fibre-reinforced structural materials
(MDPI, 2024-07-01) Kozák, Vladislav; Vala, Jiří
During the operation of structures, stress and deformation fields occur inside the materials used, which often ends in fatal damage of the entire structure. Therefore, the modelling of this damage, including the possible formation and growth of cracks, is at the forefront of numerical and applied mathematics. The finite element method (FEM) and its modification will allow us to predict the behaviour of these structural materials. Furthermore, some practical applications based on cohesive approach are tested. The main effort is devoted to composites with fibres and searching for procedures for their accurate modelling, mainly in the area where damage can be expected to occur. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements.
Bounded solutions of delay dynamic equations on time scales
(Springer Nature, 2012-10-24) Diblík, Josef; Vítovec, Jiří
In this paper we discuss the asymptotic behavior of solutions of a delay dynamic equation $$y^{\Delta}(t)=f(t,y(\tau(t)))$$ where $f\colon\mathbb{T}\times\mathbb{R}\rightarrow\mathbb{R}$, \tau\colon\T\rightarrow \T$ is a delay function and $\mathbb{T}$ is a time scale. We formulate a principle which gives the guarantee that the graph of at least one solution of above mentioned equation stays in the prescribed domain. This principle uses the idea of the retraction method and is a suitable tool for investigating the asymptotic behavior of solutions of dynamic equations. This is illustrated by an example.

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