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Application of the gradient elasticity in fracture mechanics problems
Klepáč, Jaromír ; Profant, Tomáš (referee) ; Kotoul, Michal (advisor)
The presented master’s thesis deals with the application of the gradient elasticity in fracture mechanics problems. Specifically, the displacement and stress field around the crack tip is a matter of interest. The influence of a material microstructure is considered. Introductory chapters are devoted to a brief historical overview of gradient models and definition of basic equations of dipolar gradient elasticity derived from Mindlin gradient theory form II. For comparison, relations of classical elasticity are introduced. Then a derivation of asymptotic displacement field using the Williams asymptotic technique follows. In the case of gradient elasticity, also the calculation of the J-integral is included. The mathematical formulation is reduced due to the singular nature of the problem to singular integral equations. The methods for solving integral equations in Cauchy principal value and Hadamard finite part sense are briefly introduced. For the evaluation of regular kernel, a Gauss-Chebyshev quadrature is used. There also mentioned approximate methods for solving systems of integral equations such as the weighted residual method, especially the least square method with collocation points. In the main part of the thesis the system of integral equations is derived using the Fourier transform for straight crack in an infinite body. This system is then solved numerically in the software Mathematica and the results are compared with the finite element model of ceramic foam.
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Solution of General Stress Concentrators in Anisotropic Media by Combination of FEM and the Complex Potential Theory
Ševeček, Oldřich ; Kotoul, Michal (advisor)
Disertační práce se věnuje problematice obecných koncentrátorů napětí v anisotropních prostředích. Zejména se jedná o problém trhlin končících na rozhraní dvou různých materiálů, či problém obecného více-materiálového klínu. Cílem práce je vytvořit komplexní nástroj pro posuzování obecných koncetrátorů napětí tj, popis pole napětí v jeho okolí, zahrnutí případného vlivu přemostění trhliny do výsledného pole napětí a definici lomových kritérií pro obecný koncentrátor v anisotropním prostředí. U popisu pole napětí je využit tzv. Lechnického-Strohův formalismus a technika spojitě rozložených dislokací využívající teorii komplexních potenciálů. V práci je rovněž široce uplatněn tzv. dvoustavový "psí"-integrál (pro výpočet různých součinitelů asymptotického rozvoje pro napětí), založený na Bettiho recipročním teorému v kombinaci s metodou konečných prvků. Pro formulaci lomových kritérií je použita teorie tzv. „konečné lomové mechaniky“ a teorie sdružených asymptotických rozvojů. Studován je především vztah mezi ohybem trhliny podél rozhraní a její případnou pentrací do základního materiálu. Veškeré potřebné výpočty jsou prováděny v matematických softwarech MAPLE 10.0, MATLAB 7.1 a konečnoprvkovém systému ANSYS 10.0.
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Constitutive Modelling of Composites with Elastomer Matrix and Fibres with Significant Bending Stiffness
Fedorova, Svitlana ; Kotoul, Michal (referee) ; Menzel, Andreas (referee) ; Burša, Jiří (advisor)
Constitutive modelling of fibre reinforced solids is the focus of this work. To account for the resulting anisotropy of material, the corresponding strain energy function contains additional terms. Thus, tensile stiffness in the fibre direction is characterised by additional strain invariant and respective material constant. In this way deformation in the fibre direction is penalised. Following this logic, the model investigated in this work includes the term that penalises change in curvature in the fibre direction. The model is based on the large strain anisotropic formulation involving couple stresses, also referred to as “polar elasticity for fibre reinforced solids”. The need of such formulation arises when the size effect becomes significant. Mechanical tests are carried out to confirm the limits of applicability of the classical elasticity for constitutive description of composites with thick fibres. Classical unimaterial models fail to take into account the size affect of fibres and their bending stiffness contribution. The specific simplified model is chosen, which involves new kinematic quantities related to fibre curvature and the corresponding material stiffness parameters. In particular, additional constant k3 (associated with the fibre bending stiffness) is considered. Within the small strains framework, k3 is analytically linked to the geometric and material properties of the composite and can serve as a parameter augmenting the integral stiffness of the whole plate. The numerical tests using the updated finite element code for couple stress theory confirm the relevance of this approach. An analytical study is also carried out, extending the existing solution by Farhat and Soldatos for the fibre-reinforced plate, by including additional extra moduli into constitutive description. Solution for a pure bending problem is extended analytically for couple stress theory. Size effect of fibres is observed analytically. Verification of the new constitutive model and the updated code is carried out using new exact solution for the anisotropic couple stress continuum with the incompressibility constraint. Perfect agreement is achieved for small strain case. Large strain problem is considered by finite element method only qualitatively. Three cases of kinematic constraints on transversely isotropic material are considered in the last section: incompressibility, inextensibility and the double constraint case. They are compared with a general material formulation in which the independent elastic constants are manipulated in order to converge the solution to the “constraint” formulation solution. The problem of a thick plate under sinusoidal load is used as a test problem. The inclusion of couple stresses and additional bending stiffness constant is considered as well. The scheme of determination of the additional constant d31 is suggested by using mechanical tests combined with the analytical procedure.
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Implementation, Calibration and Application of Ductile Fracture Conditions in FEM Programs
Kubík, Petr ; Plánička, František (referee) ; Kotoul, Michal (referee) ; Petruška, Jindřich (advisor)
The presented work is concerning with ductile fracture problematic under monotonic loading which is result of gradual material degradation at large plastic deformation. At present, a large number of models, which calibration is not trivial, are used for its prediction. Ductile fracture mechanisms and cut-off region are described in the literature search part. Next, there is given a summary of criteria which are based on different approaches to ductile fracture. There were proposed and used KHPS and KHPS2 criteria within this PhD thesis. Procedure of assessment of the stress-strain curve and ductile fracture criteria calibration are described in last part of the literature search. A summary of tests, which were done in order to calibrate chosen criteria, is given in experimental part. Various stress states within broad range of stress triaxiality and Lode parameter were reached using these tests. Special type of specimen, by which very low stress triaxiality value is reached, was designed within this work. All specimens were made from the steel 12 050. Wrought rods of one melt with circular cross-section of 27 mm in diameter were used as semi-product. Calibration of chosen ductile fracture criteria was performed using foregoing tests. These criteria were implemented by author into explicit finite element software ABAQUS/Explicit using user subroutine VUMAT. Chosen criteria were used for simulation of multistep extrusion at which formation of internal central cracks occurs. These criteria were also used for simulation of cutting of circular cross-section rods. Results from simulation were compared to experiments which were done by industrial partner J-VST.
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Effects of Defects on Composite Structures Load Carrying Capacity: Delaminations at a Bi-Material Interface
Matěják, Vladimír ; Kotoul, Michal (referee) ; Růžička, Milan (referee) ; Juračka, Jaroslav (advisor)
Kompozitní materiály se projevují komplexním způsobem porušování, které může být dále ovlivněno přítomností různých poruch plynoucích z výrobních processů nebo se vyskytujících v průběhu života součásti. Důkladné porozmění procesů porušování a jejich okolností je nezbytné pro navrhování kompozitních konstrukcí, jenž budou bezpečnější, trvanlivější a ekonomičtější. V první části disertační práce jsou popsány způsoby porušování kompozitů a uvedeny současné matematické metody pro analýzu a výpočet únosnosti. Dále jsou zde vyjmenovány hlavní druhy vad a stručně diskutován jejich vliv na vlastnosti kompozitních materiálů. Zvláštní důraz je kladen na delaminace, společně se základními principy lomové mechaniky a jejich uplatnění při výpočtech a zkoušení kompozitů. Druhá část je zaměřena na delaminace na rozhranní dvou různých materialů. Lomová houževnatost byla experimantálně měřena ve třech typech zatížení za účelem stanovení poruchového kritéria založeného na podílu módu I a módu II. Během tohoto experimentu byla vyvinuta nová metoda měření délky trhliny pomocí digitáního zpracování obrazu a rovněž byla navržena nová definice počátku šíření trhliny. Analytické vztahy pro výpočet míry uvolnění deformační energie z naměřených dat byly rozšířeny o vliv rozdílných elastické parametrů materiálů na rozhranní. Podrobnější prozkoumání analytických vztahů a výpočet metodou konečných prvků odhalil, že podíl módu I a módu II je závislý na vzdálenosti od čela trhliny a poruchové kritérium založené na podílu smíšenosti tak nemůže být použito.
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Polymeric metamaterials with advanced mechanical properties
Štaffová, Martina ; Lehocký,, Marian (referee) ; Kotoul, Michal (referee) ; Jančář, Josef (advisor)
Práce se zabývá 3D tištěnými polymerními auxetickými strukturami s důrazem na jejich mechanické vlastnosti. Pro zlepšení mechanických vlastností, zejména poměru schopnosti absorpce energie k hmotnosti při tlakovém zatížení, byly použity různé návrhy auxetických re-entrantních struktur. Struktury byly vytištěny pomocí stereolitografické 3D tiskárny. Byl vyvinut a systematicky demonstrován nový nástroj pro komplexní charakterizaci 3D tištěných těles s využitím komerční fotopolymerizační pryskyřice s volnými radikály. Metoda se opírá o statický a oscilační mechanický test kombinující měření teploty průhybu při zatížení (HDT) s dynamickou mechanickou analýzou (DMA) v jediném testu pro rychlou a spolehlivou charakterizaci parametrů určujících chování fotopolymeru při vytvrzování. Byly zkoumány podmínky tisku a doba vytvrzení, aby byl objasněn jejich vliv na vnitřní napětí v 3D tištěných tělesech. Auxetické struktury byly vytištěny pomocí flexibilní a tuhé pryskyřice a byl zkoumán vliv poréznosti, velikosti buněk a strukturálních gradientů na mechanické vlastnosti. Výsledky byly vyhodnoceny z konstrukčního a materiálového hlediska pro fotopryskyřice nad a pod Tg. Nejlepší účinnost absorbování energie byla zjištěna u biaxiálně orientované struktury s nejvyšší lokální porozitou umístěnou ve středu. Prezentovaná data přispívají k základnímu pochopení účinků designu buněk na deformační odezvu auxetických struktur a mohou pomoct při návrhu nových struktur s lepšími mechanickými vlastnostmi. Výsledky zároveň demonstrují všestrannost 3D tisku při realizaci komplexních, přírodou inspirovaných struktur.
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Mechanical Reinforcement of Bioglass®-Based Scaffolds
Bertolla, Luca ; Prof. Dr.-Ing. habil. Aldo R. Boccaccini (referee) ; Kotoul, Michal (referee) ; Pabst, Willi (referee) ; Dlouhý, Ivo (advisor)
Bioactive glasses exhibit unique characteristics as a material for bone tissue engineering. Unfortunately, their extensive application for the repair of load-bearing bone defects is still limited by low mechanical strength and fracture toughness. The main aim of this work was two-fold: the reinforcement of brittle Bioglass®-based porous scaffolds and the production of bulk Bioglass® samples exhibiting enhanced mechanical properties. For the first task, scaffolds were coated by composite coating constituted by polyvinyl alcohol (PVA) and microfibrillated cellulose (MFC). The addition of PVA/MFC coating led to a 10 fold increase of compressive strength and a 20 fold increase of tensile strength in comparison with non-coated scaffolds. SEM observations of broken struts surfaces proved the reinforcing and toughening mechanism of the composite coating which was ascribed to crack bridging and fracture of cellulose fibrils. The mechanical properties of the coating material were investigated by tensile testing of PVA/MFC stand–alone specimens. The stirring time of the PVA/MFC solution came out as a crucial parameter in order to achieve a more homogeneous dispersion of the fibres and consequently enhanced strength and stiffness. Numerical simulation of a PVA coated Bioglass® strut revealed the infiltration depth of the coating until the crack tip as the most effective criterion for the struts strengthening. Contact angle and linear viscosity measurements of PVA/MFC solutions showed that MFC causes a reduction in contact angle and a drastic increase in viscosity, indicating that a balance between these opposing effects must be achieved. Concerning the production of bulk samples, conventional furnace and spark plasma sintering technique was used. Spark plasma sintering performed without the assistance of mechanical pressure and at heating rates ranging from 100 to 300°C /min led to a material having density close to theoretical one and fracture toughness nearly 4 times higher in comparison with conventional sintering. Fractographic analysis revealed the crack deflection as the main toughening mechanisms acting in the bulk Bioglass®. Time–dependent crack healing process was also observed. The further investigation on the non-equilibrium phases crystallized is required. All obtained results are discussed in detail and general recommendations for scaffolds with enhanced mechanical resistance are served.
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Modeling of rigid body rotation movement in space
Komůrková, Lucia ; Kotoul, Michal (referee) ; Franců, Jan (advisor)
This bachelor thesis is focused on rotation movement of rigid body in space. It is described rotation matrices and their characteristics. It is devoted to Euler angles representation of rotation in space and introduced Euler kinematic and dynamic equations. Moreover, it indicates the numerical and analytical method for solving these equations for special types of flywheels.
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A study of the stress distribution around the bimaterial notch tip in the terms of the generalized stress intensity factor
Hrstka, Miroslav ; Kotoul, Michal (referee) ; Profant, Tomáš (advisor)
The presented diploma thesis deals with a problem of a generalized stress intensity factor determination and a consecutive study of stress distribution around the bimaterial notch tip, combining analytical and numerical methods. This task is possible to sectionalize into three parts. The first part is dedicated to the fundamentals of the linear fracture mechanics and the mechanics of composite materials. The second part deals with methods of anisotropic plane elasticity solution. Pursuant to the solution the computational models in the third part are created. The first model makes for determination of a singularity exponent eigenvalue by dint of Lekhnitskii-Eshelby-Stroh formalism. The second model makes for determination of the generalized stress intensity factor using psi-integral method, which is based on the Betti reciprocal theorem. All needed calculation are performed in the software ANSYS 12, Maple 12 and Silverforst FTN95. Results will be compared with the values obtained from a direct method of the generalised stress intensity factor determination.
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