50 research outputs found

    Formability of Aluminum Alloy Sheet at Elevated Temperature

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    An experimental and numerical study of the isothermal and non-isothermal warm formability of an AA3003 aluminum alloy brazing sheet is presented. Forming limit diagrams were determined using warm limiting dome height (LDH) experiments with in situ strain measurement based on digital image correlation (DIC) techniques. Forming limit curves (FLCs) were developed at several temperature levels (room temperature, 100ÂșC, 200ÂșC, 250ÂșC, and 300ÂșC) and strain-rates (0.003, 0.018, and 0.1s-1). The formability experiments demonstrated that temperature has a significant effect on formability, whereas forming speed has a mild effect within the studied range. Elevating the temperature to 250C improved the formability more than 200% compared to room temperature forming, while forming at lower speeds increased the limiting strains by 10% and 17% at room temperature and 250ÂșC, respectively. Non-isothermal deep draw experiments were developed considering an automotive heat exchanger plate. A parametric study of the effects of die temperature, punch speed, and blank holder force on the formability of the part was conducted. The introduction of non-isothermal conditions in which the punch is cooled and the flange region is heated to 250C resulted in a 61% increase in draw depth relative to room temperature forming. In order to develop effective numerical models of warm forming processes, a constitutive model is proposed for aluminum alloy sheet to account for temperature and strain rate dependency, as well as plastic anisotropy. The model combines the Barlat YLD2000 yield criterion (Barlat et al., 2003) to capture sheet anisotropy and the Bergstrom (1982) hardening rule to account for temperature and strain rate dependency. Stress-strain curves for AA3003 aluminum alloy brazing sheet tested at elevated temperatures and a range of strain rates were used to fit the Bergstrom parameters, while measured R-values were used to fit the yield function parameters. The combined constitutive model was implemented within a user defined material subroutine that was linked to the LS-DYNA finite element code. Finite element models were developed based on the proposed material model and the results were compared with experimental data. Isothermal uniaxial tensile tests were simulated and the predicted responses were compared with measured data. The tensile test simulations accurately predicted material behaviour. The user material subroutine and forming limit criteria were then applied to simulate the isothermal warm LDH tests, as well as isothermal and non-isothermal warm deep drawing experiments. Two deep draw geometries were considered, the heat exchanger plate experiments developed as part of this research and the 100 mm cylindrical cup draw experiments performed by McKinley et al. (2010). The strain distributions, punch forces and failure location predicted for all three forming operations were in good agreement with the experimental results. Using the warm forming limit curves, the models were able to accurately predict the punch depths to failure as well as the location of failure initiation for both the isothermal and non-isothermal deep draw operations

    Development of Polymer Nanocomposites from Renewable Cellulose Nanocrystals

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    L'utilisation de la cellulose nanocristalline (CNC) comme agent de renfort dans les polymĂšres est en croissance; toutefois, cela reste limitĂ© Ă  quelques polymĂšres hydrosolubles ou sous forme de latex. La plupart des polymĂšres industriels communs Ă©tant insolubles dans l'eau, par consĂ©quent, la CNC hydrophile ne peut ĂȘtre dispersĂ©e dans des matrices polymĂšres hydrophobes. Une compatibilisation ou une modification de surface sont donc nĂ©cessaires pour obtenir une bonne dispersion et, de ce fait, amĂ©liorer les propriĂ©tĂ©s des polymĂšres. Dans ce projet de recherche, notre objectif Ă©tait de dĂ©velopper des nanocomposites polymĂšre-CNC avec une structure bien dispersĂ©e et des propriĂ©tĂ©s amĂ©liorĂ©es, comparativement aux polymĂšres non modifiĂ©s ou avec de faibles charges de CNC. À cet effet, nous avons choisi le polypropylĂšne (PP), une des polyolĂ©fines les plus communĂ©ment utilisĂ©es dans l'industrie, et le polylactide (PLA), le biopolymĂšre ayant suscitĂ© le plus d’intĂ©rĂȘt dans l'industrie. Par consĂ©quent, notre Ă©tude a Ă©tĂ© divisĂ©e en deux grandes parties. La premiĂšre phase du projet portait sur le dĂ©veloppement des nanocomposites PP-CNC. Pour favoriser une meilleure dispersion et mouillage des CNC hydrophiles dans la matrice de PP hydrophobe, un poly(Ă©thylĂšne-co-vinyle alcool) a Ă©tĂ© utilisĂ© comme compatibilisant via une mĂ©thode de prĂ©paration d’un mélange maĂźtre (masterbatch). Deux mĂ©langes maĂźtres diffĂ©rents ont Ă©tĂ© prĂ©parĂ©s, l’un en solution et l'autre Ă  l'Ă©tat fondu, avec un rapport CNC/compatibilisant de 1/3. Les nanocomposites ont Ă©tĂ© ensuite prĂ©parĂ©s par homogĂ©nĂ©isation Ă  l'Ă©tat fondu des mĂ©langes maĂźtres avec le PP, et avec une teneur en CNC de 5% en poids. Un mĂ©langeur interne a Ă©tĂ© utilisĂ© Ă  cet effet. Pour des fins de comparaison, du PP contenant 15% en poids d'agent compatibilisant et du PP contenant 5% en poids de CNC ont Ă©galement Ă©tĂ© prĂ©parĂ©s par mĂ©lange direct Ă  l'Ă©tat fondu. Par la suite, l'effet de la CNC sur les propriĂ©tĂ©s rhĂ©ologiques, mĂ©caniques et thermiques du PP a Ă©tĂ© Ă©tudiĂ©. Les observations en microscopie Ă©lectronique Ă  balayage ont montrĂ© la disparition de gros agglomĂ©rats aprĂšs ajout du compatibilisant. La viscositĂ© complexe et le module de conservation des Ă©chantillons nanocomposites compatibilisĂ©s et obtenus avec la mĂ©thode en solution ont Ă©tĂ© augmentĂ©s Ă  basses frĂ©quences, comparativement aux nanocomposites non compatibilisĂ©s. De plus, un maximum suivi d’une décroissance (overshoot) prononcĂ© de la viscositĂ© transitoire, dans un test de croissance de la contrainte en cisaillement a Ă©tĂ© observĂ© pour vi le nanocomposite compatibilisĂ©. Le module d'Young a augmentĂ© jusqu'Ă  prĂšs de 47% par rapport au PP seul; la rĂ©sistance Ă  la rupture en traction est restĂ©e inchangĂ©e et l’allongement Ă  la rupture a diminuĂ© pour le nanocomposite compatibilisĂ©. Le module de conservation du PP Ă  tempĂ©rature ambiante dans une analyse thermique mĂ©canique dynamique (DMTA) a augmentĂ© jusqu'Ă  prĂšs de 60%, quand une bonne dispersion a Ă©tĂ© atteinte en utilisant l'agent compatibilisant. L'effet de nuclĂ©ation de la CNC sur la cristallisation du PP a Ă©tĂ© confirmĂ© par l’augmentation de la cristallinitĂ© du nanocomposite PP et par l’augmentation de la tempĂ©rature de cristallisation aprĂšs l’ajout du compatibilisant. Ces observations ont Ă©tĂ© attribuĂ©es Ă  l'efficacitĂ© du poly(Ă©thylĂšne-co-vinyle alcool) comme un agent de compatibilisation et ce, en particulier lorsque le mĂ©lange maĂźtre a Ă©tĂ© prĂ©parĂ© en solution. La deuxiĂšme phase du projet a focalisĂ© sur le dĂ©veloppement de bio-nanocomposites PLA-CNC Ă  haute performance. Une simple et nouvelle mĂ©thode de coulĂ©e utilisant un solvant polaire (N,N-dimĂ©thylformamide) a Ă©tĂ© utilisĂ©e pour favoriser une bonne dispersion de la CNC hydrophile dans une matrice PLA Ă  diffĂ©rentes charges de CNC, sans toutefois la nĂ©cessitĂ© de modifier la CNC ni d’utiliser un compatibilisant. Par la suite, l'effet de la CNC sur les propriĂ©tĂ©s du PLA a Ă©tĂ© Ă©tudiĂ© Ă  l’état fondu et Ă  l’état solide. La microscopie Ă©lectronique Ă  transmission a rĂ©vĂ©lĂ© l'existence d'une structure bien dispersĂ©e de la CNC dans la matrice de PLA. La viscositĂ© complexe et les modules de conservation et de perte ont augmentĂ© de maniĂšre significative par l'incorporation de 1% en poids et plus de CNC, en particulier Ă  basses frĂ©quences. Ces amĂ©liorations ont Ă©tĂ© attribuĂ©es Ă  la formation d'un rĂ©seau interconnectĂ© de la CNC dans la matrice PLA. Le seuil de percolation rhĂ©ologique a Ă©tĂ© Ă©tabli Ă  0,68% en poids de CNC. En outre, la viscositĂ© en rĂ©gime permanent, la contrainte de cisaillement et la premiĂšre diffĂ©rence des contraintes normales ont augmentĂ©s. Le seuil apparent des contraintes a Ă©galement augmentĂ© avec la teneur en CNC, aussi bien en SAOS qu’en cisaillement en régime permanent. La rĂšgle de Cox-Merz n’était pas valable pour les nanocomposites PLA-CNC; Toutefois, pour les nanocomposites Ă  contenu Ă©levĂ© en CNC, la rĂšgle de Cox-Merz Ă©tendue Ă©tait applicable. Des maxima prononcĂ©s suivis d’une décroissance (overshoots) ont Ă©tĂ© observĂ©s en cisaillement transitoire et pour des Ă©coulements dans un sens et le sens inverse. Ces maxima suivis d’une dĂ©croissance (overshoots) ainsi que la reprise de structure suite Ă  des prĂ©-cisaillements ont Ă©tĂ© expliqués par la formation d’un réseau et par le rĂŽle important du mouvement Brownien pour vii rĂ©tablir la structure. Les nanocomposites moins concentrĂ©s prĂ©-cisaillĂ©s Ă  des taux plus Ă©levĂ©s ont nécessité des temps plus longs pour rétablir leur structure initiale, après l’arrĂȘt du cisaillement. Les propriĂ©tĂ©s rhĂ©ologiques ont dĂ©montrĂ© l'efficacitĂ© de la sonication pour disperser la CNC et par consĂ©quent, pour former un rĂ©seau interconnectĂ© de CNC dans la matrice PLA. Le module d'Young des nanocomposites a augmentĂ© jusqu'Ă  environ 23% comparativement au PLA seul; cependant, la dĂ©formation Ă  la rupture a lĂ©gĂšrement diminuĂ© et la contrainte en traction quant Ă  elle est demeurĂ©e inchangĂ©e. En DMTA, le module de conservation des nanocomposites PLA-CNC a augmentĂ© jusqu'Ă  prĂšs de 74% dans la rĂ©gion vitreuse et jusqu'Ă  prĂšs de 490% dans la rĂ©gion caoutchoutique. L’augmentation de la cristallinitĂ© du PLA dans les nanocomposites et le dĂ©placement de la tempĂ©rature de cristallisation vers des valeurs plus Ă©levĂ©es ont Ă©tĂ© attribuĂ©s Ă  l'effet de nuclĂ©ation de la CNC sur la cristallisation du PLA. A notre connaissance, il s’agit de la première fois qu’une aussi bonne dispersion de la CNC dans un polymĂšre non hydrosoluble est obtenue et que de telles amĂ©liorations dans les propriĂ©tĂ©s des nanocomposites Ă  l'Ă©tat solide et fondu sont atteintes, sans aucun changement dans la structure morphologique de la CNC ou l'utilisation de compatibilisants. En outre, au meilleur de notre connaissance, ceci est la premiĂšre Ă©tude fondamentale du comportement rhĂ©ologique des nanocomposites PLA-CNC dans diffĂ©rents modes d'Ă©coulement en cisaillement. ---------- The application of cellulose nanocrystals (CNCs) as reinforcing agents in polymers have grown, but limited to a few water-soluble polymers or polymers in latex form. However, most of the common polymers in industry are non water-soluble. Therefore, the hydrophilic CNCs cannot be dispersed in hydrophobic polymer matrices and surface modification or compatibilization are necessary to achieve a good dispersion and, consequently, to enhance the properties of polymers. In this research project our objective was to develop polymer-CNC nanocomposites with a dispersed structure and enhanced properties, relative to the neat polymers, at low CNC loadings. To this end, we selected as matrices polypropylene (PP), which is one the most commonly polyolefins, and polylactide (PLA), which has attracted the largest interest among the biopolymers in industry. The research project was conducted in two phases. The first phase of the project dealt with the development of the PP-CNC nanocomposite. To favor a better dispersion and wetting of hydrophilic CNCs within the hydrophobic PP matrix a poly(ethylene-co-vinyl alcohol) was used as a compatibilizer via a masterbatch preparation method. Two different masterbatches were prepared, one in solution and the other in molten state, with the CNC to compatibilizer ratio of 1/3. Then, the nanocomposites were prepared by melt-mixing of the masterbatches with PP using an internal mixer at a CNC content of 5 wt%. For sake of comparison, PP containing 15 wt% compatibilizer and PP containing 5 wt% CNCs were also prepared via direct melt mixing. Then, the effect of the CNCs on the rheological, mechanical and thermal properties of PP was investigated. Scanning electron microscopy showed the disappearance of large agglomerates when the compatibilizer was employed. The complex viscosity and storage modulus of the nanocomposite sample containing the compatibilizer prepared by a solution method were increased at low frequencies, compared to those of the non-compatibilized nanocomposite samples. Moreover, a pronounced overshoot for the transient viscosity in a shear stress growth experiment was observed for the compatibilized nanocomposite. The Young modulus was enhanced, up to ca. 47 % compared to the neat PP; the tensile strength remained unchanged and strain at break was decreased for the compatibilized nanocomposite. The storage modulus of the PP at room temperature in dynamic mechanical thermal analysis (DMTA) was increased up to ca. 60% when a good state of dispersion was achieved using the compatibilizer. The nucleation effect of the CNCs on the crystallization of the ix PP was confirmed by the enhanced crystalline content of the PP nanocomposites and increased crystallization temperature using the compatibilizer. These observations were ascribed to the efficiency of the poly(ethylene-co-vinyl alcohol) as a compatibilizer, especially when the masterbatch was prepared in solution. The second phase focused on the development of high performance PLA-CNC bionanocomposites. A simple and novel solution casting method with a polar solvent (i.e. N,N-dimethylformamide) was used to favor a good dispersion of hydrophilic CNCs within a PLA matrix at different CNC loadings without the need of CNC modification or use of any compatibilizer. Then, the effect of the CNCs on the properties of the PLA for both molten and solid states was investigated. Transmission electron microscopy revealed the existence of well-dispersed structure of CNCs within the PLA matrix. The complex viscosity and the storage and loss moduli were increased significantly, especially at low frequencies, by the incorporation of 1 wt% CNCs and larger contents. These enhancements were ascribed to the formation of an interconnected network of CNCs within the PLA matrix. The rheological percolation threshold was determined to be 0.68 wt% CNC. Furthermore, the steady-state viscosity, shear stress and first normal stress difference were increased. The apparent yield stress increased with CNC content, for both SAOS and steady-shear experiments. The Cox-Merz rule was not valid for the PLA-CNC nanocomposites; however, for nanocomposites at large CNC contents the extended Cox-Merz rule was applicable. Pronounced overshoots in transient shear for forward and reverse flows and the structure build-up of samples after pre-shearing were explained by a network formation and the important role of the Brownian motion for structure recovery. The less concentrated nanocomposites pre-sheared at larger rates needed longer times after the cessation of the shear flow to recover their initial structure. The rheological properties demonstrated the efficiency of the sonicator in solution mixing to disperse the CNCs with the consequent formation of an interconnected network of CNCs within the PLA matrix. The Young modulus of the nanocomposites increased up to ca. 23% compared to the neat PLA; however, the strain at break slightly decreased and tensile strength remained unchanged. In DMTA, the storage modulus of the PLA-CNC naonocomposites increased up to ca. 74% in glassy region and up to ca. 490% in the rubbery region. The observation of increased crystalline content of the PLA in the nanocomposites and shift in the crystallization temperature x towards higher temperatures, were attributed to the nucleation effect of the CNCs on the crystallization of PLA. To our knowledge, this is the first time that a very good dispersion of CNCs in a non-hydrosoluble polymer together with huge increases in the properties of the corresponding solid and molten state nanocomposite is achieved, without any change in the morphological structure of the CNCs or use of any compatibilizer. Moreover, to the best of our knowledge, this is the first systematic investigation of the rheological behavior of PLA-CNC nanocomposites in different shear flow fields

    Comportement rhĂ©ologique Ă  l’état fondu de nanocomposites Ă  base de nanocristaux de cellulose (CNCs)

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    RĂ©sumĂ© Les propriĂ©tĂ©s rhĂ©ologiques de suspensions de nanocristaux de cellulose (CNCs), dans le polyĂ©thylĂšne glycol (PEG) et le polylactide (PLA), ont Ă©tĂ© Ă©tudiĂ©es Ă  l’état fondu. Les CNCs, obtenus par une hydrolyse Ă  l'acide sulfurique de pulpe de bois, sont sĂ©chĂ©s par pulvĂ©risation ou lyophilisation. L’ultrasonication a Ă©tĂ© utilisĂ©e pour disperser les CNCs en solvants favorables aux polymĂšres respectifs. Ces suspensions sont ensuite mĂ©langĂ©es Ă  des solutions de PEG ou PLA. Le sĂ©chage sous vide des Ă©chantillons et leur moulage par compression permet l’obtention de nanocomposites. Ces derniers possĂšdent, Ă  l'Ă©tat fondu, un comportement Ă  seuil dĂšs une faible fraction volumique de CNCs : ~0,2 et 0,55% vol dans les matrices PEG et PLA, respectivement. Ces rĂ©sultats sont caractĂ©ristiques d'un rĂ©seau de percolation et suggĂšrent une trĂšs bonne dispersion des CNCs dans ces matrices thermoplastiques, en accord avec les observations effectuĂ©es en microscopie Ă©lectronique en transmission. L’évolution du module Ă©lastique relatif et de la viscositĂ© complexe relative suggĂšrent que l’état de dispersion des CNCs est meilleur dans le PEG que dans le PLA. Ces indicateurs rhĂ©ologiques sont comparĂ©s aux prĂ©dictions thermodynamiques, liĂ©es aux paramĂštres de solubilitĂ© d’Hansen, pouvant ĂȘtre tirĂ©s du graphique de Teas des CNCs. ---------- Abstract The rheological properties of cellulose nanocrystals (CNCs) suspensions in molten polyethylene glycol (PEG) and polylactic acid (PLA) were investigated. The CNCs were obtained from sulfuric acid hydrolysis of wood pulp and were supplied after a spray- or freeze-drying process. Ultrasonication was used to achieve the CNC dispersion in solution and the suspensions were subsequently mixed with PEG and PLA, with a corresponding good solvent. The samples were subsequently dried under vacuum and molded by compression. In the molten state, the PEG/CNC nanocomposites exhibited a yield behavior and a percolation network from a low volume fraction of ~ 0.2vol % and 0.55vol % in PEG and PLA, respectively. This suggests a very good dispersion of CNCs in these thermoplastic matrices, in agreement with transmission electron microscopy observations. A comparaison of the relative elastic modulus and viscosity suggest a better dispersion state of CNCs in PEG. These rheological indices were compared to the thermodynamic predictions based on the Hansen solubility parameters, that may be drawn from the Teas graph of CNCs

    Development and characterisation of polyhydroxybutyrate from selected bacterial species

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    In this study Serratia sp. NCIMB 40259 and R. eutropha H16 were utilised for the accumulation of Polyhydroxybutyrate(PHB). Serratia sp. with high phosphatase activity (HPA)or low phosphatase activity (LPA) were fermented in order to accumulate intracellular PHB and biomineralise hydroxyapatite (HA) on the cell surface. The optimum conditions for the accumulation of PHB, in this study, were found to be after 216 h fermentation, producing a PHB yield of 14.6 %w/w and 3.3 %w/w for LPA and HPA respectively. The molecular and thermal properties of PHB were observed to be largely unaffected by the phosphatase activity of the cells and appeared to be dependent on the fermentation period. R. eutropha cells were fermented with the substrates olive oil and rapeseed oil and compared with glucose. The use of either olive oil or rapeseed oil as carbon sources resulted in the greatest accumulation of PHB. The thermal and molecular properties of the PHB samples were found to be almost identical. The use of oils in the production of PHB would make a viable replacement for the use of sugars as substrates. Electrospun fibres of PHB from glucose, olive oil and rapeseed oil were found to have diameters dependant on their solution concentrations. By adjusting the electrospinning parameters it may be possible to control the crystallinity and the diameters the PHB fibres

    Analyse et conception de métasurfaces symétriques supérieures

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    Les mĂ©tasurfaces plus symĂ©triques montrent un comportement dispersif intĂ©ressant par rapport aux structures sans symĂ©trie supĂ©rieure. Par exemple, ils assurent une propagation non dispersive dans leur bande passante et une forte attĂ©nuation dans leur bande d'arrĂȘt. En raison de ces comportements dispersifs intĂ©ressants, des structures symĂ©triques supĂ©rieures peuvent ĂȘtre utilisĂ©es dans de nombreuses applications telles que la communication mobile 5G. Dans cette thĂšse, nous dĂ©veloppons et discutons une mĂ©thode d'analyse pour les structures avec des symĂ©tries supĂ©rieurs. Nous appliquons Ă©galement cette mĂ©thode pour concevoir une technologie de guide d'ondes reconfigurable pouvant ĂȘtre utilisĂ©e dans les commutateurs Ă  ondes mm.Higher-Symmetric metasurfaces show interesting dispersive behavior compared to structures with no higher symmetry. For instance, they provide non-dispersive propagation in their passband and high attenuation in their stopband. Due to these interesting dispersive behaviors, higher-symmetric structures can be used in many applications such as in 5G mobile communication. In this thesis, we develop and discuss an analysis method for structures with higher symmetries. We also apply this method to design a reconfigurable waveguide technology that can be used in mm-wave switches

    Analyse et conception de métasurfaces symétriques supérieures

    No full text
    Higher-Symmetric metasurfaces show interesting dispersive behavior compared to structures with no higher symmetry. For instance, they provide non-dispersive propagation in their passband and high attenuation in their stopband. Due to these interesting dispersive behaviors, higher-symmetric structures can be used in many applications such as in 5G mobile communication. In this thesis, we develop and discuss an analysis method for structures with higher symmetries. We also apply this method to design a reconfigurable waveguide technology that can be used in mm-wave switches.Les mĂ©tasurfaces plus symĂ©triques montrent un comportement dispersif intĂ©ressant par rapport aux structures sans symĂ©trie supĂ©rieure. Par exemple, ils assurent une propagation non dispersive dans leur bande passante et une forte attĂ©nuation dans leur bande d'arrĂȘt. En raison de ces comportements dispersifs intĂ©ressants, des structures symĂ©triques supĂ©rieures peuvent ĂȘtre utilisĂ©es dans de nombreuses applications telles que la communication mobile 5G. Dans cette thĂšse, nous dĂ©veloppons et discutons une mĂ©thode d'analyse pour les structures avec des symĂ©tries supĂ©rieurs. Nous appliquons Ă©galement cette mĂ©thode pour concevoir une technologie de guide d'ondes reconfigurable pouvant ĂȘtre utilisĂ©e dans les commutateurs Ă  ondes mm

    Analyse et conception de métasurfaces symétriques supérieures

    No full text
    Higher-Symmetric metasurfaces show interesting dispersive behavior compared to structures with no higher symmetry. For instance, they provide non-dispersive propagation in their passband and high attenuation in their stopband. Due to these interesting dispersive behaviors, higher-symmetric structures can be used in many applications such as in 5G mobile communication. In this thesis, we develop and discuss an analysis method for structures with higher symmetries. We also apply this method to design a reconfigurable waveguide technology that can be used in mm-wave switches.Les mĂ©tasurfaces plus symĂ©triques montrent un comportement dispersif intĂ©ressant par rapport aux structures sans symĂ©trie supĂ©rieure. Par exemple, ils assurent une propagation non dispersive dans leur bande passante et une forte attĂ©nuation dans leur bande d'arrĂȘt. En raison de ces comportements dispersifs intĂ©ressants, des structures symĂ©triques supĂ©rieures peuvent ĂȘtre utilisĂ©es dans de nombreuses applications telles que la communication mobile 5G. Dans cette thĂšse, nous dĂ©veloppons et discutons une mĂ©thode d'analyse pour les structures avec des symĂ©tries supĂ©rieurs. Nous appliquons Ă©galement cette mĂ©thode pour concevoir une technologie de guide d'ondes reconfigurable pouvant ĂȘtre utilisĂ©e dans les commutateurs Ă  ondes mm

    Bloch Analysis of Electromagnetic Waves in Twist-Symmetric Lines

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    We discuss here under which conditions a periodic line with a twist-symmetric shape can be replaced by an equivalent non-twist symmetric structure having the same dispersive behavior. To this aim, we explain the effect of twist symmetry in terms of coupling among adjacent cells through higher-order waveguide modes. We use several waveguide modes to accurately derive the dispersion diagram of a line through a multimodal transmission matrix. With this method, we can calculate both the phase and attenuation constants of Bloch modes, both in shielded and open structures. In addition, we use the higher symmetry of these structures to further reduce the computational cost by restricting the analysis to a subunit cell of the structure instead of the entire unit cell. We confirm the validity of our analysis by comparing our results with those of a commercial software
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