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<p>Neuronal current MRI (ncMRI) is a field of study to directly map electrical activity in the brain using MRI, which has many benefits over functional MRI. One potential ncMRI method, Lorentz effect imaging (LEI), has shown promise but needs a better theoretical understanding to improve its use.</p><p>We develop three computational models to simulate the LEI experiments of an electrolyte filled phantom subject to a current dipole based on: ion flow, particle drift, and electromagnetohydrodynamics (EMHD). With comparative experimental results, we use the EMHD model to better understand the Lorentz effect over a range of current strengths. We also quantify the LEI experimental images and assess ways to measure the underlying current strength, which would greatly benefit comparative brain mapping.</p><p>EMHD is a good predictor of LEI signal loss. We can measure the underlying current strength and polarity in the phantom using LEI images. We can also use trends from the EMHD model results to predict the required current density for signal detection in future LEI experiments. We can also infer the electric field strength, flow velocity, displacement, and pressure from the predicted current magnitude in an LEI experiment.</p><p>The EMHD model provides information that greatly improves the utility and understanding of LEI. Future study with our EMHD model should be performed using shorter dipole lengths, higher density and lower strength of current sources, and varying current source frequencies to understand LEI in the setting of mapping brain activity.</p> / Dissertation

A new neutron Electric Dipole Moment (nEDM) experiment was proposed to be commissioned at the Fundamental Neutron Physics Beamline at the Spallation Neutron Source (SNS) of the Oak Ridge National Laboratory (ORNL). The underlying theme of this experiment (first conceived by Golub and Lamoreaux in 1994) is the search for new physics beyond the Standard Model of particle physics. The discovery of a non-zero nEDM would be of revolutionary importance to physics, with the discovery of such providing for evidence for new-beyond-the-Standard-Model physics required for a resolution to the unresolved puzzle of why the universe is dominated by matter, as opposed to anti-matter. A first demonstration of a new magnetic field monitoring system for a neutron electric dipole moment experiment is presented. The system is designed to reconstruct the vector components of the magnetic field in the interior measurement region solely from exterior measurements. The results highlight the potential for the implementation of an improved system in an upcoming neutron electric dipole moment experiment to be carried out at the Spallation Neutron Source at Oak Ridge National Laboratory.

This thesis explores the role mutualist pollinators and their symbionts play in the genetic structuring and speciation of their host plants along an elevational gradient in Papua New Guinea. Using the fig and fig-wasp mutualism as a model system, we employed high-throughput sequencing techniques to explore fine-scale population genomics of both fig and wasps along their elevational range. We found there to be clear lowland and highland clustering of tree populations along the gradient, often with a mid-elevation contact zone. In the case of the pollinating wasps, we retrieved the same clustering except in this case, the genetic difference between clusters was high enough as to consider them as separate species. This result supports evidence from other studies challenging the cospeciation paradigm of one wasp species per fig species. In addition, we explore ecological traits which may promote, or at least, maintain, reproductive isolation between fig (sub)species along with behavioural preference tests from pollinating wasps. In order to further investigate the mechanisms promoting wasp speciation along the gradient, we describe Wolbachia infection status as well as strain type. Wolbachia-induced cytoplasmic incompatibility (CI) is often invoked as a possible speciation agent since it can rapidly provoke and maintain reproductive isolation between otherwise freely interbreeding insect populations. Finally, we explore non-pollinating fig wasp (NPFW) diversity along the gradient for a subset of our focal species. Our study reveals that there is a tight relationship between NPFW diversity and host species, and a mid-elevation peak.

O desempenho das Máquinas de Medir a Três Coordenadas (MM3Cs) fica limitado por diversos fatores, que atuam de maneira conjunta gerando os denominados erros volumétricos. Para a temperatura de 20ºC os erros geométricos podem ser considerados constantes, uma vez que variam muito lentamente com o tempo. Porém, se a temperatura é alterada estes erros mudam em grandeza e comportamento, gerando os denominados erros térmicos. Alguns trabalhos têm sido desenvolvidos com o objetivo de estudar e modelar os erros térmicos, porém os resultados alcançados são, ainda, incipientes. Este trabalho apresenta o equacionamento das componentes do erro volumétrico das MM3Cs considerando as influências térmicas. A medelagem foi aplicada a uma MM3C do tipo \"Ponte Móvel\" e combina transformações homogêneas, técnicas de regressão e mínimos quadrados. As grandezas dos erros geométricos e das variações termicamente induzidas destes erros foram coletadas utilizando-se do interferômetro laser, do esquadro mecânico, do nível eletrônico, etc. Os valores das temperaturas foram monitorados através de termopares do tipo T (Cobre-Constantan). Verificou-se que a Máquina não experimenta deformações, além, das provocadas pela livre dilatação dos seus componentes. A partir do modelo proposto foram sintetizadas as componentes do erro volumétrico, os resultados foram discutidos e comparados com aqueles obtidos através da medição de um anel padrão, constatando-se a excelente capacidade do modelo na previsão do erro volumétrico da máquina. No caso, erros da ordem de grandeza de 10 &#956m foram reduzidos em pelo menos 75%, enquanto que para erros maiores que 10 &#956m a eficiência do modelo foi de 90%. / Performance of coordinate measuring machines (CMMs) is limited by numerous factors that operate simultaneously and generate volumetric errors. The most significant portion of the volumetric error is produced by geometric errors. At the temperature of 20ºC, geometric errors can be considered at steady states, once their variation in time is considerably slow. However, if temperature is modified, these errors change in magnitude and behaviour, generating the thermal induced errors. Some work has been developed aiming to study and model the thermal errors, but the achieved results are still incipient. This work presents the derivation of the components of the volumetric error considering its thermal influences. The method was employed and applied to moving bridge CMM and combines homogeneous transformation, regression techniques and least squares methods. The magnitudes of the geometric errors and its thermally induced variations were collected by means of a laser interferometer system, mechanical square, electronic level, etc. Temperature data were monitored by means of T-type thermocouples (copper-constantan). It was verified that the CMM was not susceptible to deformations other than the ones due to the dilatation of its components. From the proposed model, the components of volumetric error were synthesized; the results were discussed and compared to the ones obtained from the measurement of a ring plug, observing the outstanding ability of the model to predict the volumetric error of the machine. Errors of 10 &#956m in magnitude were reduced in at least 75%, whilst errors greater than 10 &#956m, presented a reduction efficiency of 90%. It was verified that the CMM was not susceptible to deformations other than the ones due to the dilatation of its components.

Etudier le séchage est un moyen original de caractériser les propriétés de fluides complexes. Cette technique permet de concentrer continûment des fluides : d'un état dilué à un état sec. A l'échelle microfluidique, la manipulation, les observations et les processus qui entrent en jeu sont simplifiés. Ce travail de thèse s'attache à décrire le séchage de ces fluides et plus particulièrement le cas de dispersions colloïdales. Ces travaux présentent deux méthodes pour étudier l'extraction du solvant d'un fluide à l'échelle microfluidique : la micropervaporation et la goutte confinée. Ces techniques ont notamment permis de réaliser des estimations précises de coefficients de diffusion collective sur toute la gamme de concentrations pour un mélange eau/glycérol et pour une dispersion colloïdale de nanoparticules de silice chargées. Par ailleurs, le séchage induit des contraintes mécaniques conséquentes. Ces contraintes peuvent générer des déformations importantes, des phénomènes de délamination ou de fracturation du matériau solidifié. Une méthode originale de mesure in situ de contraintes a été mise en place pendant ces travaux. Les mesures réalisées avec une dispersion colloïdale modèle permettent de mettre en évidence expérimentalement l'apparition de contraintes mécaniques au moment de la transition sol/gel de la dispersion. L'augmentation de la contrainte est ensuite associée au séchage d'un gel poroélastique. / Drying complex fluids is an original technique to study their properties. Solvent extraction enables the continuous concentration of fluids from a dilute to a solid state. The use of the microfluidic scale allows one to limit side effects and simplify experiments, observations and modeling. This project mainly describes the drying of colloidal dispersions in two confined geometries: microfluidic channels and confined droplets between two plates. With these two techniques, we estimate collective diffusion for a water/glycerol mixture and a model dispersion of charged silica nanoparticles over the whole concentration range. Moreover, the drying of complex fluids often induces mechanical stresses which are the root for deformation, delamination phenomena and cracks. We developed an original technique to measure these stresses in situ. For a model colloidal dispersion, we evidenced experimentally that these forces arise from a liquid to solid state transition. The increase of these stresses is then associated with the drying of a poroelastic gel.

Submitted by Marcos Anselmo (marcos.anselmo@unipampa.edu.br) on 2018-09-28T17:39:10Z No. of bitstreams: 1 Daniele Missio.pdf: 779272 bytes, checksum: 6b39f5d8e9f05222e1d9034c1daa6e30 (MD5) / Approved for entry into archive by Marcos Anselmo (marcos.anselmo@unipampa.edu.br) on 2018-09-28T17:39:27Z (GMT) No. of bitstreams: 1 Daniele Missio.pdf: 779272 bytes, checksum: 6b39f5d8e9f05222e1d9034c1daa6e30 (MD5) / Made available in DSpace on 2018-09-28T17:39:27Z (GMT). No. of bitstreams: 1 Daniele Missio.pdf: 779272 bytes, checksum: 6b39f5d8e9f05222e1d9034c1daa6e30 (MD5) Previous issue date: 2017 / Na produção in vitro de embriões (PIV) bovinos são utilizadas diferentes técnicas de seleção espermática, a fim de recuperar o maior número de espermatozoides com boa qualidade e remover componentes do sêmen e diluente. Dentre essas técnicas, a seleção por gradientes descontínuos de Percoll, é a mais utilizada atualmente em laboratórios de PIV de bovinos. Porém essa técnica vem sendo modificada principalmente com o aumento da utilização do sêmen sexado. O sêmen sexado possui características distintas do sêmen convencional, como baixo número de células por dose e baixa motilidade pós-descongelamento. Assim, o desenvolvimento de um método eficiente para a seleção espermática, que aumente a taxa de recuperação sem prejudicar a qualidade dos espermatozoides é essencial. O objetivo do presente estudo foi avaliar os efeitos do volume de Percoll na recuperação espermática, qualidade dos espermatozoides, estresse oxidativo e cinética do desenvolvimento embrionário de embriões bovinos PIV. Palhetas de sêmen convencional ou sexado foram descongeladas a 35º C por 20 s e distribuídas em alíquotas iguais em três diferentes volumes de Percoll: 300 μL de cada gradiente de Percoll (90, 60 e 30%), Controle; 100 μL de cada gradiente de Percoll, P100; e 200 μL de cada gradiente de viii Percoll, P200. Qualidade espermática, taxa de fecundação e cinética do desenvolvimento embrionário até 48 horas pós-inseminação foram avaliadas. Para o sêmen convencional, a motilidade, o vigor e a taxa de recuperação foram maiores nos tratamentos P100 e P200 comparado ao Controle (P < 0,05), enquanto os níveis de espécies reativas de oxigênio (ROS), peroxidação lipídica e atividade da enzima superóxido dismutase (SOD) não foram influenciados pelos tratamentos. Para o sêmen sexado, P100 aumentou a velocidade curvilínea, velocidade média da trajetória e amplitude de deslocamento lateral da cabeça (P < 0,05). A taxa de recuperação foi maior em P100 do que no Controle e P200 (P < 0,05). A formação de ROS foi menor em P100 do que em P200 e Controle, e a atividade SOD foi menor em P100 do que no Controle. As taxas de fecundação e clivagem, momento da primeira clivagem e número de células foram similares entre P100 e Controle (P > 0,05). Portanto, o volume de Percoll de 100 μL aumentou a taxa de recuperação espermática sem prejudicar a qualidade dos espermatozoides e desenvolvimento embrionário, permitindo assim, aumentar o número de gotas inseminadas/ dose e diminuir custos. / For in vitro production (IVP) of bovine embryos, different sperm selection techniques are used to recover the highest number of sperm with good quality and to remove components of the semen and diluent. Among these techniques, the selection by discontinuous Percoll gradients is currently the most used in bovine IVP laboratories. However, this technique has been modified mainly with the increase of the use of sex-sorted semen. The sex-sorted semen has distinct characteristics of conventional semen, such as low number of cells per dose and low post-thaw motility. Thus, the development of an efficient method for sperm selection, which increases the rate of recovery without impairing the quality of sperm is essential. The aim of the present study was to evaluate the effects of Percoll volume on recovery rate, sperm quality, oxidative stress and embryo development kinetics in bovine embryos IVP. Conventional or sex-sorted semen straws were thawed at 35°C for 20 s and distributed in equal aliquots into three different Percoll volume: 300 μL of each Percoll gradient (90, 60, and 30%), Control; 100 μL of each Percoll gradient, P100; and 200 μL of each Percoll gradient, P200. Sperm quality, fertilization rate, and embryonic development kinetics up to 48 h post-insemination were evaluated. For conventional semen, motility, vigor, and recovery x rate improved in the P100 and P200 treatments compared to the Control (P < 0.05), whereas reactive oxygen species (ROS) levels, lipid peroxidation, and superoxide dismutase enzyme (SOD) activity were not influenced by treatments. For sex-sorted semen, P100 increased sperm curvilinear velocity, average path velocity, and amplitude of lateral head displacement (P < 0.05). Recovery rate was higher in P100 than in the Control and P200 (P < 0.05), formation of ROS was lower in P100 than in the Control and P200, and SOD activity was lower in P100 than in the Control. Fertilization and cleavage rates, moment of first cleavage, and cell number were similar between P100 and the Control (P > 0.05). Therefore, a Percoll volume of 100 μL increased sperm recovery rate without damage to sperm quality or early embryonic development, allowing to increase the number of inseminated drops / dose and to reduce costs.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-04-16T15:52:11Z No. of bitstreams: 2 Dissertação - Carolina Ramos Caiado Gomes - 2014.pdf: 1331727 bytes, checksum: 59847313ee21fdd7bb1bffc8147d497d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-04-17T10:55:24Z (GMT) No. of bitstreams: 2 Dissertação - Carolina Ramos Caiado Gomes - 2014.pdf: 1331727 bytes, checksum: 59847313ee21fdd7bb1bffc8147d497d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-04-17T10:55:24Z (GMT). No. of bitstreams: 2 Dissertação - Carolina Ramos Caiado Gomes - 2014.pdf: 1331727 bytes, checksum: 59847313ee21fdd7bb1bffc8147d497d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2014-05-23 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Nestedness is a particular pattern of species distribution in metacommunities in which a group of species found in poorer sites is a subset of the group of species found in richer sites. In the beta diversity partition context, nestedness is considered one of beta diversity components, jointly with species turnover. However, it is clear now that this term has been used in a wrong way instead of beta diversity due to richness differences. In specific cases that such richness differences reflect an ordered gain or loss of species between sites, then the nested pattern emerges. In the present work I used the beta diversity partition approach, focusing on the richness differences component, combined with a specific metric of nestedness, the NODF, to explore situations in which the richness differences between sites occur in a nested way considering different systems and scales of study. In the first chapter I use aquatic macroinvertebrates communities to show the importance of spatial position of patches of the same microhabitat in generating nestedness in riffles. I found that patches and riffle sites located in the beginning of the riffles are poorer then patches and riffle sites at the end of the same riffles, and that initial sites are nested in final sites in a same riffle. In the second chapter I use birds and mammals communities in the New World to assess how nestedness varies in latitudinal and longitudinal gradients. Nestedness emerged in several regions in both gradients, and it is always related to richness differences in such gradients combined with directional processes that cause an ordered loss or gain of species. / O aninhamento é um padrão ecológico particular de distribuição de espécies em metacomunidades em que o grupo de espécies encontradas em sítios menos ricos é subconjunto daquele encontrado em sítios mais ricos. No contexto de partição de diversidade beta, o aninhamento era considerado, juntamente com a substituição de espécies, um componente da diversidade beta. Porém, atualmente está claro que este termo estava sendo equivocadamente utilizado para se referir à diversidade beta devido a diferenças de riqueza. Em casos específicos em que tais diferenças de riqueza refletem uma perda ou ganho ordenado de espécies entre sítios emerge, então, o aninhamento. No presente trabalho utilizei a abordagem de partição de diversidade beta, focando no componente das diferenças de riqueza, combinada com o uso de uma métrica específica de aninhamento, o NODF, a fim de explorar situações em que a diferença de riqueza entre sítios ocorre de maneira aninhada, considerando diferentes sistemas e escalas de estudo. No primeiro capítulo utilizo comunidades de macroinvertebrados aquáticos para evidenciar a importância da posição espacial de um mesmo micro-habitat na geração de aninhamento em corredeiras. Encontrei que manchas e trechos no início de corredeiras são menos ricos do que no final de corredeiras, e que para trechos há aninhamento da fauna encontrada no trecho inicial em relação à fauna encontrada no trecho final de uma mesma corredeira. No segundo capítulo utilizo comunidades de aves e mamíferos considerando uma escala que abrange todo o Novo Mundo para avaliar como o aninhamento varia ao longo de gradientes latitudinais e longitudinais. O padrão aninhado emergiu em várias regiões de ambos os gradientes, e está sempre relacionado a diferenças de riquezas existentes nesses gradientes combinada com processos direcionais que levam a perda ou ganho ordenado de espécies.

Submitted by Marcos Anselmo (marcos.anselmo@unipampa.edu.br) on 2016-09-09T19:43:53Z No. of bitstreams: 2 CECILIA ISABEL INES URQUIZA MACHADO PAVIN.pdf: 662363 bytes, checksum: ba4999290c4e1737856e3587e64e850d (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Approved for entry into archive by Marcos Anselmo (marcos.anselmo@unipampa.edu.br) on 2016-09-09T19:44:24Z (GMT) No. of bitstreams: 2 CECILIA ISABEL INES URQUIZA MACHADO PAVIN.pdf: 662363 bytes, checksum: ba4999290c4e1737856e3587e64e850d (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) / Made available in DSpace on 2016-09-09T19:44:24Z (GMT). No. of bitstreams: 2 CECILIA ISABEL INES URQUIZA MACHADO PAVIN.pdf: 662363 bytes, checksum: ba4999290c4e1737856e3587e64e850d (MD5) license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Previous issue date: 2016-08-05 / Os métodos de seleção espermática são utilizados com o objetivo de isolar uma maior população de espermatozoides morfologicamente melhor, além de permitir a capacitação destes, incrementando a taxa de embriões produzidos in vitro. Entre os métodos de seleção, o mais utilizado comercialmente na produção in vitro (PIV) de embriões bovinos é o método de Gradiente Descontínuo de Percoll® (GDP). No entanto, apesar da praticidade desta técnica, a etapa de centrifugação é prejudicial, sendo responsável por danos estruturais as células espermáticas, bem como a diminuição da motilidade e da taxa de recuperação, interferindo assim, nas taxas de fecundação in vitro (FIV). Este estudo objetivou avaliar a centrifugação com amortecimento utilizando uma solução de iodixanol durante a seleção espermática para a PIV de embriões bovinos. No experimento I, o sêmen foi descongelado e dividido em quatro grupos, e os espermatozoides submetidos a separação espermática pelo método de GDP com ou sem a solução colóide; sendo o amortecimento testado na primeira (C1), segunda (C2) ou ambas centrifugações (C1-2). Em seguida, foram avaliadas a taxa de recuperação, a cinética espermática, e a produção de espécies reativas de oxigênio (EROs). No experimento II foi avaliada a taxa de fecundação e a cinética de desenvolvimento embrionário dos grupos C e C1, que foram os tratamentos que demonstraram os melhores resultados para a cinética espermática, taxa de recuperação e análise bioquímica no primeiro experimento. Neste estudo, o grupo com amortecimento obteve um aumento na taxa de fecundação e clivagem, quando comparado ao grupo controle (60,5 vs 48,55 e 80,0 vs 64,7%, respectivamente). Com base nestes resultados foi possível concluir que o uso de 9 uma solução de amortecimento durante a etapa de centrifugação para a seleção espermática permitiu a preservação da cinética e da integridade da membrana dos espermatozoides, sem reduzir a taxa de recuperação. Adicionalmente, foi demonstrado pela primeira vez que o uso da centrifugação com amortecimento incrementa, não apenas a taxa de fecundação e clivagem, mas também o número de embriões que clivam mais cedo. / In order to optimize bovine in vitro fertilization (IVF), sperm selection methods have been used to isolate sperm subpopulations with high fertilizing capacity for use in animal breeding. Among the methods of sperm selection, the most common method used is the Discontinuous Percoll® Gradients (DPG). However, again despite the practicality of this technique, the centrifugation step is harmful, accounting for structural damage in sperm and decrease in motility and sperm recovery rate, as well as interfering with in vitro fertilization rates. This study aimed to evaluate the cushioned centrifugation using an iodixanol solution on sperm selection for in vitro production (IVP) of bovine embryos. In experiment I, the thawed semen was divided into four groups and the sperm subjected to the separation method by discontinuous Percoll® gradients with and without a colloid solution, where the cushioning was tested in the first (C1), second (C2), and both centrifugations (C1-2). Then, the recovery rate, sperm kinetics, and production of reactive oxygen species (ROS) were evaluated. Experiment II evaluated the fertilization rate and embryonic development kinetics of C and C1 groups; who were the treatments that have shown the best results for motion kinetics, recovery rate, and biochemical analysis in Experiment I. In this assay, the group with cushioning obtained an increase in the fertilization and cleavage rate when compared with the control (60.5 vs 48.55 and 80.0 vs 64.7%, respectively). With this study, we can conclude that the use of a cushion solution during centrifugation on sperm selection had preserved the motion properties of spermatozoa and the 11 integrity of the sperm membrane without lowering the recovery rate. Furthermore, it was demonstrated for the first time that the use of cushioned centrifugation increases not only the cleavage rate but the number of embryos previously cleaved.

Cette thèse traite d’une approche pragmatique de l'endommagement par fatigue des structures sous un chargement cyclique. Un modèle anisotrope de dommage par fatigue a été développé. L'évolution de la dégradation des propriétés du matériau dépend d’un critère de fatigue (choisi) et de ses gradients. La dégradation du matériau anisotrope guidera la propagation des dommages. La propagation des dommages dépend principalement des crêtes de la «surface» du critère (gradients zéro). L'approche proposée décrit l'amorçage, la propagation des dommages et la rupture structurale sous un chargement multiaxial en fatigue. Pour chaque élément fini, des distributions anisotropes non homogènes des propriétés du matériau sont associées. Schématiquement, il apparaît comme un «surfer» matériel sur la «surface» du critère et les dommages suivent la crête de la «surface» du critère (niveau et gradient). Un critère d'approche globale, basé sur des invariants du tenseur des contraintes, est adopté. La réduction des propriétés des matériaux est affectée à un certain nombre de cycles et à un niveau global de contraintes, en utilisant une courbe de Wöhler expérimentale. Deux formes simplifiées du modèle sont proposées et les résultats sont comparés avec un exemple de référence expérimental (plaque cruciforme) et un cas industriel (pale d’une éolienne d’EDF). Une cartographie avec le critère de Dang Van est également calculée pour analyser les résultats numériques. / A new practical engineering methodology for the analysis of structures under cyclic loading is proposed in this work . A new anisotropic fatigue damage model is developed. The evolution of material properties degradation depends on a failure criterion and its gradient. The anisotropic material degradation will guide the damage propagation. The propagation of damage is mainly depending on the ridges of the criterion’s « surface » (zero gradients). The proposed approach can describe the initiation and propagation of the damage until the structural failure under fatigue loading. For each finite element, non-homogeneous anisotropic distributions of material properties are associated. Schematically, it seems like a material « surfing » on the criterion’s « surface » and damages follow the crest of the criterion’s « surface » (level and gradient). A global approach criterion, based on invariants of the stress tensor, is adopted. The reduction of material properties is assigned to a number of cycles and a global level of stresses, using an experimental Wöhler curve. Two simplified forms of the model are proposed and results are compared with a cruciform experimental reference example and an industrial case. A mapping with Dang Van criterion is also computed to analyze the numerical results.