A series of measurements aimed at comparing the magneto-elastic response and non-Joulian properties (magneto-volume) of single crystals of the iron-aluminum alloys Fe_81.8 Al_18.2 with different thermal treatments were per formed; the results of which confirmed net volume expansion in all field orientations. The magnitude of this expansion varied largely across various magnetic field alignments, with a greater than four times larger expansion for fields parallel to the [010] crystal axis than the [100] axis. The magnitude and direction of the change in volumetric expansion from quenched crystals to slow cooled crystals also depended heavily on the orientation of the magnetic field, with reduced magneto-volume responses ranging from ~2.4 to 0.5 times smaller for fields parallel to the [100] and [110] crystal axes respectively. These results present evidence of the non-Joulian effect and places crystals of the iron aluminum alloy Fe_81.8 Al_18.2 within the realm of functional non-Joulian magnets. / Mechanical Engineering

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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 e-mail 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 non-quasiconvexity 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 hysteresis-modified stationary problems (the...

La technique du contrôle non destructif (CND) par courants de Foucault (CF) est une solution envisageable pour caractériser l’état de contraintes dans un matériau magnétique. En effet, les propriétés magnétiques d’un matériau magnétique dépendent sensiblement de l’état de contraintes et, par ailleurs, les CF dépendent de la perméabilité magnétique. La technique des CF est potentiellement intéressante comparée à d’autres méthodes de CND, telles que la diffraction des rayons X, car elle est simple à mettre en œuvre, automatisable et peu coûteuse. Dans l'objectif de permettre in fine l’identification inverse de l’état de contraintes, un modèle prédisant l'évolution du signal fourni par un capteur à CF en fonction de l’état de contraintes du matériau ferromagnétique contrôlé a été élaboré dans cette thèse. Ceci implique la mise en place d’une double modélisation. D’une part, l’effet des contraintes sur la perméabilité magnétique a été modélisé par un modèle de comportement magnéto-élastique simplifié dérivé d'une approche multi-échelle. Cette approche permet de décrire la perméabilité magnétique d’un matériau soumis à un chargement multiaxial, en incluant notamment les effets d'anisotropie induite. D’autre part, un modèle reposant sur la méthode des éléments finis a été développé afin de prédire le signal fourni par un capteur à CF en fonction de la perméabilité anisotrope du matériau inspecté. Afin de valider la démarche de modélisation, un protocole expérimental de caractérisation magnétique et par CF a été mis en place. Les résultats de mesure obtenus présentent un bon accord qualitatif avec la modélisation, en l’absence de tout étalonnage. Une procédure d’étalonnage s'appuyant sur une mesure sous contraintes est nécessaire pour atteindre un accord quantitatif. Le modèle développé pourrait être utilisé pour concevoir des sondes à CF idoines et identifier les conditions opératoires optimales pour l'estimation de contraintes dans les matériaux magnétiques. Les procédures d'inversion à mettre en œuvre restent cependant un défi à relever / The non-destructive evaluation (NDE) technique by eddy current (EC) is a conceivable solution to characterize the stress state in magnetic materials. The approach relies on the high sensitivity of eddy current (EC) signals to the magnetic permeability, itself highly dependent on stress. The EC technique is potentially attractive compared to other NDE methods such as X-ray diffraction, due to its simple practical implementation, easiness of automation and low cost. In order to allow eventually the inverse identification of stress states in magnetic materials, a predictive model for the evolution of an EC probe signal as a function of stress has been developed during this thesis. The modelling is done in two steps. First, the effect of stress on the magnetic permeability is described using a simplified version of a multiscale model for magneto-elastic behaviour. This approach allows describing the effect of multiaxial mechanical loadings on the magnetic behaviour of materials including induced anisotropy effects. Then, the EC probe signal is determined as a function of the anisotropic permeability of the stressed material using the finite element method (FEM). In order to validate the modelling approach, an experimental setup for magnetic characterisation and EC measurements was developed. The measurements show a good qualitative accordance with the modelling results, in absence of any calibration. A calibration procedure based on a measurement under stress is necessary to obtain a quantitative agreement. The proposed model can be used to design efficient EC probes and to define optimal operating conditions to evaluate stress in magnetic materials. The development of inversion procedures, however, remains a challenge.

No presente trabalho, foi construído um transdutor para medir as variações da indução magnética (B) e magnetostricção (?) de materiais ferromagnéticos macios em função de campo magnético aplicado (H) (voltagem V) e tensões mecânicas (?). O transdutor possui duas bobinas de excitação ligadas a uma fonte de corrente/voltagem (para aplicar o campo magnético no elemento magnetostrictivo) e um núcleo magnético, para concentrar o fluxo magnético no material ferromagnético. A magnetostricção é medida por extensômetria, a tensão mecânica é aplicada por uma máquina de ensaios universal (EMIC), a indução magnética é medida utilizando uma bobina de captura conectada a um fluxímetro e a aquisição dos dados foi realizada por um programa em Labview. O sistema mede curvas de M vs V e ? vs V para ? constante e também M vs ? e ? vs ? para V constante. A partir dessas curvas, é possível adquirir a sensibilidade do material e analisar a sua viabilidade em aplicações como sensores e atuadores. Utilizando este transdutor, foi possível caracterizar uma liga de FeAlB e comparar com uma amostra de GALFENOL, que é o material mais usado desta classe. As ligas apresentam sensibilidades (acima de 10 MPa) da ordem de 0,020 T/MPa para o GALFENOL e 0,012 T/MPa para a liga de FeAlB. Considerando que a liga de FeAlB não apresenta a melhor composição química, há muitas coisas para estudar ainda, como diferentes composições químicas e o processamento adequado para obter a textura desejada. / In the present research, a transducer was built to measure the changes of magnetic induction (B) and magnetostriction (?) for soft ferromagnetic materials in function of applied magnetic field (H) (voltage V) and mechanical stress (?).The transducer comprises two excitation coils connected at a current/voltage supply (to apply the magnetic field in the magnetostrictive element) and a magnetic core, to concentrate the magnetic flux in the ferromagnetic material. The magnetostriction is measured by strain gauges, the mechanical stress is applied by a material test system (EMIC), the magnetic induction is measured using a pick-up coil connected to a fluxmeter and the acquisition of the data was measured by a Labview software. These systems measure the curve B vs. V and ? vs. V for constant ? and B vs. ? e ? vs. ? for constant V. From this curves, it is possible to acquire the material´s sensibility and analyse the feasibility of then in application like sensor and actuators. Using this transducer, it was possible to characterize a FeAlB alloy and to compare it with a GALFENOL sample, which is the most used material of this class. The alloys show sensibility (up to 10 MPa) about 0,020 T/MPa to the GALFENOL and 0,012 T/MPa to FeAlB alloy. If we consider that the FeAlB alloy does not have the best chemical composition to reach the higher magnetostriction, there is a lot o thing to study, like different composition and the appropriate processing to obtain the desired texture.

Trabalhos publicados a partir do ano 2000 mostraram que a adição de Ga e Al ao Fe resulta em ligas com elevada magnetoestricção em campos magnéticos baixos, além de boas propriedades mecânicas e baixo custo associado (CLARK, 2001; GURUSWAMY, 2004). A partir daí os esforços foram direcionados para a descoberta de novas ligas magnetoestrictivas à base de Fe com propriedades similares às encontradas nas ligas Fe-Ga e Fe-Al. De fato, altos valores de magnetoestricção são de grande interesse para aplicações principalmente em sensores e atuadores, mas valores de magnetoestricção muito baixos ou nulos também proporcionam uma propriedade notável e desejável para muitas aplicações como em transformadores elétricos e sensores de fluxo. Este fenômeno de magnetoestricção ocorre em materiais que exibem deformação elástica na presença de um campo magnético aplicado. No presente trabalho, foi adicionado o titânio ao ferro para avaliar a influência em ligas de Fe-Ti com diferentes frações volumétricas da fase Fe2Ti na magnetoestricção e a relação com a microestrutura das ligas. Foi descoberto que as amostras de Fe100-xTix para x < 34,0 at.% apresentam uma magnetoestricção muito pequena. O Ti tem uma pequena solubilidade no ?-Fe, de aproximadamente 8,5% a 1200 ºC e diminui a 5% a 1000 ºC formando uma fase que também é ferromagnética. Assim, quando a solubilidade de Ti em ?-Fe é excedida há então a formação desta fase ferromagnética ordenada Fe2Ti chamada de Laves, com estrutura cristalina C14. Como a quantidade de Ti aumenta nas ligas estudadas, a fração volumétrica da fase Laves também aumenta e assim foi possível avaliar a magnetoestricção e a microestrutura das ligas formadas. Para isso, foram produzidas por fusão a arco as ligas com 1,61% a 34,0 % de Ti. Todas estas ligas foram submetidas a tratamentos térmicos em 1200 °C por 60 h para fim de homogeneização destas. A magnetoestricção de saturação (?S) medida nas amostras foi muito pequena. Para as ligas com 20 e 28,0% Ti, a ?S foi positiva nas e para as demais, ?S foi negativa. Concluiu-se que a substituição do Fe por Ti de um modo geral melhorou a magnetoestricção do Fe, porém a melhora não foi tão relevante quanto à observada pela substituição por Ga. / Since 2000, studies indicated that the addition of Ga and Al in Fe results in alloys with high magnetostriction at low magnetics fields as well as good mechanical properties and low cost associated (CLARK, 2001; GURUSWAMY, 2004). Since that, efforts have been directed to the discovery of new magnetostrictive Fe-based alloys with similar properties to those found in Fe-Ga and Fe-Al alloys.In fact, high values of magntetostriction are of great interest for applications, mainly in electric sensors and actuators but low values or zero magnetostriction is also a remarkable property, desirable for many applications such as electric transformers and fluxgate sensor cores. This phenomenon of magnetostriction occurs in materials that exhibit elastic strain in the presence of an applied magnetic field. In the present work, titanium was added to iron with the purpose of evaluating its influence on the magnetostriction of Fe. The variation on the magnetostriction was investigated as a function of different volumetric fractions of ordered phase Fe2Ti that coexist with the ?-phase. It was found that the samples of Fe100-xTix for x < 34,0 at.% has a very small magnetostriction. The Ti has a small solubility in ?-Fe, of approximately 7,0 % 1200 ºC and reduces at 5% 1000 ºC forming a phase that also is ferromagnetic. Thus, when the solubility of Ti in ?-Fe is exceeded there is the formation of this ordered phase Fe2Ti, a Laves phase, with crystalline structure C14. As the amount of Ti increases in the alloys, the fraction of volume of the Laves phase also increases and then it was possible to evaluate the magnetostriction and the microstructure of the alloys formed. Therefore, Fe-Ti alloys were produced by arc melting with 1,61% to 34,0 % of Ti. All the Fe-Ti alloys were heat-treated at the temperatures of 1200 °C for 60 h intending to homonezation of the alloy. The saturation magnetostriction (?S) measured in the samples was small. For alloys with 20 and 28% of Ti ,?S was positive and for the other samples, ?S was negative. We concluded that the substitution of Fe by Ti generally improved the magnetostriction of Fe, but the improvement was not as relevant as that observed by Ga substitution.

Neste trabalho foram obtidas barras da liga (Fe0,80Al0,20)97B3 por solidificação direcional (SD), nos diâmetros de 3 e 4 mm, com o objetivo de melhorar a magnetostricção desse material através da introdução de textura. As etapas de produção das barras envolveram a fabricação de lingotes da liga por fusão a arco L1, L2 e L6, seguida de usinagem (U) ou forjamento rotativo (F) a quente. O maior comprimento de barra obtido foi de 50 mm. Através de microscopia eletrônica de varredura e análise de composição por espectroscopia de energia dispersiva, detectou-se a presença de boro nas amostras pela visualização do microconstituinte eutético formado da fase ? e da fase Fe2B. Resultados obtidos por difração de raios X indicam que a amostra da liga L1 para o diâmetro de 4 mm apresentou orientação preferencial nas direções , o que causou uma diminução expressiva da magnetostricção se comparada a da amostra do mesmo lingote para o diâmetro de 3 mm. Medidas de magnetização em função do campo magnético aplicado (Ha) indicaram que a magnetização de saturação das amostras que têm boro são em média 181 Am2/kg, sugerindo que as amostras tem ~ 2% de B. A barras obtidas por SD, tratadas termicamente (CTT) a 1100°C por 6 h, alcançaram uma maior magnetostricção (? = ?l/l), comparada as amostras sem tratamento térmico (STT), no qual o maior valor foi de 73 ? 10-6 na direção longitudinal da barra L1 forjada no diâmetro de 3 mm (L1F3-CTT). O maior fator de sensibilidade d?/dH obtido em função do comprimento da barra L1F3 foi de d?/dH = 0,9 x 10-9 m/A para a razão de aspecto ? = 12. Valor da constante anisotrópica K1 calculada a partir dos resultados experimentais se comparadas com dados da literatura, sugerem que em algumas amostras ocorre ordenamento parcial da fase cúbica da matriz, havendo assim a coexistência das fases Fe3Al (D03) e ?. / In this work, bars of the alloy (Fe0,80Al0,20)97B3 were obtained by directional solidification (DS) in diameters of 3 and 4 mm, with the goal of improving the magnetostriction of the material through the introduction of texture. Production steps of the bars involved the manufacture of the alloy ingots, L1, L2, L6, by arc melting, then machining (M) or hot swaging (F). The greatest length that was achieved in a bar was 50 mm. The presence of boron in the samples was identified by scanning electron microscopy and energy dispersive spectroscopy, due to the presence of an eutectic microstructure that has dendrites and a microconstituent formed of phases ? and Fe2B. Results obtained by X-ray diffraction indicates that the bar L1 at the diameter of 4 mm showed preferential orientation in the directions , which caused a significant decrease of the magnetostriction compared to the same sample of the bar of diameter 3 mm. Magnetization measurements as a function of applied magnetic field (Ha) reveal that the saturation magnetization of the samples having boron is in average 181 Am2/kg, suggesting that the samples have ~ 2% of boron. The bars obtained by DS, thermally treated (CTT ) at 1100 ° C for 6 h, achieved higher magnetostrictions (? = ?l/l) compared to the samples without heat treatment.The highest value was 73 ? 10-6 was obtained in the longitudinal direction of the bar L1F in the diameter of 3 mm (L1F3-CTT). The maximum sensitivity factor d?/dH found as a function of the bars lengths was d?/dH = 0.9 x 10-9 m/A for an aspect ratio of ? = 12. Values of anisotropic constant K1, calculated using experimental results suggest that in some samples occur partial ordering of the cubic phase matrix , so there is the coexistence of phases Fe3Al ( D03 ) and ?.

No presente trabalho, foi construído um transdutor para medir as variações da indução magnética (B) e magnetostricção (?) de materiais ferromagnéticos macios em função de campo magnético aplicado (H) (voltagem V) e tensões mecânicas (?). O transdutor possui duas bobinas de excitação ligadas a uma fonte de corrente/voltagem (para aplicar o campo magnético no elemento magnetostrictivo) e um núcleo magnético, para concentrar o fluxo magnético no material ferromagnético. A magnetostricção é medida por extensômetria, a tensão mecânica é aplicada por uma máquina de ensaios universal (EMIC), a indução magnética é medida utilizando uma bobina de captura conectada a um fluxímetro e a aquisição dos dados foi realizada por um programa em Labview. O sistema mede curvas de M vs V e ? vs V para ? constante e também M vs ? e ? vs ? para V constante. A partir dessas curvas, é possível adquirir a sensibilidade do material e analisar a sua viabilidade em aplicações como sensores e atuadores. Utilizando este transdutor, foi possível caracterizar uma liga de FeAlB e comparar com uma amostra de GALFENOL, que é o material mais usado desta classe. As ligas apresentam sensibilidades (acima de 10 MPa) da ordem de 0,020 T/MPa para o GALFENOL e 0,012 T/MPa para a liga de FeAlB. Considerando que a liga de FeAlB não apresenta a melhor composição química, há muitas coisas para estudar ainda, como diferentes composições químicas e o processamento adequado para obter a textura desejada. / In the present research, a transducer was built to measure the changes of magnetic induction (B) and magnetostriction (?) for soft ferromagnetic materials in function of applied magnetic field (H) (voltage V) and mechanical stress (?).The transducer comprises two excitation coils connected at a current/voltage supply (to apply the magnetic field in the magnetostrictive element) and a magnetic core, to concentrate the magnetic flux in the ferromagnetic material. The magnetostriction is measured by strain gauges, the mechanical stress is applied by a material test system (EMIC), the magnetic induction is measured using a pick-up coil connected to a fluxmeter and the acquisition of the data was measured by a Labview software. These systems measure the curve B vs. V and ? vs. V for constant ? and B vs. ? e ? vs. ? for constant V. From this curves, it is possible to acquire the material´s sensibility and analyse the feasibility of then in application like sensor and actuators. Using this transducer, it was possible to characterize a FeAlB alloy and to compare it with a GALFENOL sample, which is the most used material of this class. The alloys show sensibility (up to 10 MPa) about 0,020 T/MPa to the GALFENOL and 0,012 T/MPa to FeAlB alloy. If we consider that the FeAlB alloy does not have the best chemical composition to reach the higher magnetostriction, there is a lot o thing to study, like different composition and the appropriate processing to obtain the desired texture.

Trabalhos publicados a partir do ano 2000 mostraram que a adição de Ga e Al ao Fe resulta em ligas com elevada magnetoestricção em campos magnéticos baixos, além de boas propriedades mecânicas e baixo custo associado (CLARK, 2001; GURUSWAMY, 2004). A partir daí os esforços foram direcionados para a descoberta de novas ligas magnetoestrictivas à base de Fe com propriedades similares às encontradas nas ligas Fe-Ga e Fe-Al. De fato, altos valores de magnetoestricção são de grande interesse para aplicações principalmente em sensores e atuadores, mas valores de magnetoestricção muito baixos ou nulos também proporcionam uma propriedade notável e desejável para muitas aplicações como em transformadores elétricos e sensores de fluxo. Este fenômeno de magnetoestricção ocorre em materiais que exibem deformação elástica na presença de um campo magnético aplicado. No presente trabalho, foi adicionado o titânio ao ferro para avaliar a influência em ligas de Fe-Ti com diferentes frações volumétricas da fase Fe2Ti na magnetoestricção e a relação com a microestrutura das ligas. Foi descoberto que as amostras de Fe100-xTix para x < 34,0 at.% apresentam uma magnetoestricção muito pequena. O Ti tem uma pequena solubilidade no ?-Fe, de aproximadamente 8,5% a 1200 ºC e diminui a 5% a 1000 ºC formando uma fase que também é ferromagnética. Assim, quando a solubilidade de Ti em ?-Fe é excedida há então a formação desta fase ferromagnética ordenada Fe2Ti chamada de Laves, com estrutura cristalina C14. Como a quantidade de Ti aumenta nas ligas estudadas, a fração volumétrica da fase Laves também aumenta e assim foi possível avaliar a magnetoestricção e a microestrutura das ligas formadas. Para isso, foram produzidas por fusão a arco as ligas com 1,61% a 34,0 % de Ti. Todas estas ligas foram submetidas a tratamentos térmicos em 1200 °C por 60 h para fim de homogeneização destas. A magnetoestricção de saturação (?S) medida nas amostras foi muito pequena. Para as ligas com 20 e 28,0% Ti, a ?S foi positiva nas e para as demais, ?S foi negativa. Concluiu-se que a substituição do Fe por Ti de um modo geral melhorou a magnetoestricção do Fe, porém a melhora não foi tão relevante quanto à observada pela substituição por Ga. / Since 2000, studies indicated that the addition of Ga and Al in Fe results in alloys with high magnetostriction at low magnetics fields as well as good mechanical properties and low cost associated (CLARK, 2001; GURUSWAMY, 2004). Since that, efforts have been directed to the discovery of new magnetostrictive Fe-based alloys with similar properties to those found in Fe-Ga and Fe-Al alloys.In fact, high values of magntetostriction are of great interest for applications, mainly in electric sensors and actuators but low values or zero magnetostriction is also a remarkable property, desirable for many applications such as electric transformers and fluxgate sensor cores. This phenomenon of magnetostriction occurs in materials that exhibit elastic strain in the presence of an applied magnetic field. In the present work, titanium was added to iron with the purpose of evaluating its influence on the magnetostriction of Fe. The variation on the magnetostriction was investigated as a function of different volumetric fractions of ordered phase Fe2Ti that coexist with the ?-phase. It was found that the samples of Fe100-xTix for x < 34,0 at.% has a very small magnetostriction. The Ti has a small solubility in ?-Fe, of approximately 7,0 % 1200 ºC and reduces at 5% 1000 ºC forming a phase that also is ferromagnetic. Thus, when the solubility of Ti in ?-Fe is exceeded there is the formation of this ordered phase Fe2Ti, a Laves phase, with crystalline structure C14. As the amount of Ti increases in the alloys, the fraction of volume of the Laves phase also increases and then it was possible to evaluate the magnetostriction and the microstructure of the alloys formed. Therefore, Fe-Ti alloys were produced by arc melting with 1,61% to 34,0 % of Ti. All the Fe-Ti alloys were heat-treated at the temperatures of 1200 °C for 60 h intending to homonezation of the alloy. The saturation magnetostriction (?S) measured in the samples was small. For alloys with 20 and 28% of Ti ,?S was positive and for the other samples, ?S was negative. We concluded that the substitution of Fe by Ti generally improved the magnetostriction of Fe, but the improvement was not as relevant as that observed by Ga substitution.

Most smart actuators exhibit rate-dependant hysteresis when the working frequency is higher than 5Hz. Although the Preisach model has been a very powerful tool to model the static hysteresis, it cannot be directly used to model the dynamic hysteresis. Some researchers have proposed various generalizations of the Preisach operator to model the rate-dependant hysteresis, however, most of them are application-dependant and only valid for low frequency range. In this thesis, a first-order dynamic relay operator is proposed. It is then used to build a novel dynamic Preisach model. It can be used to model general dynamic hysteresis and is valid for a large frequency range. Real experiment data of magnetostrictive actuator is used to test the proposed model. Experiments have shown that the proposed model can predict all the static major and minor loops very well and at the same time give an accurate prediction for the dynamic hysteresis loops. The controller design using the proposed model is also studied. An inversion algorithm is developed and a PID controller with inverse hysteresis compensation is proposed and tested through simulations. The results show that the PID controller with inverse compensation is good at regulating control; its tracking performance is really limited (average error is 10 micron), especially for high frequency signals. Hence, a simplified predictive control scheme is developed to improve the tracking performance. It is proved through experiments that the proposed predictive controller can reduce the average tracking error to 2 micron while preserve a good regulating performance.