Variational methods in materials science

Three problems are being investigated in this thesis. The first two relate to the modelling and analysis of martensitic phase transitions, while the third is concerned with some mathematical tools used in this setting. After a short introduction (Chapter 1) and overviews of the calculus of variations and martensitic phase transformations (Chapter 2), the research part of this thesis is divided into three chapters. We show in Chapter 3 that for the two wells $\mathrm{SO}(3)U$ and $\mathrm{SO}(3)V$ to be rank-one connected, where the $3\times 3$ symmetric positive definite $U$ and $V$ have the same eigenvalues, it is necessary and sufficient that $\mathrm{det}(U-V)=0$, a result that does not hold in higher dimensions. Using this criterion and a result of Gurtin, formulae for the twinning plane and the shearing vector are obtained, which yield an extremely simple condition for the occurrence of so-called compound twins. Our results also provide a simple classification of the twinning mode of the two wells by looking at the crystallographic properties of the eigenvectors of the difference $U-V$. As an illustration, we apply our results to cubic-to-tetra gonal,tetragonal-to-monoclinic and cubic-to-monoclinic transitions. Chapter 4 focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation $y_1(x)= U_1x$ is the stable state for `small' loads while $y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that $y_1(x)=U_1x$ remains metastable i.e., a local - as opposed to global - minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound for when metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this chapter, we rigorously justify the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the chapter is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply. Finally, Chapter 5 investigates the structure of the solutions to the two-well problem. Restricting ourselves to the subset $K=\{H\}\cup \mathrm{SO}(2)V \subset\mathrm{SO}(2)U\cup\mathrm{SO}(2)V$ and assuming the two wells to be compatible, we let $T_1$ and $T_2$ denote the two (not necessarily distinct) twins of $H$ on $\mathrm{SO}(2)V$ and ask the following question: if $\nu_x$ is a non-trivial gradient Young measure almost everywhere supported on $K$, does its support necessarily contain a pair of rank-one connected matrices on a set of positive measure? Although we do not provide a solution for the general case, we show that this is true whenever (a) $\nu_x\equiv \nu$ is homogeneous and $\mathrm{supp}\nu\cap \mathrm{SO}(2)V$ is connected, (b) $\nu_x\equiv \nu$ is homogeneous and $T_1=T_2$ i.e., when the two wells are trivially rank-one connected) or (c) $\mathrm{supp}\nu_x \subset F$ a.e., for some finite set $F$. We also establish a more general case provided a strong `rigidity' conjecture holds

L’expérience de la différence religieuse dans l'Europe moderne (XVIe – XVIIIe siè-cles)

This collective work explores the themes of religious coexistence and the broaching of religious frontiers in Europe during the 16th, 17th and 18th Centuries. It intends to open new areas for discussion on a European level through the bias of a dia-logue between researchers subscribing to highly different national traditions (France, Netherlands, Switzerland, and Germany). It acts through the contributions gathered here to analyse the possibilities and different methods of interfaith con-tact by using as a departure point the experience of religious differences: The ex-perience garnered by diverse social actors – ecclesiastical, magisterial, academics and nobility.Cet ouvrage collectif aborde les thématiques de la coexistence confessionnelle et du dépassement des frontières religieuses dans l’Europe des XVIe, XVIIe et XVIIIe siècles. Il entend ouvrir de nouvelles pistes de discussion à l’échelle européenne, par le biais d’un dialogue entre des chercheurs s’inscrivant dans des traditions nationales bien différentes (France, Pays-Bas, Suisse, Allemagne). Il s’agit dans les contributions rassemblées ici d’analyser les possibilités et les modalités des contacts interconfessionnels, en prenant comme point de départ l’expérience de la différence religieuse : l’expérience vécue d’acteurs sociaux très divers – ecclésiastiques, magistrats, lettrés ou bourgeois – mais aussi l’expérience tentée par les autorités politiques. Les approches utilisées sont diverses : étude de trajectoires individuelles, histoire intellectuelle et culturelle, histoire sociale et politique. Les recherches présentées, dans leur diversité, s’inscrivent dans un contexte historiographique marqué par la remise en cause des paradigmes insistant sur l’étanchéité des frontières confessionnelles

L’expérience de la différence religieuse dans l'Europe moderne (XVIe – XVIIIe siè-cles)

This collective work explores the themes of religious coexistence and the broaching of religious frontiers in Europe during the 16th, 17th and 18th Centuries. It intends to open new areas for discussion on a European level through the bias of a dia-logue between researchers subscribing to highly different national traditions (France, Netherlands, Switzerland, and Germany). It acts through the contributions gathered here to analyse the possibilities and different methods of interfaith con-tact by using as a departure point the experience of religious differences: The ex-perience garnered by diverse social actors – ecclesiastical, magisterial, academics and nobility

Las repercusiones de la política peronista hacia América Latina luego del golpe de estado (1955-1958)

El presente proyecto se propone investigar las reacciones que despertó la política de difusión del modelo peronista en América latina, fenómeno que adquirió mayor visibilidad inmediatamente durante el gobierno de facto que lo derrocó en 1955 y cuya gestión finalizó en 1958. Al respecto, la tarea de propaganda sumó a la capacidad de Perón para comunicarse con públicos masivos, la utilización de diferentes técnicas y la creación de organismos para propagar su modelo de conducción política en América Latina, durante la gestión 1946-1955. En tal sentido, numerosos discursos y prácticas fueron también entendidos como un modo de volver imprecisos los contornos de los espacios nacionales en la región. La recepción de esta política regional despertó posiciones antinómicas, que luego del golpe de estado de 1955, salieron más claramente a la superficie. Así, entre 1955 y 1958 tuvieron una influencia significativa en: los intentos de Perón por mantener su liderazgo a la distancia, las vinculaciones que entabló con diversos representantes de las dirigencias políticas continentales, la acción de los grupos peronistas dispersos en el Latinoamérica y la actitud ante las presiones ejercidas por gobierno de facto. Entretanto, para Perón el problema que significaba asegurarse la efectiva conducción del movimiento, sin su presencia física, era un tema prioritario y para lograrlo utilizó diversas estrategias. Este tema ha sido estudiado exhaustivamente por diversos autores para el caso argentino, no obstante son escasos y dispersos los estudios relativos a Latinoamérica. Si bien, existen algunos intentos de ordenar correspondencia y otros testimonios, todavía no se ha logrado reunir en un corpus sistemático las variadas fuentes, cuyas evidencias se encuentran sumamente fragmentadas y son de difícil acceso por sus condiciones de producción En definitiva, por ahora cada hallazgo constituye un indicio que nos orienta hacia posteriores exploraciones y permite visualizar su proyección posterior. En este contexto, nuestra hipótesis es que luego el golpe de 1955, los grupos proclives y opositores a la política regional del presiente derrocado constituyeron actores importantes en la posible comunicación con los grupos partidarios diseminados en América Latina, la intermediación con el movimiento en Argentina y la reconstrucción de la imagen del líder exilado, todo ello obstaculizado por la el proceso de desperonización, encarado por la Revolución Libertadora.Fil: Otero, Delia del Pilar. Universidad Católica de Córdoba. Facultad de Ciencia Política y Relaciones Internacionales; ArgentinaFil: Ledesma, Leandro David. Universidad Católica de Córdoba. Facultad de Ciencia Política y Relaciones Internacionales; Argentin

Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Mechano-electrical contribution of ventricular contractions to the complexity of atrial fibrillation electrograms

Purpose: Intracardiac organization indices (OI) have been used to track the efficiency of stepwise radiofrequency catheter ablation (step-CA) of persistent atrial fibrillation (pers-AF). A better understanding of the components and complexity of AF electrograms (EGMs) is fundamental for tracking the organization of AF. It remains unknown whether ventricular contractions affect the complexity of AF EGMs by means of mechano-eletrical feedback. Our study is aimed at developing new methods to quantify the potential mechano-electrical contribution of ventricular contractions on AF complexity. Methods: During step-CA, a quadripolar catheter was placed into the right atrial appendage (RAA). Robust maximum positive peak detection was applied. The time difference between two adjacent peaks was regularly resampled resulting in the intra-cardiac variability (ICV) signal. ICV reflects the intrinsic atrial activation time variability as well as the potential mechano-electrical feedback of ventricular contractions (VC). A lowpass version impulses series of R wave locations was used as input to an adaptive interference canceller in order to suppress the VC contribution to ICV (VC-ICV). Results: Step-CA (including PVI, CFAE, roof and mitral isthmus lines) terminated 2/3 pers-AF into flutter. The mean contribution of the mechano-electrical feedback on AF complexity achieved 37%. The figure shows a representative example where the VC component (2.03 Hz) has been removed from the ICV, resulting in the atrial-ICV. All other components have been preserved. Conclusions: Our preliminary findings suggest that by means of mechano-electrical feedback, VC contribute up to 37% of the atrial complexity during pers-AF. It is also a first step in the elaboration of new organization indices free of ventricular contribution

Contribution of Left and Right Atrial Appendage Activities to ECG Fibrillation Waves

It was recently shown that atrial fibrillation (AF) waves on chest lead V1 adequately reflect right atrial appendage (RAA) activity during long standing persistent AF (pers-AF). The contribution of the left atrial (LA) activity to chest leads AF waves, however, remains unknown. Our study is aimed at evaluating the respective contribution of the RA and LA depolarization to ECG chest leads AF waves during pers-AF. Methods: Catheters (CAT) were introduced in 10 consecutive patients (60±5 y, AF duration 22±14 m) prior to ablation: 1) a quadripolar CAT in the RAA, 2) a decapolar CAT in the coronary sinus (CS) and 3) a duodecapolar CAT in the LA appendage (LAA). Local activation times were extracted from bipolar recordings using sliding windows. Chest lead V6 was placed in the back (V6b). Mean AF cycle length (AFCL) of leads V1 to V6b were computed as the inverse of the dominant frequency of ECG spectra after QRST cancellation, and compared to intracardiac RAA, LAA and CS AFCL using Pearson’s correlation coefficient. Results: The figure shows that the correlation between RAA and chest leads AFCL was maximal for V1 and progressively dropped till V5, with a moderate rise for V6b. LAA AFCL showed the opposite pattern with the highest correlation in V6B and the lowest one in V2. The correlation of CS AFCL was similar to the LAA one, but of lower magnitude. Conclusion: Our preliminary results suggest that the respective contribution of RAA and LAA activities can be estimated using a modified surface ECG. Whether this technique has the potential to guide ablation of LA and RA drivers in pers-AF needs further validation

Morphological Study of Intracardiac Signals as a New Tool to Track the Efficiency of Stepwise Ablation of Persistent Atrial Fibrillation

Intracardiac organization indices such as atrial fibrillation (AF) cycle length (AFCL) have been used to track the efficiency of stepwise catheter ablation (step-CA) of longstanding persistent AF, however with limited success. The morphology of AF activation waves reflects the underlying activation patterns. Its temporal evolution is a local organization indicator that could be potentially used for tracking the efficiency of step-CA. We report a new method for characterizing the structure of the temporal evolution of activation wave morphology. Using recurrence plots, novel organization indices are proposed. By computing their relative evolution during the first step of ablation vs baseline, we found that these new parameters are superior to AFCL to track the effect of step-CA “en route” to AF termination