In den vergangenen Jahrzehnten rückte das häufige Auftreten von multiplen Polylogarithmen und multiplen Zeta-Werten, in Feynman-Diagramm Rechnungen niedriger Ordnung, verstärkt in den wissenschaftlichen Fokus. Hierbei offenbarte sich eine Verbindung zu den mathematischen Theorien der Perioden und der iterierten Integrale von Chen. Eine ähnliche Allgegenwärtigkeit von multiplen Zeta-Werten wurde jüngst auch in der α'-Entwicklung von Genus-Null Stringtheorie Amplituden beobachtet. Davon inspiriert befasst sich diese Arbeit mit der Systematik der iterierten Integralen in den Streuamplituden der offenen Stringtheorie. Unser Fokus liegt insbesondere auf der Genus-Eins Amplitude, welche sich vollständig durch iterierte Integrale, die bezüglich einer punktierten elliptischen Kurve definiert sind, ausdrücken lässt. Wir führen den Begriff der getwisteten elliptischen multiplen Zeta-Werte ein. Dieser Begriff beschreibt eine Klasse von iterierten Integralen, die auf einer elliptischen Kurve definiert sind, bei welcher ein rationales Gitter entfernt wurde. Anschließend zeigen wir, dass die Entwicklung eines jeden getwisteten elliptischen multiplen Zeta-Wertes, bezüglich des modularen Parameters τ, durch ein Anfangswertproblem beschrieben wird. Weiterhin präsentieren wir ein Argument dafür, dass sich im Limes τ→i∞ jeder getwistete elliptische multiple Zeta-Wert durch zyklotomische multiple Zeta-Werte ausdrücken lässt. Schließlich beschreiben wir wie sich Genus-Eins Amplituden in offener Stringtheorie mithilfe von getwisteten elliptischen multiplen Zeta-Werten ausdrücken lassen und illustrieren dies für die Vier-Punkt Amplitude. Hierbei zeigt es sich, dass bis zu dritter Ordnung in α' alle Beiträge durch die Unterklasse der elliptischen multiplen Zeta-Werte ausgedrückt werden können, was wiederum äquivalent zu der Abwesenheit unphysikalischer Pole in Gliozzi-Scherk-Olive projizierter Superstringtheorie ist. / Over the last few decades the prevalence of multiple polylogarithms and multiple zeta values in low order Feynman diagram computations of quantum field theory has received increased attention, revealing a link to the mathematical theories of Chen’s iterated integrals and periods. More recently, a similar ubiquity of multiple zeta values was observed in the α'-expansion of genus-zero string theory amplitudes. Inspired by these developments, this work is concerned with the systematic appearance of iterated integrals in scattering amplitudes of open superstring theory. In particular, the focus will be on studying the genus-one amplitude, which requires the notion of iterated integrals defined on punctured elliptic curves. We introduce the notion of twisted elliptic multiple zeta values that are defined as a class of iterated integrals naturally associated to an elliptic curve with a rational lattice removed. Subsequently, we establish an initial value problem that determines the expansions of twisted elliptic multiple zeta values in terms of the modular parameter τ of the elliptic curve. Any twisted elliptic multiple zeta value degenerates to cyclotomic multiple zeta values at the cusp τ → i∞, with the corresponding limit serving as the initial condition of the initial value problem. Finally, we describe how to express genus-one open-string amplitudes in terms of twisted elliptic multiple zeta values and study the four-point genus-one open-string amplitude as an example. For this example we find that up to third order in α' all possible contributions in fact belong to the subclass formed by elliptic multiple zeta values, which is equivalent to the absence of unphysical poles in Gliozzi-Scherk-Olive projected superstring theory.

The desire for energy and resource savings brings magnesium alloys as lightweight materials with high specific strength more and more into the focus. Most structural components are subjected to cyclic loading. In the course of computer aided product development, a numerical prediction of the fatigue life under these conditions must be provided. For this reason, the mechanical properties of the considered material must be examined in detail. Wrought magnesium semifinished products, e.g. magnesium sheet metals, typically reveal strong basal textures and thus, the mechanical behavior considerably differs from that of the well-established magnesium die castings. Magnesium sheet metals reveal a distinct difference in the tensile and compressive yield stress, leading to non-symmetric sigmoidal hysteresis loops within the elasto-plastic load range. These unusual hysteresis shapes are caused by cyclic twinning and detwinning. Furthermore, wrought magnesium alloys reveal pseudoelastic behavior, leading to nonlinear unloading curves. Another interesting effect is the formation of local twin bands during compressive loading. Nevertheless, only little information can be found on the numerical fatigue analysis of wrought magnesium alloys up to now. The aim of this thesis is the investigation of the mechanical properties of wrought magnesium alloys and the development of an appropriate fatigue model. For this purpose, twin roll cast AM50 as well as AZ31B sheet metals and extruded ME21 sheet metals were used. Mechanical tests were carried out to present a comprehensive overview of the quasi-static and cyclic material behavior. The microstructure was captured on sheet metals before and after loading to evaluate the correlation between the microstructure, the texture, and the mechanical properties. Stress- and strain-controlled loading ratios and strain-controlled experiments with variable amplitudes were performed. Tests were carried out along and transverse to the manufacturing direction to consider the influence of the anisotropy. Special focus was given to sigmoidal hysteresis loops and their influence on the fatigue life. A detailed numerical description of hysteresis loops is necessary for numerical fatigue analyses. For this, a one-dimensional phenomenological model was developed for elasto-plastic strain-controlled constant and variable amplitude loading. This model consists of a three-component equation, which considers elastic, plastic, and pseudoelastic strain components. Considering different magnesium alloys, good correlation is reached between numerically and experimentally determined hysteresis loops by means of different constant and variable amplitude load-time functions. For a numerical fatigue life analysis, an energy based fatigue parameter has been developed. It is denoted by “combined strain energy density per cycle” and consists of a summation of the plastic strain energy density per cycle and the 25 % weighted tensile elastic strain energy density per cycle. The weighting represents the material specific mean stress sensitivity. Applying the energy based fatigue parameter on modeled hysteresis loops, the fatigue life is predicted adequately for constant and variable amplitude loading including mean strain and mean stress effects. The combined strain energy density per cycle achieves significantly better results in comparison to conventional fatigue models such as the Smith-Watson-Topper model. The developed phenomenological model in combination with the combined strain energy density per cycle is able to carry out numerical fatigue life analyses on magnesium sheet metals.

The desire for energy and resource savings brings magnesium alloys as lightweight materials with high specific strength more and more into the focus. Most structural components are subjected to cyclic loading. In the course of computer aided product development, a numerical prediction of the fatigue life under these conditions must be provided. For this reason, the mechanical properties of the considered material must be examined in detail. Wrought magnesium semifinished products, e.g. magnesium sheet metals, typically reveal strong basal textures and thus, the mechanical behavior considerably differs from that of the well-established magnesium die castings. Magnesium sheet metals reveal a distinct difference in the tensile and compressive yield stress, leading to non-symmetric sigmoidal hysteresis loops within the elasto-plastic load range. These unusual hysteresis shapes are caused by cyclic twinning and detwinning. Furthermore, wrought magnesium alloys reveal pseudoelastic behavior, leading to nonlinear unloading curves. Another interesting effect is the formation of local twin bands during compressive loading. Nevertheless, only little information can be found on the numerical fatigue analysis of wrought magnesium alloys up to now. The aim of this thesis is the investigation of the mechanical properties of wrought magnesium alloys and the development of an appropriate fatigue model. For this purpose, twin roll cast AM50 as well as AZ31B sheet metals and extruded ME21 sheet metals were used. Mechanical tests were carried out to present a comprehensive overview of the quasi-static and cyclic material behavior. The microstructure was captured on sheet metals before and after loading to evaluate the correlation between the microstructure, the texture, and the mechanical properties. Stress- and strain-controlled loading ratios and strain-controlled experiments with variable amplitudes were performed. Tests were carried out along and transverse to the manufacturing direction to consider the influence of the anisotropy. Special focus was given to sigmoidal hysteresis loops and their influence on the fatigue life. A detailed numerical description of hysteresis loops is necessary for numerical fatigue analyses. For this, a one-dimensional phenomenological model was developed for elasto-plastic strain-controlled constant and variable amplitude loading. This model consists of a three-component equation, which considers elastic, plastic, and pseudoelastic strain components. Considering different magnesium alloys, good correlation is reached between numerically and experimentally determined hysteresis loops by means of different constant and variable amplitude load-time functions. For a numerical fatigue life analysis, an energy based fatigue parameter has been developed. It is denoted by “combined strain energy density per cycle” and consists of a summation of the plastic strain energy density per cycle and the 25 % weighted tensile elastic strain energy density per cycle. The weighting represents the material specific mean stress sensitivity. Applying the energy based fatigue parameter on modeled hysteresis loops, the fatigue life is predicted adequately for constant and variable amplitude loading including mean strain and mean stress effects. The combined strain energy density per cycle achieves significantly better results in comparison to conventional fatigue models such as the Smith-Watson-Topper model. The developed phenomenological model in combination with the combined strain energy density per cycle is able to carry out numerical fatigue life analyses on magnesium sheet metals.

In dieser Arbeit betrachten wir zwei Ansätze zur Untersuchung von Korrelationsfunktionen in eindimensionalen konformen Feldtheorien mit Defekten (dCFT1), insbesondere solche, die durch 1/2-BPS-Wilson-Linien-Defekte in den drei- und vierdimensionalen superkonformen Theorien definiert sind, die für die AdS/CFT-Korrespondenz relevant sind. Zunächst verwenden wir den analytischen konformen Bootstrap, um zwei Beispiele von Defektkorrelatoren auszuwerten. Der Vier-Punkt-Korrelator des Verschiebungs-Supermultipletts, das auf der 1/2-BPS-Wilson-Linie in der ABJM-Theorie eingefügt ist, wird bis zur dritten Ordnung in einer starken Kopplungsexpansion berechnet und reproduziert die expliziten Witten-Diagramm-Berechnungen erster Ordnung. Anschließend wird der Fünf-Punkt-Korrelator von 1/2-BPS-Operatoren, die auf der 1/2-BPS-Wilson-Linie in N=4 Super-Yang-Mills eingefügt sind, untersucht und in einer starken Kopplungsexpansion bis zur ersten Ordnung gebootstrapped. Anschließend werden die CFT1-Daten extrahiert, die bestätigen, dass das Mischen von Operatoren die anomale Dimension erster Ordnung nicht beeinflusst. Der zweite Ansatz betrachtet die allgemeine Struktur von Korrelatoren in effektiven Theorien in AdS2. Es werden alle skalaren n-Punkt-Kontakt-Witten-Diagramme für externe Operatoren mit ganzzahligem konformem Gewicht berechnet. Effektive Theorien in AdS2, die durch eine Wechselwirkungslagrange mit einer beliebigen Anzahl von Ableitungen definiert sind, werden dann betrachtet und mit Hilfe eines neuen Formalismus der Mellin-Amplituden für 1d-CFTs bis zur ersten Ordnung gelöst. Schließlich wird die diskretisierte Wirkung der Cusped-Wilson-Linie als alternative Möglichkeit zur Gewinnung nicht-perturbativer Daten vorgestellt: durch die Gitterfeldtheorie. / In this thesis, we consider two approaches to the study of correlation functions in one-dimensional defect Conformal Field Theories (dCFT1), in particular those defined by 1/2-BPS Wilson line defects in the three- and four-dimensional superconformal theories relevant in the AdS/CFT correspondence. In the first approach, we use the analytic conformal bootstrap to evaluate two examples of defect correlators. The four-point correlator of the displacement supermultiplet inserted on the 1/2-BPS Wilson line in ABJM theory is computed to the third order in a strong-coupling expansion and reproduces the explicit first-order Witten diagram calculations. The CFT1 data are then extracted from this correlator, and the operator mixing is solved at first order. Consequently, all-order results are derived for the part of the correlator with the highest logarithm power, uniquely determining the double-scaling limit. Then, the five-point correlator of 1/2-BPS operators inserted on the 1/2-BPS Wilson line in =4 super Yang-Mills are studied. The superblocks are derived for all channels of the OPE, and the five-point correlator is bootstrapped to first order in a strong coupling expansion. The CFT1 data are then extracted, confirming that operator mixing does not affect the first-order anomalous dimension. The second approach considers the general structure of correlators in effective theories in AdS2. All scalar n-point contact Witten diagrams for external operators of integer conformal weight are computed. Effective theories in AdS2 defined by an interaction Lagrangian with an arbitrary number of derivatives are then considered and solved to first order using a new formalism of Mellin amplitudes for 1d CFTs. Finally, the cusped Wilson line discretised action is presented as an alternative way to obtain non-perturbative data: through Lattice Field Theory.