Elastic-plastic design sensitivities based on variational analysis and applications in optimal specimen design
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Date
2021
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Abstract
Voranschreitender technologischer Fortschritt fordert hohe Ansprüche an ingenieurtechnische
Strukturen und Materialien. Dabei ist es wichtig, die Produktionskosten gering zu halten
und gleichzeitig höchste Sicherheit in der praktischen Anwendung zu gewährleisten. Auch
logistische und ökologische Aspekte spielen eine wichtige Rolle. Die richtige Wahl und Ausnutzung
des Potentials der verwendeten Materialien ist in diesen Zusammenhängen enorm
wichtig. Die computergestützte Optimierung von Bauteilen und Materialien ist dem Gebiet
der Strukturoptimierung zuzuordnen, welche es ermöglicht, mechanische Strukturen hinsichtlich
gewählter Eigenschaften zu verbessern und gleichzeitig wichtige Einschränkungen zu berücksichtigen.
Dafür muss das zu verändernde Design definiert und gewünschte Zielwerte und
Nebenbedingungen mathematisch formuliert werden. Dies erfordert zum einen Wissen über
den Grad der Beanspruchung des analysierten Bauteils und zum anderen müssen die mechanischen
Vorgänge innerhalb des verwendeten Materials gut verstanden und im Rahmen mathematischer
Modelle abbildbar sein. Vor allem ungünstige Phänomene, wie z.B. plastisches
Fließen, Schädigung oder Ermüdung, die letztendlich zum Strukturversagen führen können,
sind es wert detailliert analysiert zu werden. Die Entwicklung von Modellen zur Beschreibung
mechanischer Phänomene ist langjährige wissenschaftliche Tradition und wird stets weiterentwickelt.
Die Durchführung wissenschaftlicher Experimente ist in diesem Zusammenhang
unabdinglich und sollte auf das zu untersuchende mechanische Phänomen zugeschnitten sein.
In dieser Arbeit geht es um die Formoptimierung biaxialer Versuchskörper zur Charakterisierung
von Schädigung duktiler metallischer Werkstoffe. Duktile Schädigung und die treibenden
mikromechanischen Mechanismen sind abhängig vom Spannungszustand. Daher soll
die Form der Biaxialprobe so verändert werden, dass sich während der Versuche bestimmte
und möglichst homogene Spannungszustände einstellen. Hierzu wird ein effizientes computergestütztes
Modell aufbereitet, das große elastoplastische Deformationen berücksichtigt und
im Rahmen eines gradientenbasierten Optimierungsverfahrens die benötigten Sensitivitätsinformationen
bezüglich der Formänderung der Probe liefert. Die Bestimmung der Sensitivitätsinformationen
erfolgt mittels eines variationellen Ansatzes und erfordert tiefes Verständnis
der Grundgleichungen des mechanischen Modells. Zur Validierung der resultierenden optimalen
Geometrien werden Experimente durchgeführt und mittels digitaler Bildkorrelation
aufgezeichnet. Anschließend werden hochauflösende Bilder der Bruchflächen untersucht, die
mittels eines Rasterelektronenmikroskops aufgenommen wurden, um Rückschlüsse auf den
Spannungszustand kurz vor dem Versagen zu ziehen.
Advancing technological progress places high demands on engineering structures and materials. It is important to keep the production costs low and at the same time to ensure the highest level of safety in practical use. Logistical and ecological aspects also play an important role. The right choice and utilization of the potential of the used materials is extremely important in this context. Computer-aided optimization of components and materials is assigned to the field of structural optimization, which enables improvements of mechanical structures with regard to selected properties and at the same time taking important restrictions into account. For this, the design to be changed must be defined and the desired objective values and constraints have to be formulated mathematically. On the one hand, this requires knowledge of the stress and strain intensity occurring in the analyzed component and, on the other hand, the mechanical processes within the used material must be well understood and captured by mathematical models. Especially unfavorable phenomena, such as plastic yielding, damage or fatigue, which can ultimately lead to structural failure, are worth to be analyzed in detail. The development of models to describe mechanical phenomena is a longstanding scientific tradition and is constantly being further developed. The performance of scientific experiments is essential in this context and should be tailored to the mechanical phenomenon to be investigated. This thesis is about the shape optimization of biaxial test specimen for damage characterization of ductile metallic materials. Ductile damage and the driving micromechanical mechanisms highly depend on the stress state. Therefore, the shape of the biaxial specimen should be changed in such a way that certain and as homogeneous stress states as possible arise during the tests. For this purpose, an efficient computer-aided model is prepared that covers large elastoplastic deformations and, as part of a gradient-based optimization process, provides the necessary sensitivity information with regard to the change in shape of the specimen. The determination of the sensitivity information is carried out using a variational approach and requires deep understanding of the governing equations of the mechanical model. For validation of the resulting optimal geometries, experiments are carried out and monitored using digital image correlation. Furthermore, high-resolution images of the fracture surfaces, which were recorded using a scanning electron microscope, are examined in order to draw conclusions about the state of stress shortly before failure.
Advancing technological progress places high demands on engineering structures and materials. It is important to keep the production costs low and at the same time to ensure the highest level of safety in practical use. Logistical and ecological aspects also play an important role. The right choice and utilization of the potential of the used materials is extremely important in this context. Computer-aided optimization of components and materials is assigned to the field of structural optimization, which enables improvements of mechanical structures with regard to selected properties and at the same time taking important restrictions into account. For this, the design to be changed must be defined and the desired objective values and constraints have to be formulated mathematically. On the one hand, this requires knowledge of the stress and strain intensity occurring in the analyzed component and, on the other hand, the mechanical processes within the used material must be well understood and captured by mathematical models. Especially unfavorable phenomena, such as plastic yielding, damage or fatigue, which can ultimately lead to structural failure, are worth to be analyzed in detail. The development of models to describe mechanical phenomena is a longstanding scientific tradition and is constantly being further developed. The performance of scientific experiments is essential in this context and should be tailored to the mechanical phenomenon to be investigated. This thesis is about the shape optimization of biaxial test specimen for damage characterization of ductile metallic materials. Ductile damage and the driving micromechanical mechanisms highly depend on the stress state. Therefore, the shape of the biaxial specimen should be changed in such a way that certain and as homogeneous stress states as possible arise during the tests. For this purpose, an efficient computer-aided model is prepared that covers large elastoplastic deformations and, as part of a gradient-based optimization process, provides the necessary sensitivity information with regard to the change in shape of the specimen. The determination of the sensitivity information is carried out using a variational approach and requires deep understanding of the governing equations of the mechanical model. For validation of the resulting optimal geometries, experiments are carried out and monitored using digital image correlation. Furthermore, high-resolution images of the fracture surfaces, which were recorded using a scanning electron microscope, are examined in order to draw conclusions about the state of stress shortly before failure.
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Keywords
Specimen shape optimization, Variational sensitivity analysis, Finite element method