Veröffentlichungen

Zusammenfassung

In the present paper, theoretical foundations of redundancy in spatially continuous, elastostatic, and linear representations of structures are derived. Adopting an operator-theoretical perspective, the redundancy operator is introduced, inspired by the concept of redundancy matrices, previously described for spatially discrete representations of structures. Studying symmetry, trace, rank, and spectral properties of this operator as well as revealing the relation to the concept of statical indeterminacy, a continuum-mechanical theory of redundancy is proposed. Here, the notion “continuum-mechanical” refers to the representation being spatially continuous. Apart from the theory itself, the novel outcome is a clear concept providing information on the distribution of statical indeterminacy in space and with respect to different load carrying mechanisms. The theoretical findings are confirmed and illustrated within exemplary rod, plane beam, and plane frame structures. The additional insight into the load carrying behavior may be valuable in numerous applications, including robust design of structures, quantification of imperfection sensitivity, assessment of adaptability, as well as actuator placement and optimized control in adaptive structures.

Zusammenfassung

(1) Significance of geometry for bio-inspired hygroscopically actuated bilayer structures is well studied and can be used to fine-tune curvatures in many existent material systems. We developed a material design space to find new material combinations that takes into account unequal effective widths of the layers, as commonly used in fused filament fabrication, and deflections under self-weight. (2) For this purpose, we adapted Timoshenko’s model for the curvature of bilayer strips and used an established hygromorphic 4D-printed bilayer system to validate its ability to predict curvatures in various experiments. (3) The combination of curvature evaluation with simple, linear beam deflection calculations leads to an analytical solution space to study influences of Young’s moduli, swelling strains and densities on deflection under self-weight and curvature under hygroscopic swelling. It shows that the choice of the ratio of Young’s moduli can be crucial for achieving a solution that is stable against production errors. (4) Under the assumption of linear material behavior, the presented development of a material design space allows selection or design of a suited material combination for application-specific, bio-inspired bilayer systems with unequal layer widths.

Zusammenfassung

Adaptive structures are characterized by their ability to adjust their geometrical and other properties to changing loads or requirements during service. This contribution deals with a method for the design of quasi‐static motions of structures between two prescribed geometrical configurations that are optimal with regard to a specified quality function while taking large deformations into account. It is based on a variational formulation and the solution by two finite element discretizations, the spatial discretization (the standard finite element mesh) and an additional discretization of the deformation path or trajectory. For the investigations, an exemplary objective function, the minimization of the internal energy, integrated along the deformation path, is used. The method for motion design presented herein uses the Newton‐Raphson method as a second‐order optimization algorithm and allows for analytical sensitivity analysis. The proposed method is verified and its properties are investigated by benchmark examples including rigid body motions, instability phenomena and determination of inextensible deformations of shells.

Zusammenfassung

The design of adaptive structures is one method to improve sustainability of buildings. Adaptive structures are able to adapt to different loading and environmental conditions or to changing requirements by either small or large shape changes. In the latter case, also the mechanics and properties of the deformation process play a role for the structure’s energy efficiency. The method of variational motion design, previously developed in the group of the authors, allows to identify deformation paths between two given geometrical configurations that are optimal with respect to a defined quality function. In a preliminary, academic setting this method assumes that every single degree of freedom is accessible to arbitrary external actuation forces that realize the optimized motion. These (nodal) forces can be recovered a posteriori. The present contribution deals with an extension of the method of motion design by the constraint that the motion is to be realized by a predefined set of actuation forces. These can be either external forces or prescribed length chances of discrete, internal actuator elements. As an additional constraint, static stability of each intermediate configuration during the motion is taken into account. It can be accomplished by enforcing a positive determinant of the stiffness matrix.

Zusammenfassung

Die „Große Beschleunigung“ bei Bevölkerungszahlen, klimaschädlichen Emissionen, Wasserverbrauch und vielem anderen stellt die gesamte Menschheit vor große Herausforderungen. Dies trifft besonders auf das Bauschaffen zu. Es gilt, zukünftig für mehr Menschen mit weniger Material emissionsfrei zu bauen. Hierfür muss unsere Art des Planens, Bauens und Nutzens von Bauwerken neu gedacht und neu konzipiert werden. Auf der bautechnischen Seite bedeutet dies die konsequente flächendeckende Umsetzung von Leichtbaustrategien. Zu diesen zählt neben dem klassischen Leichtbau und den Gradientenbauweisen auch das Bauen mit adaptiven Hüllen und Strukturen. Unter Adaptivität sind dabei unterschiedliche Veränderungen der Geometrie, der physikalischen Eigenschaften von einzelnen Bauteilen oder von ganzen Bauwerken zu verstehen. Durch Adaption können Spannungsfelder homogenisiert, Bauteilverformungen reduziert und bauphysikalische Verhalten von Bauteilen verändert werden. All dies verringert nicht nur den Materialbedarf, sondern liefert auch einen wesentlichen Beitrag zur Steigerung des Nutzerkomforts. Adaptivität im weiteren Sinne bezeichnet einen ganzheitlichen Ansatz, in dem die Anpassung sozialer, kultureller und räumlicher Erfahrungen sowie architektonischer und planerischer Handlungsweisen eng mit den technologischen Entwicklungen verknüpft wird. Die Zusammenführung dieser Perspektiven ist Anspruch des SFB, um ganzheitliche Lösungen für eine zukünftige gebaute Umwelt zu finden.

Zusammenfassung

Hammering tools uses damping elements of various geometries and hardness levels not only to reduce the reaction forces and vibrations, but also to protect the structural parts from failure. This makes damping elements a key component during the design phase of any tool. Since many design quantities initially rely on the results obtained via simulations for optimization and improvement purposes, it makes it essential to have an appropriate material model which could capture the behaviour of the damping elements accurately. Damping elements are made up of elastomeric compounds showcasing highly non-linear behaviour especially when subjected to very high strains and strain rates when mounted in a tool. This aim of this work is to study various phenomena regarding the rubber material behaviour and develop a user-defined material model in LS-DYNA which provides error-free stress updates at a given strain level for elastomers. The fundamental concepts of hyperelastic and viscoelastic constitutive theories are emphasized in the beginning as these theories are more suitable to model elastomeric behaviour. Material parameter identification procedure is also highlighted for both hyperelastic and linear viscoleastic material models. An Ogden-based linear viscoelastic model is programmed which is extended with strain-level based non-linearity included in the material model and later validated with the experimental results obtained with a gas-gun test fixture and O-ring specimens. Discussions regarding energy dissipation is focussed towards the end as it plays a crucial role in understanding the behaviour of the damping element.

Zusammenfassung

The degree of statical indeterminacy as a fundamental property in structural mechanics is today mainly known as a property of a complete system without any information about its spatial distribution. The redundancy matrix provides information about the distribution of statical indeterminacy in the system and by this gives an additional valuable insight into the load-bearing behaviour. The derivation and definition of the redundancy matrix are presented based on truss systems and its mathematical properties and their mechanical interpretations are provided. The definition of the redundancy matrix is extended to other discrete systems like beam structures and a definition of the redundancy density is given for the continuous 1D case. Potential applications of the concept include robust design of structures, quantification of imperfection sensitivity as well as assessment of optimal actuator placement in adaptive structures.

Zusammenfassung

Part I deals with material modelling of woven fabric membranes. Due to their structure of crossed yarns embedded in coating, woven fabric membranes are characterised by a highly nonlinear stress-strain behaviour. In order to determine an accurate structural response of membrane structures, a suitable description of the material behaviour is required. A linear elastic orthotropic model approach, which is current practice, only allows a relative coarse approximation of the material behaviour. The present work focuses on two different material approaches: A first approach becomes evident by focusing on the meso-scale. The inhomogeneous, however periodic structure of woven fabrics motivates for microstructural modelling. An established microstructural model is considered and enhanced with regard to the coating stiffness. Secondly, an anisotropic hyperelastic material model for woven fabric membranes is considered. By performing inverse processes of parameter identification, fits of the two different material models w.r.t. measured data from a common biaxial test are shown. The results of the inversely parametrised material models are compared and discussed.

Zusammenfassung

Taking advantage of adaptivity in the field of civil engineering is a subject of ongoing research. Integration of adaptive elements in load-bearing structures is already well-established in many other engineering fields, albeit mostly for different purposes than withstanding predominantly static loads. Initial investigations have demonstrated potential for substantial material and energy savings also in the field of civil engineering, especially for high-rise buildings and wide-span structures, such as roofs or bridges. Adaptive civil structures show promise in tackling current challenges arising from emissions and shortages of materials. In this study, we compare the possible minimum-weight designs for different actuator placement approaches and for different structural topologies that satisfy various constraints for high-rise buildings. We use case studies as illustrative examples to show which advantages and disadvantages can be expected from a specific design. The overarching aim is to learn how truss and beam structures should be designed to perform well as adaptive structures.

Zusammenfassung

Die Antwort einer Struktur auf Aktuierung wird maßgeblich vom Grad der statischen Unbestimmtheit und deren Verteilung in der Struktur beeinflusst. Die Redundanzmatrix enthält diese Informationen über das Tragwerk. Aus ihr können daher Rückschlüsse für die Aktorplatzierung und für die Bewertung der Aktuierbarkeit von Strukturen gezogen werden. Die Untersuchung eines Beispieltragwerks veranschaulicht einerseits die Einsatzmöglichkeiten der Redundanzmatrix und andererseits das Masseneinsparungspotential, das mithilfe des Einsatzes aktiver Elemente in passiven Strukturen erschlossen werden kann.

Zusammenfassung

Taking advantage of adaptivity in the field of civil engineering is an ongoing research topic. Integration of adaptive elements in the load-bearing structure is already well established in many other engineering fields. First investigations promise large material saving potentials also in the field of civil engineering, especially when it comes to high-rise buildings or wide spanned structures like roofs or bridges. In times of emission problems and shortage of materials, the potentials of adaptive civil structures open various new possibilities. In the design and optimization process of adaptive civil structures, we address the differences between classical approaches for passive systems and new practices considering adaptivity. By using a suitable actuator placement, it is possible to manipulate the displacements of the structure as well as the force distribution within the structure. Both material and energy savings can be accomplished with an integrated design of the adaptive structure taking into account the actuation, suitable combination of structural design and actuator placement. For demonstration of the differences in the design process and in the resulting optimized structure, we use a small case study on a truss structure, which is inspired by a high-rise building, and consider static loads.

Zusammenfassung

The enhanced assumed strain (EAS) method is one of the most frequently used methods to avoid locking in solid and structural finite elements. One issue of EAS elements in the context of geometrically non‐linear analyses is their lack of robustness in the Newton‐Raphson scheme, which is characterized by the necessity of small load increments and large numbers of iterations. In the present work we extend the recently proposed mixed integration point (MIP) method to EAS elements in order to overcome this drawback in numerous applications. Furthermore, the MIP method is generalized to generic material models, which makes this simple method easily applicable for a broad class of problems. In the numerical simulations in this work, we compare standard strain based EAS elements and their MIP improved versions to elements based on the assumed stress method in order to explain when and why the MIP method allows to improve robustness. A further novelty in the present work is an inverse stress‐strain relation for a Neo‐Hookean material model.

Zusammenfassung

Die wissenschaftliche Auseinandersetzung mit der Stabilität von Tragwerken reicht in der Geschichte weit zurück. Leonhard Euler befasste sich bereits im Jahr 1744 mit dem Stabilitätsversagen druckbelasteter Stäbe. Bis heute ist die Frage nach der Stabilität in vielen Bereichen, vom Bauwesen bis zur Luft- und Raumfahrt, relevant für die Bemessung und Konstruktion einzelner Bauteile sowie gesamter Tragstrukturen. Die Analyse eines Tragwerks hinsichtlich seiner Stabilität kann dabei in drei Abschnitte untergliedert werden: erstens die Untersuchung des Systems vor dem Stabilitätsversagen, zweitens die Eigenschaften des Systems an einem kritischen Punkt sowie drittens die Untersuchung des Verhaltens im Nachbeulbereich. Für eine geometrisch nichtlineare Analyse des Vor- und Nachbeulbereichs existiert eine Vielzahl inkrementell-iterativer Lösungsverfahren, die seit den 1970er Jahren entwickelt wurden. Sie ermöglichen die zuverlässige Berechnung nichtlinearer Gleichgewichtspfade einer Struktur, auch wenn diese in Last oder Verschiebung rückläufig sind. In den vergangen Jahrzehnten wurden diese Methoden stetig weiterentwickelt und verbessert, etwa durch die Erweiterung auf mehrere Parameter. So lassen sich für Systeme mit mehreren Veränderlichen die resultierenden Gleichgewichtspfade zu Gleichgewichtsflächen erweitern. Mithilfe begleitender Maßnahmen kann zudem an jedem Gleichgewichtspunkt eine Aussage über die Stabilität des Systems getroffen werden. Der Fokus der vorliegenden Arbeit liegt auf dem zweiten der oben genannten Punkte. Es werden Methoden präsentiert, die eine direkte Berechnung von Durchschlags- und Verzweigungspunkten für Systeme ermöglichen, die durch eine von außen aufgebrachte Knotenverschiebung belastet werden. Hierfür wird die für Kraftlastfälle bereits verfügbare Methode der erweiterten Systeme, die in der Literatur meistens mit dem englischen Begriff Extended Systems bezeichnet wird, modifiziert und ergänzt. Für die direkte Berechnung von Durchschlagspunkten wird die Residuumsgleichung für das Gleichgewicht um eine Stabilitätsbedingung erweitert. Diese ist ausschließlich an kritischen Punkten erfüllt und lässt sich aus dem Indifferenzkriterium nach Euler herleiten. Eine Nebenbedingung, die eine triviale Lösung vermeidet und die Anzahl der Gleichungen und die Anzahl der Unbekannten ausgleicht, ergibt sich aus der Normierung der Länge des Eigenvektors. Für die direkte Ermittlung von Verzweigungspunkten muss zusätzlich das Kriterium zur Unterscheidung von Durchschlags- und Verzweigungspunkten für den Fall inhomogener Dirichlet-Randbedingungen neu hergeleitet werden. Nach einer konsistenten Linearisierung des nichtlinearen Gleichungssystems lässt sich die Lösung mittels inkrementell-iterativer Methoden ermitteln. Mit den präsentierten Methoden wird das Anfangsnachbeulverhalten für Systeme mit inhomogenen Dirichlet-Randbedingungen numerisch analysiert, wodurch die bisherige Kategorisierung nach Koiter (1967) erweitert wird. Zur Linearisierung des nichtlinearen Gleichungssystems ist die Ermittlung einer Richtungsableitung der Steifigkeitsmatrix in Richtung des Beulvektors erforderlich. In der bisherigen Literatur wird die Ableitung mittels Differenzenquotienten angenähert. Diese Methoden benötigen eine geringe Rechenzeit, ihre Eigenschaften hängen jedoch von einem Parameter, der Größe des Differenzenschritts, ab. Wird dieser zu groß oder zu klein gewählt, ist der resultierende Fehler zu groß und die iterative Berechnung konvergiert nur langsam oder gar nicht zur gesuchten Lösung. In dieser Arbeit werden erstmals Methoden zur Ermittlung totaler und partieller Ableitungen, die auf hyperkomplexen Zahlen basieren, mit der Methode der Extended Systems kombiniert. Diese Methoden sind unabhängig von der Parameterwahl und ermöglichen eine exakte Berechnung erster und zweiter Ableitungen. In der vorliegenden Arbeit werden die beschriebenen Methoden auf die Ermittlung der Richtungsableitung übertragen. Bei der Untersuchung der Tragfähigkeit einer stabilitätsgefährdeten Struktur haben Imperfektionen einen großen Einfluss auf die Traglast. Bereits kleinste Abweichungen in der Geometrie eines Tragwerks oder Eigenspannungen im Material können, besonders bei schlanken Tragwerken, zu einer deutlichen Abminderung der Traglast führen. Um den Einfluss geometrischer Imperfektionen genauer untersuchen zu können, werden in dieser Arbeit zwei Methoden präsentiert, die sich in ihrem Umgang mit Imperfektionen grundlegend voneinander unterscheiden. Bei der ersten Methode ist die Imperfektionsform eine gegebene Größe. Durch eine effiziente, sich wiederholende Berechnung kritischer Punkte für eine Vielzahl an Imperfektionsformen oder -amplituden lassen sich damit kritische Pfade ermitteln. Dies geschieht in der vorliegenden Arbeit durch eine geeignete Wahl der Prädiktoren für die unterschiedlichen Imperfektionsformen bzw. über eine zusätzliche Erweiterung der Gleichungssysteme zur direkten Berechnung kritischer Punkte um eine Zusatzgleichung zur Pfadverfolgung. Bei der zweiten Methode werden die Imperfektionen als Unbekannte in das Gleichungssystem eingebracht. Durch eine zusätzliche Bedingung, die sich aus der Variation des Potentials nach den Imperfektionen ergibt, sowie das Einbinden der Nebenbedingung über eine Straffunktion lassen sich mit dieser Methode ungünstigste Imperfektionsformen finden. Diese Methode ist eine Weiterentwicklung der von Deml und Wunderlich (1997) vorgschlagenen Vorgehensweise. Abschließend werden die hier vorgestellten Methoden und Algorithmen an einer Auswahl an numerischen Experimenten demonstriert. Anhand der Beispiele wird u. a. gezeigt, dass die Richtungsableitung für eine Vielzahl an Elementformulierungen effizient ermittelt werden kann. Hierbei kommen vom einfachen ebenen Stabelement über isogeometrische Schalenelemente bis hin zu Volumenschalenelementen, die durch eine Erweiterung des Verzerrungsansatzes künstliche Versteifungseffekte reduzieren, zum Einsatz. Zudem wird die Anwendbarkeit der in der vorliegenden Arbeit entwickelten Methoden auf nichttriviale Systeme im Rahmen der Finite-Elemente-Methode mit vielen Freiheitsgraden nachgewiesen.

Zusammenfassung

In this thesis, a novel approach to support the design of motions for adaptive structures is presented and gradually developed: the so-called method of motion design. It is based on the observation that, depending on the control of the actuation, the same deformation state of a structure can be reached through various motion processes. The method of motion design allows to calculate optimal deformation paths with defined properties between the initial geometry and a given deformed end geometry of a structure in a formalized way. In order to motivate the efficiency of a movement and to make it mathematically quantifiable, the so-called cost of deformation is introduced as an exemplary target value based on the strain energy. By integration over the deformation path, the motion process is considered in its entirety in this optimization problem. The method of motion design is developed based on a variational formulation using the cost of deformation as underlying functional and the displacement field as the unknown function. One of the decisive features in this work is the discretization of the motion path, i.e., the deformation process. Due to the special structure of the functional with the integration of the strain energy, analytical sensitivities can be calculated by using quantities that are generally available in finite element software. The presented basic method is particularly well suited for the identification and design of kinematic and energy-minimal motion mechanisms, which emphasizes the potential for application to deployable shape changing structures. The motion design method is extended by the use of constraints such that the actuation can be prescribed, e.g., by actuator elements, or the entire motion can be stabilized. Finally, possibilities for enhancement of the motion design method and combinations with other methods to increase the efficiency of adaptive structures are investigated. They include a combination with shape optimization of the initial geometry, an integration within an actuator placement algorithm and variations of the underlying objective function.

Zusammenfassung

The mechanical principles for fast snapping in the iconic Venus flytrap are not yet fully understood. In this study, we obtained time-resolved strain distributions via three-dimensional digital image correlation (DIC) for the outer and inner trap-lobe surfaces throughout the closing motion. In combination with finite element models, the various possible contributions of the trap tissue layers were investigated with respect to the trap’s movement behavior and the amount of strain required for snapping. Supported by in vivo experiments, we show that full trap turgescence is a mechanical–physiological prerequisite for successful (fast and geometrically correct) snapping, driven by differential tissue changes (swelling, shrinking, or no contribution). These are probably the result of the previous accumulation of internal hydrostatic pressure (prestress), which is released after trap triggering. Our research leads to an in-depth mechanical understanding of a complex plant movement incorporating various actuation principles.

Zusammenfassung

Finite elements with Allman’s rotations provide good computational efficiency for explicit codes exhibiting less locking than linear elements and lower computational cost than quadratic finite elements. One way to further raise their efficiency is to increase the feasible time step or increase the accuracy of the lowest eigenfrequencies via reciprocal mass matrices. This paper presents a formulation for variationally scaled reciprocal mass matrices and an efficient estimator for the feasible time step for finite elements with Allman’s rotations. These developments take special care of two core features of such elements: existence of spurious zero‐energy rotation modes implying the incompleteness of the ansatz spaces, and the presence of mixeddimensional degrees of freedom. The former feature excludes construction of dual bases used in the standard variational derivation of reciprocal mass matrices. The latter feature destroys the efficiency of the existing nodal‐based time step estimators stemming from the Gershgorin’s eigenvalue bound. Finally, the developments are tested for standard benchmarks and triangular, quadrilateral and tetrahedral finite elements with Allman’s rotations.

Zusammenfassung

Customization of finite elements for low dispersion error through grid dispersion analysis requires a symbolic expansion of a determinant of a representative dynamic stiffness matrix. Such an expansion turns out to be a bottleneck for many practical cases with the size of the representative matrix greater than eight or ten even if the modern computer algebra systems are applied. In this contribution, we propose an alternative approach for low-dispersion customization that avoids explicit determinant expansion. This approach reduces the customization problem to a series of quadratic programming problems and consist of two main steps. Firstly, the customization problem is reformulated as a rank minimization problem for the representative dynamic stiffness matrix evaluated at several discrete pairs of wavenumbers and frequencies. Secondly, the rank minimization problem is solved approximately via log-det heuristic. Examples for customization of reciprocal mass matrices illustrate capabilities of the proposed approach.

Zusammenfassung

We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.