Four simplified gradient elasticity models for the simulation of dispersive wave propagation

Gradient elasticity theories can be used to simulate dispersive wave propagation as it occurs in heterogeneous materials. Compared to the second-order partial differential equations of classical elasticity, in its most general format gradient elasticity also contains fourth-order spatial, temporal as well as mixed spatial temporal derivatives. The inclusion of the various higher-order terms has been motivated through arguments of causality and asymptotic accuracy, but for numerical implementations it is also important that standard discretization tools can be used for the interpolation in space and the integration in time. In this paper, we will formulate four different simplifications of the general gradient elasticity theory. We will study the dispersive properties of the models, their causality according to Einstein and their behavior in simple initial/boundary value problems

A new multi-scale dispersive gradient elasticity modelwith micro-inertia: Formulation and C0-finiteelement implementation

Motivated by nano-scale experimental evidence on the dispersion characteristics of materials with a lattice structure, a new multi-scale gradient elasticity model is developed. In the framework of gradient elasticity, the simultaneous presence of acceleration and strain gradients has been denoted as dynamic consistency. This model represents an extension of an earlier dynamically consistent model with an additional micro-inertia contribution to improve the dispersion behaviour. The model can therefore be seen as an enhanced dynamic extension of the Aifantis' 1992 strain-gradient theory for statics obtained by including two acceleration gradients in addition to the strain gradient. Compared with the previous dynamically consistent model, the additional micro-inertia term is found to improve the prediction of wave dispersion significantly and, more importantly, requires no extra computational cost. The fourth-order equations are rewritten in two sets of symmetric second-order equations so that C0-continuity is sufficient in the finite element implementation. Two sets of unknowns are identified as the microstructural and macrostructural displacements, thus highlighting the multi-scale nature of the present formulation. The associated energy functionals and variationally consistent boundary conditions are presented, after which the finite element equations are derived. Considerable improvements over previous gradient models are observed as confirmed by two numerical examples

Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes

Flexural wave propagation in carbon nanotubes (CNTs) can be described through higher-order elasticity theories so as to capture the dispersive behavior induced by the inherent nanoscale heterogeneity. Motivated by experimental dispersion characteristics of metal nano-structured crystals, a new three-length-scale gradient formulation has been recently developed by the authors. In addition to the Laplacian of the strain, this model incorporates two higher-order inertia gradients for an improved dispersion behavior. A discrete medium with lumped masses at multiple scales of observation and combination of lumped mass and distributed mass at the macro-scale is introduced here to provide a micro-mechanical background to the proposed three-length-scale gradient model. The next aim of this paper is to assess the ability of this model to simulate flexural wave dispersion occurring in CNTs. We employ gradient-enriched Euler-Bernoulli and Timoshenko beam theories incorporating either stress gradients, or a combination of both strain gradients and inertia gradients – the latter leading to novel gradient-enriched beam theories. It is demonstrated that the proposed three-length-scale gradient elasticity formulation is able to capture the wave dispersion characteristics of armchair single-walled (5,5) and (10,10) CNTs arising from Molecular Dynamics simulations with high accuracy for a wide range of wave numbers. Advantages over alternative formulations of higher-order beam theories with stress gradients or combined strain-inertia gradient enrichments are discussed for comparative purposes

A Gradient-based Constitutive Model to Predict Size Effects in the Response of Shape Memory Alloys

Shape memory alloys (SMAs) show size effect in their response because the behavior of small-scale SMA structures deviates from that of the bulk material. Ni-Fe-Ga ferromagnetic SMA micropillars, for example, demonstrated a significantly increased hardening in their compressive stress-strain response as their diameter approached micron and submicron scales. This response cannot be modeled using conventional theories that lack an intrinsic length scale in their constitutive models. Constitutive models, however, are crucial for the design and simulation of SMA components at nano and micron scales as in NEMS and MEMS. Therefore, to capture such a size effect, a gradient-based thermodynamically consistent constitutive framework is established. We assume the existence of generalized surface and body forces that contribute to the free energy as work conjugates to the generalized variables of martensite volume fraction, transformation strain tensor, and their spatial gradients. The rates of evolution of the generalized variables are obtained by invoking the principal of maximum dissipation after assuming a transformation surface. This approach is compared to the theories that use a configurational force balance law. The developed constitutive model includes various energetic and dissipative length scales that can be calibrated experimentally. To demonstrate the capabilities of this model, a series of boundary value problems are solved. The boundary value problems contain the differential equation for the transformation surface as well as the equilibrium equation and are solved analytically and numerically. Example problems include pure bending of SMA beams, simple torsion of SMA cylindrical bars, and compression of SMA micro/nanopillars. The simplest version of the model, containing only the additional gradient of martensite volume fraction, predicts a response with greater hardening for smaller structures. Also once calibrated, the model can qualitatively predict the experimentally observed response of Ni-Fe-Ga micropillars under compression

Ultrafast Electron Diffraction at Surfaces

In der vorliegenden Arbeit wird der Aufbau eines Experimentes zur Untersuchung der atomaren Dynamik an Kristalloberflächen mittels zeitaufgelöster Elektronenbeugung beschrieben. Dabei wird die zu untersuchende Probe mit Hilfe von 50 fs kurzen Infrarot-Laserpulsen optisch angeregt. Nach einer variablen Zeitspanne von einigen ps bis wenigen ns vor oder nach der optischen Anregung erfolgt die Abfrage des Momentanzustandes der Probe durch Streuung eines wenige ps kurzen Elektronenpulses an der Oberfläche. Aus der Intensitätsverteilung des dabei entstehenden Beugungsbildes lassen sich Rückschlüsse auf die Größe und Symmetrie der Einheitszelle, sowie auf die Temperatur der Oberfläche ziehen. Durch eine Aneinanderreihung derartiger Momentaufnahmen ist es möglich, die Relaxation des Kristallgitters nach der optischen Anregung der Oberfläche zu rekonstruieren. Das Kernstück des experimentellen Aufbaus bildet eine Elektronenkanone, in der ein 50 fs-kurzer Ultraviolett-Laserpuls durch Photoemission aus einem wenige nm dünnen Au-Film in einen ps-Elektronenpuls konvertiert wird. Die eingehende Charakterisierung der Photokathode, mit der die Konversion realisiert wird, lieferte Erkenntnisse, die auch für den Aufbau nachfolgender, verbesserter Elektronenkanonen wertvoll sein können. Die erreichbare Zeitauflösung der beschriebenen Beugungsexperimente beträgt etwa 20-30 ps. Sie ist bedingt durch den flachen Einfallswinkel der Elektronen auf die Probe, der erforderlich ist, um die Oberflächenempfindlichkeit der Messung zu gewährleisten. Als erstes Untersuchungsobjekt für die zeitaufgelösten Beugungsexperimente diente ein 5.5 nm dünner, epitaktischer Bi-Film auf einem Si(001)-Substrat, das während der Messung auf 85 K abgekühlt wurde. Die zeitliche Entwicklung der Oberflächentemperatur nach der Absorption des optischen Anregungspulses folgt keinem simplen Wärmeleitungsmodell. Stattdessen läßt sich die beobachtete exponentielle Relaxation der Oberflächentemperatur mit einer Zeitkonstante von etwa 640 ps qualitativ durch die Existenz einer endlichen Grenzflächenwärmeleitfähigkeit zwischen dem Bi-Film und dem Si-Substrat erklären. Diese ist auf die Unstetigkeit der Schallgeschwindigkeiten und Massendichten von Bi und Si zurückzuführen, die zur Reflexion eines Großteils der Phononen führen, die aus dem angeregten Bi-Film kommend auf die Grenzfläche treffen. Die Wärmeleitfähigkeit der Bi/Si-Grenzfläche wurde im Rahmen zweier einfacher Modelle berechnet und mit dem experimentell bestimmten Wert verglichen. Dabei betrug die Abweichung zwischen Experiment und Modell nur 30%, was – verglichen mit der Gesamtheit der Untersuchungen zu dieser Thematik – eine recht gute Übereinstimmung darstellt. Dieser Umstand wird auf die abrupte und glatte Bi/Si-Grenzfläche und die geringe Dichte von Gitterfehlern im Bi-Film zurückgeführt. Dünne Bi-Filme auf Si(001) stellen daher ein Modellsystem dar, an dem die Phononendynamik in Nanostrukturen ohne unnötige Komplikationen studiert werden kann. Da im vorliegenden Fall die Bi-Schichtdicke geringer als die mittlere freie Weglänge der Phononen ist, stellt sich für weitergehende Untersuchungen die Frage nach dem Einfluss der verschiedenen Phononen-Streuprozesse auf die Transmission der Phononen über die Bi/Si-Grenzfläche und somit auf das Abkühlverhalten des Bi-Filmes.In this thesis, the construction of a time-resolved electron diffraction experiment for studies of crystal surfaces is described. The setup is used to examine the atomic dynamics at surfaces after an initial optical excitation by a 50 fs-infrared laser pulse. To detect the transient state of the excited surface, a picosecond-electron pulse is directed onto the surface with a variable delay in the range of a few picoseconds up to some nanoseconds before or after the pump pulse. From the resulting diffraction pattern one can deduce information about the symmetry and size of the unit cell as well as the thermal motion of the surface atoms. Based on a series of diffraction patterns it is possible to reconstruct the transient evolution of the surface after the optical excitation. The core item of the experimental setup is the electron gun in which a 50 fs-ultraviolet laser pulse is converted into an electron bunch by photoemission from a thin Au film. The thorough characterisation of the Au photocathode delivered results which are be valuable for the construction of future improved electron guns. The achievable time resolution in the diffraction experiments reported in this thesis is in the range of 20-30 ps. It is limited by the grazing incidence of the electrons onto the crystal surface. However, a shallow incidence angle is necessary to achieve the desired surface-sensitivity of the diffraction experiment. In the first time-resolved studies the transient heating of a 5.5 nm thin, epitaxial Bi film on a Si(001) substrate was examined. The sample was held at a base temperature of 85 K during these experiments. The resulting transient surface temperature cannot be described by a simple heat conduction model. The surface temperature rather decreases exponentially with a time-constant of 640 ps, which can be explained qualitatively by the existence of a finite thermal boundary conductivity between the Bi film and the Si substrate. This barrier is caused by the abrupt change of the phonon velocities and mass densities at the interface, which leads to total internal reflection of the majority of phonons impinging onto the interface, starting from the Bi film. For a comparison with the experimental results, the thermal boundary conductivity was calculated based on two simple models. The deviation between the experiment and the two models was only about 30 %, which is a very good agreement, compared to many other studies in this field. This agreement can be explained by the smooth and abrupt Bi/Si interface and the low defect density in the Bi film. Due to the absence of many complicating effects, thin Bi films can be regarded as a model system to study the phonon dynamics in nanostructures. In the experiments described in this thesis, the phonons' mean free path in the Bi film was larger than the film thickness. This leads to the question in which way phonon scattering processes affect the phonon flux across the Bi/Si interface and thus the cooling of the Bi film