A new approximation scheme for Griffith's theory of brittle fracture is presented. The energy functional is approximated by a family of functionals depending on a small parameter and two fields: the displacement field and an eigendeformation describing the fractures that may occur in the body. The eigendeformations allow the displacement field to develop jumps that cost fracture energy depending on the $\epsilon$-neighborhoods of their supports. The mathematical outline of the method is briefly sketched, discussing $\Gamma$-convergence of the eigendeformation functional sequence to Griffith's energy. Numerical examples concerned with benchmark problems of brittle fracture illustrate the main numerical features of the proposed approach.
In this talk I will present some of our recent applications of diffuse-interface modelling (DIM). This technique is an elegant approach to track moving interfaces in time, including topological changes in multi-phase systems like breakup and coalescence. Briefly, the theory behind DIM will be reviewed and some details will be discussed on the numerical methods involved.The following applications will be discussed: The rheology and morphology of polymer blends in shear flow. Micron-sized drop spreading and impact on smooth and non-smooth substrates. Phase separating blends in two-component injection moulding. Marangoni convection of polymer-polymer drops (a ternary system)
First, we propose to briefly recall the basic principles of classical brittle fracture, and the link between that theory and the variational theory. After reviewing the main results and drawbacks of the variational approach, we will discuss the approximation techniques used in the computation of the obtained variational evolutions and the challenging mathematical issues that arise in attempting any kind of implementation.
We will conclude with very recent and still incomplete results on crack branching.
The flow behavior of cells and vesicles is important in many applications in biology and medicine. For example, the flow properties of blood in micro-vessels is determined by the rheological properties of the red blood cells. Furthermore, microfluidic devices have been developed recently, which allow the manipulation of small amounts of suspensions of particles or cells.
While the membrane of vesicles just consist of a fluid lipid bilayer, red blood cells have a composite membrane which has in addition an anchored polymer network. This implies that the elastic properties of vesicles and red blood cells are very different.
Due to the large length- and time-scale gap between the atomic and the mesoscopic domain in soft matter systems, several mesoscale simulation techniques have been developed in recent years to study their hydrodynamic behavior. We have investigated one of these techniques, multi-particle-collision dynamics [1], in some detail. In particular, it has been shown that the method properly describes hydrodynamic interactions at low Reynolds and high Schmidt numbers, if the parameters are chosen appropriately [2]. This method has then be applied to study the dynamical bevavior of fluid vesicles and model red blood cells both in shear and capillary flows [3-5]. Several types of dynamical behaviors as well as shape transformations occur as a function of shear rate (or flow velocity), membrane viscosity and internal viscosity, which will be discussed in some detail.
[1] A. Malevanets and R. Kapral, J. Chem. Phys. 110, 8605 (1999).[2] M. Ripoll, K. Mussawisade, R.G. Winkler and G. Gompper, Europhys. Lett. 68, 106 (2004). [3] H. Noguchi and G. Gompper, Phys. Rev. Lett. 93, 258102 (2004); Phys. Rev. E 72, 011901 (2005). [4] H. Noguchi and G. Gompper, Proc. Natl. Acad. Sci. USA 102, 14159 (2005). [5] H. Noguchi and G. Gompper, Phys. Rev. Lett. 98, 128103 (2007).
We briefly repeat some basics, i.e. what defines a martensitic phase transformation (MPT) in general, what are the characteristic features, and historic and modern applications. We then show, how a systematic order can be achieved in the broad spectrum of martensitic systems, at least the metallic ones. We discuss the physical background of the lattice instabilities and the characteristic differences – and similarities - of Fe-based systems on the one hand, and non-magnetic and ferromagnetic Hume-Rothery systems on the other hand on a microscopic length scale. We will show that the key to martensite lies in the understanding of the electronic structure of the respective materials. Phonon softening at certain q-vectors of the Brillouin Zone at temperatures above Ms is not a necessary condition for the appearance of the martensitic transition. At the end we will briefly highlight in general the shape memory behavior, the one way-, the two way-, and the magnetic shape memory-effect and briefly touch the present microscopic knowledge of these effects.
The talk will discuss the modeling of multi-phase fluid membranes surrounded by a viscous fluid with a particular emphasis on the inner flow--the motion of the lipids within the membrane surface. For this purpose, we obtain the equations of motion of a two-dimensional viscous fluid flowing on a curved surface that evolves in time. These equations are derived from the balance laws of continuum mechanics, and a geometric form of these equations is obtained. We apply these equations to the formation of a protruding bud in a fluid membrane, as a model problem for physiological processes on the cell wall. We discuss the time and length scales that set different regimes in which the outer or inner flow are the predominant dissipative mechanism, and curvature elasticity or line tension dominate as driving forces. We compare the resulting evolution equations for the shape of the vesicle when curvature energy and internal viscous drag are operative with other flows of the curvature energy considered in the literature, e.g. the $L_2$ flow of the Willmore energy. We show through a simple example (an area constrained spherical cap vesicle) that the time evolutions predicted by these two models are radically different.
We discuss recent progress in the understanding of the static and dynamic properties of nematic liquid single crystal elastomers as revealed by new experiments. Various descriptions invoking softness, semi-softness, or non-softness, as well as broken symmetries are discussed and compared. The relation between linear and nonlinear elastic properties is considered. Special emphasis is laid on the role of relative rotations between the nematic orientation and the polymer network.
The second part of the talk deals with magnetic gels, a novel class of materials that combines the physical and material properties of a magnetic liquid or a magneto-rheological fluid with those of a polymer network. We give a general introduction to the macroscopic description of ferrogels focusing on the novel physical aspects unknown from other materials.
Complex fluids with a flow behaviour differing significantly from the behaviour of normal liquids like water or oil exhibit numerous fascinating flow effects. This becomes especially interesting if the fluids properties and flows can be controlled by external forces.
Prominent examples for fluids allowing such an external control are suspensions of magnetic nanoparticles commonly known as ferrofluids. They exhibit normal liquid behaviour coupled with superparamagnetic properties leading to the possibility to control their flow and change their properties by means of moderate magnetic fields in the order of 10mT. With such fields, which can be produced easily with small electromagnets or by means of commercial permanent magnets, forces comparable to - for example - the gravitational force, can be exhibited to ferrofluids. The comparably easy magnetic control gave rise to the development of numerous applications of ferrofluids in various technical and medical areas which partly gained importance in everyday life.
Within the talk the origin of the magnetic influence, its fluid mechanical consequences as well as the application possibilities in mechanics, medicine and other fields will be outlined and illustrated with a couple of experiments.
The derivation of robust interatomic potentials is a key step for bridging from the electronic to the atomistic modelling hierarchies in materials science. We present an analytic interatomic bond-order potential (BOP) that depends explicitly on the valence of the transition metal element [1]. This analytic potential predicts the structural trend from hcp to bcc to hcp to fcc that is observed across the non-magnetic 4d and 5d transition metal series. The potential also describes the different ferromagnetic moments of the alpha (bcc), gamma (fcc) and epsilon (hcp) phase of the 3d transition metal iron, the difference between the ferromagnetic and anti-ferromagnetic states as well as non-collinear spin-configurations. In addition, this new potential includes a correct description of alloy bonding within its remit.
In this talk we will show how the potential is derived from the tight-binding electronic structure and demonstrate that it may be regarded as a systematic extension of the second-moment Finnis-Sinclair potential to include higher moments.
[1] R. Drautz and D.G. Pettifor, Phys. Rev. B 74, 174117 (2006).
In this talk we present several examples when non-dissipative mechanical systems exhibit behavior which can be interpreted at a large scale as not only thermodynamical but also dissipative. The origin of such 'transformation' is weak rater than strong convergence with respect to a small parameter.
Crack formation is conventionally described as a nucleation phenomenon despite the fact that the temperatures necessary to overcome the nucleation barrier are far too high. In this talk we consider the possibility that cracks are created due to the presence of frozen disorder (e.g., heterogeneities or frozen dislocations). In particular we calculate the probability for the occurrence of a critical crack in a quasi-two-dimensional disordered elastic system. It turns out that this probability takes the form of an Arrhenius law (as for thermal nucleation) but with the temperature T replaced by an effective disorder temperature T_{eff} which depends on the strength of the disorder. The extension of these results to d=3 dimensions is briefly discussed.
Tailoring of micro structures and surface functionalization are key goals in the technology of materials. Starting from nature we are able to learn, that well adapted periodical microstructures in small dimensions perform superior composite effects. A new technology, allowing a quick and geometrically precise and direct structuring of long range ordered periodical microstructures at artificial material surfaces, the so-called "Laser Interference Metallurgy" is presented. Basically, the high power Laser pulse is split into several coherent sub beams which interfere on the surface of the sample. The shape and the dimensions of the interference pattern as well as its periodicity can be controlled by the angular and the intensity arrangement of the sub beams. Due to the high and localised periodical intensity distribution of the nanosecond laser pulse a redesign of the surface microstructure in terms of the polycrystal formation, phase arrangement, texture, residual stresses or topography is possible. Various examples demonstrate the versatility of this technique for long range ordered surface architectures and controlled properties.
The presence of surfactants, ubiquitous at most gas/liquid interfaces, has a pronounced effect on the surface tension, hence on the stress balance at the phase boundary: local variations of the capillary forces induce transport of momentum along the interface – so-called Marangoni effects. Surfactants are often soluble in one of the adjacent bulk phases, in which case there is also exchange of surfactant between the relevant bulk phase and the interface by adsorption and desorption. Along the interface surfactant is transported by convection and diffusion. Further, changes of the interfacial area due to compression or stretching cause corresponding changes in surfactant concentration.
We discuss the mathematical model governing the dynamics of such systems, including basic analytical questions and present a Volume-of-Fluid based numerical approach to compute the behaviour of fluid particles under the influence of surfactant.
Vesicles are closed lipid bilayer compartments with an internal water core. Giant vesicles, in contrast to conventional ones (~ 100nm), have the advantage that they can be observed using optical microscopy. These giants, being only a few tens of microns in size, are quite a handy tool for studying the mechanical properties of lipid bilayers, because the physical characteristics of the membrane can be obtained from working with individual vesicles. A large variety of techniques have been developed for assessing the elastic properties (bending modulus, stretching elasticity, spontaneous curvature) and the hydrodynamic features (shear surface viscosity) of the lipid bilayer. A few of them will be presented in this talk. Giant vesicles have been also used for direct observation and measuring the properties of lipid membranes when undergoing phase transitions. A dramatic increase of the membrane surface viscosity as well as the bending stiffness of the membrane was detected for membranes crossing the fluid-to-gel phase transition.
The interaction of electric fields with lipid membranes and cells has been extensively studied in the last decades. The phenomena of electroporation and electrofusion are of particular interest because of their widespread use in cell biology and biotechnology. However, direct optical microscopy observation of effects caused by electric DC pulses on giant vesicles is difficult because of the short duration of the pulse. Recently this difficulty has been overcome in our lab. Using a digital camera with high temporal resolution, we were able to access the dynamics of electro-deformation, -poration and -fusion of vesicles on a sub-millisecond time scale. Results from these observations will be presented.
Steps are long-lived structural defects on crystal surfaces, and as such they constitute a natural basis for the description of surface morphology and dynamics on mesoscopic scales. The step dynamical approach plays a pivotal role in any multiscaling scheme for crystal surface evolution, because it can be connected in a quantitative and rigorous fashion both to atomistic methods such as kinetic Monte Carlo simulations, and to large scale continuum height representations of the global surface morphology. Both connections will be illustrated in the talk, using as examples our recent work on step bunching instabilities and island electromigration.
In 1984, Michael Berry pointed out that a quantum system that is adiabatically transported along a closed path in the space of external parameters aquires, in addition to the familiar dynamical phase, a purely geometrical phase, which is a quantum equivalent of the rotation of the Foucault pendulum oscillation plane. Since then, the concept of the Berry phase has become a central unifying concept in quantum mechanics, and has found applications in many other fields of physics.
In the first part, the Berry phase of an anisotropic spin system that is adiabatically rotated along a closed circuit C is investigated [1]. It is shown that the Berry phase consists of two contributions: (i) a geometric contribution which can be interpreted as the flux through C of a nonquantized Dirac monopole, and (ii) a topological contribution which can be interpreted as the flux through C of a Dirac string carrying a nonquantized flux, i.e., a spin analogue of the Aharonov-Bohm effect. Various experimental consequences of this novel effect are discussed, including Berry phase of magnons [2] and of holes in III-V semiconductor heterostructures.
In a second part, a topological theory of the diabolical points (degeneracies) of anisotropic quantum magnets is presented [3]. Diabolical points are characterized by their diabolicity index, for which topological sum rules are derived. The paradox of the the missing diabolical points for Fe8 molecular magnets is clarified. A new method is also developed to provide a simple interpretation, in terms of destructive interferences due to the Berry phase, of the complete set of diabolical points found in biaxial systems such as Fe8.
[1] P. Bruno, Phys. Rev. Lett. 93, 247202 (2004).[2] V.K. Dugaev, P. Bruno, B. Canals, and C. Lacroix, Phys. Rev. B 72, 024456 (2005).[3] P. Bruno, quant-ph/0511186
It is reported about two techniques to produce nanostructures on surfaces by low-energy ion bombardment. The ability to predict and control nanostructural details on surfaces is seen as crucial. The aim of these two ion beam assisted techniques is to generate well-defined nanostructures on surfaces.
(1) Nanostructuring by ion-beam erosion. Due to different roughening mechanisms a multitude of topographies can result from surface erosion by low-energy ion beams. Under certain conditions, sputtering can roughen the surface resulting in a pronounced topography evolution in some cases producing well-ordered patterns. The underlying mechanism for the evolution of regular nanometer structures is a self-organization process caused by the interplay of roughening by curvature dependent sputtering and surface smoothing by different surface relaxation mechanisms.
(2) Nanostructuring by ion beam assisted glancing angle deposition. A new class of engineered surface nanostructures in which their shapes (multible zigzags, helices, etc.) can be changed instantaneously during their growth evolution is presented. In this technique, the flux of the sputtered atoms is incident upon a substrate from highly oblique angles. This condition is sufficient to create a state of significantly enhanced surface shadowing. By rotation the around a second axis passing perpendicular through the substrate, the resulting microstructure will be helical.
Quite generally, thin solid films can be released from a substrate surface by selective underetching and form into 3D micro- and nano-objects including various types of 2D confined channels. Here, we show that such released layers self-assemble into complex nanochannel networks, which can be entirely fluid-filled and emptied within fractions of a second.
Furthermore, we demonstrate that /single /material layers roll-up into micro- and nanotubes once they are released from a substrate. In particular, we show that well-positioned all-Si as well as Si/SiO2 tubes can be fabricated, which emit and guide light in the visible spectral range at room temperature. Quantum emitters such as InAs/GaAs quantum dot heterostructures are integrated into the wall of rolled-up microtubes. We study the emission and the waveguiding properties of such quantum dots in a tube. Finally, metal layers are rolled up into microtubes, which opens the way to realize and integrate ultra-compact coils, transformers and capacitors on a single chip.
What happens to uncooked spaghetti if you hit it at 80 km/hr? The answer to this question combines fundamental aspects of elasticity and material science, from nonequilibrium Euler buckling to the failure of brittle solids. I will present an experimental study of the dynamic buckling and fragmentation of slender rods - including pasta, teflon, glass, and steel - due to rapid impact. By combining the mathematical results of Saint-Venant with elastic beam theory, we obtain a preferred buckling wavelength from the coupled partial differential equations for stress and deformation. Full time-resolved numerical simulations support these results. Experimentally, we find that the distribution of fragment lengths has peaks near 1/2 and 1/4 of the buckling wavelength. Such preferred fragment sizes represent the influence of the deterministic buckling process on the more random fragmentation processes.
Phenomenological description of spin-torque in nanomagnets: Landau-Lifshitz-Gilbert equation with Slonczewski spintorque term. Energy balance in magnetic systems driven by spin-polarized currents and nonconservative nature of dynamics. Determination of equilibria and self-oscillations (limit cycles). Perturbation approach: spin-torque driven dynamics as a perturbation of conservative (precessional) dynamics. Melnikov technique to study existence, stability and bifurcations of self-oscillatory regimes (limit cycles). Analysis of thermal stability in spin-torque driven magnetization dynamics. Fokker-Planck equation, averaging technique, and description based on the free energy probability distribution. Thermal relaxation and stationary (non-Boltzmann) energy distribution in spintorque systems. Defintion of effective potential barrier separating self-oscillatory regimes and equilibria. Arrhenius formula for spin-driven dynamics regimes.
A review will be given on the magnetic microstructure of bulk ferromagnetic materials, the character of which is mainly determined by anisotropy, surface orientation and microstructure. The review will start with an analysis of surface and internal domains in Fe-like materials with cubic anisotropy and NdFeB as an example for strong uniaxial materials. For strongly misoriented surfaces, domain branching is the characteristic feature in each case.
It will then be shown how the domain structure changes when the structural length (grain size) decreases down to the nanometer regime. For soft magnetic nanocrystalline material, the exchange coupling between the nano-grains leads to a seemingly uniform magnetic material. Comparing samples with weak and strong induced anisotropy, significant differences in the magnetic microstructure are found, reflecting the interplay of the uniform, field-induced and the random magnetocrystalline anisotropy. In particular for weak induced anisotropy an irregular patchy modulation of magnetization on the scale of a few micrometers is found within otherwise regular domains, which can be interpreted in terms of the random anisotropy model. These differences are also reflected in the magnetization processes and in the domain behaviour at elevated temperatures. In high-anisotropy materials with grain sizes in the 100 nm regime (generally known as nanocrystalline permanent magnets), so-called interaction domains are observed due to the predominance of magnetostatic interactions between the grains.
Shape memory alloys like Nickel-Titanium (NiTi) exhibit exceptional mechanical properties due to their ability to undergo diffusionless solid-solid phase transformations. Distinguishing features are the shape memory effect and superelasticity, also referred to as pseudoelasticity. NiTi in particular additionally exhibits high strength and high transformation stresses around room temperature. These properties already led to a lot of new applications using NiTi. The talk gives an overview about different kinds of experimental observations with specimens made from NiTi: First the macroscopic behaviour of NiTi-wires in tension tests under different loading conditions is shown. Then transformation kinetics and thermo-mechanical coupling are examined using optical methods and thermography. Whereas copper-based alloys transform homogeneously, NiTi shows a distinct generation of transformation bands, which has to be taken into consideration during modelling. A simple one-dimensional model has been developed to show exemplarily methods accounting for these transformation bands. The model is based on plasticity but accounts for the thermo-mechanical coupling.
Plastic deformation of metals is mainly caused by crystallographic slip of dislocations. Therefore, physical constitutive models use dislocation densities as state variables in opposition to empirical models which mostly use the accumulated strain as state variable. In the talk it is demonstrated how evolution laws for the dislocation densities can be derived and how the stress-strain relationship can be evaluated on the basis of the Orowan equation. The framework of crystal plasticity FEM offers a link between the physical process of dislocation slip and the most powerful tool in continuum mechanics the FEM. It will therefore also be shown how a dislocation density based model can be extended for the incorporation into crystal plasticity FE simulations. The model is applied to the deformation of Aluminium single and bicrystals. In these examples the importance of the treatment of strain gradients and the grain boundary becomes obvious. Therefore, the local dislocation density model is extended to a nonlocal one in a physically sound manner. The final model is capable of predicting the local deformation behaviour of the Al bicrystals much more precisely than a phenomenological viscoplastic crystal plasticity model.
Plants possess many structural and functional properties that have a potential to serve as paragons for the development of new technical materials and structures. Three examples from nature serve to demonstrate how light-weight-materials should be designed to withstand high dynamic loading. Particular emphasis is given to the problem of oscillation damping, that minimizes the danger of a resonance catastrophe and is therefore essential for the survival of plants and for the stability of technical materials.
Magnetic high-quality materials with specific material parameters and microstructures guarantee optimum magnetic properties. This correlation especially becomes obvious for nanocrystalline and nanostructured supermagnets (intermetallic 3d-4f compounds). The detailed analysis of high-sensitive magnetization measurements and high-resolution electron microscopy techniques in combination with the theory of micromagnetism allowed a quantitative interpretation of the occurring magnetization processes. This enables now a quite specific tailoring of optimized magnetic properties.
Also in the modern information and sensor technology single-domain nanoparticles and thin films are of steadily growing interest for realizing high recording densities (up to Terabit/inch2), ultrashort switching times (sub-nanosec) and long-time stability against thermal fluctuations. These characteristics sensitively depend on the magnetic material parameters, the particle's shape and dimension and on the applied magnetic field. This correlation is for the first time systematically analyzed for hard and soft magnetic particles and thin films using micromagnetic computational techniques based on the finite element method. The results give a fundamental insight into the complex correlations between the basic properties required for high-density magnetic recording.
In this talk I will summarize some mechanical and physical properties of shape-memory alloys and perform an experiment with a shape-memory wire. Mathematical concepts in the theory of shape-memory alloys will be discussed only sketchily in this talk. Instead I will focus on basic notions and physical concepts which lie at the starting-point of the mathematical modelling.
Der Vortrag behandelt 2 Schwerpunktthemen unserer Gruppe aus dem Bereich der Kristall-Anisotropie.
Der erste Teil behandelt neue theoretische und experimentelle Verfahren im Bereich der elastisch-plastischen Ein- und Polykristallmechanik (z.B. Kristallmechanische Analyse der Nanoindentierung, Bikristallmechanik, mechanische Behandlung großer Polykristallaggregate)
Der zweite Teil behandelt die Strukturanalyse, Textur, und Mikromechanik von biologischen Nano-Verbundwerkstoffen.
Mechanische Eigenschaften sind stark größenabhängig: eingeengte Metallvolumina im Submikronbereich zeigen um Größenordnungen veränderte Fließgrenzen sowie modifiziertes Ermüdungs- und Bruchverhalten. Durch Anwendung von Dünnschichtproben konnten in den letzten Jahren wichtige Einblicke in die Grundlagen von eingeengten Deformationsprozessen gewonnen werden. Aber auch mechanische Oberflächeneffekte in der Biologie zeigen ein Skalenverhalten: Insekten, Spinnen und Geckos verdanken ihre gute Adhäsion mikro- und nanoskopischen Haftorganen. In Zusammenarbeit mit Zoologen entwickeln wir ein grundlegendes Verständnis für diese biomechanischen Phänomene, mit dem Ziel der Herstellung und Optimierung künstlicher Systeme, die in der Mikrotechnik und Medizin von Interesse sind. Dieser Vortrag gibt einen Überblick über die experimentellen Ergebnisse und die mathematischen Modelle auf diesem Gebiet, das sich zunehmend als Teilgebiet einer "Nanomechanik" etabliert.
The main source of non-linear elasticity and an-elastic behaviour is the mobility of microstruntures. The best studied class of materials in that of ferroelastics where the microstructure is related to pattern of twin boundaries including individual boundaries, needle shaped boundaries, S-shaped boundaries, tartan, and tweed structures. Their analysis in local and non-local approximations will be introduced.
Two novel developments have taken place: firstly we understand better that internal wall structures can exist. They can be decribed within the scheme of coupling theories involving elastic, electronic or chemical degrees of freedom. Secondly, local atomic-sized beeds of amorphous inclusions lead to macroscopic strain of the adjacent lattice and percolation behaviour for high concentrations. Their proper mathematical description is unknown but their physical behaviour becomes increasingly clear.
Ferroics are materials which become ferro-elastic, -magnetic, -electric at lower temperatures on cooling. They usually show a phase transformation which produces an ordered state (low entropy phase). These properties are partially due to a domain structure. The strain effects are found either by cycling around the transformation temperature (two-way effect, invar), by mechanical straining plus heating above transformation temperature (one-way memory) or by cyclic mechanical loading (pseudo- or rubber elasticity). Ferromagnetic or -electric materials can be shifted into different shapes by magnetic or electrical fields instead of mechanical stresses. In combination with martensitic transformation considerable strains are obtained in some ferromagnetic alloys: $Ni_2MnGa$, multiferroics. Polymers can show one-way behaviour like metals, ferro- and piezo-electrical behaviour like ceramics. The largest amounts or reversible strains are obtained with polymers, the smallest with ceramics. Porous charged polymers show hysteresis curves like ceramic ferroelectrics, but larger amounts of strain: ferro electrets. The coiled high entropy conformation corresponds to the high temperature $\beta$-phase (austenite) in alloys, the stretched state to martensite $\alpha$. An entropic force is the cause for a $\alpha\rightarrow\beta$ reverse transformation and the origin of shape memory. A systematic understanding of this group of materials permits to compare their abilities and limits. This in turn provides a base for the selection of the best material for certain applications.
In geophysics, the thermodynamics of recrystallization and creep of large polycrystalline rocks is of vital importance. On the other hand, physicists have since long proposed sophisticated theories for the modeling of induced anisotropy and streaming birefringence in polymers and liquid crystals. Likewise, the dynamics of structured populations (viz. populations composed of individuals classified by age or other physiological characteristics) has been a far-reaching research theme for mathematical biologists along the last decades. Evidently, the mechanics of polydisperse granular media has become a stimulating topic in civil engineering, and so do also the theories of polymerization, cracking and other complex reactions in petroleum and atmospheric chemistry. Curiously, in spite of the dissimilitude of all these problems, it happens that their modeling falls into the same mathematical structure. It is the objective of the theory of mixtures with continuous diversity to formalize and explore this common mathematical framework so as to unify the formulation of all these apparently disparate theories and serves as a bridge for the exchange of methods and ideas between distinct disciplines.
Plastic deformation of crystals is mainly driven by the motion of line-like crystal defects called dislocations. Physically founded continuum descriptions of dislocation-based crystal plasticity must be formulated in terms of dislocation densities. Classical dislocation density measures, as the Kroener-Nye tensor, can account for the kinematic evolution of dislocation systems only if they are considered on the discrete dislocation level. Upon averaging, relevant information both on the orientation of dislocation segments and the presence of 'statistically stored' dislocations of zero net Burgers vector is lost, and therefore any theory of dislocation motion and plastic flow based on the averaged Kroener-Nye tensor is bound to be incomplete. At the beginning of a three-dimensional continuum theory of dislocation motion therefore stands the definition of a dislocation density measure which retains the macroscopically relevant information about the dislocation system, together with the derivation of a kinematic evolution equation for it. A 3D dislocation density measure is proposed as a differential form on the space of directions and curvatures at each point of a crystal viewed as Riemannian manifold with a metric connectiction which is not necessarily free of torsion. From the velocity of a curved dislcoation-line, a higher order velocity on the configuration space including line-directions and curvatures is derived in this general setting. This allows for a nice crystallographic interpretation of the torsion tensor. A kinematic evolution equation for the proposed density measure is derived from assuming the density to be invariant under the flow of the derived higher order velocity field. This evolution equation inherently accounts for line-element rotation and elongation as well as for curvature changes during the motion of curved dislocation-lines, as is shown at an exemplary numerical comparison with the motion of a single dislocation. Finally relations with other density-based models of the kinematic evolution of dislocation systems are discussed.
The II-VI semiconductor ZnO can be grown at relatively low temperatures on inexpensive substrates. ZnO ranks among the best materials for ultraviolet light emitters, gas sensors, surface acoustic wave devices, and transparent contacts.
The optical room-temperature band gap of undoped ZnO of 3.3 eV can be varied in the whole energy range from 3.18 eV up to 4.5 eV by cationic alloying with Cd or Mg, respectively. The well established one-electron pseudopotential band structure theory for describing experimentally observed electronic properties of Zn(Cd,Mg)O with transferable, structural independent Zn, O, Cd, and Mg empirical pseudopotentials will be presented. Furthermore, the idea of possible calculations of phononic bandstructures utilizing the same empirical pseudopotentials obtained for the electronic bandstructure calculations will be discussed.
At room temperature ferromagnetic ZnO possesses great importance for potential applications in spintronic devices, as for example Spin-LEDs and Spin-FETs. For such spintronic applications ZnO is alloyed with 3d transition metals thus obtaining a diluted magnetic semiconductor (DMS). However, the nature of ferromagnetism in ZnO-based DMS is still under active debate. The successes and shortcomings of the theoretical models describing magnetic phenomena in DMS materials will be listed.
First an overview on the diffusive fluxes of several components in a multicomponent system is presented. Also diverse reference system established in the literature are discussed. Flux balances are derived in the bulk of a material including vacancies. Their role is discussed in a detail with respect to their production and annihilation. Furthermore, evolution laws for the fluxes are derived by application of Onsager's principle of maximum dissipation rate. Both the flux balances and their evolution laws can be considered as physically based general versions of the classical 'laws' of diffusion. In addition balance relations at interfaces are explained. This allows finally to derive the thermodynamical driving force in a material point. Also in this case the role of vacancies is taken into account considering interfaces as ideal sources and sinks of vacancies. Finally, examples from solid/solid phase transformation in metals are demonstrated. It is emphasized that the concept presented is by no means restricted to metals.
Speaker: F.D. Fischer:1 Institute of Mechanics, Montanuniversität Leoben2 Erich Schmid Institute for Materials Science, Austrian Academy of Sciences, Leoben3 Materials Center Leoben4 Christian Doppler Laboratory of Functionally Oriented Materials Design
In recent years continuum mechanics has been applied to microelectronic devices, where typical length scales may range from 0.1µm to 10µm. At these length scales experiments display strong size effects, as do simulations using discrete dislocation theory; such simulations also exhibit boundary layers. Classical crystal plasticity and, generally, most classical plasticity theories exhibit neither size effects nor boundary layers.
This talk will discuss recent work in developing a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroscopic forces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems, yield conditions that feature backstresses resulting from energy stored in dislocations. The resulting field equations are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach-Koehler force on a single dislocation.
The theory is compared numerically to discrete dislocation simulations for two boundary-value problems. The first concerns a two-dimensional composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear; the second concerns simple shear of a constrained strip. In the composite problem, the discrete dislocation solutions give rise to hardening dependent on the reinforcement morphology, to a size dependence of the overall stress-strain response, and to a strong Bauschinger effect on unloading. In the constrained layer problem, boundary layers develop. In neither problem are the qualitative features of the discrete dislocation results represented by conventional continuum theories of crystal plasticity. It will be shown that the gradient-theory calculations reproduce the behavior seen in the discrete dislocation simulations in excellent detail.
A solution for a general 3D dynamic perturbation of a crack which, when unperturbed, propagates at uniform speed in the plane $x_3 = 0 $, so that, at time t, it occupies the surface $$ \left\{ x :-\infty < x_1 < Vt , -\infty < x_2 < \infty , x_3 =0 \right\} $$ was developed in recent years by Willis and Movchan. When coupled with a fracture criterion, it permits the study of the dynamic stability of the propagating crack. This has so far provided explicit confirmation of the existence of a "crack front wave", which is an in-plane disturbance of the crack front, which propagates along it without attenuation or dispersion. Also, more recently, it has been employed in its 2D specialisation to study the stability of the crack to out-of-plane disturbance. An outline of the general perturbation solution will be presented and both of these applications will be described.
We study thin films subjected to strongly coupled stress and diffusion at small length scales. The mechanisms by which surface morphology evolves are investigated by a series of theoretical and experimental studies. The methods we use include complex variable theory of elasticity, J-integral method, anisotropic elasticity, boundary integral method, surface diffusion equation, dislocation theory and molecular dynamics method. We show that the stress drives atomic diffusion along the film surface in such a way that an initially flat film evolves into an undulating profile with cusp-like surface valleys with singular stress concentration. Modeling at atomic resolution shows that the cusp formation leads to creation of various nanoscale defects in the form of dislocations and deformation twins. Controlled annealing experiments are performed to validate theoretical predictions. Future studies will involve combination of continuum and atomistic analysis to model the growth and evolution of structures at nanoscale.
Solid foams is a term commonly used to describe materials with a highly disperse solid phase arranged into cells which can be either open or closed. These materials can be found in many natural systems such as cork, wood, cancelleous bone and soft tissue among many others. Also, manufacturing of artificial foams such as honeycombs, foamed polymers, ceramics and metals has been promoted by some of distinctive characteristics of these systems including an excellent strength-to-density ratio. The topological arrangement of the cell structure in conjunction with the material behavior of the solid phase determine the macroscopic response of the foam. Since the early efforts of Gent and Thomas~(1959), who proposed one of the first models for cellular materials, many studies have followed correlating the micro to macro properties. Extensive reviews on the mechanics of cellular materials include Hilyard (1982), Hilyard and Cunningham (1994), and Gibson and Ashby (1997). With the advent of novel manufacturing processes which can produce foams with controlled topological features at small scale (see for example Jackman et al. 1998), more detailed formulations correlating explicitly foam structure and solid phase behavior to the constitutive relation are required in order to tailor the design of the micro features for specific applications.
In this talk, we present a hyperelastic model for light and compliant open cell foams with an explicit correlation between microstrucure and macroscopic behavior. The model describes a large number of three dimensional structures with regular and irregular cells. The theory is based on the formulation of general strain-energy function which accounts for localized bending and stretching. Within the same framework, however, general bending, shear and twisting energies can also be incorporated. The formulation incorporates nonlinear kinematics which traces the evolution of the structure during loading process and its effects on the constitutive behavior, including the cases where configurational transformations are present leading to non-convex strain-energy functions. The implications of the non-convexity are further explored by theoretical and experimental means to elucidate the spatially heterogeneous distributions of local stretch during compressive loading.
