Oscillatory Microstructure Research Articles

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions $u(x)$ which are nonconvex in the gradient $\nabla u$ and possibly also in $u$. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.

On oscillatory microstructure during cellular growth of directionally solidified Sn-36at.%Ni peritectic alloy.

An oscillatory microstructure has been observed during deep-cellular growth of directionally solidified Sn–36at.%Ni hyperperitectic alloy containing intermetallic compounds with narrow solubility range. This oscillatory microstructure with a dimension of tens of micrometers has been observed for the first time. The morphology of this wave-like oscillatory structure is similar to secondary dendrite arms, and can be observed only in some local positions of the sample. Through analysis such as successive sectioning of the sample, it can be concluded that this oscillatory microstructure is caused by oscillatory convection of the mushy zone during solidification. And the influence of convection on this oscillatory microstructure was characterized through comparison between experimental and calculations results on the wavelength. Besides, the change in morphology of this oscillatory microstructure has been proved to be caused by peritectic transformation during solidification. Furthermore, the melt concentration increases continuously during solidification of intermetallic compounds with narrow solubility range, which helps formation of this oscillatory microstructure.

Modeling of curing processes based on a multi-field potential. Single- and multiscale aspects

This paper provides a continuum mechanical model for the curing of polymers, including the incompressibility effects arising at the late stages of the process. For this purpose, the free energy density functional is split into a deviatoric and a volumetric part, and a multifield formulation is inserted. An integral formulation of the functional is used to depict the time-dependent material behavior. The model is also coupled with the multiscale finite element method, a numerical approach serving for the modeling of heterogeneous materials with a highly oscillatory microstructure. The effects of the proposed extensions are illustrated on the basis of several numerical examples concerned with the study of the influence of Poisson’s ratio on the curing process and the behavior of the microheterogeneous polymers.

Contribution of the reflection to the attenuation properties of cancellous bone

The article deals with the contribution of reflection effects to the attenuation properties of cancellous bone. The bone behaviour is simulated by the multiscale finite element method, a numerical homogenization approach, suitable for the modelling of heterogeneous material with a highly oscillatory microstructure. The focus is on the modelling of a novel type of the representative volume element, which apart from the solid framework filled with fluid marrow also includes an infinitesimally thin ‘transition’ layer at the contact of the phases. The mentioned layer is implemented in order to simulate the amplitude transformation of an incident wave and to take the loss of energy caused by the reflection into account. The given numerical examples consider the simulation of wave propagation through a sample while the excitation frequency is varied. The numerical values are compared with the results which are determined without considering the reflection, in order to point out the contribution of the newly introduced phenomenon.

Lagrangian and Hamiltonian two-scale reduction

Studying spatially extended Hamiltonian systems with coherent microstructure, it is an important and challenging problem to identify reduced Hamiltonian models that describe the effective dynamics on large spatial and temporal scales. Such models require macroscopically varying, deterministic initial data which can possess a well-organized microstructure. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can be derived from the microscopic system. In the first part we develop a general approach by considering noncanonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices [or Hamiltonian partial differential equations (PDEs)] and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, and (iii) an elementary model reduction that is based on a principle of consistent expansions. The result is always a reduced Hamiltonian PDE that governs the macroscopic dynamics, provided that the initial data are chosen consistently with the underlying two-scale ansatz. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long-wave motion or describe macroscopic modulations of oscillatory microstructures.

Modelling of microstructure evolution in peritectic systems

Theoretical models for directional solidification microstructures are largely developed for steady-state growth conditions. In the two-phase region of peritectic system, a rich variety of two-phase microstructures form in which the two phases often form under non-steady-state conditions and the microstructure evolution depends on the nucleation characteristics of the two phases as well as on the competitive growth of the two phases under dynamical growth conditions. Theoretical models, based on nucleation and diffusive growth, are presented to establish the mechanisms that lead to different microstructure formation in peritectic systems. The critical role played by nucleation in the selection of oscillatory microstructures is discussed, and experimental studies to select different nucleation sites are described. Experimental studies in the Sn–Cd system are carried out in thin capillary samples to minimize fluid flow effects that are dominant in most experiments. Critical conditions for the observation of composite microstructures are evaluated. These results will be presented to support the predictions of the models.

Development of Powered Resonance-Tube Actuators for Aircraft Flow Control Applications

The present paper addresses both active-flow-control actuator technology development and the demonstration of the effectiveness of actuators that could be easily integrated into practical aircraft applications. The actuator used is an adaptation of the Hartmann oscillator. Demonstration experiments that illustrate the effectiveness of this actuator include cavity tone suppression at transonic speeds and the reduction of jet-impingement tones. The actuator concept is based on a high-speed jet aimed at the mouth of a cylindrical tube closed at the other end. The result is a high-amplitude self-sustaining fluctuating field accompanied by an intense narrowband tone, all in the region between the supply jet and the resonance tube. Using unsteady pressure sensors and flow visualization, we explored the effect of varying actuator parameters such as the spacing between the power jet and the resonance tube, supply pressure, resonance-tube depth, diameter, shape, and lateral spacing. By varying the depth of the tube, the frequency could be varied from about 1.6 kHz to over 10 kHz and amplitudes as high as 156 dB (microphone location dependent) were obtained in the vicinity of actuation. To integrate this concept into practical aircraft applications, two generations of a more complex version of this device known as the powered resonance-tube bank (PRTB) were developed and demonstrated. Results indicate that by using high-frequency excitation at 5-kHz suppression levels in excess of 20 dB were consistently obtained over a range of operating conditions in both cavity and impingement flow situations. Based on our results, we have grounds to believe that a properly designed PRTB has significant advantages over conventional actuators such as acoustic, piezo, and oscillatory microstructures.

Relaxation of rate-independent evolution problems

The relaxation of certain time-evolution problems is investigated. As a conceptually simple example, we study elastically deformable bodies that undergo martensitic phase transformations. The movement of the phase boundaries is hindered by dry friction. The fundamental problem is that the phase distribution forms a highly oscillatory microstructure in space. Therefore, it is desirable to derive a coarse-grained system that describes the effective properties. We introduce a concept of relaxation of the evolution system and apply it to the case where only two phases occur and the elastic energy is quadratic. Finally, we present a candidate for the relaxation in the general case.

Dynamics of microstructure formation in the two-phase region of peritectic systems

Directional solidification experiments have been carried out in the two-phase region of the Sn–Cd peritectic system. A variety of different microstructures have been shown to form under intrinsically non-steady-state growth conditions in both the hyperperitectic and hypoperitectic compositions. The selection of microstructure is shown to be governed by the relative effects of convection, nucleation undercooling, nucleation rate and the growth competition between the two phases. Convection effects have been found to give significantly different microstructures than those predicted by the diffusive growth models. Experiments are thus carried out in samples of different diameters in which convection effects are shown to be reduced as the diameter of the sample is decreased. It is shown that diffusive growth condition is achieved in this alloy system when the sample diameter is <1.0 mm. Experimental studies are then carried out in hypoperitectic alloys in a composition range for which banded microstructures have been predicted by the diffusive growth model. This composition window is established experimentally, and the values of nucleation undercoolings of the primary and peritectic phase have been obtained. The experiments also confirmed the prediction of the model that the banding cycle occurs above and below the peritectic temperature and bands of primary and peritectic phases form by the alternate nucleation and lateral spreading of one phase onto the other. It is also found that when the sample diameter is reduced further, or growth conditions changed, several new dynamical oscillatory microstructures evolve and the formation of these microstructures is shown to be controlled by the competitive growth processes between the nuclei and the parent phase. This competitive growth is shown to be influenced by the distance between the nuclei or the nucleation rate. It is shown that the directional solidification of peritectic alloys provides a means to quantitatively study the complex process of dynamical evolution of two-phase microstructures in which fluid flow, nucleation undercoolings, nucleation rates of the two phases and the competitive growth of the two phases control the evolution of complex two-phase microstructures.

Eutectic colony formation: a stability analysis.

Experiments have widely shown that a steady-state lamellar eutectic solidification front is destabilized on a scale much larger than the lamellar spacing by the rejection of a dilute ternary impurity and forms two-phase cells commonly referred to as "eutectic colonies." We extend the stability analysis of Datye and Langer [V. Datye and J. S. Langer, Phys. Rev. B 24, 4155 (1981)] for a binary eutectic to include the effect of a ternary impurity. We find that the expressions for the critical onset velocity and morphological instability wavelength are analogous to those for the classic Mullins-Sekerka instability of a monophase planar interface, albeit with an effective surface tension that depends on the geometry of the lamellar interface and, nontrivially, on interlamellar diffusion. A qualitatively new aspect of this instability is the occurrence of oscillatory modes due to the interplay between the destabilizing effect of the ternary impurity and the dynamical feedback of the local change in lamellar spacing on the front motion. In a transient regime, these modes lead to the formation of large scale oscillatory microstructures for which there is recent experimental evidence in a transparent organic system. Moreover, it is shown that the eutectic front dynamics on a scale larger than the lamellar spacing can be formulated as an effective monophase interface free boundary problem with a modified Gibbs-Thomson condition that is coupled to a slow evolution equation for the lamellar spacing. This formulation provides additional physical insights into the nature of the instability and a simple means to calculate an approximate stability spectrum. Finally, we investigate the influence of the ternary impurity on a short wavelength oscillatory instability that is already present at off-eutectic compositions in binary eutectics.

Morphological instabilities of lamellar eutectics

We present the results of a numerical study based on the boundary integral technique of interfacial pattern formation in directional solidification of thin-film lamellar eutectics at low velocity. Microstructure selection maps that identify the stability domains of various steady-state and nonsteady-state growth morphologies in the spacing-composition (λ –C 0) plane are constructed for the transparent organic alloy CBr4-C2Cl6 and for a model eutectic alloy with two solid phases of identical physical properties. In CBr4-C2Cl6, the basic set of instabilities that limit steady-state growth is richer than expected. It consists of three primary instabilities, two of which are oscillatory, which bound the domain of the commonly observed axisymmetric lamellar morphology, and two secondary oscillatory instabilities, which bound the domain of the nonaxisymmetric (tilted) lamellar morphology. The latter is predicted to occur over a hypereutectic range of composition which coincides well with experiment. Moreover, the steady tilt bifurcation lies between but does not directly bound either of these two domains, which are consequentlydisjoint. Four stable oscillatory microstructures, at least three of which have been seen experimentally, are predicted to occur in unstable regimes. In the model alloy, the structure is qualitatively similar, except that a stable domain of tilted steady-state growth is not found, in agreement with previous random-walk simulations. Furthermore, the composition range of stability of the axisymmetric morphology decreases sharply with increasing spacing away from minimum undercooling but extends further off-eutectic than predicted by the competitive growth criterion. In addition, oscillations with a wavelength equal to two λ lead to lamella termination at a small distance above the onset of instability. The implications of these two features for the eutectic to dendrite transition are examined with the conclusion that in the absence of heterogeneous nucleation, this transition should be histeritic at small velocity and temperature gradient.