Structural characterization of two-dimensional granular systems during the compaction

We examine numerically the density relaxation of frictional hard disks in two dimensions (2D), subjected to vertical shaking. Dynamical recompression of the packing under the action of gravity is based on an efficient event-driven molecular-dynamics algorithm. To quantify the changes in the internal structure of packing during the compaction, we use the Voronoï tessellation and a certain shape factor which is a clear indicator of the presence of different domains in the packing. It is found that the narrowing of the probability distribution of the shape factor during the compaction is in accordance with the fact that the packings of monodisperse hard disks spontaneously assemble into regions of local crystalline order. An interpretation of the memory effects observed for a sudden perturbation of the tapping intensity is provided by the analysis the accompanying transformations of disk packings at a “microscopic” level. In addition, we investigate the distributions of the shape factor in a 2D granular system of metallic disks experimentally, and compare them with the simulation results.     

Experimental analysis of overtaking behavior in disks falling in a low-density particle bed

Cooperative behavior displayed by five steel disks falling in a low-density particle bed involves the formation of upward and downward convex configurations, which resembles the flying pattern of a flock of birds. In this study, we focused on overtaking behavior in two falling disks, which causes the cooperative behavior, and we investigated the effects of differences in the disk release time and the initial disk separation distance on the falling behavior of the disks experimentally. Expanded polystyrene (EPS) particles (diameter 5.08 mm, mass 1.45 mg) were used as the bed particles and steel disks (diameter 25.4 mm, thickness 5.22 mm, mass 20.2 g) were used as the falling disks. We released one to five disks with various disk release time differences (0–0.154 s) and initial separation distances (0–100 mm). We recorded the disk falling behavior in the particle bed with a high-speed video camera (500 fps) and analyzed the behavior with image analysis software. Five-disk cooperative behavior similar to that reported in the literature occurred in our experimental setup. In the two-disk experiments, we observed overtaking behavior for an initial separation distance of 10 mm and release time difference of ≤ 0.076 s, and for an initial separation distance of ≤ 60 mm and release time difference of 0.02–0.03 s. The overtaking behavior arose from the decrease in the falling velocity of the first disk released. The EPS particle packing fraction in the area above the disk one disk diameter wide and a quarter of the disk diameter high determined the disk falling velocity. This mechanism was explained by the displacement behavior of EPS particles around the disks as the disks fell.     

Multiple hot images from an obscuration in an intense laser beam through cascaded Kerr medium disks

We present a theoretical investigation on the formation of hot images in an intense laser beam through cascaded Kerr medium disks, to disclose the distribution and intensity of hot images in high-power disk amplifiers. It is shown that multiple hot images from an obscuration may be formed, instead of one hot image as reported previously in the literature. This gives a clear explanation for the curious damage pattern of hot images, namely, damage sites appearing on alternating optics in periodic trains. Further analysis demonstrates that the distribution and intensity of hot images depend closely on the number of Kerr medium disks, the distance from the obscuration to the front of the first disk downstream, the space between two neighboring disks, and the thickness and B integral of each disk. Moreover, we take two cascaded Kerr medium disks for example to detail multiple hot images from an obscuration and confirm the theoretical results by numerical simulations.     

巴西圆盘中心裂纹摩擦接触问题的逐点Lagrange乘子法

采用逐点Lagrange乘子法求解巴西圆盘中心裂纹在压剪荷载作用下裂纹面可能发生的摩擦接触问题。为了避免传统的Lagrange乘子法中总刚度阵求逆的困难,将Lagrange乘子逐点转到局部坐标系下,采用Gauss-Seidel迭代法求解法向和切向乘子,同时注意在求解的过程中对切向乘子约束修正,待所有点乘子求解完成后再变换到整体坐标系下迭代求解位移。与传统接触算法相比,该算法无需对总刚度阵求逆,降低了求解规模,提高了计算效率。通过该方法计算了巴西圆盘中心裂纹两种典型情况下的应力强度因子,计算结果与文献比较,吻合良好。考虑不同荷载角和裂纹长度对位移,应力强度因子和接触区的影响,并对不同摩擦系数下应力强度因子的影响进行了分析。结果表明:忽略裂纹接触摩擦作用,应力强度因子可能被高估。     

Simplified elastic models for disk-shaped precipitates

A procedure is presented to calculate the elastic displacement fields found in the early stages of coherent precipitation in age hardenable alloys. The procedure is designed for a subsequent calculation of X-ray diffraction profiles. Displacement fields from precipitates of finite size having local order may be examined with reduced computer times. Point sources consisting of a doublet combined with a spherical field are distributed over the plane of a disk shaped precipitate. The relative strength of this combination is adjustable; however, the total strength is scaled to be in agreement with lattice parameter data. An empirical correction for elastic anisotropy, is guided by Green’s function calculations. Examples are given for calculating fields from single disks and stair step pairs. These calculations may be carried out with reduced computer time. This is made possible by reducing a large number of uniformly distributed point sources to an optimum number having weighted strengths and special locations throughout the plane of a disk. In order to assess directional effects and more readily relate the displacements to diffraction data, projected displacements are used. The tetragonality of the interplanar d-spacings is examined within zones of severe deformation located in the immediate vicinity of the precipitate. This revised version was published online in November 2006 with corrections to the Cover Date.     

Dual Functional Modification of Alkaline Amino Acids Induces the Self‐Assembly of Cylinder‐Like Tobacco Mosaic Virus Coat Proteins into Gear‐Like Architectures

Herein, the assembly of 3D uniform gear‐like architectures is demonstrated with a tobacco mosaic virus (TMV) disk as a building block. In this context, the intrinsic behavior of the TMV disk that promotes its assembly into nanotubes is altered by a synergistic effect of dual functional modifications at the 53rd arginine mutation and the introduction of lysine groups in the periphery at 1st and 158th positions of the TMV disk, which results in the formation of 3D gear‐like superstructures. Therein, the 53rd arginine moiety significantly strengthens the linkage between TMV disks in the alkaline environment through hydrogen bond interactions. The charge of lysine‐modified lateral surfaces is partially neutralized in the alkaline solution, which induces the TMV disk to form a gear‐like architecture to maintain its structural stability by exploiting the electrostatic repulsion between neighboring TMV disks. This study not only provides explicit evidence regarding the molecular‐level understanding of how the modification of site‐specific amino acid affects the assembly of resultant superstructures but also encourages the fabrication of functional protein‐based nanoarchitectures.     

Study of the bending modes in circular quartz resonators

An experimental and theoretical study of bending modes in a partially electroded circular piezoelectric quartz (AT-cut) with free edge is presented. The quartz is excited by a voltage pulse applied on the electrodes, and its surface is scanned by a laser vibrometer that measures the out-of-plane displacements. The classical theory of bending of thin disks is used to describe the flexural modes at frequencies lower than the first thickness shear resonance (6 MHz). A fairly good agreement is found between experimental and theoretical results for the forced mode shapes and for the resonance frequencies. However, it appears that the two springs used to maintain the disk in position introduce extra clamping conditions. Several source shapes were studied, among which a collection of an arbitrary number of forces is particularly useful. The two-dimensional wavenumber representation shows the presence of anisotropy related to the crystallographic axes at higher frequencies, which is not predicted by the model. The experimental phase velocities are compared to those given by the classical theory of disks and to those of Lamb A(0) mode. This study confirms the correspondence at low frequencies between the A(0) mode and the bending eigenmodes of a disk with finite size.     

Analysis of two-dimensional functionally graded rotating thick disks with variable thickness

Elasticity solutions of two-dimensional functionally graded rotating annular and solid disks with variable thickness are presented. Material properties vary through both the radial and axial directions continuously. Axisymmetric conditions are assumed for the two-dimensional functionally graded disk. The graded finite element method (GFEM) has been applied to solve the equations. The distributions of displacements and stresses in radial and axial directions for four different thickness profiles (constant, linear, concave and convex) and various power law exponents have been investigated. The achieved results show that by the use of functionally graded materials and variable thicknesses, the stresses are reduced, so a higher capability of angular velocity can be obtained. Also, using two-dimensional functionally graded materials leads to a more flexible design in comparison with conventional one-dimensional functionally graded materials. The GFEM solution of a functionally graded thin rotating annular disk has been compared with the published literature and it shows good agreement.     

Adjustable balancer with electromagnetic release of balancing members

Adjustable balancers are useful in applications where rotational imbalance is frequently varied. For example, in media level servo track writers used in the hard disk drive industry, a set of blank disks is loaded on a spindle motor for the purpose of servo writing. Since servo writing is performed at high speeds, imbalance introduced by disks and spacers results in significant vibrations that can affect the quality of the servo writing process. This paper proposes a new configuration of adjustable balancer that enables to compensate for rotational imbalance each time a new set of blank disks is loaded into the servo track writing machine. The adjustable balancer is ring shaped with a number of grooves displaced around axis of rotation. Steel balls are placed inside the grooves and held by permanent magnets. Imbalance compensation is achieved by electromagnetically activated release of the steel balls. The main advantage of the proposed configuration is its large compensation range. Presented three-dimensional finite-element analysis results and experimental results demonstrate and prove the effectiveness of this novel configuration.     

PERIODICITY AND FINE STRUCTURE IN TWO-DIMENSIONAL GRAVITATIONAL RANDOM PACKING

ABSTRACT

Gravitational random packing of equal disks can be viewed as the evolution of a Markov chain in which a state is the shape of the front line formed by joining the centers of touching disks at the upper boundary. The placement of a new disk adds two straight-line segments and deletes one or more old ones. The replaced segments plus the two new ones form a polygon. Triangles, rhombi, and pentagons are most common, but hexagons and heptagons have been observed. The latter appear only in the early stages of packing while the numbers of triangles and pentagons in horizontal strips decay exponentially with height. Thus, the packing evolves to a purely rhombic structure.

The appearance of triangles, pentagons, or higher polygons indicates a choice between mutually exclusive sites. These branch points, which disrupt the rhombic structure, will occur whenever certain constraints are not satisfied. If the rhombic structure persists long enough, a diamond-shaped repeating unit is formed. The limiting packing density is simply the density of this repeating unit. Since units may differ greatly in their density, estimates should be based on the repeating units from many different runs.     

PERIODICITY AND FINE STRUCTURE IN TWO-DIMENSIONAL GRAVITATIONAL RANDOM PACKING

Gravitational random packing of equal disks can be viewed as the evolution of a Markov chain in which a state is the shape of the front line formed by joining the centers of touching disks at the upper boundary. The placement of a new disk adds two straight-line segments and deletes one or more old ones. The replaced segments plus the two new ones form a polygon. Triangles, rhombi, and pentagons are most common, but hexagons and heptagons have been observed. The latter appear only in the early stages of packing while the numbers of triangles and pentagons in horizontal strips decay exponentially with height. Thus, the packing evolves to a purely rhombic structure.

The appearance of triangles, pentagons, or higher polygons indicates a choice between mutually exclusive sites. These branch points, which disrupt the rhombic structure, will occur whenever certain constraints are not satisfied. If the rhombic structure persists long enough, a diamond-shaped repeating unit is formed. The limiting packing density is simply the density of this repeating unit. Since units may differ greatly in their density, estimates should be based on the repeating units from many different runs.     

Structural properties of wet granular beds subjected to tapping

Based on a simple numerical model for wet granular beds, we study the structural properties of wet particles subjected to tapping in terms of the global anisotropy and angular distribution presented by their contacts. The model allows to generate 2-dimensional packings of disks that can form capillary bridges due to the presence of interstitial liquid. A pseudodynamic simulation of adhesive hard disks has been implemented. The bed is subjected to a tapping-like excitation and we study the evolution of structural anisotropy of the packing with the number of taps. We also analyse the behavior of the angular distribution of contacts and anisotropy as a function of tapping intensity and liquid content. Present results help to better understand the behavior found in a previous work for the packing fraction of these systems. They also demonstrate that anisotropy alone not always helps to completely understand the behavior of the structural properties of wet particles.     

Vibrations and static responses of asymmetric bimorph disks of piezoelectric ceramics

In an earlier article, the flexural vibrations in bimorph disks and extensional vibrations in homogeneous disks of piezoelectric ceramics were studied. In the present paper, the coupled flexural and extensional vibrations and static responses in an asymmetric bimorph disk, which is formed by bonding together two piezoelectric ceramic disks of unequal thickness and opposite polarization, are investigated. Governing equations of coupled motions for asymmetric bimorphs are deduced from the recently derived 2-D, first-order equations for piezoelectric crystal plates with thickness-graded material properties. Then, closed form solutions of these equations for circular disks are obtained for free vibrations, piezoelectrically forced vibrations, and responses under static voltage difference. Resonance frequencies, distribution of displacements and surface charges, impedances, and static responses are calculated for asymmetric bimorph disks of various thickness ratios and diameter-to-thickness ratios. Experimental data on resonances and impedances are obtained for asymmetric bimorph disks of PZT-857 for different thickness ratios. Comparisons of predicted and measured results show that the agreements are close.     

Tilt correction in an optical disk system

A servo system for the correction of disk tilt in optical disk data storage is proposed, and its basic concepts are demonstrated by the use of a static system in which the disk does not spin. Because disk tilt produces primarily coma in the beam focused onto the disk, the system uses a variable coma generator to produce an equal and opposite amount of coma as that caused by the tilted disk. The magnitude and direction of disk tilt are detected by the use of the light reflected from the front facet of the disk substrate.     

头/盘界面均匀化Reynolds方程及其高效数值求解

当硬盘工作时,磁头滑块飞行在高速旋转的磁盘上方,头/盘界面产生了一层只有几个纳米的气膜,该气膜润滑方程为考虑气体稀薄效应的修正Reynolds方程。为了提高磁记录密度,人们正在考虑用离散磁道（Discrete Track Recording,DTR）磁盘取代传统的光滑磁盘。由于这种DTR磁盘表面磁道和沟槽的影响,在数值求解修正Reynolds方程的过程中,就需要足够多的离散网格来分辨出DTR磁盘表面几何形状,从而带来了计算效率问题。基于均匀化理论和修正Reynolds方程的线性流率（Linearized Flow Rate,LFR）模型,该文推导了一个适合分析DTR磁盘气体润滑特性的均匀化Reynolds方程,并采用有限体积法对均匀化Reynolds方程进行了求解和验证。其结果显示:相对于修正Reynolds方程,求解均匀化Reynolds方程只需要很少的离散网格,从而节省了大量的计算时间,大幅提高了计算效率,且两者压力分布、压力中心和承载力的最大相对误差都小于3.5%。     

Finite element analysis of thermoelastic contact problem in functionally graded axisymmetric brake disks

An analysis of thermoelastic contact problem of functionally graded (FG) rotating brake disk with heat source due to contact friction is presented. Finite element method (FEM) is used. The material properties of disk are assumed to be represented by power-law distributions in the radial direction. The inner and outer surfaces considered are metal and ceramic, respectively. Pure material is considered for the brake pad. Coulomb contact friction is assumed as the heat source. It is divided into two equal parts between pad and brake disk which leads to thermal stresses. Mechanical response of FG disks are compared and verified with the known results from the literatures. The results show that the maximum value of radial displacement in mounted FG brake disk is not at outer surface. It is found that the all areas between pad and brake disk is in full-contact status when the ratio of pad thickness to brake disk thickness is 0.66. It is observed that the total strain due to thermomechanical load is negative for some parts of the disks, whereas, the thermal strains are always positive. It can be concluded that gradation index of the metal-ceramic has significant effect in the thermomechanical response of FG disks.     

Free transverse vibration of rotating blades in a bladed disk assembly

This paper investigates the natural frequency of free transverse vibration of blades in rotating disks to examine the relationship of natural frequencies, blade stiffness and nodal diameters to study how neighboring blades react upon each other and affect blade natural frequency. With the use of elastic hinge theory and a cantilever beam model subjected to either a transverse concentrated force or a bending moment at the free end, the force-deflection stiffness/moment-rotation stiffness of the beam have been developed. Thereafter, the reaction forces and moments from the neighboring blades have been determined without the need for an exact solution of large deformation of cantilever beams including geometrical nonlinearity effects. With the use of the energy conservation principle and modal theory, the natural frequency of free transverse vibration of blades in rotating disks has been determined for any nodal diameter. A comparison of the analytical and finite element solutions for a bladed disk with uniform aerofoils shows that the analytical method presented in this paper is accurate.     

The effects of loading waveform and microstructure on the fatigue response of titanium aero-engine compressor disk alloys

Microstructural variations produced from manufacturing processes and their influence on fatigue crack growth in titanium disks were investigated. Charpy‐tests on titanium disk material were performed and materials with fracture energy values in the range of 3.8–19.1 J/cm² were selected for tests under cyclic loads. Results of Charpy‐tests were compared with fractographic features related to fatigue crack growth in Ti?6Al?3Mo?0.4Si and Ti?6Al?3Mo?2Cr alloys with a two‐phase (α + β) lamellar structure under various cyclic waveforms using specimens made from compressor disks. The material sensitivity to cyclic load waveform can be seen for in‐service disks using a criteria based on fracture energy values determined in Charpy‐tests. A difference in fatigue crack growth periods of 2.5 times was discovered for specimens made from the disk with a filament type microstructure and the mainly globular two‐phase structure of the Ti?6Al?3Mo?0.4Si alloy. The shorter crack growth period correlated with the mainly facetted pattern formation with local zones of fatigue striations when fatigue crack growth is along the planes of the filaments. Fatigue striations are the major fracture surface relief when crack growth occurs in the perpendicular direction to the plane of the filaments. A quantitative fractographic method for estimating the crack growth period for in‐service failed disks was performed for the case of crack development along planes of such microstructural filaments created during the manufacturing process. Specimen tests involving a hold‐time in the cyclic loads are recommended for in‐service accepted titanium disks using a criteria based on the fracture energy value. Selection of disks based on these criteria can indicate a material sensitivity to cyclic load waveforms.     

Modelling of cutting tool – soil interaction – part II: macromechanical model and upscaling

In the foregoing paper we have proposed a strategy for soil modelling based on the discrete approach. By considering the soil as an assembly of rigid disks we have developed a local model. The model concentrates at the contact level between the disks the real mechanical behaviour of the soil. For this purpose suitable contact models have been developed, where specific elasto-plastic laws have been implemented in the node-to-segment contact formulation.In this second part after investigating the mechanical behaviour of the local model we reproduce, with an assembly of disks (global model), the real behaviour of stiff soils and rocks under standard loading paths. The change of scale from the local to global level is carried out through dimensional analysis. The behaviour of different soils is simulated by regular and irregular monodisperse packings of disks and compared with real laboratory tests.     

FREQUENCY DISTRIBUTION OF CONTACT ANGLES IN RANDOM PACKINGS OF GRANULAR MATERIALS

For a certain spectrum of stable grain configurations in randomly packed granular aggregates it is possible to determine the frequency distribution of relative contact angles among neighboring grains. This possibility is explored in the present paper for both two-dimensional and three-dimensional random packings of granular materials. Two relationships are first derived for the local void ratios of any stable “Voronoi Cell“ within the spectrum of stable configurations for the two and the three dimensional random packings, respectively. These relationships depend on the local distribution of relative contact angles, i.e., directions of contact normals. These local void ratios are then related to the gross void ratios of the random 2-D and 3-D assemblies through two integral equations of the Fredholm type of the first kind, their arguments being the frequency distribution functions. The first integral equation corresponding to a two-dimensional random disk packing is solved, exactly by a set of exponential functional transformations. These exact distributions are shown to be generally Maxwellian, with tails favorably biased towards the population of denser “Voronoi Cells” that are statistically more stable. A discussion on the uniform solutions of the integral equation for three-dimensional random packing of spheres is also presented. However, its general solution is left for a future work.