Lower Length Scale Research Articles

Effect of material’s randomness on scaling of crack propagation in ceramics

Crack propagation in real (quasi)brittle materials demonstrates various signs of stochasticity; a tortuous character of fracture surfaces, multiple cracking and crack branching observed in experiments are a vivid confirmation of it. Traditional approaches of fracture mechanics represent cracks as geometrically smooth objects with straight (or curved) crack fronts, thus usually neglecting morphology of real cracks. An introduction of a direct account for stochastic features of brittle materials into modelling schemes leads to a more adequate description of real fracture processes. The effect of the material’s randomness on crack propagation in ceramics is studied based on the approach, combining these random features with continuum damage mechanics (CDM) and fracture mechanics. CDM describes a macroscopic manifestation of various failure processes developing at lower length scales. The numerical mode-I fracture analysis, based on discretization of a specimen’s cross-section, containing a sharp notch, into rectangular elements, provides detailed information on slow crack propagation. A necessity to describe a crack with its length changing along the front presupposes a transition from a unique stress-intensity factor to a set of its local values. A computational procedure for simulation of crack-damage interaction and crack propagation in alumina specimens at tension is suggested on the basis of a modification of a lattice scheme unified with ideas of CDM and local stress-intensity factors. Inhomogeneity of material properties is modelled in terms of various random spatial distributions of the initial damage in the specimen’s cross-section. Characterization of complicated morphology of cracks is implemented by means of scaling analysis of the crack-front shape.

Vortex method for simulation of a 3D round jet in a cross-stream

In this paper, the three-dimensional (3D) flow of a jet in a crossflow was investigated. This is a numerical study based on a vortex particle method. In order to account for the lower length scales, a crude LES turbulence model was implemented together with a regridding procedure. The numerical simulation was done with an injection ratio R inj = W jet/U ∞ = 4.0. The code was run until an averaged steady state of the jet was reached. The resulting vortex structures were shown to be consistent with previously published analyses of the flow, from both experiments and numerical simulations. In particular, the onset of counter-rotating vortex pairs (CVP) was also investigated.

Multiscale modelling for equations of state

When modelling the shock response of a material it is imperative to have a physically-based equation of state. These can be expensive and difficult to measure. Existing formulations can also be somewhat lacking if strength considerations need to be included. Modifications to materials are difficult to account for quickly within a program of work. An hierarchical approach is suggested that overcomes these obstacles and also takes into account the different approaches needed for different materials. We find good predictions for simple crystals using quantum mechanics codes. Further up the hierarchy of length scales it is seen that molecular dynamics and molecular mechanics is useful in determining pressure-volume-energy relations for polymers and organic macromolecules. A simplified mean field approach using atomistic parameters can give analytic functionality and serves as a bridge between the lower length scales and the thermodynamic length scales. Composite models can be formulated from models of the constituents. The approach provides results that can be implemented into hydrocodes in tabulated form and then validated from a few experimental points. The tools with which to carry out this hierarchical approach are readily available in the literature and in commercially-available software.

Stability of thermal convection in two superimposed miscible viscous fluids

The stability of two-layer thermal convection in high-Prandtl-number fluids is investigated using laboratory experiments and marginal stability analysis. The two fluids have different densities and viscosities but there is no surface tension and chemical diffusion at the interface is so slow that it is negligible. The density stratification is stable. A wide range of viscosity and layer depth ratios is studied. The onset of convection can be either stationary or oscillatory depending on the buoyancy number B, the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly: when B is lower than a critical value (a function of the viscosity and layer depth ratios), the oscillatory regime develops, with a deformed interface and convective patterns oscillating over the whole tank depth; when B is larger than this critical value, the stratified regime develops, with a flat interface and layers convecting separately. Experiments agree well with the marginal stability results. At low Rayleigh number, characteristic time and length scales are well-predicted by the linear theory. At higher Rayleigh number, the linear theory still determines which convective regime will start first, using local values of the Rayleigh and buoyancy numbers, and which regime will persist, using global values of these parameters.

Application of image processing for simulation of mechanical response of multi-length scale microstructures of engineering alloys

Microstructures of engineering alloys often contain features at widely different length scales. In this contribution, a digital image processing technique is presented to incorporate the effect of features at higher length scales on the damage evolution and local fracture processes occurring at lower length scales. The method is called M-SLIP: Microstructural Scale Linking by Image Processing. The technique also enables incorporation of the real microstructure at different length scales in the finite element (FE)-based simulations. The practical application of the method is demonstrated via FE analysis on the microstructure of an aluminum cast alloy (A356), where the length scales of micropores and silicon particles differ by two orders of magnitude. The simulation captures the effect of nonuniformly distributed micropores at length scales of 200 to 500 µm on the local stresses and strains around silicon particles that are at the length scales of 3 to 5 µm. The procedure does not involve any simplifying assumptions regarding the microstructural geometry, and therefore, it is useful to model the mechanical response of the real multi-length scale microstructures of metals and alloys.

Drumlinization and drumlin-forming instabilities: viscous till mechanisms

AbstractGlacially induced flow naturally tends to thin and extended till cover through shock formation, even in the absence of longitudinal gradients in the applied stress. Thicker till cover has an increased effective pressure at its surface and base, a lower sliding velocity or deformation rate and above a critical thickness, a decrease in wave velocity with thickness, leading to reverse-facing shocks moving downstream. For sliding and for some theologies of internal deformation, a decrease in sediment flux with thickness occurs, implying backward-moving kinematic waves and reverse-facing, reverse-moving shock.Downstream-facing shocks are also formed which move upstream if the till is sliding and downstream if the till is deforming internally. Eventually, shocks coalesce, leaving an upstream-lacing shock for sliding and a downstream-facing shock for internal deformation. It is observed that some drumlins have downstream blunt ends only.Fairly realistic three-dimensional drumlin shapes can be produced from symmetric sediment bodies and barchan shapes can be produced from linear forms perpendicular to the ice-sheet flow.The fact that viscous theories produce drumlinoid forms suggests that on this scale till behaves viscously and the the lower length scale for drumlins represents the plastic/viscous transition scale.

Drumlinization and drumlin-forming instabilities: viscous till mechanisms

AbstractGlacially induced flow naturally tends to thin and extended till cover through shock formation, even in the absence of longitudinal gradients in the applied stress. Thicker till cover has an increased effective pressure at its surface and base, a lower sliding velocity or deformation rate and above a critical thickness, a decrease in wave velocity with thickness, leading to reverse-facing shocks moving downstream. For sliding and for some theologies of internal deformation, a decrease in sediment flux with thickness occurs, implying backward-moving kinematic waves and reverse-facing, reverse-moving shock.Downstream-facing shocks are also formed which move upstream if the till is sliding and downstream if the till is deforming internally. Eventually, shocks coalesce, leaving an upstream-lacing shock for sliding and a downstream-facing shock for internal deformation. It is observed that some drumlins have downstream blunt ends only.Fairly realistic three-dimensional drumlin shapes can be produced from symmetric sediment bodies and barchan shapes can be produced from linear forms perpendicular to the ice-sheet flow.The fact that viscous theories produce drumlinoid forms suggests that on this scale till behaves viscously and the the lower length scale for drumlins represents the plastic/viscous transition scale.

Are channel networks statistically self-similar?

Scaling properties of both field-mapped and threshold-delineated channel networks were studied by applying the box-counting method to three drainage basins in the western United States. This method involves (1) examination of power-law relations between the box size, ϵ, and the number of boxes, N , that intersect channel segments across a range of box sizes appropriate for the method and then (2) examining the standardized residuals for the least squares linear regressions of log N vs. log ϵ used to calculate a fractal dimension ( D ). For each channel network, the slope of the log N vs. log ϵ relation varies from 1 at small length scales to 2 at large length scales, a range that defines the limits to the applicability of the box-counting method. At length scales below which this slope equals 1, the plots simply record the linear aspect of streams; the length scale defining an upper limit to the application of the box-counting method corresponds to a box size large enough to intersect a channel in each box. Although a fractal dimension may be meaningfully defined only between these upper and lower length scales, neither the field-mapped nor the artificially delineated networks that we examined exhibit discrete fractal dimensions within this range. Instead, the slope of the log-log plot systematically varied with box size. The consistent lack of log-linear plots for the networks that we examined violates a fundamental requirement for fractal geometry and contrasts with general assertions about the fractal nature of river networks. A strong correlation between mean source-area size and the length scale above which the slope of plots implies D = 2 indicates that, although channel networks are not statistically self-similar, they are space filling at length scales greater than twice the mean source-basin length.

Basic issues in the mechanics of high cycle metal fatigue

Mechanics issues related to the formation and growth of cracks ranging from subgrain dimension to up to the order of one mm are considered under high cycle fatigue (HCF) conditions for metallic materials. Further efforts to improve the accuracy of life estimation in the HCF regime must consider various factors that are not presently addressed by traditional linear elastic fracture mechanics (LEFM) approaches, nor by conventional HCF design tools such as the S-N curve, modified Goodman diagram and fatigue limit. A fundamental consideration is that a threshold level for ΔK for small/short cracks may be considerably lower than that for long cracks, leading to non-conservative life predictions using the traditional LEFM approach. Extension of damage tolerance concepts to lower length scales and small cracks relies critically on deeper understanding of (a) small crack behavior including interactions with microstructure, (b) heterogeneity and anisotropy of cyclic slip processes associated with the orientation distribution of grains, and (c) development of reliable small crack monitoring techniques. The basic technology is not yet sufficiently advanced in any of these areas to implement damage tolerant design for HCF. The lack of consistency of existing crack initiation and fracture mechanics approaches for HCF leads to significant reservations concerning application of existing technology to damage tolerant design of aircraft gas turbine engines, for example. The intent of this paper is to focus on various aspects of the propagation of small cracks which merit further research to enhance the accuracy of HCF life prediction. Predominant concern will rest with polycrystalline metals, and most of the issues pertain to wide classes of alloys.

Small-Angle Neutron Scattering of Poly(γ-benzyl l-glutamate) in Deuterated Benzyl Alcohol

Small-angle neutron scattering has been used to study poly(γ-benzyl L-glutamate) (PBLG) in deuterated benzyl alcohol (DBA) as the solution is brought from the isotropic phase into the gel phase. The results show that an aggregation of PBLG molecules exists up to 80°C even in the isotropic phase and the size of the aggregate does not change with temperature within the isotropic phase for a given concentration. However, the size of the aggregate increases with concentration or as the gelation threshold is crossed. The measurement of fractal dimension at intermediate length scales of the aggregates in the isotropic phase shows that the aggregates are less dense at higher polymer concentrations. But at lower length scales of ∼50 A, the mesh size of the aggregates in the isotropic phase decreases with increasing polymer concentration. Studies of the fractal dimension of the resultant gels show that the thermal history of the sample in the isotropic phase influences the gel microstructure. The fractal dimension of the PBLG-DBA gel obtained by a step quench procedure is higher than that of a gel obtained by a sudden quench. We have also observed that the local concentration gradients become sharper as the polymer concentration increases in the gel phase.

Numerical and analytical study to estimate the effect of two length scales upon the permeability of a fibrous porous medium

A fibrous porous medium with two length scales is modeled as a bed of porous cylinders aligned perpendicular to the flow of viscous fluid. The flow behavior is described using Stokes and Darcy flow equations in the regions around (higher length scale) and within the cylinders (lower length scale) respectively. The typical ratio of higher and lower length-scale regions enable us to invoke lubrication approximation and simplify the equations to develop a closed form solution for the overall permeability of this dual-scale porous medium. A parametric analysis is performed to explore the dependence of permeability on factors such as the volumetric ratio of higher and lower length-scale regions, permeability and size of inclusions in the smaller length-scale region. The analytical model is compared with the numerical results and the trend is compared with the experiments.

Symptoms of chaos in observed oscillations near a bifurcation with noise

We examine an experimental transition from periodic to aperiodic and back to periodic dynamics in the combustion of acetaldehyde(ACH) in a continuous stirred tank reactor (CSTR) with power spectra, autocorrelation functions, phase portraits, Poincaŕe sections, the Wolf–Swift–Swinney–Vastano (WSSV) method for determining the largest Lyapounov exponent, and the Grassberger–Procaccia (GP) method for determining correlation dimension. Each technique gives some indications of a transition to chaos, but there are discrepancies in that the largest Lyapounov exponent is positive but does not converge and the GP method results in a correlation dimension between one and two for two aperiodic data sets. We explore in instructive detail possible explanations for false indications of chaos by comparing our results with calculations on the Rössler chaotic attractor and the van der Pol periodic attractor modified to examine the effects of uneven point distribution and three types of experimental noise. An uneven distribution of points results in a decreased range of length scales for convergence and a larger required embedding dimension for the GP method, but does not explain our experimental results. Observation noise (a Gaussian noise added to each term in the time series but not entering in the equations of motion) and constraint shift (the motion relaxes to an attractor but a constraint changes monotonically during the course of measurement) added to a periodic attractor both result in a low length scale cutoff below which the attractor dimension does not converge with embedding dimension, and above which it converges to 1. Constraint variation noise (a Gaussian noise is added to each term in the time series and enters into the equations of motion as a stochastic perturbation) does yield correlation dimensions between 1 and 2. The experimental transition shows many similarities to a Hopf bifurcation found in another experiment on the same system and to a theoretical Hopf bifurcation with constraint variation noise. A modification of the WSSV Lyapounov exponent analysis for this experimental transition shows the random walk separation of trajectories expected for constraint variation noise added to the dynamics of a periodic attractor with a Hopf bifurcation. We therefore identify the experimental transition as an arc in constraint space which does not cross but is nearly tangent to a Hopf bifurcation set.