Continuum Analysis Research Articles

The policy continuum–Policy authoring and conflict analysis

The policy continuum is a fundamental component of any policy-based management implementation for autonomic networking, but as of yet has no formal operational semantics. We propose a policy continuum model and accompanying policy authoring process that demonstrates the key properties that set a continuum apart from a non-hierarchical policy model. As part of the policy authoring process we present a policy conflict analysis algorithm that leverages the information model, making it applicable to arbitrary applications and continuum levels. The approach for policy conflict analysis entails analysing a candidate policy (either newly created or modified) on a pair-wise basis with already deployed policies and potential conflicts between the policies are fed back to the policy author. Central to the approach is a two-phase algorithm which firstly determines the relationships between the pair of policies and secondly applies an application specific conflict pattern to determine if the policies should be flagged as potentially conflicting. In this paper we present the formal policy continuum and two-phase conflict analysis algorithm as part of the policy authoring process, we describe an implementation where we demonstrate the detection of potential conflicts within a policy continuum.

A micromechanical characterization of angular bidirectional fibrous composites

A micromechanical numerical algorithm to efficiently determine the homogenized elastic properties of bidirectional fibrous composites is presented. A repeating unit cell (RUC) based on a pre-determined bidirectional fiber packing is assumed to represent the microstructure of the composite. For angular bidirectional fiber distribution, the symmetry lines define a parallelepiped unit cell, representing the periodic microstructure of an angular bidirectional fiber composite. The lines of symmetry extrude a volume to capture a three dimensional unit cell. Finite element analysis of this unit cell under six possible independent loading conditions is carried out to study and quantify the homogenized mechanical property of the cell. A volume averaging scheme is implemented to determine the average response as a function of loading in terms of stresses and strains. The individual elastic properties of the constituents’ materials, as well as, the composite can be assumed to be completely isotropic to completely anisotropic. The output of the analysis can determine this degree. The logic behind the selection of the unit cell and the implementation of the periodic boundary conditions as well as the constraints are presented. To verify this micromechanics algorithm, the results for four composites are presented. The results in this paper are mainly focused on the impact of the fiber cross angles on the stiffness properties of the composites chosen. The accuracy of the results from this micromechanics modeling procedure has been compared with the stiffness/compliance solutions from lamination theory. The methodology is to be accurate and efficient to the extent that periodicity of the composite material is maintained. In addition, the results will show the impact of fiber volume fraction on the material properties of the composite. This micromechanics tool could make a powerful viable algorithm for determination of many linear as well as nonlinear properties in continuum mechanics material characterization and analysis.

Three-dimensional continuum analysis for a unidirectional composite with a broken fiber

The three-dimensional problem of a periodic unidirectional composite with a penny-shaped crack traversing one of the fibers is analyzed by the continuum equations of elasticity. The solution of the crack problem is represented by a superposition of weighted unit normal displacement jump solutions, everyone of which forms a Green’s function. The Green’s functions for the unbounded periodic composite are obtained by the combined use of the representative cell method and the higher-order theory. The representative cell method, based on the triple discrete Fourier transform, allows the reduction of the problem of an infinite domain to a problem of a finite one in the transform space. This problem is solved by the higher-order theory according to which the transformed displacement vector is expressed by a second order expansion in terms of local coordinates, in conjunction with the equilibrium equations and the relevant boundary conditions. The actual elastic field is obtained by a numerical evaluation of the inverse transform. The accuracy of the suggested approach is verified by a comparison with the exact analytical solution for a penny-shaped crack embedded in a homogeneous medium. Results for a unidirectional composite with a broken fiber are given for various fiber volume fractions and fiber-to-matrix stiffness ratios. It is shown that for certain parameter combinations the use of the average stress in the fiber, as it is employed in the framework of the shear lag approach, for the prediction of composite’s strength, leads to an over estimation. To this end, the concept of “point stress concentration factor” is introduced to characterize the strength of the composite with a broken fiber. Several generalizations of the proposed approach are offered.

Crack arresting low-density porous materials with periodic microstructure

The problem of brittle fracture of a 2D periodic porous material is considered. For the considered relative density range the cellular material approximation with beam elements becomes invalid and a novel analysis technique based on the discrete Fourier transform is suggested. Mode I fracture toughness is derived from the continuum analysis of the stress state in a perforated elastic plane with a macroscopic crack composed of a number of microcracks between the neighboring voids. The crack length is proved to be sufficiently large such that the K-field of the linear elastic fracture mechanics takes place. The solution is obtained in the form of superposition of weighted Green functions every one of which corresponds to a unit displacement jump at a single microcrack. The Green function solutions are found by a combination of the representative cell technique and the finite element method. As a result the initial problem for infinite plane is solved by multiple analysis of the repetitive periodicity cell. An optimization of the voids shape for fixed relative density is performed in order to design the material with improved fracture toughness. Several types of voids arrangement are considered. In the limiting case of low density the results are compared with known data for cellular materials.

216 VAC法を用いた超弾性合金の原子-連続体結合の解析方法(OS15.電子・原子・マルチシミュレーションに基づく材料特性評価(5),オーガナイズドセッション)

216 VAC法を用いた超弾性合金の原子-連続体結合の解析方法(OS15.電子・原子・マルチシミュレーションに基づく材料特性評価(5),オーガナイズドセッション)

The size effect of nanoindentation on ZnO nanofilms

Nanoindentation behaviors of zinc oxide (ZnO) nanofilms with different film thicknesses are studied by using both molecular mechanics (MM) simulations and continuum analyses. It is found that there is a significant size effect on the indentation modulus obtained from MM simulations, which is absent in the continuum studies. The indentation modulus increases with the film thickness, and it also increases with the indentation depth; the trend of such a variation also depends on the film thickness. The contributions of the contact size effect, film thickness size effect, and microstructural size effect (surface effect) are elucidated and their couplings are explored. In addition, the substrate effect and nonlinear hyperelastic effect are incorporated to explain the size dependence of elastic indentation behaviors of ZnO nanofilms.

On scale dependence in friction: Transition from intimate to monolayer-lubricated contact

Scale dependence in friction is studied in the present paper using the newly developed mesoscale friction tester (MFT). A transition in frictional shear strength from several hundreds of MPa to several tens of MPa was observed over a very limited range of contact radii (20–30 nm) in both ambient and dry environments. Thus, a single apparatus has been able to establish these two limits which are consistent with the values previously obtained from friction experiments using atomic force microscopy (AFM) and the surface force apparatus (SFA), respectively. Consequently, it is hypothesized here that a shear strength in the hundreds of MPa results from intimate contact (solid–solid) and a shear strength in the tens of MPa results from a monolayer-lubricated contact. Furthermore, both the probe size and the normal pressure govern the interfacial conditions in the contact zone and it is these conditions, rather than the nominal environment, which in turn determine the resulting shear strengths. A continuum analysis based on the Lifshitz theory for van der Waals interactions is used to explain the quantized shear strengths which were obtained from our experiments and previous AFM and SFA friction experiments. This quantized friction behavior [J.N. Israelachvili, P.M. McGuiggan, A.M. Homola, Science 240 (1988) 189] results from the discrete separation due to the different interfacial conditions that can arise between two sliding surfaces. The consistency between the analysis and the experimental results shows that this analysis is applicable for nonwear friction with single asperity contact.

Elastic continuum analysis of the liquid crystal polarization grating

We apply elastic continuum theory to model critical parameters influencing the free-energy equilibrium configuration and the dynamic performance of a continuous and in-plane liquid crystal profile acting as a polarization grating. We present analytical expressions for the threshold voltage, critical thickness, and the dynamic switching times under strong anchoring conditions, negligible flow, and arbitrary splay, twist, and bend constants. We also study the influence of weak anchoring, and derive expressions describing a dramatic reduction of the critical thickness and voltage threshold, even for modest grating periods and surface anchoring strengths. Good correlation exists with previously reported experimental data, except in the dynamic response; we therefore show that flow effects (backflow and kickback) likely play an essential role in the fall times, presumably due to the prominent splay-bend deformation of the zero-field configuration. We consider the impact of surface pretilt, and validate our entire analysis with numerical simulations. The approximation technique we employ is likely broadly useful for many problems which include nano- or micropatterned surfaces.

Buried Pipelines Subject to Subgouge Deformations

Engineers often model pipe/soil interaction events based on the concept of subgrade reactions originally proposed by Winkler. Engineering models often utilize beam and nonlinear/plastic spring elements to represent pipelines and the surrounding soil medium, respectively. The spring formulations, defining soil resistance to deformations in three-dimensional space, are usually assumed to be independent and the responses are discrete between adjacent soil zones. However, this idealization does not truly replicate a soil medium behavior. This study presents coupled numerical analyses of pipeline for the specific problem of subgouge deformations due to ice gouge events. Three dimensional continuum analyses of coupled pipe/soil/ice keel interaction using an explicit arbitrary Lagrangian finite- element approach were performed. The study compares the continuum finite-element results with Winkler-type analysis for the specific analyzed problem. A Lagrangian adaptive meshing technique was employed to model very la...

Nanoscale Jet Collision and Mixing Dynamics

Micro- and macroscale investigations have shown that colliding drops always coalesce for small values of the Weber number We = rhoU2d/sigma. Our molecular dynamic simulations show that nanojets always recoil following head-on collision even though We --> 0. The duration between collision and recoil is a function of the nanojet impact velocity Uo and the nature of intermolecular interactions. Evaporation, which promotes mixing, occurs during recoil and is enhanced by reducing intermolecular interactions. Thereafter, mixing occurs through diffusion. The mixing dynamics are independent of Uo and the orifice shape. Consistent with a continuum analysis, the characteristic nanojet diameter at stagnation ds,1 proportional to Uo, recoil time following collision tau proportional to Uo-2, and the number of evaporating molecules N proportional to Uo.

Electrokinetic transport and separations in fluidic nanochannels

This article presents a summary of theory, experimental studies, and results for the electrokinetic transport in small fluidic nanochannels. The main focus is on the effect of the electric double layer on the EOF, electric current, and electrophoresis of charged analytes. The double layer thickness can be of the same order as the width of the nanochannels, which has an impact on the transport by shaping the fluid velocity profile, local distributions of the electrolytes, and charged analytes. Our theoretical consideration is limited to continuum analysis where the equations of classical hydrodynamics and electrodynamics still apply. We show that small channels may lead to qualitatively new effects like selective ionic transport based on charge number as well as different modes for molecular separation. These new possibilities together with the rapid development of nanofabrication capabilities lead to an extensive experimental effort to utilize nanochannels for a variety of applications, which are also discussed and analyzed in this review.

Atomistic-continuum and ab initio estimation of the elastic moduli of single-walled carbon nanotubes

This paper deals with the calculation of elastic moduli and stress–strain curves for single-walled carbon nanotubes (SWNTs) using a computationally efficient, atomistically enriched continuum analysis. This approach is adopted to estimate shear and Young’s moduli and obtain stress–strain curves for carbon nanotubes (CNTs) subject to coupled extension and twist deformations. This accounts for the effects of natural extension-twist coupling [K. Chandraseker, S. Mukherjee, ASME J. Appl. Mech. 73 (2) (2006) 315–326; K. Chandraseker, S. Mukherjee, Y.X. Mukherjee, Int. J. Solids Struct. 43 (2006) 7128–7144] on SWNT constitutive properties. The constitutive properties are evaluated by assuming a cylindrical reference configuration [Chandraseker and Mukherjee, 2006; Chandraseker et al., 2006] rather than a planar graphene sheet [P. Zhang, Y. Huang, P.H. Geubelle, P.A. Klein, K.C. Hwang, Int. J. Solids Struct. 39 (2002) 3893–3906; M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415] thereby allowing for the anisotropy and change in strain energy that results from the finite deformation required to roll up a graphene sheet into a nanotube [M. Arroyo, T. Belytschko, Phys. Rev. B 69 (2004) 115415]. The Tersoff–Brenner multi-body empirical interatomic potential for carbon [J. Tersoff, Phys. Rev. B 37 (1988) 6991–7000; D.W. Brenner, Phys. Rev. B 42 (1990) 9458–9471] is used to model the C–C bond energies in this work. This enables exact analytic evaluation of the derivatives of the strain energy density rather than a numerical approach. Consistent values obtained corresponding to these material properties indicate that they do not depend strongly on the chirality of the nanotube [R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998 [7]]. The relative magnitudes of the Young’s and shear moduli obtained from this approach fall within the well known range in classical elasticity theory in most cases, and the computed values for the moduli agree well with existing experimental results and atomistic studies that employ the same interatomic potential. Further, in the present work, the moduli are also evaluated using a more accurate, albeit computationally expensive, ab initio density-functional-theoretic (DFT) approach (see for e.g., [D. Sáncez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejón, Phys. Rev. B 59 (1999) 12678; K.N. Kudin, G.E. Scuseria, B.I. Yakobson, Phys. Rev. B 64 (2001) 235406]). A comparison between these values and the ones from the atomistic-continuum analysis brings to notice some of the advantages and limitations of both these approaches.

Modelling of mixed mode debonding in externally FRP reinforced beams

In this paper a refined model able to analyze edge debonding problems in beams strengthened with externally bonded composite laminated plates, is presented. The structural system is viewed as composed by three physical different components: the base beam (made of steel or concrete), the adhesive layer and the bonded plate. Each component may be comprised by one or several mathematical layers which adopts the first-order shear deformation laminate theory. Bonding and continuity conditions between different layers are simulated by using the interface modelling technique. Strong and collapsed interface models are introduced in order to capture stress singularities and to reduce the complexity of the analysis, respectively. Governing equations for displacement fields complemented with boundary and continuity conditions, are obtained by a variational approach. According to a fracture mechanics approach, the analysis is carried out by evaluating the total and individual mode components of energy release rate (ERR). Applications for typical strengthened systems, carried out by numerical integration procedures, are proposed in which the energy release rates are evaluated by means of interface displacement jumps, leading to a very efficient numerical procedure. The approximations introduced in the model with respect to the adopted number of mathematical layers are analyzed and comparisons with existent models are given. For the simpler two-layer model of the structure, comparisons are given with the closed-form solutions obtained in [Greco F, Nevone Blasi P, Lonetti P. An analytical investigation of debonding problems in beams strengthened using composite plates. Eng Fract Mech 2006, in press]. The convergence to the results from continuum analysis is investigated when a refined assembly of layers is adopted, by means of comparisons with predictions from very careful FE solutions. Finally, the effect of different debonding modes on the overall behaviour of the structural system is analyzed. These results show the capability and the accuracy of the proposed approach to predict debonding failure behaviour in both steel and concrete strengthened beams. As a matter of fact, the proposed approach involves reduced computational cost with respect to FE solutions based on 2D continuum elements and the use of a multi-layer structural model leads to avoid some complexities related to the classical elasticity theory for bimaterial interface cracks.

Mass Transfer Limitations at Crystallizing Interfaces in an Atomic Force Microscopy Fluid Cell: A Finite Element Analysis

Although atomic force microscopy (AFM) has emerged as the preeminent experimental tool for real-time in situ measurements of crystal growth processes in solution, relatively little is known about the mass transfer limitations that may impact these measurements. We present a continuum analysis of flow and mass transfer in an atomic force microscope fluid cell during crystal growth, using data acquired from calcium oxalate monohydrate (COM) crystal growth measurements as a comparison. Steady-state flows and solute concentration fields are computed using a three-dimensional, finite element method implemented on a parallel supercomputer. Steady-state flow results are compared with flow visualization experiments to validate the model. Computations of the flow field demonstrate how nonlinear momentum transport alters the spatial structure of the flow with increasing flow volume, altering mass transport conditions near the AFM cantilever and tip. The simulations demonstrate that the combination of solute depletion from crystal growth and mass transfer resistance lowers the solute concentration in the region between the tip and the crystal compared with the solute concentration at the inlet of the AFM cell. For example, using experimentally measured growth rates for COM, the solute concentration in this region is 3.1% lower than the inlet value because the solute consumed by crystal growth beneath the AFM tip cannot be replenished fully due to mass transport limitations. The simulations also reveal that increasing the flow rate through the cell does not affect this difference significantly because of the inherent shielding by the AFM tip in proximity with the crystal surface. Models such as the one presented here, used in conjunction with AFM measurements, promise more precise interpretations of measurement data.

Buckling of Multiwall Carbon Nanotubes under Axial Compression

We have investigated the axial buckling of multiwall nanotubes under the axial compression using nanomanipulation experiments and molecular dynamics (MD) simulations. Experimentally, Young's moduli of nanotubes with different inner hollow diameters for the same outer diameters are consistent with the Eulers buckling model based on the continuum analysis. The MD simulations for the buckling behavior of triple- and double-walled nanotubes are also consistent with the continuum analysis. This good agreement indicates that Euler's buckling model is applicable to the analysis of the axial buckling behavior of the multiwall nanotubes.

Cracking and adhesion at small scales: atomistic and continuum studies of flaw tolerant nanostructures

Once the characteristic size of materials reaches nanoscale, the mechanical properties may change drastically and classical mechanisms of materials failure may cease to hold. In this paper, we focus on joint atomistic-continuum studies of failure and deformation of nanoscale materials. In the first part of the paper, we discuss the size dependence of brittle fracture. We illustrate that if the characteristic dimension of a material is below a critical length scale that can be on the order of several nanometres, the classical Griffith theory of fracture no longer holds. An important consequence of this finding is that materials with nano-substructures may become flaw-tolerant, as the stress concentration at crack tips disappears and failure always occurs at the theoretical strength of materials, regardless of defects. Our atomistic simulations complement recent continuum analysis (Gao et al 2003 Proc. Natl Acad. Sci. USA 100 5597–600) and reveal a smooth transition between Griffith modes of failure via crack propagation to uniform bond rupture at theoretical strength below a nanometre critical length. Our results may have consequences for understanding failure of many small-scale materials. In the second part of this paper, we focus on the size dependence of adhesion systems. We demonstrate that optimal adhesion can be achieved by either length scale reduction, or by optimization of the shape of the surface of the adhesion element. We find that whereas change in shape can lead to optimal adhesion strength, those systems are not robust against small deviations from the optimal shape. In contrast, reducing the dimensions of the adhesion system results in robust adhesion devices that fail at their theoretical strength, regardless of the presence of flaws. An important consequence of this finding is that even under the presence of surface roughness, optimal adhesion is possible provided the size of contact elements is sufficiently small. Our atomistic results corroborate earlier theoretical modelling at the continuum scale (Gao and Yao 2004 Proc. Natl Acad. Sci. USA 101 7851–6). We discuss the relevance of our studies with respect to nature's design of bone nanostructures and nanoscale adhesion elements in geckos.

Mechanisms of nanoindentation on single-walled carbon nanotubes: The effect of nanotube length

The mechanisms of nanoindentation on single-walled carbon nanotubes (SWCNTs) have been studied by using molecular dynamics simulation and continuum analysis during which a flat layer of diamond atoms is pressed down incrementally on a vertically aligned SWCNT. SWCNTs are divided into three distinct categories based on their aspect ratios, such that the nanotube behavior transits from a shell (short tube) to a beam (long tube). Molecular dynamics simulations are used to explore the diverse indentation characteristics in each domain, where the relationships between the strain energy and indentation depth during loading, unloading, and reloading are continuously recorded. The nanoindentation mechanisms are characterized by the critical indentation depth, maximum strain energy and force associated with buckling, as well as with the evolution of carbon bond length and morphology of the SWCNTs. Bifurcation behaviors are explored by investigating the loading-unloading-reloading behaviors of the nanotubes. Parallel finite element simulations are also used to study the pre- and post-buckling behaviors of SWCNT by incorporating the van der Waals interaction into the continuum code. It is found that, for the most part, continuum analysis can effectively capture the overall indentation characteristics, yet some details related to the atomic characteristics of nanoindentation may only be revealed by molecular dynamics simulation. Finally, an indentation mechanism map is derived by comparing behaviors of SWCNTs with different aspect and section ratios. Focusing on the effects of nanotube length, this paper is the first of a series of numerical studies on the indentation mechanisms of carbon nanotubes, which may be used to determine the intrinsic mechanical properties of SWCNTs by means of nanoindentation.

Analysis of jacking forces during microtunnelling in limestone

This paper is intended to describe an Italian case study of a microtunnelling project where the boring machine got stuck during jacking of a 760 mm pipe in a limestone formation. Reference is made to rock mass characterisation, including site investigations and laboratory tests. The machine’s performance is compared to prediction. To allow for a better understanding of the conditions which led to the unexpectedly high jacking forces, continuum and discontinuum numerical analyses have been used. These analyses are shown to be essential at the design stage, when dealing with microtunnelling in a rock mass, in order to obtain a good prediction of the jacking forces.

DEVELOPMENT OF SEEPAGE FAILURE ANALYSIS METHOD OF GROUND WITH SMOOTHED PARTICLE HYDRODYNAMICS

Seepage failure including flowage deformation and hydraulic fracture, plays an important role on damage of dyke under flood, flow of ground caused by liquefaction and improvement of ground by injection and/or penetration procedures. These problems must be solved with interaction among three phases: solid (soil), liquid (water) and gas (air). Discrete analysis (e.g. DEM) is adapted to abruption, failure and flowage, but unsuitable procedure to analysis domain of large scale. Continuum analysis has opposite properties to that.In this paper, there is a new attempt to develop the procedure which fuses discrete and continuum analyses by ‘Smoothing Particle Hydrodynamics (SPH)’ with account for the interaction among three phases.

P-167: FDTD and Elastic Continuum Analysis of the Liquid Crystal Polarization Grating

liquid crystal polarization grating (LCPG) has advantages of polarization independence of the zero diffraction order, nearly 100% diffraction efficiency in the first orders, and higher resolution capability over previously reported binary LC gratings. Here we analyze the LCPG, an electrically controlled, polarization-independent light modulator using the finite- difference time-domain (FDTD) method and the elastic continuum theory. The optical performance is studied and critical electrical parameters for a LCPG cell are presented. 1. Introduction of liquid crystal (LC) light modulators operating on unpolarized light have been proposed in an effort to overcome the substantial losses in conventional LC displays that require polarized light. Most prominently, Bos and coworkers suggested several potentially efficient and polarization-independent diffraction gratings using liquid crystals (1-4). These approaches, however, face crucial limitations for real applications due to their binary nature: fabrication difficulty and the easy appearance of defects. They therefore tend to have inherently large periods that result in many diffraction orders with small diffraction angles. A newly demonstrated liquid crystal polarization grating (LCPG) (5) is a switchable diffractive optical element with a continuously varying, periodic, anisotropic index profile. As shown in Fig. 1(a), this can be embodied as a nematic director that follows (6): n(x) = ˆ cos(x / ) + ˆ y sin(x /) + ˆ (0) (1)