Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses:discrete and continuum models

The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses.The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix.The inclusions are idealized by lumped masses,and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses.Additionally,the model is capable of depicting the wave propagation in bi-material bars,wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses.The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered,and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method(LPM).Next,a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique.The subsequent use of the second-order method of multiple scales(MMS)facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form.The novelties of the present study consist of(i)considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains;(ii)developing the second-order LPM for the wave propagation in the discrete chains;and(iii)deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses.Finally,a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models.These parameters include the ratio of the spring mass to the lumped mass,the nonlinear stiffness coefficient of the spring,and the wave amplitude.

Verification of a laboratory-based dilation model for in situ conditions using continuum models

With respect to constitutive models for continuum modeling applications, the post-yield domain remainsthe area of greatest uncertainty. Recent studies based on laboratory testing have led to thedevelopment of a number of models for brittle rock dilation, which account for both the plastic shearstrain and confining stress dependencies of this phenomenon. Although these models are useful inproviding an improved understanding of how dilatancy evolves during a compression test, there hasbeen relatively little work performed examining their validity for modeling brittle rock yield in situ. Inthis study, different constitutive models for rock dilation are reviewed and then tested, in the context of anumber of case studies, using a continuum finite-difference approach （FLAC）. The uncertainty associatedwith the modeling of brittle fracture localization is addressed, and the overall ability of mobilizeddilation models to replicate in situ deformation measurements and yield patterns is evaluated. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.

Using fracture-based continuum modeling of coupled geomechanical-hydrological processes for numerical simulation of hydraulic fracturing

This paper describes numerical simulation of hydraulic fracturing using fracture-based continuum modeling(FBCM)of coupled geomechanical-hydrological processes to evaluate a technique for high-density fracturing and fracture caging.The simulations are innovative because of modeling discrete fractures explicitly in continuum analysis.A key advantage of FBCM is that fracture initiation and propagation are modeled explicitly without changing the domain grid(i.e.no re-meshing).Further,multiple realizations of a preexisting fracture distribution can be analyzed using the same domain grid.The simulated hydraulic fracturing technique consists of pressurizing multiple wells simultaneously:initially without permeating fluids into the rock,to seed fractures uniformly and at high density in the wall rock of the wells;followed by fluid injection to propagate the seeded fracture density hydraulically.FBCM combines the ease of continuum modeling with the potential accuracy of modeling discrete fractures and fracturing explicitly.Fractures are modeled as piecewise planar based on intersections with domain elements;fracture geometry stored as continuum properties is used to calculate parameters needed to model individual fractures;and rock behavior is modeled through tensorial aggregation of the behavior of discrete fractures and unfractured rock.Simulations are presented for previously unfractured rock and for rock with preexisting fractures of horizontal,shallow-dipping,steeply dipping,or vertical orientation.Simulations of a single-well model are used to determine the pattern and spacing for a multiple-well design.The results illustrate high-density fracturing and fracture caging through simultaneous fluid injection in multiple wells:for previously unfractured rock or rock with preexisting shallow-dipping or horizontal fractures,and in situ vertical compressive stress greater than horizontal.If preexisting fractures are steeply dipping or vertical,and considering the same in situ stress condition,well pressurization without fluid permeation appears to be the only practical way to induce new fractures and contain fracturing within the target domain.

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.

A model of continuum damage mechanics for high cycle fatigue of metallic materials

A non-linear continuum damage model was presented based on the irreversible thermodynamics framework developed by LEMAITRE and CHABOCHE. The proposed model was formulated by taking into account the influence of loading frequency on fatigue life. The parameters H and c are constants for frequency-independent materials, but functions of cyclic frequency for frequency-dependent materials. In addition, the expression of the model was discussed in detail at different stress ratios （R）. Fatigue test data of AlZnMgCu1.5 aluminium alloy and AMg6N alloy were used to verify the proposed model. The results showed that the model possesses a good ability of predicting fatigue life at different loading frequencies and stress ratios.

Numerical simulation for macrosegregation in direct-chill casting of 2024 aluminum alloy with an extended continuum mixture model

An extended continuum mixture model for macrosegregation is applied to predicting Cu and Mg segregation in large-size ingot of 2024 aluminum alloy during direct chill casting（DC）. A microsegregation model using the approximate phase diagram data was coupled with macroscopic transport equations for macrosegregation profiles. Then, the impacts of transport mechanisms on the formation of macrosegregation were discussed. It is found that copper and magnesium have a similar segregation configuration from the billet center to surface. Negative segregation is observed in the centerline and subsurface, whereas positive segregation is obtained in the surface and somewhat underestimated positive segregation in the middle radius. Further, the discrepancy between the predicted and experimental results was discussed in detail. The results show that the magnesium to some extent alleviates the copper segregation in ternary alloy, compared with that in binary alloy. The predicted results show good agreement with measured experimental data obtained from literatures.

Aero-engine Blade Fatigue Analysis Based on Nonlinear Continuum Damage Model Using Neural Networks

Fatigue life and reliability of aero-engine blade are always of important significance to flight safety.The establishment of damage model is one of the key factors in blade fatigue research.Conventional linear Miner＇s sum method is not suitable for aero-engine because of its low accuracy.A back propagation neutral network（BPNN） based on the combination of Levenberg-Marquardt（LM） and finite element method（FEM） is used to describe process of nonlinear damage accumulation behavior in material and predict fatigue life of the blade.Fatigue tests of standard specimen made from TC4 are carried out to obtain material fatigue parameters and S-N curve.A nonlinear continuum damage model（CDM）,based on the BPNN with one hidden layer and ten neurons,is built to investigate the nonlinear damage accumulation behavior,in which the results from the tests are used as training set.Comparing with linear models and previous nonlinear models,BPNN has the lowest calculation error in full load range.It has significant accuracy when the load is below 500 MPa.Especially,when the load is 350 MPa,the calculation error of the BPNN is only 0.4%.The accurate model of the blade is built by using 3D coordinate measurement technology.The loading cycle in fatigue analysis is defined from takeoff to cruise in 10 min,and the load history is obtained from finite element analysis（FEA）.Then the fatigue life of the compressor blade is predicted by using the BPNN model.The final fatigue life of the aero-engine blade is 6.55 104 cycles（10 916 h） based on the BPNN model,which is effective for the virtual design of aero-engine blade.

A CONTINUUM MODEL FOR AXIAL-STRAIN-INDUCED TORSION IN SINGLE-WALL CARBON NANOTUBES

Using the atomistic-based finite-deformation shell theory, we analytically investigate the coupling between the axial deformation and the torsion in single-wall carbon nanotubes. We find that the axial-strain-induced torsion（ASIT） response is limited only to chiral nanotubes. This response is affected by chiralities and radii of carbon nanotubes. Our results are similar to that of molecular dynamic simulations reported in the literatures.

Continuum Constitutive Modeling for Isotropic Hyperelastic Materials

The partial differential equation for isotropic hyperelastic constitutive models has been postulated and derived from the balance between stored energy and stress work done. The partial differential equation as a function of three invariants has then been solved by Lie group methods. With geometric meanings of deformations, the general solution boils down to a particular three-term solution. The particular solution has been applied for several isotropic hyperelastic materials. For incompressible materials, vulcanized rubber containing 8% sulfur and Entec Enflex S4035A thermoplastic elastomer, three coefficients have been determined from uniaxial tension data and applied to predict the pure shear and equibiaxial tension modes. For a slightly compressible rubber material, the coefficients have also been extracted from the confined volumetric test data.

Distinct element modelling of fracture plan control in continuum and jointed rock mass in presplitting method of surface mining

Controlled blasting techniques are used to control overbreak and to aid in the stability of the remaining rock formation. Presplitting is one of the most common methods which is used in many open pit mining and surface blast design. The purpose of presplitting is to form a fracture plane across which the radial cracks from the production blast cannot travel. The purpose of this study is to investigate of effect of presplitting on the generation of a smooth wall in continuum and jointed rock mass. The 2D distinct element code was used to simulate the presplitting in a rock slope. The blast load history as a function of time was applied to the inner wall of each blasthole. Important parameters that were considered in the analysis were stress tensor and fracturing pattern. The blast loading magnitude and blasthole spacing and jointing pattern were found to be very significant in the final results.

Study on micronozzle flow and propulsion performance using DSMC and continuum methods

In this paper, both DSMC and Navier-Stokes computational approaches were applied to study micronozzle flow. The effects of inlet condition, wall boundary condition, Reynolds number, micronozzle geometry and Knudsen number on the micronozzle flow field and propulsion performance were studied in detail. It is found that within the Knudsen number range under consideration, both the methods work to predict flow characteristics inside micronozzles. The continuum method with slip boundary conditions has shown good performance in simulating the formation of a boundary layer inside the nozzle. However, in the nozzle exit lip region, the DSMC method is better due to gas rapid expansion. It is found that with decreasing the inlet pressure, the difference between the continuum model and DSMC results increases due to the enhanced rarefaction effect. The coefficient of discharge and the thrust efficiency increase with increasing the Reynolds number. Thrust is almost proportional to the nozzle width. With dimension enlarged, the nozzle performance becomes better while the rarefaction effects would be somewhat weakened.

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries （KdV） equation, and the soliton solution is derived.

Numerical Simulation of Transport Phenomena in Solidification of Multicomponent Ingot Using a Continuum Model

A continuum model proposed for dendrite solidification of multicomponent alloys, with any partial solid back diffusion, was used to numerically simulate the macroscopic solidification transport phenomena and macrosegregations in an upwards directionally solidified plain carbon steel ingot. The computational results of each macroscopic field of the physical variables involved in the solidification process at a middle solidification stage were presented.

A viscous continuum traffic flow model with consideration of the coupling effect for two-lane freeways

In this paper, the viscous continuum traffic flow model for a single lane is extended to the traffic flow for two-lane freeways. The proposed model is a higher-order continuum model considering the coupling and lane changing effects of the vehicles on two adjacent lanes. It results from integrating the Taylor series expansion of the viscous continuum traffic flow model proposed by Ge （2006 Physiea A 371 667） into the multi-lane model presented by Daganzo （1997 Transpn. Res. B 31 83）. Our proposed model may be used to describe non-anisotropic behaviour because of lane changing in multi-lane traffic. A linear stability analysis is given and the neutral stability condition is obtained. Also, issues related to lane changing, shock waves and rarefaction waves, local clustering and phase transition are investigated through a simulation experiment. The simulation results show that the proposed model is capable of explaining some particular traffic phenomena commonly observable in real world traffic flow.

Non-classical solutions of a continuum model for rock descriptions

The strain-gradient and non-Euclidean continuum theories are employed for construction of non- classical solutions of continuum models. The linear approximation of both models＇ results in identical structures in terms of their kinematic and stress characteristics. The solutions obtained in this study exhibit a critical behaviour with respect to the external loading parameter. The conclusions are obtained based on an investigation of the solution for the scalar curvature in the non-Euclidean continuum theory, The proposed analysis enables us to use different theoretical approaches for description of rock critical behaviour under different loading conditions.

Modeling continuous traffic flow with the average velocity effect of multiple vehicles ahead on gyroidal roads

In the future connected vehicle environment,the information of multiple vehicles ahead can be readily collected in real-time,such as the velocity or headway,which provides more opportunities for information exchange and cooperative control.Meanwhile,gyroidal roads are one of the fundamental road patterns prevalent in mountainous areas.To effectively control the system,it is therefore significant to explore the evolution mechanism of traffic flow on gyroidal roads under a connected vehicle environment.In this paper,we present a new continuum model with the average velocity of multiple vehicles ahead on gyroidal roads.The stability criterion and KdV-Burger equation are deduced via linear and nonlinear stability analysis,respectively.Solving the above KdV-Burger equation yields the density wave solution,which explores the formation and propagation property of traffic jams near the neutral stability curve.Simulation examples verify that the model can reproduce complex phenomena,such as shock waves and rarefaction waves.The analysis of the local cluster effect shows that the number of vehicles ahead and the radius information,and the slope information of gyroidal roads can exert a great influence on traffic jams.The effect of the first and second terms are positive,while the last term is negative.

Predicting effects of tunnel throttling of annular freeway vehicular flow by a continuum model

Fluid flow throttling is common in industrial and building services engineering.Similar tunnel throttling of vehicular flow is caused by the abrupt number reduction of roadway lane,as the tunnel has a lower lane number than in the roadway normal segment.To predict the effects of tunnel throttling of annular freeway vehicular flow,a three-lane continuum model is developed.LaneⅢof the tunnel is completely blocked due to the need of tunnel rehabilitation,etc.There exists mandatory net lane-changing rate from laneⅢto laneⅡjust upstream of the tunnel entrance,which is described by a model of random number generated through a golden section analysis.The net-changing rate between adjacent lanes is modeled using a lane-changing time expressed explicitly in algebraic form.This paper assumes that the annular freeway has a total length of 100 km,a two-lane tunnel of length 2 km with a speed limit of 80 km/h.The free flow speeds on lanesⅠ,ⅡandⅢare assumed to be 110,100 and 90 km/h respectively.Based on the three-lane continuum model,numerical simulations of vehicular flows on the annular freeway with such a tunnel are conducted with a reliable numerical method of 3rd-order accuracy.Numerical results reveal that the vehicular flow has a smaller threshold of traffic jam formation in comparison with the case without tunnel throttling.Vehicle fuel consumption can be estimated by interpolation with time averaged grid traffic speed and an assumed curve of vehicle performance.The vehicle fuel consumption is lane number dependent,distributes with initial density concavely,ranging from 5.56 to 8.00 L.Tunnel throttling leads to an earlier traffic jam formation in comparison with the case without tunnel throttling.

Hydrolysis Mechanism of the NAMI-A-type Antitumor Complex （HL）[trans-RuCl4L（dmso-S）] （L=1-methyl-l,2,4-triazole）

The hydrolysis process of Ru（III） complex （HL）[trans-RuC14L（dmso-S）] （L=l-methyl-l,2,4- triazole and dmso-S=S-dimethyl sulfoxide） （1）, a potential antitumor complex similar to the well-known antitumor agent （Him）[trans-RuC14 （dmso-S）（im）] （NAMI-A, im=imidazole）, was investigated using density functional theory combined with the conductor-like polarizable continuum model approach. Tile structural characteristics and the detailed energy profiles for the hydrolysis processes of this complex were obtained. For the first hydrolysis step, complex 1 has slightly higher barrier energies than the reported anticancer drug NAMI-A, and the result is in accordance with the experimental evidence indicating larger half-life for complex 1. For the second hydrolysis step, the formation of cis-diaqua species is thermodynamic preferred to that of trans isomers. In addition, on the basis of the analysis of electronic characteristics of species in the hydrolysis process, the trend in nucleophilic attack abilities of hydrolysis products by pertinent biomolecules is revealed and predicted.

Effects of Rotational Isomerism and Bond Length Alternation on Optical Spectra of FTC Chromophore in Solution

Rotational isomerism effects on the optical spectra of a push-pull nonlinear optical chro-mophore 2-dicyanomethylen-3-cyano-4-f2-[E-（4-N,N-di（2-acetoxyethyl）-amino）-phenylene-（3,4-dibutyl）-thien-5]-E-vinylg-5,5-dimethyl-2,5-dihydrofuran （FTC） in a few solvents have been studied using the time-dependent density functional theory in combination with the polarizable continuum model. It is shown that the maximum absorption peaks of the ro-tamers have difference of nearly 30 nm both in vacuum and in solutions. The population of the rotamers changes a lot in different solvents. Based on the geometries optimized by Hartree-Fock method, the Maxwell-Boltzmann averaged absorption has been calculated and the maximum absorption peak is in good agreement with experiment. It indicates that the bond length alternation can have an important effect on the optical spectra.

Density Functional Study on Structures and Relative Stability of Gd（H2O）n^3＋ （n=8,9）

Density functional theory calculations were performed to study the structures and relative stability of the gadolinium complexes, Gd（H2O）n^3＋ （n=8,9）, in vacuo and in aqueous solution. The polarizable continuum model with various radii for the solute cavity was used to study the relative stability in aqueous solution. The calculated molecular geometries for n=8 and 9 obtained in vacuo are consistent with those observed in experiments. It was found that while the nona-aqua complex is favored in the gas phase, in aqueous solution the octa-aqua conformation is preferred. This result, independent of the types of cavities employed, is in agreement with the experimental observation. The reliability of the present calculation was also addressed by comparing the calculated and experimental free energy of hydration, which revealed that the UA0, UAHF, and UAKS cavities are most appropriate when only the first solvation shell is treated explicitly.