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.

A Continuum Model for Heavy-electron Metal

The exact eigenstates of the Hamiltonian of a continuum model for heavy-electron metal are constructed by using the Bethe ansatz. The Bethe ansatz equations are obtained from the periodic boundary conditions. The results show that this system is also completely integrable.

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.

Motivation in Foreign Language Learning: Applying the Time Continuum Model of Motivation in FLL

The research is: by using Wdolkowski's Time Continuum Model throughout a lesson plan enables the teacher to increase students'motivation and help them move closer to success in a learning environment. This research supports the theory that instruction is a network of interactions between the teacher and learner that promotes a successful learning experience. It identifies a three-part learning sequence-a beginning, middle and an end. Each part has two of six key motivational factors that when applied correctly by the teacher will maximize the success and continued motivation of the learner.

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 remains the area of greatest uncertainty. Recent studies based on laboratory testing have led to the development of a number of models for brittle rock dilation, which account for both the plastic shear strain and confining stress dependencies of this phenomenon. Although these models are useful in providing an improved understanding of how dilatancy evolves during a compression test, there has been relatively little work performed examining their validity for modeling brittle rock yield in situ. In this study, different constitutive models for rock dilation are reviewed and then tested, in the context of a number of case studies, using a continuum finite-difference approach(FLAC). The uncertainty associated with the modeling of brittle fracture localization is addressed, and the overall ability of mobilized dilation models to replicate in situ deformation measurements and yield patterns is evaluated.

Static analysis of ultra-thin beams based on a semi-continuum model

A linear semi-continuum model with discrete atomic layers in the thickness direction was developed to investigate the bending behaviors of ultra-thin beams with nanoscale thickness.The theoretical results show that the deflection of an ultra-thin beam may be enhanced or reduced due to different relaxation coefficients.If the relaxation coefficient is greater/less than one,the deflection of micro/nano-scale structures is enhanced/reduced in comparison with macro-scale structures.So,two opposite types of size-dependent behaviors are observed and they are mainly caused by the relaxation coefficients.Comparisons with the classical continuum model,exact nonlocal stress model and finite element model (FEM) verify the validity of the present semi-continuum model.In particular,an explanation is proposed in the debate whether the bending stiffness of a micro/nano-scale beam should be greater or weaker as compared with the macro-scale structures.The characteristics of bending stiffness are proved to be associated with the relaxation coefficients.

Seepage simulation of high concrete-faced rockfill dams based on generalized equivalent continuum model

This research focused on the three-dimensional(3 D) seepage field simulation of a high concrete-faced rockfill dam(CFRD) under complex hydraulic conditions. A generalized equivalent continuum model of fractured rock mass was used for equivalent continuous seepage field analysis based on the improved node virtual flow method. Using a high CFRD as an example, the generalized equivalent continuum range was determined, and a finite element model was established based on the terrain and geological conditions, as well as structural face characteristics of the dam area. The equivalent seepage coefficients of different material zones or positions in the dam foundation were calculated with the Snow model or inverse analysis. Then, the 3 D seepage field in the dam area was calculated under the normal water storage conditions, and the corresponding water head distribution, seepage flow, seepage gradient, and seepage characteristics in the dam area were analyzed. The results show that the generalized equivalent continuum model can effectively simulate overall seepage patterns of the CFRD under complex hydraulic conditions and provide a reference for seepage analysis of similar CFRDs.

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

用原子论底的有限变丑的壳理论，我们经分解调查在单个墙的碳 nanotubes 在轴的变丑和扭转之间联合。我们发现 axial-strain-induced 扭转(ASIT ) 反应仅仅限于 chiral nanotubes。这回答被碳 nanotubes 的 chiralities 和半径影响。我们的结果类似于在文学报导的分子的动态模拟的。

A review on the application of modified continuum models in modeling and simulation of nanostructures

Analysis of the mechanical behavior of nanostructures has been very challenging. Surface energy and nonlocal elasticity of materials have been incorporated into the traditional continuum analysis to create modified continuum mechanics models. This paper reviews recent advancements in the applications of such modified continuum models in nanostructures such as nanotubes, nanowires, nanobeams,graphenes, and nanoplates. A variety of models for these nanostructures under static and dynamic loadings are mentioned and reviewed. Applications of surface energy and nonlocal elasticity in analysis of piezoelectric nanomaterials are also mentioned. This paper provides a comprehensive introduction of the development of this area and inspires further applications of modified continuum models in modeling nanomaterials and nanostructures.

Analytical Polarizable Continuum Model for Wavelets on NURBS Patches

This article concerns the application of wavelet techniques on molecular surfaces constituted of four-sided patches. The Polarizable Continuum Model, which is governed by the Poisson-Boltzmann equation, is treated by means of boundary integral equations. The media inside and outside the molecular surface consist respectively of the solute and the solvent. For a given electrically charged molecule, the principal unknown is the electrostatic solvation energy when the permittivity is specified. The wavelet basis functions are constructed on the unit square which are subsequently mapped onto the patches that are assumed to be isotropically shaped and to admit similar surface areas. The initial transmission problem is recast as an integral equation in term of both the single and the double layers. Domain decomposition preconditioner serves as acceleration of the linear solver of the single layer which is badly conditioned.

Non-classical solutions of a continuum model for rock descriptions

The strain-gradient and non-Euclidean continuum theories are employed for construction of nonclassical 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.

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.

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.

Modes of grain growth and mechanism of dislocation reaction under applied biaxial strain:Atomistic and continuum modeling

The phase field crystal method and Continuum Modeling are applied to study the cooperative dislocation motion of the grain boundary(GB)migration,the manner of the nucleation of the grain and of the grain growth in two dimensions(2 D)under the deviatoric deformation at high temperature.Three types of the nucleation modes of new finding are observed by the phase field crystal simulation:The first mode of the nucleation is generated by the GB splitting into two sub-GBs;the second mode is of the reaction of the sub-GB dislocations,such as,the generation and annihilation of a pair of partial Frank sessile dislocation in 2 D.The process can be considered as the nucleation of dynamic recrystallization;the third mode is caused by two oncoming rows of the dislocations of these sub-GBs,crossing and passing each other to form new gap which is the nucleation place of the new deformed grain.The research is shown that due to the nucleation of different modes the mechanism of the grain growth by means of the sub-GB migration is different,and therefore,the grain growth rates are also different.Under the deviatoric deformation of the applied biaxial strain,the grain growth is faster than that of the grain growth without external applied stress.It is observed that the cooperative dislocation motion of the GB migration under the deviatoric deformation accompanies with local plastic flow and the state of the stress of the system changes sharply.When the system is in the process of recrystallized grain growth,the system energy is in an unstable state due to the release of the strain energy to cause that the reverse movement of the plastic flow occurs.The area growth of the deformed grain is approximately proportional to the strain square and also to the time square.The rule of the time square of the deformed grain growth can also be deduced by establishing the continuum dynamic equation of the biaxial strain-driven migration of the GB.The copper metal is taken as an example of the calculation,and the obtained result is a good agreement with that of the experiment.

Numerical Simulation of Macrosegregation During Steel Ingot Solidification Using Continuum Model

A continuum model is adopted to study the macrosegregation phenomena during solidification of large steel ingots.Evolution of temperature,melt velocity,and compositional concentration field during a 22 t steel ingot solidification are illustrated by using the finite volume method.Numerical results of temperature distribution are validated by experiments.The influence of local permeability relates to the friction that the melt experienced in mushy region is investigated.It is shown that the continuum model is able to predict the temperature field,and the variation of permeability obviously affects the melt flowing behavior and the final compositional distribution.

A new continuum model with driver's continuous sensory memory and preceding vehicle's taillight

Car taillights are ubiquitous during the deceleration process in real traffic,while drivers have a memory for historical information.The collective effect may greatly affect driving behavior and traffic flow performance.In this paper,we propose a continuum model with the driver's memory time and the preceding vehicle's taillight.To better reflect reality,the continuous driving process is also considered.To this end,we first develop a unique version of a car-following model.By converting micro variables into macro variables with a macro conversion method,the micro carfollowing model is transformed into a new continuum model.Based on a linear stability analysis,the stability conditions of the new continuum model are obtained.We proceed to deduce the modified KdV-Burgers equation of the model in a nonlinear stability analysis,where the solution can be used to describe the propagation and evolution characteristics of the density wave near the neutral stability curve.The results show that memory time has a negative impact on the stability of traffic flow,whereas the provision of the preceding vehicle's taillight contributes to mitigating traffic congestion and reducing energy consumption.

Hydrophobic Effect in a Continuum Model of the Lipid Bilayer

We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules.This paper extends previous work of Blom and Peletier[European J.Appl.Math.,15(2004),pp.487-508]in the following ways.(a)It formulates a more general model of the hydrophobic effect to facilitate connections with microscale simulations and first-principles analysis.(b)It clarifies the meaning and role of the model parameters.(c)It outlines a method for determining parameter values so that physically-realistic bilayer density profiles can be obtained,for example for use in macroscale simulations.Points(a)-(c)suggest that the model has potential to robustly connect some micro-and macroscale levels of multiscale blood flow simulations.The mathematical modelling in point(a)is based upon a consideration of the underlying physics of inter-molecular forces.The governing equations thus obtained are minimized by gradient flows via a novel numerical approach;this enables point(b).The numerical results are shown to behave physically in terms of the effect of background concentration,in contrast to the earlier model which is shown here to not display the expected behaviour.A“short-tail”approximation of the lipid molecules also gives an analytical tool which yields critical values of some parameters under certain conditions.Point(c)involves the first quantitative comparison of the numerical data with physical experimental results.

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.

Molecular dynamics simulation and continuum modelling of granular surface flow in rotating drums

Numerical simulations of granular flows in rotating drums operated at medium to high rates (Fr=0.1― 0.2) have been carried out by using a Molecular Dynamics (MD) algorithm that incorporates inelastic particle interactions, sliding friction and rolling friction. The results indicate that the behavior of granular flow in rotating drums can be classified into two distinct zones: a shear active layer at the bed surface and a quasi-static plug flow region adjacent to the wall. The residence time of a tracer particle in the active layer is approximately a third or a half of that in the plug flow region. The thickness of the active layer at mid-chord is about 0.57―0.61 times that of the plug flow region. It is found that all cases simulated in this work are in the rolling-cascading intermediate regime instead of the pure rolling re-gime. The simulated tangential velocity at the mid-chord is also compared with experimental results reported in the literature and good agreement has been obtained. Based on the MD simulations and experimental results, a continuum approach has also been developed. It is shown that the behavior of granular solids in the plug flow region experiences plastic deformation along the radial direction from the wall with the velocity profiles well described by an exponential function, whereas the active layer velocity follows a simple expression for the Couette shear flow. Discussion has also been made on the granular temperature and concentration profiles.