Analysis of shear bands in slow granular flows using a frictional Cosserat model

The slow flow of granular materials is often marked by the existence of narrow shear layers, adjacent to large regions that suffer little or no deformation. This behaviour, in the regime where shear stress is generated primarily by the frictional interactions between grains, has so far eluded theoretical description. In this paper, we present a rigid-plastic frictional Cosserat model that captures thin shear layers by incorporating a microscopic length scale. We treat the granular medium as a Cosserat continuum, which allows the existence of localised couple stresses and, therefore, the possibility of an asymmetric stress tensor. In addition, the local rotation is an independent field variable and is not necessarily equal to the vorticity. The angular momentum balance, which is implicitly satisfied for a classical continuum, must now be solved in conjunction with the linear momentum balances. We extend the critical state model, used in soil plasticity, for a Cosserat continuum and obtain predictions for flow in plane and cylindrical Couette devices. The velocity profile predicted by our model is in qualitative agreement with available experimental data. In addition, our model can predict scaling laws for the shear layer thickness as a function of the Couette gap, which must be verified in future experiments. Most significantly, our model can determine the velocity field in viscometric flows, which classical plasticity-based model cannot.

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

In this article, the Eringen's nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

A Revisit to Capture the Entire Behavior of Ultrasonic Wave Dispersion Characteristics of Single Walled Carbon Nanotubes Based on Nonlocal Elasticity Theory and Flugge Shell Model

In this paper, an ultrasonic wave propagation analysis in single-walled carbon nanotube (SWCNT) is re-studied using nonlocal elasticity theory, to capture the whole behaviour. The SWCNT is modeled using Flugge's shell theory, with the wall having axial, circumferential and radial degrees of freedom and also including small scale effects. Nonlocal governing equations for this system are derived and wave propagation analysis is also carried out. The revisited nonlocal elasticity calculation shows that the wavenumber tends to infinite at certain frequencies and the corresponding wave velocity tends to zero at those frequencies indicating localization and stationary behavior. This frequency is termed as escape frequency. This behavior is observed only for axial and radial waves in SWCNT. It has been shown that the circumferential waves will propagate dispersively at higher frequencies in nonlocality. The magnitudes of wave velocities of circumferential waves are smaller in nonlocal elasticity as compared to local elasticity. We also show that the explicit expressions of cut-off frequency depend on the nonlocal scaling parameter and the axial wavenumber. The effect of axial wavenumber on the ultrasonic wave behavior in SWCNTs is also discussed. The present results are compared with the corresponding results (for first mode) obtained from ab initio and 3-D elastodynamic continuum models. The acoustic phonon dispersion relation predicted by the present model is in good agreement with that obtained from literature. The results are new and can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Ultrasonic Wave Dispersion Characteristics of Fluid-Filled Single-Walled Carbon Nanotubes Based on Nonclassical Shell Model

In this paper, ultrasonic wave propagation analysis in fluid filled single-walled carbon nanotube (SWCNT) is studied using nonlocal elasticity theory. The SWCNT is modeled using Flugge's shell theory, with the wall having axial, circumferential and radial degrees of freedom and also including small scale effects. The fluid inside the SWCNT is assumed as water. Nonlocal governing equations for this system are derived and wave propagation analysis is also carried out. The presence of fluid in SWCNT alters the ultrasonic wave dispersion behavior. The wavenumber and wave velocity are smaller in presence of fluid as compared to the empty SWCNT. The nonlocal elasticity calculation shows that the wavenumber tends to reach the continuum limit at certain frequencies and the corresponding wave velocity tends to zero at those frequencies indicating localization and stationary behavior. It has been shown that the circumferential. waves will propagate non-dispersively at higher frequencies in nonlocality. The magnitudes of wave velocities of circumferential waves are smaller in nonlocal elasticity as compared to local elasticity. We also show that the cut-off frequency depend on the nonlocal scaling parameter and also on the density of the fluid inside the SWCNT, and the axial wavenumber, as the fluid becomes denser the cut-off frequency decreases. The effect of axial wavenumber on the ultrasonic wave behavior in SWCNTS filled with water is also discussed.

Continuum theory of edge states of topological insulators: variational principle and boundary conditions

We develop a continuum theory to model low energy excitations of a generic four-band time reversal invariant electronic system with boundaries. We propose a variational energy functional for the wavefunctions which allows us to derive natural boundary conditions valid for such systems. Our formulation is particularly suited for developing a continuum theory of the protected edge/surface excitations of topological insulators both in two and three dimensions. By a detailed comparison of our analytical formulation with tight binding calculations of ribbons of topological insulators modelled by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian, we show that the continuum theory with a natural boundary condition provides an appropriate description of the low energy physics.

Nonlocal continuum mechanics based ultrasonic flexural wave dispersion characteristics of a monolayer graphene embedded in polymer matrix

Wave propagation in graphene sheet embedded in elastic medium (polymer matrix) has been a topic of great interest in nanomechanics of graphene sheets, where the equivalent continuum models are widely used. In this manuscript, we examined this issue by incorporating the nonlocal theory into the classical plate model. The influence of the nonlocal scale effects has been investigated in detail. The results are qualitatively different from those obtained based on the local/classical plate theory and thus, are important for the development of monolayer graphene-based nanodevices. In the present work, the graphene sheet is modeled as an isotropic plate of one-atom thick. The chemical bonds are assumed to be formed between the graphene sheet and the elastic medium. The polymer matrix is described by a Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation of the surrounding elastic medium. When the shear effects are neglected, the model reduces to Winkler foundation model. The normal pressure or Winkler elastic foundation parameter is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs where the foundation modulus is assumed equivalent to stiffness of the springs. For this model, the nonlocal governing differential equations of motion are derived from the minimization of the total potential energy of the entire system. An ultrasonic type of flexural wave propagation model is also derived and the results of the wave dispersion analysis are shown for both local and nonlocal elasticity calculations. From this analysis we show that the elastic matrix highly affects the flexural wave mode and it rapidly increases the frequency band gap of flexural mode. The flexural wavenumbers obtained from nonlocal elasticity calculations are higher than the local elasticity calculations. The corresponding wave group speeds are smaller in nonlocal calculation as compared to local elasticity calculation. The effect of y-directional wavenumber (eta(q)) on the spectrum and dispersion relations of the graphene embedded in polymer matrix is also observed. We also show that the cut-off frequencies of flexural wave mode depends not only on the y-direction wavenumber but also on nonlocal scaling parameter (e(0)a). The effect of eta(q) and e(0)a on the cut-off frequency variation is also captured for the cases of with and without elastic matrix effect. For a given nanostructure, nonlocal small scale coefficient can be obtained by matching the results from molecular dynamics (MD) simulations and the nonlocal elasticity calculations. At that value of the nonlocal scale coefficient, the waves will propagate in the nanostructure at that cut-off frequency. In the present paper, different values of e(0)a are used. One can get the exact e(0)a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this article. (c) 2012 Elsevier Ltd. All rights reserved.

Reaction dynamics under confinement: an exact path integral treatment of a two-stage model of stochastic gene expression

Gene expression in living systems is inherently stochastic, and tends to produce varying numbers of proteins over repeated cycles of transcription and translation. In this paper, an expression is derived for the steady-state protein number distribution starting from a two-stage kinetic model of the gene expression process involving p proteins and r mRNAs. The derivation is based on an exact path integral evaluation of the joint distribution, P(p, r, t), of p and r at time t, which can be expressed in terms of the coupled Langevin equations for p and r that represent the two-stage model in continuum form. The steady-state distribution of p alone, P(p), is obtained from P(p, r, t) (a bivariate Gaussian) by integrating out the r degrees of freedom and taking the limit t -> infinity. P(p) is found to be proportional to the product of a Gaussian and a complementary error function. It provides a generally satisfactory fit to simulation data on the same two-stage process when the translational efficiency (a measure of intrinsic noise levels in the system) is relatively low; it is less successful as a model of the data when the translational efficiency (and noise levels) are high.

A spectral multiscale method for wave propagation analysis: Atomistic-continuum coupled simulation

In this paper, we present a new multiscale method which is capable of coupling atomistic and continuum domains for high frequency wave propagation analysis. The problem of non-physical wave reflection, which occurs due to the change in system description across the interface between two scales, can be satisfactorily overcome by the proposed method. We propose an efficient spectral domain decomposition of the total fine scale displacement along with a potent macroscale equation in the Laplace domain to eliminate the spurious interfacial reflection. We use Laplace transform based spectral finite element method to model the macroscale, which provides the optimum approximations for required dynamic responses of the outer atoms of the simulated microscale region very accurately. This new method shows excellent agreement between the proposed multiscale model and the full molecular dynamics (MD) results. Numerical experiments of wave propagation in a 1D harmonic lattice, a 1D lattice with Lennard-Jones potential, a 2D square Bravais lattice, and a 2D triangular lattice with microcrack demonstrate the accuracy and the robustness of the method. In addition, under certain conditions, this method can simulate complex dynamics of crystalline solids involving different spatial and/or temporal scales with sufficient accuracy and efficiency. (C) 2014 Elsevier B.V. All rights reserved.

Optimal Linear Glauber Model

Contrary to the actual nonlinear Glauber model, the linear Glauber model (LGM) is exactly solvable, although the detailed balance condition is not generally satisfied. This motivates us to address the issue of writing the transition rate () in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least. The advantage of this work is that, by studying the LGM analytically, we will be able to anticipate how the kinetic properties of an arbitrary Ising system depend on the temperature and the coupling constants. The analytical expressions for the optimal values of the parameters involved in the linear are obtained using a simple Moore-Penrose pseudoinverse matrix. This approach is quite general, in principle applicable to any system and can reproduce the exact results for one dimensional Ising system. In the continuum limit, we get a linear time-dependent Ginzburg-Landau equation from the Glauber's microscopic model of non-conservative dynamics. We analyze the critical and dynamic properties of the model, and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach. We will also show in this paper that the effect of magnetic field can easily be studied within our approach; in particular, we show that the inverse of relaxation time changes quadratically with (weak) magnetic field and that the fluctuation-dissipation theorem is valid for our model.

Notch sensitivity in nanoscale metallic glass specimens: Insights from continuum simulations

Recent experiments have shown that nano-sized metallic glass (MG) specimens subjected to tensile loading exhibit increased ductility and work hardening. Failure occurs by necking as opposed to shear banding which is seen in bulk samples. Also, the necking is generally observed at shallow notches present on the specimen surface. In this work, continuum finite element analysis of tensile loading of nano-sized notched MG specimens is conducted using a thermodynamically consistent non-local plasticity model to clearly understand the deformation behavior from a mechanics perspective. It is found that plastic zone size in front of the notch attains a saturation level at the stage when a dominant shear band forms extending across the specimen. This size scales with an intrinsic material length associated with the interaction stress between flow defects. A transition in deformation behavior from quasi-brittle to ductile becomes possible when this critical plastic zone size is larger than the uncracked ligament length. These observations corroborate with atomistic simulations and experimental results. (C) 2015 Elsevier Ltd. All rights reserved.

Continuum in the X-Z-Y Weak Bonds: Z= Main Group Elements

The Continuum in the variation of the X-Z bond length change from blue-shifting to red-shifting through zero-shifting in the X-Z---Y complex is inevitable. This has been analyzed by ab-initio molecular orbital calculations using Z= Hydrogen, Halogens, Chalcogens, and Pnicogens as prototypical examples. Our analysis revealed that, the competition between negative hyperconjugation within the donor (X-Z) molecule and Charge Transfer (CT) from the acceptor (Y) molecule is the primary reason for the X-Z bond length change. Here, we report that, the proper tuning of X-and Y-group for a particular Z-can change the blue-shifting nature of X-Z bond to zero-shifting and further to red-shifting. This observation led to the proposal of a continuum in the variation of the X-Z bond length during the formation of X-Z---Y complex. The varying number of orbitals and electrons available around the Z-atom differentiates various classes of weak interactions and leads to interactions dramatically different from the H-Bond. Our explanations based on the model of anti-bonding orbitals can be transferred from one class of weak interactions to another. We further take the idea of continuum to the nature of chemical bonding in general. (C) 2015 Wiley Periodicals, Inc.

Peierls-Nabarro model of interfacial misfit dislocation: An analytic solution

We propose a method to treat the interfacial misfit dislocation array following the original Peierls-Nabarro's ideas. A simple and exact analytic solution is derived in the extended Peierls-Nabarro's model, and this solution reflects the core structure and the energy of misfit dislocation, which depend on misfit and bond strength. We also find that only with beta < 0.2 the structure of interface can be represented by an array of singular Volterra dislocations, which conforms to those of atomic simulation. Interfacial energy and adhesive work can be estimated by inputting ab initio calculation data into the model, and this shows the method can provide a correlation between the ab initio calculations and elastic continuum theory.

3D Anisotropic Elastoplastic-Damage Model and its Application in Simulating the Behavior of Rock Materials

A 3D anisotropic elastoplastic-damage model was presented based on continuum damage mechanics theory. In this model, the tensor decomposition technique is employed. Combined with the plastic yield rule and damage evolution, the stress tensor in incremental format is obtained. The derivate eigenmodes in the proposed model are assumed to be related with the uniaxial behavior of the rock material. Each eigenmode has a corresponding damage variable due to the fact that damage is a function of the magnitude of the eigenstrain. Within an eigenmodes, different damage evolution can be used for tensile and compressive loadings. This model was also developed into finite element code in explicit format, and the code was integrated into the well-known computational environment ABAQUS using the ABAQUS/Explicit Solver. Numerical simulation of an uniaxial compressive test for a rock sample is used to examine the performance of the proposed model, and the progressive failure process of the rock sample is unveiled.

Correlative reference model and molecular dynamics simulation of dislocation emission process

A correlative reference model for computer molecular dynamics simulations is proposed. Based on this model, a flexible displacement boundary scheme is introduced and the dislocations emitted from a crack tip can continuously pass through the border of the inner discrete atomic region and pile up at the outer continuum region. The effect of the emitted dislocations within the plastic zone on the inner atomistic region can be clearly demonstrated. The simulations for a molybdinum crystal show that a full dislocation in a bcc crystal is dissociated into three partial dislocations and interaction between the crack and the emitted dislocations results in gradual decrease of the local stress intensity factor.

Study Of The Damage-Induced Anisotropy Of Quasi-Brittle Materials Using The Component Assembling Model

Damage-induced anisotropy of quasi-brittle materials is investigated using component assembling model in this study. Damage-induced anisotropy is one significant character of quasi-brittle materials coupled with nonlinearity and strain softening. Formulation of such complicated phenomena is a difficult problem till now. The present model is based on the component assembling concept, where constitutive equations of materials are formed by means of assembling two kinds of components' response functions. These two kinds of components, orientational and volumetric ones, are abstracted based on pair-functional potentials and the Cauchy - Born rule. Moreover, macroscopic damage of quasi-brittle materials can be reflected by stiffness changing of orientational components, which represent grouped atomic bonds along discrete directions. Simultaneously, anisotropic characters are captured by the naturally directional property of the orientational component. Initial damage surface in the axial-shear stress space is calculated and analyzed. Furthermore, the anisotropic quasi-brittle damage behaviors of concrete under uniaxial, proportional, and nonproportional combined loading are analyzed to elucidate the utility and limitations of the present damage model. The numerical results show good agreement with the experimental data and predicted results of the classical anisotropic damage models.