A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes

The purpose of this study is to describe the axial buckling behavior of chiral single-walled carbon nanotubes (SWCNTs) using a combined continuum-atomistic approach. To this end, the nonlocal Flugge shell theory is implemented into which the nonlocal elasticity of Eringen incorporated. Molecular mechanics is used in conjunction with density functional theory (DFT) to precisely extract the effective in-plane and bending stiffnesses and Poisson's ratio used in the developed nonlocal Flugge shell model. The Rayleigh-Ritz procedure is employed to analytically solve the problem in the context of calculus of variation. The results generated from the present hybrid model are compared with those from molecular dynamics simulations as a benchmark of good accuracy and excellent agreement is achieved. The influences of small scale factor, commonly used boundary conditions and chirality on the critical buckling load are fully explored. It is indicated that the importance of the small length scale is affected by the type of boundary conditions considered.     

Periodic,incommensurate and chaotic states in a continuum statistical mechanics model

We study the thermodynamic behavior of a continuum system with competing periodicities. We show that in addition to commensurate and incommensurate phases, there exist configurations which are chaotic in nature and exhibit no long-range order. These phases are metastable and characterized by an order parameter with a continuous spectrum. By transforming the problem of determining the ground states of the system into a classical mechanics problem, we construct a two-dimensional area-preserving map which can be used to study the qualitative nature of the orbits. Our results might be of relevance to adsorbed monolayers on periodic substrates.     

Rotating frames and continuum mechanics: A relativistic appraisal

Rotation in a general relativistic framework is examined. This concept, combined with the appropriate mechanical work concept, is used to show how the Euclidean group of transformations, serving as an invariance requirement associated with the principle of objectivity, can lead to erroneous conclusions.     

A continuum model of a multilayer nanosheet

A continuum model for describing the bending and free vibrations of a crystalline graphite sheet consisting of graphene layers is proposed. Graphene is modeled by a two-dimensional layer having a finite rigidity under extension and bending. The interval between graphene layers through which their Van-der-Waals interaction occurs is modeled by a fictitious layer with relatively low rigidity. In the solution, formulas describing the bending of a multilayer sheet with alternating rigid and soft layers are used.     

Modelling of combined creep and cyclic plasticity in a model component undergoing ratchetting using continuum damage mechanics

A thermomechanical loading component facility has been built and developed for testing model copper slag tap components under ratchetting conditions due to combined damage resulting from creep and plasticity. Model copper slag tap components have been tested under dominant plasticity conditions at temperatures up to 300°C, and under balanced creep–cyclic plasticity conditions at temperatures up to 340°C. Overall ratchet deformations have been continuously measured to failure. A cycle jumping numerical technique has been used to analyse a multi-bar model of the slag tap using viscoplastic constitutive equations embodying softening due to combined cyclic plasticity and creep damage. Excellent predictions of lifetimes, ratchet strains and ratchet rates have been achieved, despite the details of the rupture processes not being faithfully modelled.     

Hydration characteristics of Ca2+ and Mg2+: a density functional theory,polarized continuum model and molecular dynamics investigation

In this work, density functional theory, Møller–Plesset second-order perturbation theory, and ab initio molecular dynamics (AIMD) were used to investigate hydrated characteristics of Mg²⁺ and Ca²⁺ as a function of coordination number in the first hydration shell (CN) and cluster size. It is generally accepted that the CNs of Mg²⁺ and Ca²⁺ are both six. Calculations show that the hydration of Mg²⁺ generally prefers six-coordinated structures, whereas the CN value of Ca²⁺ varies from 6 to 8 as the hydration proceeds. Moreover, the first hydration of Ca²⁺ is found to be more flexible than that of Mg²⁺, as indicated by the results of transition state calculations and AIMD simulations. In addition, the constraint of Mg²⁺ on the first hydration shell is obviously stronger than that of Ca²⁺, while the constraint on the inner hydration shells fades slightly faster for Mg²⁺ than Ca²⁺. It is also found that the charge transfer from central cation to water molecules is affected only by the first hydration shell for Mg²⁺, whereas by the first and second hydration shells for Ca²⁺. Based on hydration characteristics, approximatively saturated ion hydration shells for the hydration of Mg²⁺ and Ca²⁺ were proposed.     

Crystals and quasicrystals: A continuum model

We construct the first model of particles in the plane with completely symmetric, short range, two body interactions which has quasiperiodic, but no periodic, ground states.Supported in part by NSF Grant No. DMS-8501911     

Variational methods, multisymplectic geometry and continuum mechanics

This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.     

A molecular quantum mechanics approach to evaluate molecular properties in liquid phase using statistical mechanics

A method for the calculation of molecular properties in liquid phase is proposed. The theoretical development links in a natural fashion Molecular Quantum Mechanics and Statistical Mechanics. The method can be applied to molecular liquids in thermodynamical equilibrium without strong interactions between molecules. After the Introduction (in which some fundamental ideas are discussed) the model is developed in its simplest form, giving a numerical application of interest for IR spectroscopy in liquid phase. Immediatly, this preliminar version is generalized for pure liquids taking into account the polarization effects on all the molecules of the sample. A numerical example on carbon disulphide shows the convergence of this method and its sensitivity to the structural changes in the liquid. In the last section, some future improvements are suggested and several topics related with the study of molecular liquids are considered.     

Vibrational spectrum at a water surface: a hybrid quantum mechanics/molecular mechanics molecular dynamics approach

A hybrid quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) simulation is applied to the calculation of surface orientational structure and vibrational spectrum (second-order nonlinear susceptibility) at the vapor/water interface for the first time. The surface orientational structure of the QM water molecules is consistent with the previous MD studies, and the calculated susceptibility reproduces the experimentally reported one, supporting the previous results using the classical force field MD simulation. The present QM/MM MD simulation also demonstrates that the positive sign of the imaginary part of the second-order nonlinear susceptibility at the lower hydrogen bonding OH frequency region originates not from individual molecular orientational structure, but from cooperative electronic structure through the hydrogen bonding network.     

A continuum model for the jumping sandbox

We propose a continuum model for the motion of sand oscillating vertically. We derive a nonlinear parabolic equation with a nonlinearity corresponding to a bistable potential multiplied by a switching function. This is obtained by modifying the equation for a free surface and incorporating the thermodynamics of the interface.     

Magnetic color center on a continuum model

A simple model is proposed for a trapping center in magnetic semiconductors. The model is a hydrogen-like atom immersed in a continuous paramagnetic medium (we confine ourselves to the paramagnetic region of magnetic semiconductors). The 1-s and 2-p levels are calculated for each fixed spin configuration, and are plotted in a diagram against an appropriate single variable which characterizes the configuration.     

A multiscale molecular dynamics approach to contact mechanics

The friction and adhesion between elastic bodies are strongly influenced by the roughness of the surfaces in contact. Here we develop a multiscale molecular dynamics approach to contact mechanics, which can be used also when the surfaces have roughness on many different length-scales, e.g., for self affine fractal surfaces. As an illustration we consider the contact between randomly rough surfaces, and show that the contact area varies linearly with the load for small load. We also analyze the contact morphology and the pressure distribution at different magnification, both with and without adhesion. The calculations are compared with analytical contact mechanics models based on continuum mechanics.     

A staggered six-vertex model with non-compact continuum limit

The antiferromagnetic critical point of the Potts model on the square lattice was identified by Baxter [R.J. Baxter, Proc. R. Soc. London A 383 (1982) 43] as a staggered integrable six-vertex model. In this work, we investigate the integrable structure of this model. It enables us to derive some new properties, such as the Hamiltonian limit of the model, an equivalent vertex model, and the structure resulting from the Z₂$Z_2$ symmetry. Using this material, we discuss the low-energy spectrum, and relate it to geometrical excitations. We also compute the critical exponents by solving the Bethe equations for a large lattice width N  . The results confirm that the low-energy spectrum is a collection of continua with typical exponent gaps of order (logN)⁻²${(\log N)}^{\mathrm{-2}}$.     

Diffusion-limited aggregation: A continuum mean field model

Mean field theory is used as a basis for a new approach to analyzing fractal pattern formation by diffusion-limited aggregation. A coarse time scale is introduced to take into account the discrete nature of DLA clusters. A system of equations is derived and solved numerically to determine the fractal dimension and density of a cluster as a function of distance from its center. The results obtained are in good agreement with direct numerical simulations.     

A continuum model for dephasing in mesoscopic systems

A model of continuous dephasing for one-dimensional mesoscopic conductors is proposed. The model is based on Büttiker's fictitious-probe model which is extended by attaching an infinite number of probes uniformly to various points on the conductor. The associated scattering problem is then solved. When the continuum limit is taken, it becomes possible to describe the dephasing as taking place everywhere. The dephasing rate enters into the model as an adjustable parameter. The effect of dephasing on the conductance for a double-barrier system is also studied numerically.