A multi-scale atomistic-continuum modelling of crack propagation in a two-dimensional macroscopic plate

A novel multi-scale seamless model of brittle-crack propagation is proposed and applied to the simulation of fracture growth in a two-dimensional Ag plate with macroscopic dimensions. The model represents the crack propagation at the macroscopic scale as the drift-diffusion motion of the crack tip alone. The diffusive motion is associated with the crack-tip coordinates in the position space, and reflects the oscillations observed in the crack velocity following its critical value. The model couples the crack dynamics at the macroscales and nanoscales via an intermediate mesoscale continuum. The finite-element method is employed to make the transition from the macroscale to the nanoscale by computing the continuum-based displacements of the atoms at the boundary of an atomic lattice embedded within the plate and surrounding the tip. Molecular dynamics (MD) simulation then drives the crack tip forward, producing the tip critical velocity and its diffusion constant. These are then used in the Ito stochastic calculus to make the reverse transition from the nanoscale back to the macroscale. The MD-level modelling is based on the use of a many-body potential. The model successfully reproduces the crack-velocity oscillations, roughening transitions of the crack surfaces, as well as the macroscopic crack trajectory. The implications for a 3-D modelling are discussed.

Multiscale numerical modelling of crack propagation in two-dimensional metal plate

A novel multiscale model of brittle crack propagation in an Ag plate with macroscopic dimensions has been developed. The model represents crack propagation as stochastic drift-diffusion motion of the crack tip atom through the material, and couples the dynamics across three different length scales. It integrates the nanomechanics of bond rupture at the crack tip with the displacement and stress field equations of continuum based fracture theories. The finite element method is employed to obtain the continuum based displacement and stress fields over the macroscopic plate, and these are then used to drive the crack tip forward at the atomic level using the molecular dynamics simulation method based on many-body interatomic potentials. The linkage from the nanoscopic scale back to the macroscopic scale is established via the Ito stochastic calculus, the stochastic differential equation of which advances the tip to a new position on the macroscopic scale using the crack velocity and diffusion constant obtained on the nanoscale. Well known crack characteristics, such as the roughening transitions of the crack surfaces, crack velocity oscillations, as well as the macroscopic crack trajectories, are obtained.

A multi-scale numerical modelling of crack propagation in a 2D metallic plate

A new multi-scale model of brittle fracture growth in an Ag plate with macroscopic dimensions is proposed in which the crack propagation is identified with the stochastic drift-diffusion motion of the crack-tip atom through the material. The model couples molecular dynamics simulations, based on many-body interatomic potentials, with the continuum-based theories of fracture mechanics. The Ito stochastic differential equation is used to advance the tip position on a macroscopic scale before each nano-scale simulation is performed. Well-known crack characteristics, such as the roughening transitions of the crack surfaces, as well as the macroscopic crack trajectories are obtained.

A mathematical description of the acoustic coupling of the mass/spring model

This paper describes hybrid mathematical model which couples the mechanics of the mass/spring model to the acoustic wave propagation model for use in generating the acoustic signal emitted by complex structures of paper fibres under strain. A discussion of the coupling method is presented including remarks on the errors encountered intrinsic to the discretisation scheme. The numerical results of a vibrating rubber band and a vibrating paper fibre are compared to their experimental counterparts. The fundamental frequencies of the acoustic signals are compared showing a close agreement between the experimental and numerical results

Models for magnetohydrodynamics of aluminium electrolysis cells

The electric current and the associated magnetic field in aluminium electrolysis cells create effects limiting the cell productivity and possibly cause instabilities: surface waving, ‘anode effects’, erosion of pot lining, feed material sedimentation, etc. The instructive analysis is presented via a step by step inclusion of different physical coupling factors affecting the magnetic field, electric current, velocity and wave development in the electrolysis cells. The full time dependent model couples the nonlinear turbulent fluid dynamics and the extended electromagnetic field in the cell, and the whole bus bar circuit with the ferromagnetic effects. Animated examples for the high amperage cells are presented. The theory and numerical model of the electrolysis cell is extended to the cases of variable cell bottom of aluminium layer and the variable thickness of the electrolyte due to the anode non-uniform burn-out process and the presence of the anode channels. The problem of the channel importance is well known Moreau-Evans model) for the stationary interface and the velocity field, and was validated against measurements in commercial cells, particularly with the recently published ‘benchmark’ test for the MHD models of aluminium cells [1]. The presence of electrolyte channels requires also to reconsider the previous magnetohydrodynamic instability theories and the dynamic wave development models. The results indicate the importance of a ‘sloshing’ parametrically excited MHD wave development in the aluminium production cells.

Cyclic voltammetry and spectroelectrochemical study of nickel and cobalt diphenyltetraazaannulene complexes

The absorption spectra. cyclic voltammetry and spectroelectrochemistry of [Ni(II)DPTAA] and [Co(II)DPTAA] (DPTAA = 6,13-diphenyldibenzo[b,i][1,4,8,11] tetraaza[14]annulene) complexes in DMF are reported in detail. The ligand oxidation is observed for [Ni(II)DPTAA] at +0.70 V vs. SCE whereas Ni2(+/+) occurs at - 1.60 V. For [Co(II)DPTAA], a ligand oxidation redox couple is seen at +0.56 V while the Co2+/+ and Co2+/3+ redox couples appear at -1.21 and +0.24 V, respectively. All observed redox couples are assigned to reversible one-electron processes on account of peak separations and scan-rate dependency. These processes were further investigated by spectroelectrochemistry for [Co(II)DPTAA]. For [Co(II)DPTAA], axial ligation of pyridine was found to shift the Co2+/3+ redox couple more negative. while the ligand oxidation was shifted to more positive potentials. From a spectrophotometric titration of [Co(II)DPTAA] with pyridine an equilibrium constant, K-f, was determined for the binding of pyridine to [Co(II)DPTAA]. This was found to be 10.2 dm(3) mol(-1), slightly lower than that of [Co(II)TAA], indicating the influence of the phenyl groups. From this value and shifts in the Co2+/3+ redox couple upon ligation, an equilibrium constant for the binding of pyridine to [Co(III)DPTAA], K'(f), was found to be 5.06 x 10(6) dm(3) mol(-1). (c) 2007 Elsevier B.V. All rights reserved.