3D analytical mathematical models of random star-shape particles via a combination of X-ray computed microtomography and spherical harmonic analysis

To compute any physical quantity for a random particle, one needs to know the mathematical shape of the particle. For regular particles like spheres and ellipsoids, the mathematics are straightforward. For random particles, with realistic shapes, mathematically characterizing the shape had not been generally done. But since about the year 2002, a method has been developed that combines X-ray computed tomography and spherical harmonic analysis to give analytical, differentiable mathematical functions for the three-dimensional shape of star-shape particles, which are a wide class of particles covering most industrial particles of interest, ranging from micrometer scale to millimeter scale particles. This review article describes how this is done, in some detail, and then gives examples of applications of this method, including a contact function that is suitable for these random shape particles. The purpose of this article is to make these ideas widely available for the general powder researcher who knows that particle shape is important to his/her applications, and especially for those researchers who are just starting out in their particle science and technology careers.     

On the motion of a spherical bubble deforming near a plane wall

Equations of motion are derived for an expanding spherical bubble in potential flow near a plane wall using the Lagrange-Thomson method and an extended Rayleigh dissipation function to account for the drag. This method is shown to yield the same acceleration of the bubble center as that obtained using the Lagally theorem. An extended Rayleigh-Plesset equation is derived to describe deformation in the vicinity of a plane wall, and expressions relating the drag force to the distance from the wall and the bubble growth rate are derived. The solution method for the velocity potential can also be applied to the case of non-spherical deformation.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.