Chaos, Solirons & Fractals Vol. 8, No. 2, pp. 207-220, 1997 Copyright @ 1991 Elsevier Science Ltd Printed in Great Britain. All rights reserved 096@0779/97 $17.00 + 0.00 PIk SO960-0779(96)00049-5 Fundamentals of Constructing Particle-Laden Jetflow by Fractal Point Sets and Predicting 3D Solid-Erosion Rates Z. YONG and R. KOVACEVIC Center for Robotics and Manufacturing Systems, University of Kentucky, Lexington, KY 40506-0108, USA Abstract -A theoretical framework is established for constructing arbitrary particle-laden jetflow by means of fractal sets in the sense of average scale. This novel approach involves three main coherent components, in which one (1) searches for an appropriate original point set and confirms its intrinsic properties such as chaos, symmetry and density distribution; (2) derives governing equations for the formation of new point sets with any geometric configuration and desired normal velocity profile so that they can physically and geometrically represent particle motion on the cross-section of jetflow; and (3) develops the constitutive equations of particle-target interaction with erosion histories of particles and introduces the memory-element technique to handle the chaotic penetrating capabilities of millions of particles. Based on these results, the model is applied to predict the erosion rate in a drilling operation by abrasive waterjet. Results show that theory is consistent with experiments of drilling glass and titanium. @ 1997 Elsevier Science Ltd All rights reserved 1. INTRODUCTION Recently, the attention received by three-dimensional (3D) abrasive waterjet (AWJ) machining has led to an increase in both laboratory experimentation and usage in indus- trial practice. As a result, this state-of-the art cutting tool has been promoted to a multi- functional level, and is currently used for milling, drilling and turning hard-to-machine materials, such as titanium and advanced ceramics. Extensive research by engineers and scientists in diverse disciplines [l-11] has proved the 3D machining technique to be industrially applicable and to have some remarkable advantages in comparison with traditional tools. At present, however, the purely experiment-oriented research is facing great challenges in extending the knowledge from laboratories to the shop floor due to the lack of a theoretical basis behind 3D machining. This issue arises because many machining parameters, such as material properties, jet velocity and nozzle traverse rate, to name just a few, are coherently influential on the machining result. Consequently, the current research activities must rely heavily on numerous tentative tests on a case-by-case basis, making the achievements too specific for industrial applications. To reduce the cost and improve the quality of products, a model for off-line simulation becomes imperative. Transported by high speed waterjet and air (600-900 m/s), about lo5 tiny solid particles per second go through the cross-section of a nozzle (diameter 1.5-2.5 mm) and penetrate into the material to be machined. The machining result depends on the kinetic energy of each individual particle among the millions in the multiphase flow. Thus it is crucial for an accurate model to analyze the kinematic property of every particle. It is a well-known fact that there have been great difficulties in coping with the turbulence of even a one-phase flow to date. Therefore from the point of view of fluid mechanics, it is not reasonable in a short term to expect to gain quickly a thorough understanding of the chaotic behavior of solid particles in a multiphase flow like AWJ. To 207