Ebook Casting: An analytical approach - Part 1

Part 1 of ebook Casting: An analytical approach provide readers with content about: casting of light metals; introduction to fluid dynamics; part design; low-pressure permanent mould casting; equations of motion; useful dimensionless numbers; computational fluid dynamics; design of light metal castings; bending stresses;...

Engineering Materials and Processes Series Editor Professor Brian Derby, Professor of Materials Science Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK Other titles published in this series: Fusion Bonding of Polymer Composites C Ageorges and L Ye Composite Materials D.D.L Chung Titanium G Lütjering and J.C Williams Corrosion of Metals H Kaesche Corrosion and Protection E Bardal Intelligent Macromolecules for Smart Devices L Dai Microstructure of Steels and Cast Irons M Durand-Charre Phase Diagrams and Heterogeneous Equilibria B Predel, M Hoch and M Pool Computational Mechanics of Composite Materials ´ski M Kamin Gallium Nitride Processing for Electronics, Sensors and Spintronics S.J Pearton, C.R Abernathy and F Ren Materials for Information Technology E Zschech, C Whelan and T Mikolajick Fuel Cell Technology N Sammes Computational Quantum Mechanics for Materials Engineers L Vitos Publication due August 2007 Alexandre Reikher and Michael R Barkhudarov Casting: An Analytical Approach 123 Alexandre Reikher, PhD Albany Chicago Co 8200 100th St Pleasant Prairie, WI 53158 USA Michael R Barkhudarov, PhD Flow Science, Inc 683A Harkle Road Santa Fe, NM 87505 USA British Library Cataloguing in Publication Data Reikher, Alexandre Casting : an analytical approach - (Engineering materials and processes) Founding I Title II Barkhudarov, Michael R 671.2 ISBN-13: 9781846288494 Library of Congress Control Number: 2007928128 Engineering Materials and Processes ISSN 1619-0181 ISBN 978-1-84628-849-4 e-ISBN 978-1-84628-850-0 Printed on acid-free paper © Springer-Verlag London Limited 2007 FAVOR™ and FLOW-3D® are trademarks and registered trademarks of Flow Science Inc., 683 Harkle Rd Ste A, Santa Fe, NM 87505, USA http://www.flow3d.com MATLAB® is a registered trademark of The MathWorks, Inc., Apple Hill Drive, Natick, MA 01760-2098, USA http://www.mathworks.com Visual Basic® is a registered trademark of Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399, USA http://www.microsoft.com The software disk accompanying this book and all material contained on it is supplied without any warranty of any kind The publisher accepts no liability for personal injury incurred through use or misuse of the disk Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency Enquiries concerning reproduction outside those terms should be sent to the publishers The use of registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made 987654321 Springer Science+Business Media springer.com Michael Barkhudarov: I dedicate this work to my family, Natasha, Sophia and Philip, and to my dear parents Alexandre Reikher: I dedicate this work To my dear parents To my wife Marianna And to my children Daniel and Alison Preface This book is the result of 40 years of the combined authors’ experience in mechanical and fluid dynamics engineering It gives an overview of product and process development from the analytical standpoint This book has not been intended to revolutionize the casting industry The principals of fluid dynamics and static mechanics were largely developed in the nineteenth century, but process development still largely remains a trial and error method This book is intended to underline the principals of strength of materials and fluid dynamics that are the foundation of the casting product and process development This book has been written as a resource and design tool for product and process engineers and designers who work with aluminium castings It combines many aspects of product and process development, which include the basic principals of static mechanics and fluid dynamics as well as completely developed applications allowing solving problems at every stage of the development process This book has five main parts: (1) overview of casting processes, (2) fluid dynamics, (3) strength of materials, (4) sand casting, permanent mould, and die casting process development, and (5) quality control The unique feature of this book is a combination of real life problems, which product and process engineers face every day, with user-friendly applications written in MATLAB® and Visual Basic The comprehensive unit conversion calculator as well as examples in the area of strength of materials is intended to serve as a reference for practicing engineers as well as for students who are beginning to study mechanical engineering It cab also be used by process engineers, who contribute greatly to product design Completely developed process design applications will help process engineers to take full advantage of the power that computers and software bring to engineering Product engineers can also get inside of process development procedures as well as the governing equations that are used in casting process development Examples in this book are solved with the help of MATLAB® functions, Visual Basic applications as well as a general purpose CFD code FLOW-3D® All MATLAB® functions, art work and Visual Basic applications are developed by the viii Preface authors Proper references are given for commonly available equations and theories Considerable effort has been made to avoid errors The authors would appreciate reader’s comments and suggestions for any corrections and improvements to this book Alexandre Reikher, Milwaukee, Wisconsin, USA, April 2007 Michael Barkhudarov, Los Alamos, New Mexico, USA, April 2007 Contents Casting of Light Metals 1.1 Casting Processes 1.2 Sand Casting 1.2.1 Gating System 1.2.2 Risers and Chills 1.3 Permanent Mould 1.3.1 Gravity Casting 1.3.2 Low-pressure Permanent Mould Casting 1.3.3 Counterpressure Casting 1.4 Die Casting 1.4.1 Die-cast Process 1.4.2 Die-cast Dies 1.4.3 Runner System 1.4.4 Cavity of Die-cast Die 11 1.4.5 Air Ventilation System 11 1.4.6 Ventilation Blocks 12 Introduction to Fluid Dynamics 13 2.1 Basic Concepts 13 2.1.1 Pressure 13 2.1.2 Viscosity 14 2.1.3 Temperature and Enthalpy 16 2.2 Equations of Motion 18 2.3 Boundary Conditions 19 2.3.1 Velocity Boundary Conditions at Walls 20 2.3.2 Thermal Boundary Conditions at Walls 20 2.3.3 Free Surface Boundary Conditions 21 2.4 Useful Dimensionless Numbers 23 2.4.1 Definitions 23 2.4.2 The Reynolds number 24 2.4.3 The Weber Number 24 x Contents 2.5 2.6 2.7 2.4.4 The Bond Number 25 2.4.5 The Froude Number 25 The Bernoulli Equation 26 Compressible Flow 26 2.6.1 Equation of State 27 2.6.2 Equations of Motion 28 2.6.3 Specific Heats 29 2.6.4 Adiabatic Processes 31 2.6.5 Speed of Sound 31 2.6.6 Mach Number 33 2.6.7 The Bernoulli Equation for Gases 33 Computational Fluid Dynamics 35 2.7.1 Computational Mesh 35 2.7.2 Numerical approximations 36 2.7.3 Representation of Geometry 38 2.7.4 Free Surface Tracking 39 2.7.5 Summary 41 Part Design 43 3.1 Design of Light Metal Castings 43 3.2 Static Analysis 43 3.2.1 Moment of an Area 44 3.2.2 Moment of Inertia of an Area 44 3.3 Loading 46 3.4 Stress Components 47 3.4.1 Linear Stress 48 3.4.2 Plane Stress 50 3.4.3 Mohr’s Circle 51 3.5 Hooke’s Law 55 3.6 Saint-Venant’s Principle 56 3.7 Criteria of Failure 57 3.7.1 Ductile Material Failure Theory 58 3.7.2 Brittle Material Failure Theory 58 3.8 Beam Analysis 58 3.8.1 Free Body Diagram 67 3.8.2 Reactions 67 3.9 Buckling 68 3.10 Bending Stresses 71 3.10.1 Normal Stresses in Bending 72 3.10.2 Shear Stress 74 3.11 Stresses in Cylinders 76 3.12 Press-fit Analysis 82 3.13 Thermal Stresses 86 3.14 Torque 87 3.15 Stress Concentration 93 Part Design 67 3.8.1 Free Body Diagram In real-life problems, analyzing complex structures is often necessary To simplify the analysis, a small portion of the system must be isolated All possible interactions between the isolated segment and the rest of the system are replaced by forces, moments and reactions If the system is in equilibrium, applied external loads must be in equilibrium as well When forces are applied to this segment, it is called a free body diagram 3.8.2 Reactions When external forces or moments are applied to a solid body, they exert reactions in the supports For a solid body under external loads to remain in equilibrium, the sum of external loads and reactions must be equal to zero For static analysis, the beam is replaced with its centre axis All forces and reactions are applied to the central axis of the beam The types of possible beam supports are shown in Table 3.4 Table 3.4 Static beam supports Type of support Roller Pin Fixed The types of beams are shown in Table 3.5 68 Casting: An Analytical Approach Table 3.5 Support types Types of beams Simply supported Overhanging beam Cantilever beam 3.9 Buckling When an axial compression load is applied to a long column, it retains a straight form equivalent to a certain value of the external force At some point, when the load reaches a critical value, the column buckles sideways In the 18th century, Swiss mathematician, Leonhard Euler, developed a formula that allows calculating the critical load for long axially loaded columns: PCR S EI kl , (3.36) where E = modulus of elasticity, I = area moment of inertia, l = unsupported length of a column, k = a constant whose value depends on the conditions of the end support of the column Part Design 69 Table 3.6 End restraint conditions and effective length Restriction conditions k 2 0.69 0.5 y/L 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k 1.9 1.7 1.6 1.5 1.35 1.23 1.1 70 Casting: An Analytical Approach Table 3.6 End restraint conditions and effective length (continued) Restriction conditions k y/L k 0.1 0.2 1.8 0.3 1.7 0.4 1.6 0.5 1.4 0.6 1.3 0.7 1.1 0.8 0.98 0.9 0.76 Example 3.3 Initial conditions An aluminium bar has dimensions: A = 15 mm b = 26 mm L = 200 mm Is subjected to an axial load P The modulus of elasticity is 71 000 mPa Part Design 71 Example 3.3 (continued) Calculate the cross-sectional area of the beam: A = 15 u 26 = 390 mm2 The moment of inertia of the cross-section of the beam: 26 u 15 = 7312.5 mm4 12 I Using MATLAB® function buckling, we can calculate the critical load and critical stress on the bar: Initial values: Moment of inertia I = 7312.5 mm4 Modulus of elasticity E = 71 000 mPa Length of beam L = 200 mm Area of beam cross-section A = 390 mm2 End restriction coefficient k = Calculated values: Critical load = 32026.1 N Critical Stress = 82.1182 mPa 3.10 Bending Stresses To develop relations for bending stresses, the next assumptions must be made: The material is isentropic and homogeneous The beam has an axis of symmetry in the plane of deformation The cross section of the beam remains plane after deformation The beam is subjected to pure bending The material of the beam obeys Hooke’s law The beam is straight before an external load is applied The beam has a constant cross section Relations between the dimensions of the beam are such that the beam is going to fail by bending 72 Casting: An Analytical Approach 3.10.1 Normal Stresses in Bending To simplify the development of the calculation procedure for beams subjected to a bending moment, let us examine the case of pure bending In pure bending, only moment around the “x” axis acts on the beam The condition of equilibrium of a beam under pure bending is MX ³ VydA (3.37) A A beam under pure bending is shown in Figure 3.10 When the bending moment acts in a positive direction, the upper zone of the beam is under compression stresses, and the lower zone of the beam is under tensile stresses As shown in Figure 3.11, the bending stress is changing from maximum compression in the upper zone of the beam to maximum tensile in the lower zone of the beam Stress distribution in the cross section of the beam is passing through a neutral zone, where it is equal to zero Figure 3.10 Beam under pure bending The strain in a beam can be defined as the ratio of elongation of fibres divided by its original length H dl dy (3.38) The strain of fibres at distance z from a neutral axis with an initial length dy can be found: H ( R  z )dI  dy dy ( R  z )dI  RdI RdI z R (3.39) Part Design 73 Figure 3.11 Bending stresses in a beam From Hooke’s law, shown in Equation 3.22, normal stress is V EH (3.40) Substitute the strain value from Equation 3.40 into Equation 3.41 V E z R (3.41) M EJ X (3.42) Now Equation 3.33 can be rearranged as R If Equations 3.41 and 3.42 are combined, the following is obtained: V Mz JX (3.43) Equation 3.43 shows that stress and strain are proportional, and maximum stress varies from zero on the neutral axis to the maximum value when the distance from the neutral axis is equal to zmax 74 Casting: An Analytical Approach 3.10.2 Shear Stress Pure bending, as discussed in the previous chapter, is very rare in practical application Pure bending in the case of the beam deformation was selected to simplify the development of the governing equations Stress distributions throughout the cross section of a beam remains the same whether or not shear stresses are present Relations between shear force and the bending moment are dM dy T (3.44) To visualize the shear stress distribution better in the beam’s cross section, we will cut out a section of the beam, as shown on Figure 3.12 Both the bending moment and shear force act on the section of the beam Because of the shear force acting on the beam, the bending moment is changing along the length of the beam If we refer to the bending moment at the near section as M, the bending moment on the far end of the beam will be equal to M + dM Moment M produces normal stress ı Moment M + dM will produce normal stress ı The resultant stress in the far end section is greater than that in the near side section To prevent the section of the beam from sliding in an íx direction, it must be balanced by a shear force The balanced equation of the beam will be in form N1  FS N2 , (3.45) where Wbdx FS (3.46) The normal force acting on the beam’s cross section is equal to the area of the beam’s cross section times the normal stress By designating dF as the beam’s cross section, the normal force acting on the near side section is ıdF The force acting on the beam’s farside is ıdF Now the force acting on the entire crosssectional area of the beam is N1 ³ VdF F N2 " ³ V dF ³ F ³ F My dF I ( M  dM ) y dF I M S, I M  dM S I (3.47) (3.48) Part Design 75 a c b Figure 3.12 Shear stress distribution in a beam a Simply supported beeam under distributed load; b changes in the bending moment along the length of the beam; c stresses generated by the bending moment Next, using Equations 3.46–3.48 and solving Equation 3.45 for shear stress results in W TS bI (3.49) where S = first moment of the area of the vertical face about the neutral axis b = width of the section at parallel distance y from the neutral axis I = second moment of inertia of the entire cross section about the neutral axis 76 Casting: An Analytical Approach 3.11 Stresses in Cylinders In practical applications, cylindrical parts are often subjected to tangential and radial stresses Examples can be passages that carry fluids in high-pressure, pressfit assemblies Let us examine the cylinder shown on Figure 3.13 The cylinder has an inside radius “r” and an outside radius “R.” The cylinder is subjected to external pressure p1 and internal pressure p2 To calculate radial and tangential stresses in the cross section of the cylinder, we will assume that longitudinal elongation is constant around the circumference of the cylinder This means that if the cross-sectional area of the cylinder was flat before an external force was applied, it will remain flat under stress as well Figure 3.13 Cylinder under internal and external stress Figure 3.14 Stresses in the element of a cylinder Part Design 77 Let’s cut out the element in Figure 3.14 There are radial ır and tangential ıt stresses acting on the element Since the element remains in equilibrium, we can write ¦X 0; ¦Y (3.50) Referring to Figure 3.14, the element is symmetrical relative to the X axis This means that all forces projected on the Y axis in positive and negative directions cancel each other The conditions of equilibrium in the X direction will be ¦X V r rdG  (V r  dV r )(r  dr )dG  2(V t dr sin dG ) (3.51) With a certain degree of accuracy, the condition of equilibrium in the X direction is r dV r  Vr  Vt dr (3.52) To solve Equation 3.52, we need to find the radial and tangential stresses Let’s denote the elongation of the surface with radius r as er, the elongation of the cylindrical surface with radius R as eR and the length of the element as l Strain in the radial direction is Hr der l (3.53) RdG  rdG , rdG (3.54) The tangential strain is Ht where R = r + l Substituting R in Equation 3.54 will yield Ht (r  l )dG  rdG rdG l r (3.55) Hooke’s law for radial and tangential stresses will be Vr Vt E (H r  PH t ),  P2 E (H t  PH r )  P2 (3.56) 78 Casting: An Analytical Approach Putting the values of radial and tangential strain from Equations 3.53 and 3.55 into 3.56 will yield E § der e ·  P r á, ă rạ  P â dr E Đ er de à ă  P r dr  P2 â r Vr Vt (3.57) Using the values of radial and tangential stresses from Equation 3.57 in Equation 3.52 will result in a differential equation: d er dr  e de r  r2 r dr r d ª d er r dr ôơ r dr ằẳ 0, (3.58) (3.59) The general solution of Equation 3.59 can be found by integrating it twice: er A1r  A2 r (3.60) A1 and A2 are constants of integration They can be derived from boundary conditions By combining equations 3.60 and 3.57, the following are obtained: Vr Vt E ª  P A1  2 P A2 º» ô r 1 P ẳ E ê  P ... 6 71. 2 ISBN -1 3 : 97 818 46288494 Library of Congress Control Number: 200792 812 8 Engineering Materials and Processes ISSN 16 1 9-0 18 1 ISBN 97 8 -1 -8 462 8-8 4 9-4 e-ISBN 97 8 -1 -8 462 8-8 5 0-0 Printed on acid-free... 8 Casting: An Analytical Approach Figure 1. 4 Cold chamber die-cast machine 1. 4.2 Die-cast Dies Die-cast dies (Figure 1. 6) are made from alloy tool steel and must withstand multiple cooling-heating... 1. 1 Casting Processes 1. 2 Sand Casting 1. 2 .1 Gating System 1. 2.2 Risers and Chills 1. 3 Permanent Mould 1. 3 .1 Gravity Casting 1. 3.2