Design and analysis of composite structures for automotive applications

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Design and Analysis of Composite Structures for Automotive Applications Automotive Series Advanced Battery Management Technologies for Electric Vehicles Rui Xiong, Weixiang Shen Noise and Vibration Control in Automotive Bodies Jian Pang Automotive Power Transmission Systems Yi Zhang, Chris Mi High Speed Off-Road Vehicles: Suspensions, Tracks, Wheels and Dynamics Bruce Maclaurin Hybrid Electric Vehicles: Principles and Applications with Practical Perspectives, 2nd Edition Chris Mi, M Abul Masrur Hybrid Electric Vehicle System Modeling and Control, 2nd Edition Wei Liu Thermal Management of Electric Vehicle Battery Systems Ibrahim Dincer, Halil S Hamut, Nader Javani Automotive Aerodynamics Joseph Katz The Global Automotive Industry Paul Nieuwenhuis, Peter Wells Vehicle Dynamics Martin Meywerk Modelling, Simulation and Control of Two-Wheeled Vehicles Mara Tanelli, Matteo Corno, Sergio Saveresi Vehicle Gearbox Noise and Vibration: Measurement, Signal Analysis, Signal Processing and Noise Reduction Measures Jiri Tuma Modeling and Control of Engines and Drivelines Lars Eriksson, Lars Nielsen Advanced Composite Materials for Automotive Applications: Structural Integrity and Crashworthiness Ahmed Elmarakbi Guide to Load Analysis for Durability in Vehicle Engineering P Johannesson, M Speckert Design and Analysis of Composite Structures for Automotive Applications Chassis and Drivetrain Vladimir Kobelev Department of Natural Sciences, University of Siegen, Germany This edition first published 2019 © 2019 John Wiley and Sons Ltd All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions The right of Vladimir Kobelev to be identified as the author of this work has been asserted in 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professional services The advice and strategies contained herein may not be suitable for your situation You should consult with a specialist where appropriate Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages Library of Congress Cataloging-in-Publication Data Names: Kobelev, Vladimir, 1959- author Title: Design and analysis of composite structures for automotive applications : chassis and drivetrain / Vladimir Kobelev, Department of Natural Sciences, University of Siegen, Germany Description: First edition | Hoboken, NJ : Wiley, 2019 | Series: Automotive series | Includes bibliographical references and index | Identifiers: LCCN 2019005286 (print) | LCCN 2019011866 (ebook) | ISBN 9781119513841 (Adobe PDF) | ISBN 9781119513865 (ePub) | ISBN 9781119513858 (hardback) Subjects: LCSH: Automobiles–Chassis | Automobiles–Power trains | Automobiles–Design and construction Classification: LCC TL255 (ebook) | LCC TL255 K635 2019 (print) | DDC 629.2/4–dc23 LC record available at https://lccn.loc.gov/2019005286 Cover Design: Wiley Cover Images: © Vladimir Kobelev, Background: © solarseven/ShuWerstock Set in 10/12pt WarnockPro by SPi Global, Chennai, India Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY 10 v Contents Foreword xiii Series Preface xv List of Symbols and Abbreviations xvii Introduction xxiii About the Companion Website xxxv Elastic Anisotropic Behavior of Composite Materials 1.1 1.1.1 1.1.2 1.1.3 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 Anisotropic Elasticity of Composite Materials Fourth Rank Tensor Notation of Hooke’s Law Voigt’s Matrix Notation of Hooke’s Law Kelvin’s Matrix Notation of Hooke’s Law Unidirectional Fiber Bundle Components of a Unidirectional Fiber Bundle Elastic Properties of a Unidirectional Fiber Bundle Effective Elastic Constants of Unidirectional Composites Rotational Transformations of Material Laws, Stress and Strain 10 Rotation of Fourth Rank Elasticity Tensors 11 Rotation of Elasticity Matrices in Voigt’s Notation 11 Rotation of Elasticity Matrices in Kelvin’s Notation 13 Elasticity Matrices for Laminated Plates 14 Voigt’s Matrix Notation for Anisotropic Plates 14 Rotation of Matrices in Voigt’s Notation 15 Kelvin’s Matrix Notation for Anisotropic Plates 15 Rotation of Matrices in Kelvin’s Notation 16 Coupling Effects of Anisotropic Laminates 17 Orthotropic Laminate Without Coupling 17 Anisotropic Laminate Without Coupling 17 Anisotropic Laminate With Coupling 17 Coupling Effects in Laminated Thin-Walled Sections 18 Conclusions 18 References 19 Phenomenological Failure Criteria of Composites 21 2.1 2.1.1 2.1.2 2.1.3 Phenomenological Failure Criteria 21 Criteria for Static Failure Behavior 21 Stress Failure Criteria for Isotropic Homogenous Materials 21 Phenomenological Failure Criteria for Composites 22 vi Contents 2.1.4 2.1.4.1 2.1.4.2 2.1.5 2.1.5.1 2.1.5.2 2.1.5.3 2.1.5.4 2.1.5.5 2.2 2.2.1 2.2.2 2.2.3 2.3 2.3.1 2.3.2 2.4 2.5 Phenomenological Criteria Without Stress Coupling 23 Criterion of Maximum Averaged Stresses 23 Criterion of Maximum Averaged Strains 24 Phenomenological Criteria with Stress Coupling 24 Mises–Hill Anisotropic Failure Criterion 24 Pressure-Sensitive Mises–Hill Anisotropic Failure Criterion 26 Tensor-Polynomial Failure Criterion 27 Tsai–Wu Criterion 30 Assessment of Coefficients in Tensor-Polynomial Criteria 30 Differentiating Criteria 33 Fiber and Intermediate Break Criteria 33 Hashin Strength Criterion 33 Delamination Criteria 35 Physically Based Failure Criteria 35 Puck Criterion 35 Cuntze Criterion 36 Rotational Transformation of Anisotropic Failure Criteria 37 Conclusions 40 References 40 Micromechanical Failure Criteria of Composites 45 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.4.1 3.1.4.2 Pullout of Fibers from the Elastic-Plastic Matrix 45 Axial Tension of Fiber and Matrix 45 Shear Stresses in Matrix Cylinders 51 Coupled Elongation of Fibers and Matrix 53 Failures in Matrix and Fibers 54 Equations for Mean Axial Displacements of Fibers and Matrix 54 Solutions of Equations for Mean Axial Displacements of Fibers and Matrix 56 Rupture of Matrix and Pullout of Fibers from Crack Edges in a Matrix 57 Elastic Elongation (Case I) 57 Plastic Sliding on the Fiber Surface (Case II) 58 Fiber Breakage (Case III) 58 Rupture of Fibers, Matrix Joints and Crack Edges 59 Crack Bridging in Elastic-Plastic Unidirectional Composites 60 Crack Bridging in Unidirectional Fiber-Reinforced Composites 60 Matrix Crack Growth 61 Fiber Crack Growth 62 Penny-Shaped Crack 65 Crack in a Transversal-Isotropic Medium 65 Mechanisms of the Fracture Process 66 Crack Bridging in an Orthotropic Body With Disk Crack 66 Solution to an Axially Symmetric Crack Problem 68 Plane Crack Problem 72 Equations of the Plane Crack Problem 72 Solution to the Plane Crack Problem 74 Debonding of Fibers in Unidirectional Composites 75 3.1.5 3.1.5.1 3.1.5.2 3.1.5.3 3.1.6 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.4.1 3.2.4.2 3.2.4.3 3.2.4.4 3.2.5 3.2.5.1 3.2.5.2 3.3 Contents 3.3.1 3.3.2 3.3.2.1 3.3.2.2 3.3.2.3 3.3.3 3.3.3.1 3.3.3.2 3.3.3.3 3.3.3.4 3.3.3.5 3.3.4 3.3.5 3.3.6 3.4 Axial Deformation of Unidirectional Fiber Composites 75 Stresses in Unidirectional Composite in Cases of Ideal Debonding or Adhesion 79 Equations of an Axially Loaded Unidirectional Compound Medium (A) 79 Total Debonding (B) 82 Ideal Adhesion (C) 83 Stresses in a Unidirectional Composite in a Case of Partial Debonding 84 Partial Radial Load on the Fiber Surface 84 Partial Radial Load on the Matrix Cavity Surface 84 Partial Debonding With Central Adhesion Region (D) 85 Partial Debonding With Central Debonding Region (E) 88 Semi-Infinite Debonding With Central Debonding Region (F) 89 Contact Problem for a Finite Adhesion Region 89 Debonding of a Semi-Infinite Adhesion Region 93 Debonding of Fibers from a Matrix Under Cyclic Deformation 95 Conclusions 98 References 98 Optimization Principles for Structural Elements Made of Composites 105 4.1 4.1.1 4.1.2 4.1.3 4.1.4 4.2 Stiffness Optimization of Anisotropic Structural Elements 105 Optimization Problem 105 Optimality Conditions 106 Optimal Solutions in Anti-Plane Elasticity 109 Optimal Solutions in Plane Elasticity 109 Optimization of Strength and Loading Capacity of Anisotropic Elements 110 Optimization Problem 110 Optimality Conditions 113 Optimal Solutions in Anti-Plane Elasticity 114 Optimal Solutions in Plane Elasticity 114 Optimization of Accumulated Elastic Energy in Flexible Anisotropic Elements 116 Optimization Problem 116 Optimality Conditions 117 Optimal Solutions in Anti-Plane Elasticity 118 Optimal Solutions in Plane Elasticity 119 Optimal Anisotropy in a Twisted Rod 119 Optimal Anisotropy of Bending Console 122 Optimization of Plates in Bending 123 Conclusions 125 References 125 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.5 4.6 4.7 5.1 5.2 5.2.1 129 Torsion of Anisotropic Shafts With Solid Cross-Sections 129 Thin-Walled Anisotropic Driveshaft with Closed Profile 132 Geometry of Cross-Section 132 Optimization of Composite Driveshaft vii viii Contents 5.2.2 5.3 5.3.1 5.3.2 5.3.2.1 5.3.2.2 5.3.2.3 5.3.3 5.3.3.1 5.3.3.2 5.3.4 5.4 5.4.1 5.4.2 5.4.3 5.5 5.6 Main Kinematic Hypothesis 133 Deformation of a Composite Thin-Walled Rod 135 Equations of Deformation of a Anisotropic Thin-Walled Rod 135 Boundary Conditions 138 Ideal Fixing 138 Ideally Free End 138 Boundary Conditions of the Intermediate Type 140 Governing Equations in Special Cases of Symmetry 140 Orthotropic Material 140 Constant Elastic Properties Along the Arc of a Cross-Section 140 Symmetry of Section 140 Buckling of Composite Driveshafts Under a Twist Moment 141 Greenhill’s Buckling of Driveshafts 141 Optimal Shape of the Solid Cross-Section for Driveshaft 143 Hollow Circular and Triangular Cross-Sections 144 Patents for Composite Driveshafts 146 Conclusions 150 References 150 Dynamics of a Vehicle with Rigid Structural Elements of Chassis 155 6.1 6.1.1 6.1.2 6.1.3 6.2 6.2.1 6.2.2 6.2.2.1 6.2.2.2 6.2.2.3 6.2.2.4 6.2.3 6.2.4 6.2.5 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.4.4.1 6.4.4.2 6.4.4.3 6.4.5 6.4.5.1 Classification of Wheel Suspensions 155 Common Designs of Suspensions 155 Types of Twist-Beam Axles 156 Kinematics of Wheel Suspensions 157 Fundamental Models in Vehicle Dynamics 159 Basic Variables of Vehicle Dynamics 159 Coordinate Systems of Vehicle and Local Coordinate Systems 161 Earth-Fixed Coordinate System 161 Vehicle-Fixed Coordinate System 162 Horizontal Coordinate System 162 Wheel Coordinate System 162 Angle Definitions 162 Components of Force and Moments in Car Dynamics 163 Degrees of Freedom of a Vehicle 163 Forces Between Tires and Road 167 Tire Slip 167 Side Slip Curve and Lateral Force Properties 168 Dynamic Equations of a Single-Track Model 170 Hypotheses of a Single-Track Model 170 Moments and Forces in a Single-Track Model 171 Balance of Forces and Moments in a Single-Track Model 173 Steady Cornering 174 Necessary Steer Angle for Steady Cornering 174 Yaw Gain Factor and Steer Angle Gradient 175 Classification of Self-Steering Behavior 176 Non-Steady Cornering 179 Equations of Non-Stationary Cornering 179 Contents 6.4.5.2 6.4.6 6.5 Oscillatory Behavior of Vehicle During Non-Steady Cornering 180 Anti-Roll Bars Made of Composite Materials 181 Conclusions 182 References 182 Dynamics of a Vehicle With Flexible, Anisotropic Structural Elements of Chassis 183 7.1 7.1.1 7.1.2 7.1.3 Effects of Body and Chassis Elasticity on Vehicle Dynamics 183 Influence of Body Stiffness on Vehicle Dynamics 183 Lateral Dynamics of Vehicles With Stiff Rear Axles 184 Induced Effects on Wheel Orientation and Positioning of Vehicles with Flexible Rear Axle 185 Self-Steering Behavior of a Vehicle With Coupling of Bending and Torsion 188 Countersteering for Vehicles with Twist-Beam Axles 188 Countersteering Mechanisms 188 Countersteering by Anisotropic Coupling of Bending and Torsion 190 Bending-Twist Coupling of a Countersteering Twist-Beam Axle 192 Roll Angle of Vehicle 193 Relationship Between Roll Angle and Centrifugal Force 193 Lateral Reaction Forces on Wheels 193 Steer Angles on Front Wheels 194 Steer Angles on Rear Wheels 194 Steady Cornering of a Flexible Vehicle 196 Stationary Cornering of a Car With a Flexible Chassis 196 Necessary Steer Angles for Coupling and Flexibility of Chassis 196 Limit Case: Lateral Acceleration Vanishes 196 Absolutely Rigid Front and Rear Wheel Suspensions 197 Bending and Torsion of a Twist Member Completely Decoupled 197 General Case of Coupling Between Bending and Torsion of a Twist Member 198 Neutral Steering Caused by Coupling Between Bending and Torsion of a Twist Member 198 Estimation of Coupling Constant for a Twist Member 199 Coupling Between Vehicle Roll Angle and Twist of Cross-Member 199 Stiffness Parameters of a Twist-Beam Axle 200 Roll Spring Rate 200 Lateral Stiffness 201 Camber Stiffness 203 Design of the Countersteering Twist-Beam Axle 203 Requirements for a Countersteering Twist-Beam Axle 203 Selection and Calculation of the Cross-Section for the Cross-Member 205 Elements of a Countersteering Twist-Beam Axle 208 Patents on Twist-Beam Axles 211 Conclusions 214 References 214 7.2 7.2.1 7.2.1.1 7.2.1.2 7.2.2 7.2.3 7.2.3.1 7.2.3.2 7.2.3.3 7.2.3.4 7.3 7.3.1 7.3.2 7.3.2.1 7.3.2.2 7.3.2.3 7.3.2.4 7.3.2.5 7.4 7.4.1 7.4.2 7.4.2.1 7.4.2.2 7.4.2.3 7.5 7.5.1 7.5.2 7.5.3 7.6 7.7 ix 340 Appendix B Anisotropic Elasticity where Qij = Cij − Ci3 Cj3 C33 (B.17) B.4 Generalized Airy Stress Function B.4.1 Plane Stress State The equilibrium conditions (B.20) for the plane stress state are identically fulfilled by the scalar function F(x, y) (the Airy stress function): 𝜎xx = 𝜕2F , 𝜕y2 𝜎yy = 𝜕2F , 𝜕x2 𝜏xy = − 𝜕2F 𝜕x𝜕y From the two-dimensional compatibility condition follows the equation for the Airy stress function F(x, y) for the plane stress state: S11 𝜕4F 𝜕4F 𝜕4F + (2S12 + S66 ) 2 + S22 = 𝜕y 𝜕x 𝜕y 𝜕x (B.18) B.4.2 Plane Strain State The generalized Airy equation for plane strain state is: q11 𝜕4F 𝜕4F 𝜕4F + (2q12 + q66 ) 2 + q22 = 𝜕y 𝜕x 𝜕y 𝜕x (B.19) B.4.3 Rotationally Symmetric Elasticity Problems For another important class of rotationally symmetric spatial elasticity problems, the displacements and stresses depend only on two cylindrical coordinates (r, z) The basic equations for this result from the general relationships for the three-dimensional continuum For this purpose, set: ur = u(r, z), u𝜃 = 0, w = w(r, z) (B.20) Between the distortion components 𝜕u u 𝜕w 𝜕u 𝜕w , 𝜀 = , 𝜀z = , 𝛾r z = + , 𝛾 = 𝛾𝜃 r = 𝜕r 𝜃 r 𝜕z 𝜕z 𝜕r z 𝜃 there are two compatibility conditions: 𝜀r = 𝜀r − 𝜕 (r 𝜀𝜃 ) = 0, 𝜕r 𝜕 𝛾r z 𝜕 𝜀r 𝜕 𝜀z + − = 𝜕z2 𝜕r2 𝜕r𝜕z (B.21) Appendix B Anisotropic Elasticity The equilibrium conditions are 𝜕 𝜎r 𝜕 𝜏r z 𝜎r − 𝜎𝜃 + + = 0, (B.22) 𝜕r 𝜕z r 𝜕 𝜎z 𝜕 𝜏r z 𝜏r z + + =0 (B.23) 𝜕z 𝜕r r For transversal-isotropic homogeneous bodies, the generalized Hooke’s law applies: 𝜀r = S11 𝜎r + S12 𝜎𝜃 + S13 𝜎z , 𝜀𝜃 = S21 𝜎r + S11 𝜎𝜃 + S13 𝜎z , 𝜀z = S31 (𝜎r + 𝜎𝜃 ) + S33 𝜎z , 𝛾r z = 𝛾z r = S44 𝜏r z = S44 𝜏z r , 𝛾z 𝜃 = 𝛾𝜃 z = S44 𝜏z 𝜃 = S44 𝜏𝜃 z , 𝛾r 𝜃 = 𝛾𝜃 r = 2(S11 − S12 )𝜏r 𝜃 (B.24) 341 343 Appendix C Integral Transforms in Elasticity C.1 One-Dimensional Integral Transform A general integral transform of the function 𝜑(x) is defined by: b J{𝜑(x), x → 𝜉} = 𝜑∗ (𝜉) = ∫a 𝜑(x)K(𝜉, x)dx, (C.1) where K(𝜉, x) is the transformation kernel Complex Fourier transform: ∞ ℑ{𝜑(x), x → 𝜉} = 𝜑∗ (𝜉) = √ 𝜑(x)ei𝜉x dx ∫ 2𝜋 −∞ Fourier-SINUS- and Fourier-COSINUS transforms: √ ∞ ∗ ℑS {𝜑(x), x → 𝜉} = 𝜑 (𝜉) = 𝜑(x) sin(𝜉 x)dx, 𝜋∫ √ −∞∞ 𝜑(x) cos(𝜉 x)dx ℑC {𝜑(x), x → 𝜉} = 𝜑∗ (𝜉) = 𝜋 ∫−∞ The inversion of Fourier transforms: ∞ 𝜑∗ (𝜉)e−i 𝜉 x d𝜉, ℑ−1 {𝜑∗ (𝜉), 𝜉 → x} = 𝜑(x) = √ ∫ −∞ √2𝜋 ∞ ℑS −1 {𝜑∗ (𝜉), 𝜉 → x} = 𝜑(x) = 𝜑∗ (𝜉) sin(𝜉 x) d𝜉, 2𝜋 ∫−∞ √ ∞ −1 ∗ 𝜑∗ (𝜉) cos(𝜉 x) d𝜉 ℑC {𝜑 (𝜉), 𝜉 → x} = 𝜑(x) = 2𝜋 ∫−∞ The transform of the derivative reads } { n d 𝜑(x), x → 𝜉 = (𝜑(n) (𝜉))∗ = (−i𝜉)n 𝜑∗ (𝜉) ℑ dxn Design and Analysis of Composite Structures for Automotive Applications: Chassis and Drivetrain, First Edition Vladimir Kobelev © 2019 John Wiley & Sons Ltd Published 2019 by John Wiley & Sons Ltd Companion website: www.wiley.com/go/kobelev/automotive_suspensions (C.2) (C.3) (C.4) (C.5) (C.6) (C.7) (C.8) 344 Appendix C Integral Transforms in Elasticity C.2 Two-Dimensional Fourier Transform Two-dimensional Fourier transform of the function 𝜑(x, y): ∞ ℑ{𝜑(x, y), x → 𝜉} = 𝜑∗ (𝜉, y) = √ 𝜑(x, y)ei𝜉x dx, (C.9) 2𝜋 ∫−∞ ∞ ℑ{𝜑(x, y), y → 𝜂} = 𝜑∗ (x, 𝜂) = √ 𝜑(x, y)eiy𝜂 dy, (C.10) 2𝜋 ∫−∞ ∞ ∞ 𝜑(x, y)ei(y𝜂+x𝜉) dx dy (C.11) ℑ{𝜑(x, y), x → 𝜉, y → 𝜂} = 𝜑∗ (𝜉, 𝜂) = √ 2𝜋 ∫−∞ ∫−∞ The corresponding retransformation formulas are: ∞ ℑ{𝜑∗ (𝜉, y), 𝜉 → x} = 𝜑(x, y) = √ 𝜑∗ (𝜉, y)e−i 𝜉 x d𝜉, 2𝜋 ∫−∞ ∞ ℑ{𝜑∗ (x, 𝜂), 𝜂 → y} = 𝜑(x, y) = √ 𝜑∗ (x, 𝜂)e−i 𝜂 y d𝜂, 2𝜋 ∫−∞ ∞ (C.12) (C.13) ∞ ℑ{𝜑∗ (𝜉, 𝜂), 𝜉 → x, 𝜂 → y} = 𝜑(x, y) = √ 𝜑∗ (𝜉, 𝜂)e−i(y 𝜂+x 𝜉) d𝜂 d𝜉 ∫ ∫ −∞ −∞ 2𝜋 (C.14) C.3 Potential Functions for Plane Elasticity Problems For plane elasticity problems of an orthotropic medium, the displacements, distortions and stresses depend only on two coordinates in the plane These equations can be represented as follows: LE [F] = 0, (C.15) where a generalized biharmonic operator for plane stress state reads: LE [F] = C11 𝜕4F 𝜕4F 𝜕4F + (2C + C ) + C =0 12 66 22 𝜕y4 𝜕x2 𝜕y2 𝜕x4 For a plane strain state, the coefficients should be recalculated according to formulas from Appendix B The strains after the substitution read: 𝜕2F 𝜕2F 𝜕u = C11 + C12 , 𝜕x 𝜕y 𝜕x 𝜕v 𝜕2F 𝜕2F = C21 + C22 𝜕y 𝜕y 𝜕x 𝜕u 𝜕v 𝜕 F + = −C66 (C.16) 𝜕y 𝜕x 𝜕x 𝜕y The polynomial LA [𝜇] = C 11 𝜇4 + (2C 12 + C 66 )𝜇2 + c22 Is the characteristic polynomial to (C.15), and the corresponding equation is: LA [𝜇] = (C.17) Appendix C Integral Transforms in Elasticity The roots of (C.17) 𝜇1 ,𝜇2 ,𝜇3 and 𝜇4 are imaginary: 𝜇1 = −𝜇3 = ip1 , p1,2 𝜇2 = −𝜇4 = ip2 , √ 2C12 + C66 ± (2C11 + C66 )2 − 4C11 C22 = > 2C11 (C.18) From the positivity of LA [𝜇] for real argument, the condition follows: (2C11 + C66 )2 − 4C11 C22 > The substitution: 𝜕2 𝜕2 ΔE1 = + p21 , 𝜕y 𝜕x ΔE2 = 𝜕2 𝜕 + p 𝜕y2 𝜕x2 leads to factorization of the differential Eq (C.17): LE [F] = ΔE1 ΔE1 F = (C.19) The characteristic polynomial of the factorized differential Eq (C.19) is: LA [𝜇] = C11 (𝜇 − ip1 )(𝜇 + ip1 )(𝜇 − ip2 )(𝜇 + ip2 ) = (C.20) By comparing (C.17) with (C.20) it can be concluded that: p21 p22 = C22 ∕C11 , p21 + p22 = (2C12 + C66 )∕C11 (C.21) For the isotropic body p1 = p2 = and 𝜇1 = 𝜇3 = i, 𝜇2 = 𝜇4 = − i The general representation of a real solution (C.19) using two complex analytical functions U1 (z1 ) and U2 (z2 ) the complex variables z1 = x + 𝜇1 y, z2 = x + 𝜇2 y is: (C.22) F(x, y) = 2Re[U1 (z1 ) + U2 (z2 )] The plane problem of the elasticity theory of the orthotropic body is thus attributed to the determination of both analytical functions U1 (z1 ) and U2 (z2 ) The relationship between the stress and displacement components and the complex stress functions applies: 𝜎xx = 2Re [𝜇12 Φ′ (z1 ) + 𝜇22 Ψ′ (z2 )], 𝜎yy = 2Re [Φ′ (z1 ) + Ψ′ (z2 )] 𝜏xy = −2Re [𝜇1 Φ′ (z1 ) + 𝜇2 Ψ′ (z2 )], u = 2Re [P1 Φ(z1 ) + P2 Ψ(z2 )], v = 2Re [Q1 Φ(z1 ) + Q2 Ψ(z2 )] In these equations Pi = C11 𝜇i2 + C12 − C66 𝜇i , Φ(z1 ) = dU(z1 ) , dz1 Ψ(z1 ) = Qi = C12 𝜇i2 − C66 + C22 ∕𝜇i , dU(z2 ) dz2 The boundary condition plays an important role in determining the two stress U1 (z1 ), U2 (z2 ) functions In a complex representation for the first boundary value problem, this reads as follows: (1 − p1 )Φ(z1 ) + (1 − p2 )Ψ(z2 )+ (1 + p1 )Φ(z1 ) + (1 + p2 )Ψ(z2 ) = f1 + if2 345 346 Appendix C Integral Transforms in Elasticity where l f1 (l) + i f2 (l) = i ∫l0 (Fx + iFy ) ds It is useful to represent the function F(x, y) as the sum of two functions: F(x, y) = 𝜓1 (x, y) + 𝜓2 (x, y) (C.23) The functions 𝜓 (x, y), 𝜓 (x, y) are the solutions of the following equations: 2 𝜕 𝜓1 𝜕 𝜓1 𝜕 𝜓1 E 𝜕 𝜓1 + p = 0, Δ 𝜓 = + p =0 1 1 𝜕y2 𝜕x2 𝜕y2 𝜕x2 Inserting (B.27) in (B.22) and (C.16) returns: ΔE1 𝜓1 = (C.24) 2 𝜕 𝜓1 𝜕 𝜓2 − , p21 𝜕y2 p22 𝜕y2 𝜕 𝜓1 𝜕 𝜓2 𝜏xy = − − , 𝜕x𝜕y 𝜕x𝜕y 𝜕𝜓 𝜕𝜓 v = 𝛿1 + 𝛿2 , 𝜕x 𝜕y (C.25) 𝜎yy = − with 𝛿1 = C21 + C22 p−2 , 𝛿2 = C21 + C22 p−2 C.4 Rotationally Symmetric, Spatial Elasticity Problems For another important class of rotationally symmetric, spatial elasticity problem, the displacements and stresses depend only on two cylindrical coordinates (r, z) The basic equations for this result from the general relationships for the three-dimensional continuum The equilibrium conditions (B.22–B.23) and the compatibility conditions (B.21) are determined by the following approach: ) ( 𝜕 𝜕2F b 𝜕 F 𝜕2F + +a , 𝜎r = − 𝜕z 𝜕r2 r 𝜕r 𝜕z ) ( 𝜕 𝜕 F 1𝜕F 𝜕2F 𝜎𝜃 = − b + +a , 𝜕z 𝜕r r 𝜕r 𝜕z ) ( 2 𝜕 𝜕 F c𝜕F 𝜕 F 𝜎z = c + +d , 𝜕z 𝜕r r 𝜕r 𝜕z ) ( 𝜕 𝜕 F 1𝜕F 𝜕2F + 𝜏rz = (C.26) +a 𝜕r 𝜕r2 r 𝜕r 𝜕z The equilibrium equations are satisfied by of the scalar stress function F(r, z) The coefficients read: S (S + S44 ) − S12 S33 S (S − S12 ) , b = 13 13 , a = 13 11 D D S (S − S12 ) + S11 S44 S − S12 , d = 11 , c = 13 11 D D D = S11 S33 − S13 (C.27) Appendix C Integral Transforms in Elasticity From the second equilibrium condition (B.21) follows the basic equation of the rotationally symmetric, spatial elasticity problem: ( )( ) 𝜕 𝜕 𝜕 F 1𝜕F 𝜕2F R L [F] = + + +a 𝜕r2 r 𝜕r 𝜕r2 r 𝜕r 𝜕z ( ) 2 𝜕 𝜕 F c𝜕F 𝜕 F + c + (C.28) + d = 𝜕z 𝜕r r 𝜕r 𝜕z The substitution ( ) 𝜕 𝜕2 𝜕 r , + q12 𝜕z r 𝜕r 𝜕r ( ) 𝜕 𝜕2 𝜕 ΔR2 = + q22 r 𝜕z r 𝜕r 𝜕r ΔR1 = with { d q12 = a + c + d q22 = a + c − √ √ (C.29) (C.30) (a + c)2 − 4d (C.31) (a + c)2 − 4d Leads to the following representation of (C.28): LR [F] = d ΔR1 ΔR2 F = (C.32) It is useful to represent the stress function F(r, z) as an indefinite integral of the sum of two functions 𝜑1 (r, z) 𝜑2 (r, z): z F(r, z) = ∫0 [𝜑1 (r, Z) + 𝜑2 (r, Z)] dZ (C.33) The functions fulfill the equations ΔR1 𝜑1 = 0, ΔR2 𝜑2 = (C.34) Substitution of (C.33) into (C.26) provides: 𝜎zz = 𝛼1 𝜕 𝜑1 𝜕 𝜑2 + 𝛼 , 𝜕 z2 𝜕 z2 𝜕 𝜑1 𝜕 𝜑2 + 𝛽2 , 𝜕r 𝜕z 𝜕r 𝜕z 𝜕𝜑 𝜕𝜑 w = 𝛾1 + 𝛾 2 𝜕z 𝜕z 𝜏rz = 𝛽1 (C.35) with 𝛼i = a − c𝜔2i , 𝜔i = qi−1 , 𝛽i = a − 𝜔2i , 𝛾i = S13 (1 − 2aq2i + b) + S33 (aq2i − c) qi2 This transformation is similar to the transformation (C.25) used in the plane elasticity problem 347 348 Appendix C Integral Transforms in Elasticity C.5 Application of the Fourier Transformation to Plane Elasticity Problems The application of the Fourier transformation with respect to one of the variables allows the partial differential slip for the Airy stress function to be reduced to a common differential equation With the Fourier transform G(𝜉, y) = ℑ{ LE [F(x, y)], x → 𝜉 } (C.36) follows the equation: ∞ LE [F(x, y)]ei𝜉x dx = ∫−∞ ] ∞ [ 𝜕4 𝜕2 F(x, y)ei𝜉x dx C11 + (2C12 + C66 )𝜉 2 + C22 𝜉 ∫−∞ 𝜕y 𝜕y Thus, from the partial differential Eq (C.19) follows the ordinary differential equation for G(𝜉, y): C11 d4 G 2d G + (2C + C )𝜉 + C22 𝜉 G = 12 66 dy dy (C.37) The solution of (C.37) reads: G(𝜉, y) = A(𝜉)e−p1 𝜉y + B(𝜉)e−p2 𝜉y + C(𝜉)ep1 𝜉y + D(𝜉)ep2 𝜉y (C.38) The integration constants A(𝜉), B(𝜉), C(𝜉) and D(𝜉) are determined from the boundary conditions of the problem In the case of a semi-infinite medium in which all stresses and displacements decay for y → ∞, the integration constants C(𝜉) and D(𝜉) disappear With the retransformation of (C.36) we finally get an integral representation for the Airy stress function: ∞ G(𝜉, y)e−i𝜉x dx F(x, y) = √ ∫ −∞ 2𝜋 (C.39) The stress and displacement components can also be expressed by the function G(𝜉, y) and its derivatives It follows: ∞ d2 G(𝜉, y)e−i𝜉x dx, 𝜎xx (x, y) = √ ∫ dy 2𝜋 −∞ ∞ 𝜎yy (x, y) = √ 𝜉 G(𝜉, y)e−i𝜉x dx, ∫ −∞ 2𝜋 ∞ d 𝜏xy (x, y) = √ 𝜉 G(𝜉, y)e−i𝜉x dx ∫ dy 2𝜋 −∞ (C.40) Appendix C Integral Transforms in Elasticity and ∞ u(x, y) = √ 2𝜋 ∫−∞ ∞ v(x, y) = √ 2𝜋 ∫−∞ ] [ ( ) −i𝜉x i d2 C11 G(𝜉, y) − C12 𝜉G 𝜉, y e dx, 𝜉 dy [ ] ( ) −i𝜉x d3 d C11 G(𝜉, y) − (C12 + C66 )𝜉 G 𝜉, y e dx 𝜉2 dy dy (C.41) C.6 Application of the Hankel Transformation to Spatial, Rotation-Symmetric Elasticity Problems The application of the Hankel transformation with respect to one of the variables allows the partial differential equation for the stress function F(r, z) to be reduced to an ordinary differential equation This partial differential equation is obtained by introducing the Hankel transformation of zero order with respect to the coordinate r to a common differential equation in z, in which the role of the coordinate r is represented by a parameter 𝜉: ∞ G(𝜉, z) = H0 { LR [F(r, z)], r → 𝜉 } = ∫0 F(r, z)rJ0 (𝜉r)dr The procedure shows parallels to the plane problem when the Fourier transformations used there are replaced by the Hankel transformation For the function F(r, z), < r < ∞ the formula is valid: ∞ LR [F(r, z)]rJ0 (𝜉r)dr = ∫0 [ ] ∞ d d d − (a + c)𝜉 +𝜉 F(r, z)rJ0 (𝜉r)dr (C.42) ∫0 dz dz2 From the partial differential Eq (C.28) for F(r, z) follows the ordinary differential equation for G(𝜉, z): [ ] d d2 d − (a + c)𝜉 2 + 𝜉 G(𝜉, z) = (C.43) dz dz The integration of (C.43) provides: G(𝜉, z) = A(𝜉)e−q1 𝜉 z + B(𝜉)e−q2 𝜉 z + C(𝜉)eq1 𝜉 z + D(𝜉)eq2 𝜉 z (C.44) 349 351 Index a absolute lowest mass stress at solid length 230 angle definitions 162 chamber 160 toe-in 160 anisotropic creep constant shear strain 321 uniaxial strain 321 b bending optimization 121 c camber change due to flexibility of rear axis 187 car body bending stiffness 183 dive 183 lateral stiffness 183 local stiffness 183 squat behavior 183 twist stiffness 183 Castigliano’s method 225 compliance steer angle 185 composite material effective modules engineering constants non-woven fabric 334 unidirectional 334 volume fractions composite twisted driveshaft Greenhill’s problem 141 torsional buckling 141 conical composite spring 236 anisotropic shell equations 236 axial force 239 circumferential elastic modulus 238 circumferential moment 238 deformed state 235 effective circumferential elastic modulus, orthotropic 242 extesional and flexural energies 238 free state 232 optimization of stored energy 239 optimizazion of orthotropic spring 241 optimization, orthotropic 243 spring force, orthotropic 243 variational principle 237 constitutive law for laminates anisotropic laminate with coupling 17 anisotropic laminate without coupling 17 Kelvin’s notation 15 orthotropic laminate without coupling 17 Voigt’s notation 14 coordinate system earth fixed 161 horizontal 162 local 161 vehicle 161 vehicle fixed 162 wheel 162 Design and Analysis of Composite Structures for Automotive Applications: Chassis and Drivetrain, First Edition Vladimir Kobelev © 2019 John Wiley & Sons Ltd Published 2019 by John Wiley & Sons Ltd Companion website: www.wiley.com/go/kobelev/automotive_suspensions 352 Index cornering, rollover and suspension isolation 167 coupling constant roll angle to steer angle 195 torsion angle to steer angle 194 twist angle to steer angle 200 crack bridging 60 energy release rate 61 extension resistance 61 fibers rupture 62 fracture toughness 61 matrix rupture 61 creep 319 anisotropic creep function 319 equivalent anisotropic creep stress 321 Norton–Bailey, anisotropic 321 phenomenological hereditary laws 318 plastic potential, Mises–Hill 320 strain, deviatoric component 320 stress, deviatoric component 320 cross-section bending moment 219 circular 226, 273 elliptic 226 rectangular 226 cyclic load stress ratio 97 d debonding ideal adhesion, free state A 77 ideal contact, stressed state C 78 length of the cylindrical crack in matrix 78, 84 matrix and fibers 75 partial debonding, stressed state E 79 partial debonding, stressed state D 78 stress intensity factor 92 total debonding, stressed state B 78 degrees of freedom of vehicle 163 directrix 132, 232, 290 drapability 334 dynamic stability method 265 e energy release rate fibers rupture 62 matrix rupture 62 equivalent beam 255 bending moment 258 Bernoully-type model 255 centerline 255 deflection, dimensionless 263 degree of compression 263 displacement due to bending moment 256 due to shear force 256 dynamic equations 258 external force 258 length, dimensionless 263 meander spring 292 shear force 258 slenderness ratio 263 static equations 257 stiffness bending 257 shear 257 Timoshenko-type model 255 torque 258 f Failure criterion compatibility conditions 32 composites 22 Cuntze 36 delamination criterion 35 differentiating criterion 33 fiber break 33 Goldenblat–Kopnov 27 Hashin 33 intermediate fiber break criterion 33 isotropic homogenous materials 21 maximum averaged stresses 23 Mises–Hill, Kelvin’s notation 26 Mises–Hill, pressure sensitive, Kelvin’s notation 26 Mises–Hill, pressure sensitive, Voigt’s notation 26 Mises–Hill, Voigt’s notation 24 Puck 35 tensor and polynomial 24 Index tensor-polynomial 27 Tsai–Wu 30 Zacharov 30 fatigue crack propagation 96 initiation 96 Paris–Erdogan law 96, 98 stages of fracture 96 fibers aramid 331 axial forces 50 axial stresses 50 carbon 331 cross-section 50 glass 331 pull-out 50 elastic region 54 plastic region 54 rupture 59 shear stress on interface 51 volume concentration 50 Filon problem contact fiber to matrix 85 infinite cylinder, fiber material 84 infinite cylindrical cavity in matrix 85 flexibility constant front axle 194 rear axle 195 forces and moments, on vehicle 169 front wheel steer 166 Ackermann angle 185 g generatrix 132, 232, 290 geometry parameter 97 h helical composite spring buckling 265 closed-coiled 220 compressed length 223 creep 324 critical compression 265 design value for spring constant 228 dimensionless length 263 dynamical equations 231 effective constants 256, 263 elastic energy 225 energy capacity 223 fundamental frequencies, axial/twist vibrations, coupled 232 fundamental frequencies, axial/twist vibrations, decoupled 232 fundamental frequencies, transverce 263 ideal stress at full stroke 228 ideal stress at solid height 228 mass 224 non-cylindrical 225 mass 226 spring rate 226 optimization problem 228 rate 224 relaxation 324 released length 223 slenderness ratio 263, 265 solid length 223 spring rate axial 226 twist 226 stiffness 224 symmetric stacking 264 buckling 266 frequency 264 torsion 219 travel 224 work of applied forces 223 helical composite spring wire length 225 mass 225 optimal diameter 228 hereditary mechanics 317 Hooke’s law isotropic materials Kelvin’s notation tensor notation tetragonal elastic syngony transversal isotropic materials 341 unidirectional composite material, Kelvin’s notation unidirectional composite material, Voigt’s notation Voigt’s notation 353 354 Index i initial length of crack 98 sinusoidal parameter 299 spring rate, compression 292 modulus, shear 224 l layered anisotropic plate anisotropic laminate with coupling 17 without coupling 17 orthotropic laminate 17 layered anisotropic thin-walled beam closed cross-section 132 open cross-section 18 leaf composite spring 269, 297 absolute lowest mass 275 circular cross-section 273 rectangular cross-section 273 creep 326 cross-section, rectangular 272 elastic energy stored 272 mass 272 relaxation 325 simply supported 271 spring rate 272 ultimate bending stress 272 wheel-guided transverse 280 leaf-tension composite spring 275 force 277 stress 278 length of plastic zone 55 m material density 224 matrix termoplastics 333 termosetting plastics 333 maximization of loading capacity 113 meander spring bending stiffness 303 buckling stability 305 directrix 294 effective constants 293 equivalent beam 292 fiber orientation 293 multiarc design 294 multiarc parameter 295 sinusoidal design 299 n natural frequency 263 o optimization elastic energy density 110 specific 110 elastic energy maximum 106 minimum 106 plate pure bending 124 specific ultimate 118, 119 strength, maximum 112 strength, maximum, plate pure bending 124 ultimate elastic energy 117 plate pure bending 125 ultimate stored energy 116 plate pure bending 124 ultimate strength 114 specific 114 oversteer due to flexibility of rear axis 187 p Pacejka magic formula 169 path radius Ackermann radius 184 flexible rear axis 185 rigid rear axis 184 plastic potential Mises–Hill, anisotropic 320 propagation function unified 98 propagation law common form 97 Paris–Erdogan 98 r rear thread 192, 202, 282 rear track width 192, 202, 282 rear wheel steering induced angle due to side forces 187 Index relaxation function bending 326 helical spring 324 twist, circular shaft 323 roll angle 162, 193 center 160 spring rate 193, 201, 201, 282 rolling moment 192, 193 rotary motion effect 255 rotationally symmetric crack 65 rupture plane 51 s section modulus torsion 225 separation surface 52 sideslip angle coupling 196 of vehicle 173 skew compliance of tire 193 stiffness of tire 170, 193 slip 167 angle of wheel 167, 170, 173, 193 circumferential 167 longitudinal 168 𝜇-slip curve 168 total 168 transverse 168 spring rate, axle lateral 202 roll 201, 282 vertical 281 steer compliance 185 steer angle Ackermann 197 coupling 196 and flexible stiffness 197 flexible side stiffness of axles 197 full compensation 199 Stiffness configuration Circumferentially Asymmetric (CAS) 18, 190, 205 Circumferentially Uniform (CUS) 18, 190, 205 stiffness constant front axle 194 rear axle 195 stress basic 224 uncorrected 224 stress intensity factor debonding 94 maximum 97 mean value 97 minimum 97 range 97 stress per cycle maximum 97 minimum 97 range 97 surface developable 290 ruled 290 scroll 290 suspension independent axle 155 kinematics 159 semi-independent axle 155 solid axle 155 t textiles 334 thin-walled beam with closed cross-section boundary conditions 138 ideal fixing 138 ideally free end 138 intermediate type 140 cases of symmetry 140 directrix 132 elastic energy 136 equations of deformation 135 equations of equilibrium 136 generatrix 132 governing equations 137 kinematic hypothesis 133 torque, optimization 121 track radius absolutely rigid suspension 187 flexible rear axle 187 trailing arm length 202 transformation of Hooke’s law 355 356 Index transformation of Hooke’s law (contd.) 2nd rank matrix notation 12 2nd rank tensor notation 13 4th rank tensor notation 11 Kelvin’s notation 13 Voigt’s notation 12 transformation of Mises–Hill criterion Kelvin’s notation 37 Voigt’s notation 38 transformation of plate stiffness 2nd rank matrix notation 15 2nd rank tensor notation 16 Kelvin’s notation 16 Voigt’s notation 15 transverse leaf composite spring elastic energy stored, roll 282 elastic energy, symmetric 280 roll spring rate 282 vertical spring rate 281 transverse vibrations, 258 twist angle of cross-member 200 twist-beam axle camber stiffness 202 countersteering, anisotropic 191 countersteering, mechanical 189 lateral stiffness 202 twisted shaft, relaxation 322 v vehicle dynamics bicycle model 170, 186 linear 174 classification of self-steering behavior 176 dedicated bicycle model 187, 191 dedicated single-track model 187, 191 oscillatory instability 181 oscillatory stability 180 required steer angle 175 single-track model 170, 186 linear 174 steer angle gradient 175 viscoelasticity 318 w wheel base 174 wire cross-section area 256 circular 256 mass per unit length 256 stiffness bending in binormal direction 256 in normal direction 256 twist 256