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Advances in Geophysical and Environmental Mechanics and Mathematics

Ishan Sharma

Shapes and Dynamics of Granular Minor PlanetsThe Dynamics of Deformable Bodies Appliedto Granular Objects in the Solar System

Advances in Geophysical and EnvironmentalMechanics and Mathematics

Series editors

Kolumban Hutter, Zürich, SwitzerlandHolger Steeb, Stuttgart, Germany

More information about this series at http://www.springer.com/series/7540

Ishan Sharma

Shapes and Dynamicsof Granular Minor PlanetsThe Dynamics of Deformable BodiesApplied to Granular Objectsin the Solar System

123

Ishan SharmaMechanics and Applied Mathematics GroupDepartment of Mechanical EngineeringIndian Institute of Technology KanpurKanpur, Uttar PradeshIndia

ISSN 1866-8348 ISSN 1866-8356 (electronic)Advances in Geophysical and Environmental Mechanics and MathematicsISBN 978-3-319-40489-9 ISBN 978-3-319-40490-5 (eBook)DOI 10.1007/978-3-319-40490-5

Library of Congress Control Number: 2016943448

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This book is dedicated to Indian Astronomy,whose science continues to surprise.

Foreword

This book was born when advances in astronomical techniques permitted, forexample, the shapes and spins of asteroids to be determined using radar (see, e.g.,Ostro et al. 2002), just as interest in the mechanics and physics of granular materialswas being renewed in the mid-1980s. The much-publicized tidal fragmentation ofcomet Shoemaker–Levy in 1994 as it passed Jupiter stimulated the development ofnumerical simulations. Asteroids, by their sheer number and variety, provide anatural laboratory in which the translational and rotational motion of deformablesolid bodies can be investigated. Relatively high-resolution dynamic imaging madeit clear that at least some and, perhaps, many near-Earth asteroids were notmonolithic rocks, but likely consisted of discrete solid elements held together bytheir mutual gravity. The advances in the mechanics of granular materials provideda means to treat the collisional interactions between the elements that transferredmomentum and dissipated the energy associated with them. The advances in bothsubjects made it possible for the two of us to develop our common interest in theirdynamics.

Ishan Sharma, a gifted graduate student in the Department of Theoretical andApplied Mechanics at Cornell University, was the intellectual agent of thisdevelopment. He shared our interest and enthusiasm for the subject and we bene-fited from his intelligence and energy. Starting with his doctoral work with us,Sharma restricted attention to affine deformations of extended bodies and employeda volume-averaged approach to determine their equations of motion. In doing this,he followed Chandrasekhar (1969), who introduced this technique in his famouswork on the equilibrium shapes of spinning fluid ellipsoids. In a series of researchpapers in Icarus initially with us and, later, independently, Sharma also adopted andmade more transparent elements of the dynamics of deforming bodies introducedby Cohen and Muncaster (1988). Sharma (2004) first determined the equilibriumand failure of a spinning asteroid and placed existing results by Holsapple (2001) ina dynamical context. Sharma also phrased and numerically solved the equationsthat describe planetary fly-bys of asteroids of a less tightly packed granular

ix

aggregate in which the elements interacted through collisions, so as to obtain resultssimilar to those of the molecular dynamics simulations of Richardson et al.(1998) and others.

Since completing his dissertation (2004), Sharma has extended the results on tothe equilibrium and failure of an exhaustive list of asteroids and satellites. He hasalso made important steps in characterizing the stability of their equilibrium states.Finally, he has completed a refined analysis of planetary fly-bys. These elements arecollected in this volume. However, the resulting volume is much more than this. Itis, also, a compact introduction to continuum mechanics of deformable bodies and,further, a rather complete treatment of the dynamics of self-gravitating deformablebodies, when they are treated, in first approximation, as having uniform materialproperties and deforming homogeneously. This makes the volume, on the one hand,a valuable general introduction to the dynamics of deformable bodies and, on theother hand, a detailed treatment of the multitude of objects in the solar system forwhich dynamics is likely to be coupled with deformation. We are proud to havebeen involved in the beginning of the scholarly activity that led to this manuscript.We believe it to be a worthy successor to the classic work of Chandrasekhar (1969).

Ithaca, NY, USA Jim JenkinsSeptember 2015 Joe Burns

References

S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale Univ. Press, New Haven, CT, 1969)H. Cohen, R.G. Muncaster, The Theory of Pseudo-Rigid Bodies (Springer-Verlag, New York,

1988)K.A. Holsapple, Equilibrium configurations of solid cohesionless bodies. Icarus 154, 432–448

(2001)S.J. Ostro et al. Asteroid radar astronomy, in Asteroids III ed. by W.F. Bottkeet al. (U. Arizona

Press, 2002), pp. 151–168D.C. Richardson, W. F. Bottke Jr., S.G. Love, Tidal distortion and disruption of Earth–crossing

asteroids. Icarus 134, 47–76 (1998)I. Sharma, Rotational dynamics of deformable ellipsoids with application to asteroids. Ph.D.

dissertation, (Cornell University, 2004)

x Foreword

Preface

Starting about twenty years ago, astronomers gradually realized that many of thesmall bodies in the solar system (asteroids, comets and satellites) are rubble piles,i.e., granular aggregates. The first unequivocal evidence for this came when thecomet Shoemaker–Levy nine broke apart, apparently by tides, into dozens of piecesas it passed close to Jupiter. Numerical simulations of self-gravitating granularaggregates were developed and they exhibited such fragmentation during closeplanetary encounters. Around the same time, researchers recognized that very fewasteroids were found with spin periods of less than a few hours, and that this couldbe understood simply as the consequence of the fragility of fast-spinning bodies tocentrifugal breakup. Furthermore, other asteroids and a few close-on satellites of thegiant planets were observed in radar “images” and spacecraft images, respectively,to have smooth elongated shapes, suggesting rotational and tidal distortion.

Around the same time, the masses of dozens of asteroids began to be measured,usually by observing the orbital periods of binary asteroids, or by mutual gravita-tional perturbations of distant asteroids on Mars or another asteroid, or by space-craft flybys. For those asteroids, comets and a few satellites that had known sizes,their densities were immediately available. More often than not, these measureddensities were remarkably low, sometimes less than 0.5 g/cm3 for comets and smallsatellites, or often 1–2 g/cm3 for asteroids and satellites. Because the likely con-stituents of these bodies (water, ice and rock) have greater densities, the low bulkdensities required significant pore space and, accordingly, implied granularaggregates held together primarily by self-gravity, rather than monolithic rocks.Such a loose character is not unexpected for objects that accreted gravitationally ina cold environment.

This book investigates the equilibrium, stability and dynamics of these rubblesolar-system bodies. It is clear that any careful investigation will need to considerthese bodies as objects with finite extent, and not as mere point masses, and as agranular medium, distinct in its constitutive response from solids and fluids. Thisbook pays particular attention to these aspects.

xi

In this book, we develop a framework for analyzing the dynamics of rotatingcomplex materials; this is predicated on the systematic approximations of thesystem’s kinematics. We apply the method to investigate rotating andself-gravitating granular aggregates in space. Here, we limit the kinematicapproximation to, at most, an affine deformation, so that the most general shape thatan object can take is that of an ellipsoid. Necessary governing equations may beobtained by a variety of methods, but we prefer to follow the virial method, orvolume-averaging, employed by Chandrasekhar (1969). We do this primarily forhistorical continuity and for greater general familiarity with that method, but alsobecause the current research was motivated to a great extent by Chandrasekhar’streatise that explored similar questions in the context of inviscid fluids.

The constitutive model that we employ depends on the situation. For example,when considering equilibrium, or its stability, the granular aggregate is modeled asa rigid-perfectly-plastic material obeying a pressure-dependent yield criterion, e.g.,the Drucker–Prager yield criterion, and deforming post-yield as per an appropriateflow rule. However, when studying the disruptive effects of a tidal flyby, theaggregate is taken to be an ensemble of dissipative spheres whose macroscopicbehavior is determined through an application of kinetic theory. The currentframework has the advantage that it allows us to improve the kinematic approxi-mation in a structured manner as well as to explore a wide variety of constitutivelaws.

The book is divided into four parts. The first part introduces the necessarymathematics and continuum mechanics, as well as, describes affine dynamics thatforms the basis of all subsequent development. Part II investigates the equilibriumof rubble asteroids, satellites, and binaries, and applies it to known or suspectedcases. Equilibrium, here, refers to possible ellipsoidal shapes that a rubble asteroidcan take, and to both shape and orbital separation for granular satellites andbinaries. In Part III, we develop a linear stability criterion specifically for rotatinggranular aggregates, which is then applied to the equilibria obtained in Part I.Finally, in Part IV we provide a pair of examples of dynamical evolution. Theserelate to the disruption and possible re-agglomeration of rubble piles during tidalflybys.

Finally, I confess to some nervousness. It is dangerous to write a book that maybe viewed by some as a would-be successor to Chandrasekhar’s classic treatise.Followers in the footsteps of giants risk sinking or getting lost. But then, there areworse ways to go.

Kanpur, India Ishan SharmaSeptember 2015

Reference

S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale Univ. Press, New Haven, CT, 1969)

xii Preface

Acknowledgements

This book would not have been possible without the help and guidance of Profs.Joseph A. Burns and James T. Jenkins of Theoretical & Applied Mechanics atCornell University. It was they who introduced me to this research area during myPh.D., and later inspired me to write this book. They also read through the book andprovided critical feedback that has helped improve this work significantly. The partsof this book that may appeal to the reader are directly a result of their academicinfluence; the errors, of course, are entirely my own.

I am also grateful to Prof. Kolumban Hutter who, as editor, gave me theopportunity to write this book as part of the Springer series on Advances inGeophysical and Environmental Mechanics and Mathematics. Not only did hisregular prompting help me complete this book, but his thorough review of the finalmanuscript was extremely helpful.

It is also a pleasure to thank many colleagues from the Mechanics & AppliedMathematics Group (www.iitk.ac.in/mam) at the Indian Institute of TechnologyKanpur: S. Mahesh, Shakti Singh Gupta, Sovan Lal Das, Anindya Chatterjee,Chandrashekhar Upadhyay, Prakash M. Dixit, P. Venkitinarayanan, Sumit Basu,Pankaj Wahi, Anurag Gupta, Mahendra K. Verma, V. Shankar, and Basant LalSharma. Several of them have, possibly unknowingly, helped this work since ourtime together at the Department of Theoretical & Applied Mechanics, Cornell.Other member of the larger IIT Kanpur fraternity whose support has been criticalthrough the last decade while I was writing this book include, in no particular order,Profs. Asok Mallik, K. Muralidhar, (Late) Himanshu Hatwal, V. Eswaran, B.N.Banerjee, Amitabha Ghosh, C. Venkatesan, P.K. Panigrahi, N.N. Kishore,Muthukumar T., Jayant K. Singh, Koumudi P. Patil, Sunil Simon, Preena Samuel,Akash Anand, Shikha Prasad, Anandh Subrahmaniam, Brijesh Eshpuniyani, Dr.Bhuvana T., Shirolly Anand, and Dr. Ashish Bhateja. I would also like to mentionthe help and support of Satya Prakash Mishra, Bharat Singh, Ram Lakhan, andLakshmi.

xiii

There are several others who have, directly or indirectly, contributed towards thecompletion of this book. They are, again in no particular order: Dr. Anil Bhardwaj(Director, Space Physics Laboratory, Vikram Sarabhai Space Centre, Indian SpaceResearch Organization); Prof. Devang V. Khakhar (IIT Bombay); Prof. IndranilManna (IIT Kharagpur); Prof. C.-Y. Hui, Prof. Subrata Mukherjee, Prof. Andy L.Ruina and Prof. Timothy J. Healey (T&AM, Cornell U.); Prof. S.V.S. Murty andProf. Debabrata Banerjee (PLANEX, Physical Research Laboratory, Ahmedabad);Prof. E. John Hinch, Prof. M. Grae Worster, Prof. Herbert E. Huppert and Prof.John E. Willis (DAMTP, U. Cambridge); Prof. V. Kumaran and Prof. Prabhu Nott(IISc. Bangalore); Prof. P. Chandramouli and Prof. S. Narayanan (IIT Madras);Prof. Sanjeev Sanghi and Prof. S Veeravalli (IIT Delhi); Dr. Alessandro Morbidelli(Observatory of Nice); Prof. Daniel J. Scheeres (U. Colorado at Boulder); Prof.Derek C. Richardson (U. Maryland, College Park); Dr. Anthony R. Dobrovolskis(Lick Observatory, UC Santa Cruz); Dr. Michael Efroimsky (US NavalObservatory); Swati Meherishi (Springer, India); Dr. Annett Buettner, AbiramiPurushothaman, Sudeshna Das and other members of the Springer team.

I would also like to acknowledge the following grants that helped sustainresearch during the period that the book was being written: Department of Science& Technology (DST) grant SR/S3/CE/0053/2010 (G), and grantPRL/ADM-AC/PB/2013-14 under the PLANEX program from the PhysicalResearch Laboratory.

Finally, no goal may possibly be reached without the best wishes and prayers ofone’s immediate and extended family, and close friends, and I thank each and everyone of them. I mention my late grandparents – Prof. Govind Chandra Pande, Major(retd.) Gokul Chandra Sharma and Smt. Sarah Sharma – who would have been thehappiest to see this book in print, but desist from mentioning anyone else by namein order to save the publisher ten or so pages.

xiv Acknowledgements

Contents

Part I Toolbox

1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Second-Order Tensors. . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Third- and Fourth-Order Tensors . . . . . . . . . . . . . . . 9

1.4 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.1 Gradient and Divergence. Taylor’s Theorem. . . . . . . . 111.5.2 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . 121.5.3 Time-Varying Fields . . . . . . . . . . . . . . . . . . . . . . . . 12

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Motion in a Rotating Coordinate System. . . . . . . . . . . . . . . . . 16

2.2.1 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . 17

2.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Simple Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Example 1: Pure Rotation . . . . . . . . . . . . . . . . . . . . 202.4.2 Example 2: General Rigid Body Motion . . . . . . . . . . 212.4.3 Example 3: Homogeneous Motion . . . . . . . . . . . . . . 222.4.4 Example 4: Affine Motion . . . . . . . . . . . . . . . . . . . . 22

2.5 Local Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.1 Strain. Surface Change. Volume Change . . . . . . . . . . 242.5.2 Rate of Local Motion . . . . . . . . . . . . . . . . . . . . . . . 252.5.3 Transport Theorem. Mass Balance . . . . . . . . . . . . . . 27

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2.6 Further Analysis of Simple Motions . . . . . . . . . . . . . . . . . . . . 292.6.1 Example 3: Homogeneous Motion . . . . . . . . . . . . . . 292.6.2 Example 4: Affine Motion . . . . . . . . . . . . . . . . . . . . 31

2.7 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.8 Moments of the Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Power Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.10 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10.1 Rigid-Perfectly Plastic Materials . . . . . . . . . . . . . . . . 382.10.2 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . 44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Affine Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Governing Equations: Structural Motion . . . . . . . . . . . . . . . . . 53

3.2.1 Statics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Moment Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 Gravitational-Moment Tensor. . . . . . . . . . . . . . . . . . 573.3.2 Tidal-Moment Tensor . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Governing Equations: Orbital Motion . . . . . . . . . . . . . . . . . . . 623.4.1 Circular Tidally-Locked Orbits. . . . . . . . . . . . . . . . . 64

3.5 Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5.1 Angular Momentum Balance . . . . . . . . . . . . . . . . . . 673.5.2 Power Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.3 Total Energy of a Deformable Gravitating

Ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Part II Equilibrium

4 Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Average Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.2 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . 784.2.3 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Equilibrium Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.3.1 Oblate Asteroids. . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Prolate Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.3 Triaxial Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5.1 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . 884.5.2 Near-Earth Asteroid Data . . . . . . . . . . . . . . . . . . . . 884.5.3 Equilibrium Shapes. . . . . . . . . . . . . . . . . . . . . . . . . 89

xvi Contents

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.1 Average Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2.2 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.3 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . 985.2.4 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3 Example: Satellites of Oblate Primaries. . . . . . . . . . . . . . . . . . 101

5.3.1 Bð0ÞP and Bð1ÞP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3.2 The Orbital Rate ω00E . . . . . . . . . . . . . . . . . . . . . . . . 1035.3.3 Equilibrium Landscape . . . . . . . . . . . . . . . . . . . . . . 103

5.4 Application: The Roche Problem . . . . . . . . . . . . . . . . . . . . . . 1105.4.1 Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1115.4.2 Moons of Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.4.3 Alternate Yield Criteria and Previous Work . . . . . . . . 113

5.5 Application: Satellites of the Giant Planets . . . . . . . . . . . . . . . 1165.5.1 Satellite Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.5.2 Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5.3 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120


6 Binaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2.1 Average Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.2.2 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.2.3 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . 1326.2.4 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 Example: Prolate Binary System . . . . . . . . . . . . . . . . . . . . . . 1356.3.1 Bð0Þ, Bð1Þ and B(2) . . . . . . . . . . . . . . . . . . . . . . . . . 1366.3.2 The Orbital Rate ωB . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4 Equilibrium Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.5 Example: Fluid Binaries and the Roche Binary

Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.6 Application: Binary Asteroids . . . . . . . . . . . . . . . . . . . . . . . . 148

6.6.1 216 Kleopatra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.6.2 25143 Itokawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.6.3 624 Hektor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.6.4 90 Antiope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157


Contents xvii

Part III Stability

7 Granular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.2.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 1657.2.2 Energy Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 1677.2.3 Compatibility and Normality . . . . . . . . . . . . . . . . . . 1727.2.4 Stability at First-Order . . . . . . . . . . . . . . . . . . . . . . 1737.2.5 Stability at Second-Order . . . . . . . . . . . . . . . . . . . . 1757.2.6 Stability to Finite Perturbations . . . . . . . . . . . . . . . . 175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

8 Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1798.2 Asteroid Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 1828.3.2 Energy Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.4 Example: Rubble-Pile Asteroids . . . . . . . . . . . . . . . . . . . . . . . 1878.4.1 Compatible Perturbations. . . . . . . . . . . . . . . . . . . . . 1898.4.2 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.4.3 Stability to Finite Perturbations . . . . . . . . . . . . . . . . 193

8.5 Application: Near-Earth Asteroids . . . . . . . . . . . . . . . . . . . . . 1968.5.1 Near-Earth Asteroid Data . . . . . . . . . . . . . . . . . . . . 1968.5.2 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.5.3 Planetary Encounters . . . . . . . . . . . . . . . . . . . . . . . 200


9 Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.2 Satellite Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.2.1 Structural Deformation . . . . . . . . . . . . . . . . . . . . . . 2089.2.2 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

9.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 2119.3.2 Energy Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.4 Example: Rubble-Pile Planetary Satellites . . . . . . . . . . . . . . . . 2189.4.1 Orbital Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.4.2 Structural Stability . . . . . . . . . . . . . . . . . . . . . . . . . 2219.4.3 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.4.4 Stability to Finite Structural Perturbations . . . . . . . . . 225

9.5 Application: Planetary Satellites . . . . . . . . . . . . . . . . . . . . . . . 2289.5.1 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.5.2 Stability to Finite Structural Perturbations . . . . . . . . . 233

xviii Contents


10 Binaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.2 Binary Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.2.1 Structural Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.2.2 Orbital Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

10.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24210.3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 24310.3.2 Admissible Perturbations . . . . . . . . . . . . . . . . . . . . . 24510.3.3 Energy Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . 247

10.4 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.5 Example: Planar Binary with Near-Spherical,

Rigid Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.6 Example: Rigid Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

10.6.1 Orbital Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . 25410.6.2 Structural Kinetic Energy . . . . . . . . . . . . . . . . . . . . 25510.6.3 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25610.6.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.7 Example: Rubble-Pile Binaries. . . . . . . . . . . . . . . . . . . . . . . . 26110.7.1 Orbital Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . 26110.7.2 Structural Kinetic Energy . . . . . . . . . . . . . . . . . . . . 26310.7.3 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.7.4 Local Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26510.7.5 Stability to Finite Structural Perturbations . . . . . . . . . 270

10.8 Application: Near-Earth Binaries . . . . . . . . . . . . . . . . . . . . . . 27210.8.1 216 Kleopatra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27210.8.2 25143 Itokawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27510.8.3 624 Hektor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27610.8.4 90 Antiope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27610.8.5 Stability to Finite Structural Perturbations . . . . . . . . . 277


Part IV Dynamics

11 Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

11.2.1 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . 28811.2.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

11.3 Example: Prolate Asteroids . . . . . . . . . . . . . . . . . . . . . . . . . . 29011.3.1 Switching States. . . . . . . . . . . . . . . . . . . . . . . . . . . 29111.3.2 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . 292

Contents xix

11.3.3 Application: Equilibrium Shapes . . . . . . . . . . . . . . . 29311.3.4 Discussion: Dynamics of a Homogeneously

Deforming Rigid—Plastic Ellipsoid . . . . . . . . . . . . . 29811.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

12 Tidal Flybys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30712.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30712.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30812.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31012.4 Material Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

12.4.1 A Kinetic Theory Based Model for LooseGranular Aggregates . . . . . . . . . . . . . . . . . . . . . . . . 312

12.4.2 Transition Criterion . . . . . . . . . . . . . . . . . . . . . . . . 31512.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

12.5.1 Outcomes with the Tensile Criterion . . . . . . . . . . . . . 31712.5.2 The Roche Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 31912.5.3 Further Analysis of Flybys . . . . . . . . . . . . . . . . . . . 32112.5.4 Outcomes with the Mohr-Coulomb Criterion . . . . . . . 32812.5.5 The Effect of Rotation Direction. . . . . . . . . . . . . . . . 32912.5.6 Different Initial Rotation Rates . . . . . . . . . . . . . . . . . 333


Appendix A: Rate of Change of the Gravitational Shape Tensor . . . . . . 337

Appendix B: The Tidal Shape Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 341

Appendix C: Rate of Change of the Tidal Shape Tensor. . . . . . . . . . . . 345

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

xx Contents

Part IToolbox

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2 Part I: Toolbox

Chapter 1Mathematical Preliminaries

In this chapter, we quickly summarize necessary tensor algebra and calculus, andintroduce the notation employed in this text. We assume familiarity with matrixalgebra and indical notation. More information may be obtained from standard textssuch as Strang (2005) or Knowles (1998).

1.1 Coordinate Systems

We will exclusively employ right-handed cartesian coordinate systems. The coordi-nate system of choice may be stationary, translating, or rotating, or both. We willemploy calligraphic capital letters to identify coordinate systems. In this text we willtypically employ three coordinate systemsO,P andS with associated unit vectorsêi, ê

′i and ê

′′i , respectively. This will be indicated by, e.g.,

{O, êi

}.

1.2 Vectors

A vector a is represented as thus. The components of a in O will be denoted by aiand inP by a′i, so that we have the identities

a = (a · êi)êi = aiêi = (a · ê′i)ê′i = a′iê′i, (1.1)

where ‘ · ’ is the usual vector dot product. The magnitude or norm of a is

|a| = (a · a)1/2 = (aiai)1/2 =(a′ia

′i

)1/2. (1.2)

© Springer International Publishing Switzerland 2017I. Sharma, Shapes and Dynamics of Granular Minor Planets,Advances in Geophysical and Environmental Mechanics and Mathematics,DOI 10.1007/978-3-319-40490-5_1

3

4 1 Mathematical Preliminaries

We now collect several useful formulae:

a · b = |a||b| cos θ = aibi, (1.3a)a × b = −b × a = |a||b| sin θ êc = εijk êiajbk, (1.3b)

a · (b × c) = b · (c × a) = c · (a × b) = εijkaibjck (1.3c)and a × (b × c) = (a · c)b − (a · b)c = εijkεklmêiajblcm. (1.3d)

where θ is the angle between vectors a and b, êc is a unit vector normal to the planecontaining a and b, and εijk is the alternating tensor defined by

εijk =⎧⎨

⎩

1 if i, j and k are an even permutation−1 if i, j and k are an odd permutation0 otherwise;

(1.4)

cf. Sect. 1.3.2.We will typically limit ourselves to three-dimensional vectors.

1.3 Tensors

Afirst-order tensor is simply a vector.A second-order tensor is a linear transformationthat maps a vector to another vector. Third- and fourth-order tensors relating lower-order tensors to other lower-order tensors may be similarly defined.

1.3.1 Second-Order Tensors

A second-order tensor A is probed by its action ‘ · ’ on vectors. We employ the samesymbol as for the dot-product of vectors because of similarities between the twooperations. We define the resultant b of A’s operation, specifically a right-operation,on a by

b = A · a.

Similarly a left-operation may be defined. As with vectors, we will typically limitourselves to second-order tensors that operate on and result in three-dimensionalvectors.

The addition A + B and multiplication A · B of two tensors A and B result intensors C and D, respectively, that are defined in terms of how they operate on somevector a, i.e.,

(A + B) · a = C · a := A · a + B · a (1.5a)and (A · B) · a = D · a := A · (B · a). (1.5b)

1.3 Tensors 5

It is understood that the two tensors A and B relate vectors belonging to the same set.To better understand tensors, it is useful to generalize the concept of a unit vec-

tor to a tensorial basis. Such a generalization is furnished by the tensor producta ⊗ b of two vectors a and b. The entity a ⊗ b is a second-order tensor that can acton another vector c in two different ways – the left- and right- operations – to yieldanother vector:

(a ⊗ b) · c = (c · b)a and c · (a ⊗ b) = (c · a)b , (1.6)

where the ‘·’ on the left-hand sides denotes a tensor operation, and the usual vectordot product on the right-hand sides. Contrasting the computation

a ⊗ b = aiêi ⊗ bj êj = (aibj)êi ⊗ êj (1.7)

with (1.1), suggests that a tensorial basis may be constructed by taking appropriateorder tensor products of the unit vectors. We note that the above represents the linearcombination of êi ⊗ êj. Thus, a second-order tensorial basis in the coordinate systemO is given by the nine unit tensors êi ⊗ êj. A second-order tensor A may then bewritten as

A = Aij êi ⊗ êj, (1.8)

in terms of A’s components Aij in O . These components, obtained by appealing to(1.6), are given by the equations

Aij = êi · A · êj, (1.9)

that are reminiscent of analogous ones for vector components; see (1.1).Wewill referto the nine Aij’s as the “matrix of A in

{O, êi

}” denoted by [A]. In another coordinate

system, sayP , the tensorial basis is given by ê′i ⊗ ê′j, whileA′ij = ê′i · A · ê′j constitutethe “matrix of A in

{P, ê′i

}” denoted by [A]′. A second-order tensor’s interactions

with vectors and other second-order tensorsmay be obtained by repeated (if required)application of (1.6). These operations are summarized below:

A · a = Aij êi ⊗ êj · amêm = Aijaj êi, (1.10a)a · A = amêm · Aij êi ⊗ êj = aiAij êj, (1.10b)A · B = Aij êi ⊗ êj · Bmnêm ⊗ ên = AijBjnêi ⊗ ên (1.10c)

and A : B = Aij êi ⊗ êj : Bmnêm ⊗ ên = Aij êi · Bjnên = AijBji, (1.10d)

where the first two operations produce vectors, the next another second-order tensor,and the third a scalar. The double-dot product ‘:’, as its form suggests, denotes asequential application of dot products, as illustrated above. The tensor A’s actionson higher-order tensors may be analogously defined. When there is no confusion,second-order tensors are referred to simply as tensors.


The transpose AT of a tensor A is defined by the following formula

(A · a) · b = a · (AT · b), (1.11)

for any two vectors a and b. The above results in [AT ]ij = Aji = [A]ji. From the abovedefinition of a tensor’s transpose the following identities are easily proved:

(A + B)T = AT + BT and (A · B)T = BT · AT . (1.12)

The trace of a tensor Awith components Aij is obtained by contracting the indicesi and j:

trA = Aii = A11 + A22 + A33. (1.13)

We see below that the trace of a tensor is independent of the coordinate system inwhich it is computed. The following identities regarding transposes are easily proved:

trA = trAT ⇒ trn∏

i=1Ai = tr

(1∏

i=nATi

)

, (1.14a)

trn∏

i=1Ai = tr

(n∏

i=n−kAi ·

n−k−1∏

i=1Ai

)

, 0 � k � n − 1, and0∏

i=1Ai = 1, (1.14b)

where∏n

i=1 Ai = A1 · A2 · · · · · An.A tensor is said to be symmetric ifA = AT , andanti- / skew- symmetric ifA = −AT .

Given an arbitrary tensor A we define its symmetric part

sym A = 12

(A + AT ) , (1.15)

and anti-symmetric part

asym A = 12

(A − AT ) , (1.16)

so that any tensor A may be written as a sum of a symmetric and anti-symmetrictensor

A = sym A + asym A. (1.17)

An anti-symmetric tensor has at most three independent components in any coor-dinate system. Thus, for any anti-symmetric tensor W, it is possible to associate anaxial vector denoted by w with the property that for all vectors b,

W · b = w × b. (1.18)

1.3 Tensors 7

The operations of constructing anti-symmetric tensors from vectors and extract-ing axial vectors from anti-symmetric tensors are denoted by sk w (= W) andax W (= w), respectively. The relationship between w and W may be expressedin indical notation employing the alternating tensor of (1.4):

axW = w = −12εijkWjk êi and skw = W = −εijkwiêj ⊗ êk . (1.19)

Employing (1.3b), it is straightforward to check that the above prescription for axWand skw will satisfy (1.18).

For most tensors, and almost all tensors occurring in this book, it is possible tofind three unit vectors that are simply scaled under that tensor’s operation, i.e., givenA, there (almost always) exist three unit vectors v̂i and correspondingly three scalarsλi, such that

A · v̂i = λiv̂i (no sum). (1.20)

These special vectors v̂i are the eigenvectors of A, and the corresponding scalings λiare A’s eigenvalues. In the coordinate system described by the three eigenvectors, thetensor’s matrix is diagonalized with the tensor’s eigenvalues as the diagonal entries.This simple diagonal nature makes employing the eigen-coordinate system verytempting for computation. Unfortunately, there is no guarantee that the eigenvectortriad are mutually orthogonal, so that the coordinate system they describe may notbe cartesian. However, if the tensor is symmetric, it is always possible to diagonalizeit, and, moreover, the eigenvectors are orthogonal, so that the coordinate systemthey describe is frequently a convenient operational choice. Thus, given a symmetrictensor S, it is possible to find three eigenvectors v̂i and corresponding eigenvaluesλi, so that S is simply

S =3∑

i=1λiv̂i ⊗ v̂i. (1.21)

The operation of a symmetric S therefore corresponds to a linear scaling along threemutually orthogonal eigen-directions.

If in case all the eigenvalues of a symmetric tensor are non-zero and positive, thesymmetric tensor is said to be positive definite. Finally, it is important to mentionthat for any tensor the number of eigenvalues equals the dimension of the under-lying space, whether or not it is diagonalizable. For example, in three dimensions,every tensor has three eigenvalues even if it doesn’t have three eigenvectors. Theseeigenvalues are either all real, or a mixture of real and complex conjugate pairs.

While the components of a tensor depend on the coordinate system, its eigen-values do not. Therefore, functions of these eigenvalues, called principal invariants,also remain unaffected by the choice of the coordinate system; the number of theseinvariants equaling the dimension of the underlying space. In three dimensions, asecond-order tensor A with eigenvalues λi has the three invariants


IA =3∑

i=1λi = Aii = tr A, (1.22a)

IIA =∑

i �=jλiλj = 1

2

(I2A − IA2

)(1.22b)

and IIIA = =3∏

i=1λi = detA, (1.22c)

where the last invariant represents the determinant of A that may also be computedvia standard formulae after finding A’s matrix in any coordinate system. Finally, asfor vectors, it is possible to measure the magnitude of a tensor, by employing thedouble-dot product ‘ : ’ introduced in (1.10). The norm of a tensor A is defined by

|A| :=√A : AT = √IA2 =

√AijAij. (1.23)

Frequently, and again, for all tensors considered in this book, it is possible todefine associated inverse tensors, i.e., given A taking a to b, the inverse tensor A−1brings b back to a. It is easy enough to see that a tensor and its inverse share the sameeigenvectors, but inverse eigenvalues. Thus, if A has a zero eigenvalue, its inversedoes not exist. The following identities regarding inverses are easily verified:

A · A−1 = A−1 · A = 1, (1.24a)(A · B)−1 = B−1 · A−1, (1.24b)detA−1 = (detA)−1 (1.24c)

and (AT )−1 = (A−1)T = A−T . (1.24d)

An important class of tensors thatwill occur frequently in the text is the orthogonaltensor Q that has the property that given any vector a,

|Q · a| = |a|, (1.25)

i.e., Q preserves a vector’s length. From this the following properties follow:

Q−1 = QT and detQ = ±1. (1.26)

In applications to follow, all orthogonal tensors will have determinant one. Suchproper orthogonal tensors are called rotation tensors. Physically, as its name sug-gests, a rotation tensor represents rotation about the origin. It may be shown that of arotation tensor’s three eigenvalues, two are complex conjugates of norm one and thethird is unity; see e.g., Knowles (1998, p. 51). The eigenvector corresponding to theunitary eigenvalue provides the axis of rotation. The amount of rotation is providedby the argument of the complex eigenvalue.

1.3 Tensors 9

Symmetric and rotation tensors come together in the polar decomposition theorem(Knowles 1998, p. 57), which states that for any tensorAwith det A > 0, it is possibleto find a rotation tensor R and positive definite tensors U and V, so that

A = R · U = V · R (1.27)

uniquely. Thus, U = RT · V · R, and U and V share the same eigenvalues, while theireigenvectors are related through R. We recall that transformation via a symmetrictensor’s operation corresponds to linearly and independently scaling three mutuallyperpendicular directions. Any linear transformation may thus be viewed as a rotationfollowed (preceded) by three scalings along the orthogonal eigen-coordinate systemof V (U).

We have already mentioned the tensor product of two vectors in (1.7). Amongstother things, the tensor product helps in “tensorizing” the vector operations of takingdot- and cross- products, viz.,

a · b = tr a ⊗ b = a ⊗ b : 1, (1.28a)and a × b = −2 ax sk a ⊗ b. (1.28b)

Some additional identities that are easily proved, and will often be used are

a ⊗ A · b = a ⊗ b · AT , (1.29a)a · A · b = a ⊗ b : AT , (1.29b)skA : B = skA : symB + skA : skB = ax skA · ax skB, (1.29c)A · B : C = C · A : B = B · C : A (1.29d)

and S : W = tr (S · W) = 0, (1.29e)

where S and W are, respectively, symmetric and anti-symmetric tensors.

1.3.2 Third- and Fourth-Order Tensors

First, consider third-order tensors. In terms of the third-order tensorial bases, êi ⊗ êj ⊗ êkin

{O, êi

}, a third-order tensor is defined as

A = Aijk êi ⊗ êj ⊗ êk, (1.30)

so that Aijk areA ’s components in this coordinate system. The actions ofA on vectorsand other tensors of various orders are defined in a manner similar to that of a second-order tensor (1.10), e.g.,


A · a = Aijk êi ⊗ êj ⊗ êk · amêm = Aijkak êi ⊗ êj (1.31a)and a · A = amêm · Aijk êi ⊗ êj ⊗ êk = aiAijk êj ⊗ êk . (1.31b)

An important example of a third-order tensor is the alternating tensor that has alreadybeen defined by (1.4).

Fourth-order tensors are formed in a manner analogous to third-order tensors,

A = Aijkl êi ⊗ êj ⊗ êk ⊗ êl, (1.32)

and their operations on vectors and tensors of various orders may be developed byfollowing (1.10) and (1.31), for example,

A : B = Aijkl êi ⊗ êj ⊗ êk ⊗ êl : Bmnêm ⊗ ên = AijklBlk êi ⊗ êj. (1.33)

1.4 Coordinate Transformation

We will need to find the components of vectors and second-order tensors in one coordi-nate system, say

{O, êi

}, given its matrix in the other, say

{P, ê′i

}. This may be done

by expressing the unit vectors ofP in terms of those of O as,

ê′j = (êi · ê′j) êi,

and substituting this relationship into (1.1) and (1.9). It may be proved that the tensor

R = (êi · ê′j) êi ⊗ êj (1.34)

is, in fact, a rotation tensor with the property that

ê′i = R · êi. (1.35)

This represents the geometrically intuitive fact that, because both O and P are right-handed Cartesian coordinate systems, it is possible to obtain one from the other by arotation. The components of R are

Rij = êi · ê′j. (1.36)

Substituting the previous two equations into (1.1) and (1.9), we obtain the coordinatetransformation rules

[a] = [R][a]′, so that ai = Rija′j (1.37a)

and [A] = [R][A]′[R]T , so that Aij = RikA′klRjl, (1.37b)

1.4 Coordinate Transformation 11

for vector and second-order tensor components, respectively. As a special case, whenA = R, we find that

[R] = [R]′, so that Rij = R′ij, (1.38)

i.e., the rotation tensor R has the same components in the two frames that it relates.Higher-order tensor transformation formulae may be developed similarly when

required.

1.5 Calculus

1.5.1 Gradient and Divergence. Taylor’s Theorem

Let Φ(x) be a nth-order tensor field defined over three-dimensional space with x beinga position vector. The gradient of Φ is defined by

∇Φ = ∂Φ(x)∂x

= Φ,i êi, (1.39)

where the comma denotes differentiation and

∇(·) = ∂(·)∂x

= ∂(·)∂xi

êi = ∂(·)∂x1

ê1 + ∂(·)∂x2

ê2 + ∂(·)∂x3

ê3

is the gradient operator. In particular, we have the formulae

∇a = ∂a∂xi

êi, ∇b = ∂bi∂xj

êi ⊗ êj and ∇C = ∂Cij∂xk

êi ⊗ êj ⊗ êk, (1.40)

where a, b and C are, respectively, scalar, vector and tensor fields. Contracting anytwo free indices of the gradient of Φ, we obtain its divergence denoted by ∇ · Φ; this,therefore, is applicable only when Φ is not a scalar field. We obtain the formulae

∇ · b = ∂bi∂xi

= bi,i and ∇ · C = ∂Cij∂xj

êi = Cij,j êi. (1.41)

Note that in the last equation we could have alternatively defined ∇ · C as Cij,i êj. Thegradient of a field identifies the direction of steepest change through its eigenvectors,while its divergence is an estimate of the field’s local flux.

A fieldΦ(x)may be expanded in a Taylor’s series about a location x0 provided somesmoothness conditions are satisfied; see, e.g., Sokolnikoff (1980, p. 311). The expansionmay be expressed as

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