TRANSCRIPT
GEOMETRY AND MECHANICS OF GROWING,NONLINEARLY ELASTIC PLATES AND MEMBRANES
by
Joseph Brian McMahon
Copyright c© Joseph Brian McMahon 2009
A Dissertation Submitted to the Faculty of the
GRADUATE INTERDISCIPLINARY PROGRAM IN APPLIEDMATHEMATICS
In Partial Fulfillment of the RequirementsFor the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2009
UMI Number: 3387379
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THE UNIVERSITY OF ARIZONAGRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dis-sertation prepared by Joseph Brian McMahonentitled Geometry and Mechanics of Growing, Nonlinearly Elastic Plates and Mem-branesand recommend that it be accepted as fulfilling the dissertation requirement for theDegree of Doctor of Philosophy.
Date: 1 December 2009Alain Goriely
Date: 1 December 2009Michael Tabor
Date: 1 December 2009Shankar Venkataramani
Date: 1 December 2009
Date: 1 December 2009
Final approval and acceptance of this dissertation is contingent upon the candidate’ssubmission of the final copies of the dissertation to the Graduate College.I hereby certify that I have read this dissertation prepared under my direction andrecommend that it be accepted as fulfilling the dissertation requirement.
Date: 1 December 2009Dissertation Director: Alain GorielyDissertation Co-Director: Michael Tabor
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at the University of Arizona and is deposited in the UniversityLibrary to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,provided that accurate acknowledgment of source is made. Requests for permissionfor extended quotation from or reproduction of this manuscript in whole or in partmay be granted by the copyright holder.
SIGNED: Joseph Brian McMahon
4
ACKNOWLEDGEMENTS
I would like to thank The University of Arizona’s Graduate College and the Pro-gram in Applied Mathematics for their generous financial support during my firstyear in Arizona. I am indebted to the Department of Mathematics’ committeefor reviewing applications for funding through the National Science Foundation’sVertical InteGration of Research and Education (VIGRE) program, without whichmy research would have proceeded much more slowly during the last two years.I would also like to express my gratitude for support under Army Research OfficeMURI Award 50342-PH-MUR (Co-PI: Ildar Gabitov) and NSF grant DMS-0604704(Co-PIs: Alain Goriely and Michael Tabor).
I was able to focus on classwork, seminars, colloquia, and research instead ofon forms, signatures, and deadlines thanks to Linda Silverman, Anne Keyl, andespecially Stacey Wiley, all of the Program in Applied Mathematics. Stacey hasoffered an unending ball of twine through the labyrinth of University regulations.
I have grown as a student and researcher thanks to the faculty of the Program inApplied Mathematics. No one has had greater impact than my advisors, ProfessorMichael Tabor and Professor Alain Goriely, both of the Department of Mathematicsand the Program in Applied Mathematics. I could not have asked for a bettercombination of enthusiasm tempered with caution and an eye for detail matched byan eye locked on the “big picture”.
I sincerely thank you all.
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DEDICATION
This dissertation is dedicated to Dr. Herbert Wood and Mrs. Olive Lewis of St.
Anselm’s Abbey School, my early inspirations in physical science and mathematics;
to my parents, Bernadette and Jim, who encourage and support me in everything
I try; and most of all to Bridget, who stands beside me not just as wife and best
friend, but as fellow mathematician.
6
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 15
CHAPTER 2 EUCLIDEAN GEOMETRY WITH CURVILINEAR COORDI-NATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Euclidean Point Space . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Curvilinear Coordinates on a Body . . . . . . . . . . . . . . . . . . . 23
2.2.1 Example: Cylindrical Coordinates . . . . . . . . . . . . . . . . 242.3 Contravariant and Covariant Vectors . . . . . . . . . . . . . . . . . . 272.4 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Christoffel Symbols of the Second Kind . . . . . . . . . . . . . . . . . 322.6 Christoffel Symbols of the First Kind . . . . . . . . . . . . . . . . . . 342.7 Ricci’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Further Properties of the Covariant Derivative . . . . . . . . . . . . . 36
2.8.1 Covariant Derivative with Respect to a Vector Field . . . . . . 362.8.2 Covariant Derivative as a Tensor Field . . . . . . . . . . . . . 38
2.9 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.9.1 Example: Cartesian Coordinates . . . . . . . . . . . . . . . . 40
2.10 Abstract Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.10.1 Coordinates and Tangent Spaces . . . . . . . . . . . . . . . . 402.10.2 Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.10.3 Linear Connection . . . . . . . . . . . . . . . . . . . . . . . . 432.10.4 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . 44
2.11 Isometric Immersion and Embedding of a Riemannian Manifold . . . 452.11.1 Example: Immersion without Embedding . . . . . . . . . . . . 46
CHAPTER 3 KINEMATICS OF DEFORMATION . . . . . . . . . . . . . . 493.1 Deformation versus Change of Coordinates . . . . . . . . . . . . . . . 493.2 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Example: Cartesian Coordinates . . . . . . . . . . . . . . . . 533.2.2 Example: Cylindrical Deformation . . . . . . . . . . . . . . . 54
TABLE OF CONTENTS – Continued
7
3.3 Local Deformation of Euclidean Lines . . . . . . . . . . . . . . . . . . 553.3.1 Example: Cylindrical Deformation . . . . . . . . . . . . . . . 56
3.4 Local Deformation of Volumes . . . . . . . . . . . . . . . . . . . . . . 563.5 Local Deformation of Areas . . . . . . . . . . . . . . . . . . . . . . . 573.6 Differential and Pullback . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.6.1 Example in Cylindrical Coordinates . . . . . . . . . . . . . . . 62
CHAPTER 4 INCOMPATIBLE GROWTH . . . . . . . . . . . . . . . . . . 644.1 Multiplicative Decomposition of Deformation Gradient . . . . . . . . 664.2 Change of Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . 674.3 Example: Immersion without Embedding . . . . . . . . . . . . . . . . 694.4 Example: Lack of Immersion . . . . . . . . . . . . . . . . . . . . . . . 704.5 Non-Metric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
CHAPTER 5 ELEMENTARY MECHANICS OF SOLIDS . . . . . . . . . . 765.1 Material vs. Spatial Formulation; Conservation of Mass . . . . . . . . 765.2 Other Integral Balance Laws . . . . . . . . . . . . . . . . . . . . . . . 785.3 Stress Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.4 Differential Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4.1 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . . . 805.4.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . 82
CHAPTER 6 HYPERELASTICITY . . . . . . . . . . . . . . . . . . . . . . 836.1 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Stress Power and Hyperelasticity . . . . . . . . . . . . . . . . . . . . 846.4 Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.5 Isotropic Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5.1 Isotropic Functions . . . . . . . . . . . . . . . . . . . . . . . . 866.5.2 Form of the Nominal Stress . . . . . . . . . . . . . . . . . . . 87
6.6 Hyperelasticity with Growth . . . . . . . . . . . . . . . . . . . . . . . 886.6.1 Units without Growth . . . . . . . . . . . . . . . . . . . . . . 886.6.2 Units with Growth . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Example: Compressible Neo-Hookean . . . . . . . . . . . . . . . . . . 92
CHAPTER 7 CAVITATION IN THE HAUGHTON-OGDEN MEMBRANE 957.0.1 Kinematics of an Axisymmetric Membrane with Growth . . . 967.0.2 Thickness-Averaged Hyperelasticity . . . . . . . . . . . . . . . 977.0.3 Equilibrium Equation . . . . . . . . . . . . . . . . . . . . . . . 977.0.4 Growth-Induced Cavitation . . . . . . . . . . . . . . . . . . . 98
TABLE OF CONTENTS – Continued
8
7.1 Absence of Cavitation in Neo-Hookean Material . . . . . . . . . . . . 997.1.1 λ2 À λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.1.2 λ1
À∼ λ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Varga Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2.1 An Equivalent ODE System . . . . . . . . . . . . . . . . . . . 1047.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 1057.2.3 A Necessary Condition . . . . . . . . . . . . . . . . . . . . . . 1087.2.4 Constructing an Appropriate g′(J) . . . . . . . . . . . . . . . 1087.2.5 Converting to an Autonomous System . . . . . . . . . . . . . 1097.2.6 Converting to a Lotka-Volterra System . . . . . . . . . . . . . 1097.2.7 Asymptotics of the Lotka-Volterra System . . . . . . . . . . . 1107.2.8 Boundary Condition at Outer Radius . . . . . . . . . . . . . . 1147.2.9 Numerical Construction of Solutions . . . . . . . . . . . . . . 116
7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
CHAPTER 8 GLOBAL CONSTRAINTS AND THE INDUCED THEORYOF SHELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.1 Kirchhoff’s Classical Theory of Plates . . . . . . . . . . . . . . . . . . 1198.2 Foppl-von Karman Equations . . . . . . . . . . . . . . . . . . . . . . 1218.3 Plate Theory Induced from Three-Dimensional Elasticity . . . . . . . 122
8.3.1 Global Constraints . . . . . . . . . . . . . . . . . . . . . . . . 1238.3.2 Stress Decomposition . . . . . . . . . . . . . . . . . . . . . . . 1248.3.3 Weak Formulation of Balance Laws . . . . . . . . . . . . . . . 124
CHAPTER 9 THE FLAT KIRCHHOFF PLATE . . . . . . . . . . . . . . . 1269.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.1.1 Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . 1269.1.2 Incompatible Growth . . . . . . . . . . . . . . . . . . . . . . . 1269.1.3 Elastic Response . . . . . . . . . . . . . . . . . . . . . . . . . 1299.1.4 Virtual Displacements . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Compressible Neo-Hookean Constitutive Relation . . . . . . . . . . . 1329.2.1 A Restriction on γ1 and γ2 at R = 0 . . . . . . . . . . . . . . 134
9.3 Conversion to a Dynamical System . . . . . . . . . . . . . . . . . . . 1349.4 γ1(R) ≡ 1, γ′2(R) > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.4.1 Fixed Points and Linearization . . . . . . . . . . . . . . . . . 1389.4.2 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . 139
9.5 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 1449.6 γ1(R) ≡ 1, γ′2(R) < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
TABLE OF CONTENTS – Continued
9
CHAPTER 10 AXISYMMETRIC BUCKLING OF THE KIRCHHOFF PLATE15410.1 Kirchhoff Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 15410.2 Kinematics of an Axisymmetric Kirchhoff Plate . . . . . . . . . . . . 15510.3 Radial Virtual Displacement . . . . . . . . . . . . . . . . . . . . . . . 15910.4 Director Virtual Displacement . . . . . . . . . . . . . . . . . . . . . . 16210.5 Addressing Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . 16310.6 Problem As a Pair of ODEs . . . . . . . . . . . . . . . . . . . . . . . 16710.7 Applying the Constitutive Relation . . . . . . . . . . . . . . . . . . . 17110.8 The Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17410.9 Differential Equations for Bifurcation . . . . . . . . . . . . . . . . . . 17510.10Linear Boundary Conditions for Bifurcation . . . . . . . . . . . . . . 17910.11Evidence of Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 18010.12Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.13Numerical Example: Buckling Near a Bifurcation Point . . . . . . . . 18310.14Numerical Example: Buckling Due to Immersion-Preventing Growth . 18610.15Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
CHAPTER 11 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . 190
APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
APPENDIX D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10
LIST OF FIGURES
1.1 Polymerization-induced curvature in gel discs and sheets . . . . . . . 17
2.1 Cartesian vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Labeling a Euclidean point by a vector . . . . . . . . . . . . . . . . . 222.3 A differentiable manifold . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 A cylindrical coordinates chart . . . . . . . . . . . . . . . . . . . . . . 252.5 Another cylindrical coordinates chart . . . . . . . . . . . . . . . . . . 252.6 Tangent vectors in E3, R3, and B . . . . . . . . . . . . . . . . . . . . 412.7 An isometric immersion that is not an embedding . . . . . . . . . . . 47
3.1 Change of coordinates on a body in E3 . . . . . . . . . . . . . . . . . 503.2 Deformation of a body in E3 . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Local azimuthal expansion . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Stress-free “virtual configuration” . . . . . . . . . . . . . . . . . . . . 654.3 Stress-free “virtual configuration” . . . . . . . . . . . . . . . . . . . . 664.4 Isometric immersion that is not an embedding . . . . . . . . . . . . . 75
5.1 Stress tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.1 Phase portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Cavity radius as function of γ2 . . . . . . . . . . . . . . . . . . . . . . 117
8.1 A slice of a Cosserat shell . . . . . . . . . . . . . . . . . . . . . . . . 123
9.1 Isoclines in the (τ , N)-plane . . . . . . . . . . . . . . . . . . . . . . . 1409.2 Vector field of the dynamical system run in reverse . . . . . . . . . . 1429.3 Curve formed by unstable manifold orbits . . . . . . . . . . . . . . . 1439.4 Stereographic projection of curve of initial conditions . . . . . . . . . 1449.5 Radial stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.6 τ = r/R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.7 Impact of κ on radial stress profiles . . . . . . . . . . . . . . . . . . . 1479.8 Impact of κ on τ = r/R . . . . . . . . . . . . . . . . . . . . . . . . . 1479.9 Impact of rate of change of γ2 on radial stress profiles . . . . . . . . . 1489.10 Impact of rate of change of γ2 on τ = r/R . . . . . . . . . . . . . . . 1489.11 Vector field of dynamical system run in reverse . . . . . . . . . . . . . 1499.12 Radial and azimuthal stresses . . . . . . . . . . . . . . . . . . . . . . 1509.13 τ = r/R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
LIST OF FIGURES – Continued
11
9.14 Impact of κ on stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 1519.15 Impact of κ on τ = r/R . . . . . . . . . . . . . . . . . . . . . . . . . 1529.16 Doubling γ2 − 1 approximately doubles the stress . . . . . . . . . . . 1529.17 Doubling γ2 − 1 approximately doubles the expansion . . . . . . . . . 153
10.1 Kinematics of axisymmetric buckled Kirchhoff plate . . . . . . . . . . 15610.2 Stress perturbation at R = R1 . . . . . . . . . . . . . . . . . . . . . 18210.3 Moment perturbation at R = R1 . . . . . . . . . . . . . . . . . . . . 18310.4 Bifurcation value of γ1 . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.5 First three critical values of γ1 . . . . . . . . . . . . . . . . . . . . . . 18410.6 Buckling angle φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.7 Middle surface height ζ . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.8 Integrated radial stress . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.9 Integrated radial moment . . . . . . . . . . . . . . . . . . . . . . . . 18610.10Integrated stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18710.11Buckling angle φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18810.12Middle surface height ζ . . . . . . . . . . . . . . . . . . . . . . . . . . 18810.13Top view of middle surface . . . . . . . . . . . . . . . . . . . . . . . . 189
12
LIST OF TABLES
2.1 Equivalent contravariant vectors in E3 and R3 . . . . . . . . . . . . . 422.2 Equivalent covariant vectors in E3 and R3 . . . . . . . . . . . . . . . 42
3.1 Body manifold with cylindrical coordinates . . . . . . . . . . . . . . . 623.2 Secondary manifold with cylindrical coordinates . . . . . . . . . . . . 62
7.1 Fixed points in the in (z1, z2)-plane . . . . . . . . . . . . . . . . . . . 111
13
ABSTRACT
Until the twentieth century, theories of elastic rods and shells arose from collections
of geometric and mechanical assumptions and approximations. These theories often
lacked internal consistency and were appropriate for highly proscribed and some-
times unknown geometries and deformation sizes. The pioneering work of Trues-
dell, Antman, and others converted mechanical intuition into rigorous mathematical
statements about the physics and mechanics of rods and shells. The result is the
modern, geometrically exact theory of finite deformations of rods and shells.
In the latter half of the twentieth century, biomechanics became a major focus
of both experimental and theoretical mechanics. The genesis of residual stress by
non-elastic growth has significant impact on the shape and mechanical properties
of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism
found in elasto-plasticity, mechanicians have produced rigorous and applied results
on the effect of growth on finite elastic deformations of columns and hollow tubes.
Less attention has been paid to shells.
A theory of growing elastic plates has been constructed in the context of linear
elasticity. It harnessed many results in the theory of Riemann surfaces and has
produced solutions that are surprisingly similar to experimental observations. Our
intention is to provide a finite-deformation alternative by combining growth with the
geometrically exact theory of shells. Such a theory has a clearer and more rigorous
foundation, and it is applicable to thicker structures than is the case in the current
theory of growing plates.
This work presents the basic mathematical tools required to construct this alter-
native theory of finite elasticity of a shell in the presence of growth. We make clear
that classical elasticity can be viewed in terms of three-dimensional Riemannian ge-
ometry, and that finite elasticity in the presence of growth must be considered in this
14
way. We present several examples that demonstrate the viability and tractability of
this approach.
15
CHAPTER 1
INTRODUCTION
The theory of elastic shells, which includes plates and membranes, can be traced
back to Euler (1766), who derived a model of what would today be called an elastic
membrane. While French and German researchers dominated the construction of
the theory through much of the nineteenth century, around the turn of the century
Russians scientists made great contributions while applying elastic models to prob-
lems in naval architecture. The aerospace industry and modern numerical computing
have provided another major impetus for theoretical refinements and computational
solutions; see Todhunter and Pearson (1960), Ventsel and Krauthammer (2001), and
Szilard (2004). More recently, the theory of elastic solids has provided the founda-
tion for much of modern biomechanics. This should be recognized as a return of
mechanics to biology, as some nineteenth-century botanists used mechanical theory
to explain experimental observations; see Strasburger (1895).
Mechanics of soft tissues has become a focus of much experimental and theoret-
ical research. Two of the most-studied phenomena in soft tissues are the generation
of residual stress — stress that remains in a body after body forces and applied
tractions have been removed — and its role in morphogenesis and changes in me-
chanical properties, especially in cardiovascular tissues. See, for example, Rachev
et al. (1998), Lin and Taber (1995), Taber (1998a), Taber (1998b), Taber and Pe-
rucchio (2000), Humphrey (2006), and Chapter 9 of Humphrey (2002).
Soft tissues undergo significant deformations, so much research has been set in
the context of finite elasticity. Non-elastic growth has been incorporated into this
theory by considering deformations consisting of two steps. In the first, the body
undergoes non-elastic volumetric growth. This growth is described locally by a
growth tensor G that determines how infinitesimal material lines are distorted by
the growth. If the local values of the growth tensor do not form the gradient of a
16
deformation of the body, the growth is called incompatible. In the presence of incom-
patible growth, the body undergoes an instantaneous elastic response. This elastic
response is also described locally by a tensor A, which determines how “grown”
infinitesimal material lines are distorted. The product A · G is the gradient of a
true deformation of the body. The grown body is assumed to be stress-free, if not
necessarily a continuous body in Euclidean space. The elastic response distorts the
grown body so that it “fits” into Euclidean space as a solid continuum once again,
at the cost of carrying residual stress.
While much work has been devoted to importing the decomposition into soft
tissue mechanics (Klisch et al. (2003), Lubarda and Hoger (2002)), creating a the-
ory for initially residually stressed bodies (Hoger (1997), Klarbring et al. (2007)),
deriving evolution laws for the growth tensor (Rodriguez et al. (1994), DiCarlo and
Quiligotti (2002)), and clarifying the foundations of the decomposition independent
of its origin in elasto-plasticity theory (Chen and Hoger (2000)), most applications of
the theory have avoided analysis of buckling due to growth. Exceptions include the
analysis of spherical shells in Ben Amar and Goriely (2005), Goriely and Ben Amar
(2005), Goriely et al. (2006), and of cylindrical columns in Goriely et al. (2008),
Vandiver and Goriely (2008).
A theory of grown plates has been constructed and has produced solutions
that share many qualitative features of experimental results (Efrati et al. (2007a),
Efrati et al. (2007c), Efrati and Kupferman (2009), Efrati et al. (2009)). In
the experiments, thin discs of gel are formed by mixing water, monomers of N-
isopropylacrylamide (NIPA), bisacrylamide (a “cross-linker”), and polymerization
agents. The concentration of NIPA varies with radius in the disc. Due to the non-
uniformity of NIPA distribution, above a critical temperature the disc is described
as possessing a non-Euclidean metric. This is meant to indicate that intrinsic dis-
tances in the middle surface of the (three-dimensional) disc are now assigned by a
(two-dimensional) metric that precludes the middle surface from residing in two-
dimensional Euclidean space. See Figure 1.1. See Efrati et al. (2007b) for a brief
video of the buckling.
17
Figure 1.1: Discs and cylindrical sheets of gel with polymerization-induced curva-ture. Different shapes were produced by varying the thickness and distribution ofN-isopropylacrylamide, which polymerizes. From Efrati et al. (2007a). Used withthe kind permission of Eran Sharon, Racah Institute of Physics, The Hebrew Uni-versity of Jerusalem.
The mechanics in Efrati et al. (2007a) and related work is derived from linear
elasticity. In particular, the work adapts the Foppl-von Karman equations for large
deformations of plates. The energy density in this theory is a linear combination of
a bending energy quadratic in the curvatures of the middle surface and a stretching
energy quadratic in the difference between the curvatures of the middle surfaces of
the non-Euclidean plate and of a “nearby” Euclidean plate. Each of these com-
18
ponents is scaled by an appropriate power of the plate’s thickness. The choice of
Foppl-von Karman equations limits the variety of constitutive relations that can
be employed. Further, this work has considered only non-Euclidean plates that are
not even immersible (locally embeddable), neglecting the simpler case of immersible
but not globally embeddable plates. Nevertheless, this theory has made good use
of results on Riemann surfaces, and its results are qualitatively similar to a wide
variety of observed phenomena.
Recently, another example of polymerization-induced growth and elastic re-
sponse has been constructed in the laboratory of Shuji Hashimoto, of Waseda Univer-
sity’s Department of Applied Physics. The physical dimensions of the “gel caterpil-
lar”, in which the oscillatory Belousov-Zhabotinsky reaction causes polymer growth
and shrinkage, put it outside the realm of membranes, where Foppl-von Karman
is applicable, but not outside the geometrically exact theory of shells; see Simonite
(2009).
Our goal is to provide an alternative to the linearized theory of incompatibly
grown shells. We combine the growth/elastic multiplicative decomposition with the
geometrically exact theory of elastic shells. The result is not only free of Foppl-von
Karman’s limits of applicability, but also makes explicit the form of the Euclidean-
to-non-Euclidean transformation, which is obscured in a linearized approach.
In Chapter 2 we introduce the machinery of curvilinear coordinates in three-
dimensional Euclidean space. We introduce the relationship between coordinate sys-
tems and tangent vectors, covariant differentiation of vector fields, and the Riemann-
Christoffel curvature tensor. We then see that Euclidean space is an example of a
Riemannian manifold. Incompatibly grown solid bodies will be portrayed as Rie-
mannian manifolds.
Chapter 3 presents the kinematics of finite deformations of solid bodies, with a
special emphasis on the distinction between coordinate points and Euclidean points.
We emphasize that a deformation of a body in three-dimensional Euclidean space
should be viewed as the composition of a map from Euclidean points to coordinate
points, followed by a map from coordinate points to new coordinates, and completed
19
by a map from the new coordinates back into Euclidean space. This explicit division
of stages is obscured in standard introductions such as Gurtin (1981), Ogden (1984),
Antman (1995), and Liu (2002). We introduce the form and role of the deforma-
tion gradient. Once this classical material is covered, we present some elementary
notions of differential geometry and show their relation to the kinematics of finite
deformations. In particular, we see that a deformation of a solid body changes the
metric tensor on the Riemannian manifold that represents the body, so that intrinsic
arclengths of curves are different after a deformation.
In Chapter 4 we use differential geometry to show that incompatible growth,
when described via a growth tensor, causes a change of metric tensor in the same
fashion that a true deformation does. In the case of incompatible growth, the new
metric precludes the body from residing in three-dimensional Euclidean space. We
consider simple examples of incompatible growth in cylindrical coordinates.
In Chapter 5 we introduce the fundamental balance laws of solid mechanics. We
adopt the material of Lagrangian formulation of continuum mechanics and show
how differential equations are derived from integral balance laws.
Chapter 6 is an introduction to constitutive relations. In particular, we explore
hyperelastic materials and how growth is incorporated via the multiplicative de-
composition A · G. The detail devoted to the geometric roles of the deformation
gradient and the stress tensors make clear in this section why the nominal stress,
rather than the otherwise more fundamental first Piola-Kirchhoff stress, is of central
importance in the computation of stress power.
In Chapter 7 we introduce growth to a simple (and not geometrically exact)
axisymmetric model of a membrane and demonstrate that, unlike in the growthless
case, it supports solutions with cavities.
Before tackling geometrically exact models of plates, we open Chapter 8 with a
brief look at the original Kirchhoff theory of plates, the Foppl-von Karman equa-
tions, and rational mechanical objections to the latter. As an alternative, we intro-
duce the Principle of Virtual Power for plates and review the theory by which we
derive the differential equations of equilibrium for plates.
20
In Chapter 9 we examine the genesis of residual stress by incompatible growth
in a flat Kirchhoff plate. The type of growth considered precludes the grown body
from being put into three-dimensional Euclidean space, even locally, and yet we find
in our examples that the elastic response is capable of bringing the plate back into
Euclidean space, without the assistance of applied tractions.
Chapter 10 is devoted to incompatible growth-induced axisymmetric buckling of
the Kirchhoff plate. We find that incompatible growth of a type simpler than that
found in Efrati et al. (2007a), etc., leads to the existence of buckled configurations
after the elastic response to the growth, without applied tractions. We examine the
onset of buckling’s dependence on physical parameters such as thickness.
It is hoped that this work will serve not just as a first step in the geometrically
exact theory of incompatibly grown shells, but also as an introduction to finite
elasticity as an application of three-dimensional Riemannian geometry. Multiple
pages are devoted to fundamental notation and machinery because they are not
included in standard introductions, while more advanced treatises, such as Noll
et al. (1968) and Bloom (1979), lack basic examples and leapfrog directly from
Euclidean geometry to non-Riemannian geometry.
Comments on notation
Many of the quantities in this work are tensor-valued. The tensor products we
consider will be outer products of two vectors, although tensor products of other
objects can be defined. For example, the products a⊗ b can be viewed as
a⊗ b = abT =
a1
a2
a3
( b1 b2 b3 )
Given this structure, one can take a dot-product from the left or the right:
(a⊗ b) · c = (b · c) a,
d · (a⊗ b) = (d · a)b,
where b · c and d · a are scalar constants or scalar-valued functions.
21
Since constant tensors and tensor-valued functions can be viewed as linear com-
binations of such tensor products, we will also use the dot-product notation for the
action of tensors on vectors, such as F · v instead of Fv, and for inner products of
tensors, such as A ·B instead of AB.
We use the Einstein summation convention unless otherwise noted. If an “upper”
index and a “lower” index in an expression are the same, then the expression is to
be understood as the sum of that same expression over that index. This is true for
each matched pair of indices. For example,
Eij Ei ⊗Ej = E11 E1 ⊗E1 + E12 E1 ⊗E2 + E13 E1 ⊗E3
+ E21 E2 ⊗E1 + E22 E2 ⊗E2 + E23 E2 ⊗E3
+ E31 E3 ⊗E1 + E32 E3 ⊗E2 + E33 E3 ⊗E3,
as there is one upper and one lower i and one upper and one lower j. With this
convention, the Kronecker delta will have one upper index and one lower index.
If the same index label is used more than once for an upper or a lower index, a
mistake has been made, and the author will be grateful for notification.
22
CHAPTER 2
EUCLIDEAN GEOMETRY WITH CURVILINEAR COORDINATES
2.1 Euclidean Point Space
The setting for continuum mechanics is three-dimensional Euclidean point space,
denoted by E3. E3 consists of points, which may be denoted with respect to a fixed
orthogonal system of Cartesian coordinates (X,Y, Z). Each point in E3 is equipped
with a three-dimensional real inner-product space whose elements are called tangent
vectors. This vector space is called the tangent space at that point. The Cartesian
tangent vectors i, j, and k provide an orthonormal basis for the tangent space; see
Figure 2.1. All tangent vectors can be expressed as linear combinations of i, j, and
k. Tensors, to be discussed later, can be expressed in terms of the tensor products
i` ⊗ im, i` ⊗ im ⊗ in, and so on, where i1 = i, i2 = j, and i3 = k.
Z j
i
k
XY
Figure 2.1: i, j, and k are unit vectorsanchored at a point in E3, that point alongthe X-, Y -, and Z-axes, respectively.
H1,1,1L
i
j
k
X=i+ j+k
E3
Figure 2.2: We will refer to a point in E3
by the tangent vector that points from theorigin to the point.
This endows E3 with a metric: if we have two points with Cartesian coordinates
23
(x0, y0, z0) and (x1, y1, z1), then the distance vector between them is
v = (x1 − x0)i + (y1 − y0)j + (z1 − z0)k, (2.1)
where the Cartesian vectors i, j, and k are anchored at (x0, y0, z0) and point along
the x-, y-, and z-axes, respectively. This vector can be viewed as an arrow anchored
at (x0, y0, z0) and pointing to (x1, y1, z1). The distance between the two points is
the Euclidean norm of this vector:
dist ((x0, y0, z0), (x1, y1, z1)) =√
v · v=
√(x1 − x0)2 + (y1 − y0)2 + (z1 − z0)2. (2.2)
We will most often refer to a point in E3 not by its Cartesian coordinates, but
by the difference vector between the point and the origin; see Figure 2.2.
2.2 Curvilinear Coordinates on a Body
We will describe a solid body by a differentiable manifold B. B consists of material
points that can be assigned coordinates in an invertible fashion, at least locally. We
call the map Ψ : B → R3 a coordinate system on B. Each material point p ∈ B is
given a triple of real coordinates
Ψ(p) =(ξ1(p), ξ2(p), ξ3(p)
). (2.3)
Some manifolds may require different coordinate systems for different subsets. See
Figures 2.4 and 2.5 for an example in cylindrical coordinates.
The coordinate points Ψ(p) are mapped to spatial points in E3 by a position
function
X : Ψ(B) → E3 :(ξ1, ξ2, ξ3
) 7→ X(ξ1, ξ2, ξ3
). (2.4)
The map from material points to coordinate points to spatial points is illustrated
in Figure 2.3.
At each spatial point (X Ψ)(p) the tangent vector-valued partial derivatives
Ei =∂X
∂ξi(2.5)
24
B
Y YHBL
R3
A BR
Π
2 Π
Q
X`
HX`ëYLHBL
E3
Figure 2.3: The material points p ∈ B are mapped to coordinate points Ψ(p) ∈ R3 bythe coordinate system Ψ. The coordinate points Ψ(p) are mapped to spatial points
in E3 by the position function X. See Figures 2.4 and 2.5 for a caveat regardingcoordinate systems.
provide a basis for the tangent space at that point. The basis formed by the vectors
Ei is called the coordinate basis (Marsden and Hughes (1983), page 29) for the
tangent space at that point. There is a corresponding dual basis of vectors Ej
defined so that
Ej ·Ei = δji . (2.6)
The existence of the dual basis is guaranteed, even in the case of a non-orthogonal
coordinate basis E1,E2, E3, by the fact that if a, b, and c are linearly indepen-
dent, then the triple product det (abc) = a · (b× c) is non-zero. We set
Ei =Ej ×Ek
Ei · (Ej ×Ek), (2.7)
where (ijk) is a cyclic permutation of (123), so that
Ei ·Ej = Ei ·Ek = 0 and Ei ·Ei =Ei · (Ej ×Ek)
Ei · (Ej ×Ek)= 1. (2.8)
2.2.1 Example: Cylindrical Coordinates
First we demonstrate that we may need to choose different charts, or local patches
of coordinates, to put an open subset of a manifold into R3. In Figure 2.4, we use
a chart that assigns Θ-coordinates between 0 and 2π. Bo, the open interior of B, is
mapped into the open subset of R3 shown in the middle of Figure 2.4.
25
BoY YHBoL R3
0.25 0.75 R
Π
2 Π
3 ΠQ
X`
HX`ëYLHBoL
E3
Figure 2.4: One chart for cylindrical coordinates assigns Θ-coordinates between 0and 2π. The position function X for this chart sends this portion of R3 into E3.
Bo
YYHBoL
R3
0.25 0.75 R
Π
2 Π
3 ΠQ
X`
HX`ëYLHBoL
E3
Figure 2.5: Some subsets of Bo (the open interior of B) may straddle the line thatcorresponds to Θ = 0 = 2π in the chart in Figure 2.4. To avoid a disconnectedimage of the subset in R3, we choose a different chart, which assigns Θ-coordinatesbetween π and 3π. The position function X for this chart sends this portion of R3
into E3.
The case shown in Figure 2.5, though, requires a different chart. The subset
pictured would straddle the line marking Θ = 0 = 2π, and using the chart shown
in Figure 2.4 would leave the image of the subset disconnected in R3. We use a
different chart, which amounts to choosing a different branch of arctangent. In this
chart, the Θ-coordinates assigned to points in B are between π and 3π.
In the case of cylindrical coordinates (R, Θ, Z) the standard position function is
X(R, Θ, Z) = R cos Θ i + R sin Θ j + Z k, (2.9)
which can be used for both charts shown in Figures 2.4 and 2.5. The corresponding
26
coordinate basis consists of
ER =∂X
∂R= cos Θ i + sin Θ j, (2.10)
EΘ =∂X
∂Θ= −R sin Θ i + R cos Θ j, (2.11)
EZ =∂X
∂Z= k, (2.12)
and the dual basis vectors are
ER = cos Θ i + sin Θ j, (2.13)
EΘ = − 1
Rsin Θ i +
1
Rcos Θ j, (2.14)
EZ = k, (2.15)
so that Ei ·Ej = δji , the Kronecker delta.
Note that the cylindrical coordinate basis vectors do not all have the same lengths
or even the same units. To compute the arclength of a curve parametrized by
cylindrical coordinates, these tangent vectors must be weighted differently. Consider
a differentiable curve γ : s 7→ (R(s), Θ(s), Z(s)) in Ψ(B), defined for some interval
of s-values. Via the position function X, this corresponds to a curve in E3:
(X γ)(s) = R(s) cos Θ(s) i + R(s) sin Θ(s) j + Z(s) k. (2.16)
The velocity of a point tracing out this Euclidean curve is now easily computed:
(X γ)′(s) = (R′(s) cos Θ(s)−R(s)Θ′(s) sin Θ(s)) i
+ (R′(s) sin Θ(s) + R(s)Θ′(s) cos Θ(s)) j
+ Z ′(s) k, (2.17)
where ()′ indicates differentiation with respect to the parameter s. The size of this
velocity is the Euclidean norm of the right-hand side:
√(X γ
)′·(X γ
)′=
√(R′)2 + (RΘ′)2 + (Z ′)2. (2.18)
27
We see now that the parameter s does not correspond to Euclidean arclength, and
that the derivatives R′, Θ′, and Z ′ must be properly weighted to find the Euclidean
velocity.
Note the equivalent expression for the velocity of the Euclidean curve:
(X γ
)′= ERR′ + EΘΘ′ + EZZ ′ = Ei
dξi
ds, (2.19)
where (ξ1, ξ2, ξ3) = (R, Θ, Z), and where we have employed Einstein’s summation
notation. Thus the absolute velocity is
√(X γ
)′·(X γ
)′=
√(Ei
dξi
ds
)·(
Ejdξj
ds
)
=
√Ei ·Ej
dξi
ds
dξj
ds. (2.20)
When computing Euclidean speed from curvilinear velocities, the dot products Eij =
Ei ·Ej provide the proper weights for the curvilinear velocities. The 3 × 3 matrix
with entries Eij is symmetric and positive-definite, so it provides a quadratic form
for converting a triple of velocities into the corresponding total Euclidean velocity
squared: (dξ1
ds,dξ2
ds,dξ3
ds
)7→ Eij
dξi
ds
dξj
ds. (2.21)
We will explore more properties of the matrix [Eij] in the following sections.
2.3 Contravariant and Covariant Vectors
When a position function X exists, there need not be any distinction between the
spaces to which the basis vectors Ei and the dual basis vectors Ej belong. We will
eventually see, however, the distinct roles they play in more abstract geometries.1
For the moment we treat them as equivalent.
Given a point in (X Ψ)(B), we have two bases for the tangent space at that
point: E1,E2, E3 andE1,E2, E3
. Hence, each tangent vector v can be ex-
1For those familiar with Dirac’s bra-ket notation in quantum mechanics it is worth noting that
Ei and Ei are related in exactly the same fashion that |φ〉 and 〈φ| are related.
28
pressed with respect to either basis:
v = viEi = viEi. (2.22)
We will show that under a change of variables, the coefficients vi and vi change in
distinct ways.
Suppose Φ = (η1, η2, η3) is another coordinate system on B and that the position
function for this coordinate system is X. Then (η1, η2, η3) can be viewed as functions
of (ξ1, ξ2, ξ3) such that
X(η1
(ξ1, ξ2, ξ3
), η2
(ξ1, ξ2, ξ3
), η3
(ξ1, ξ2, ξ3
))= X
(ξ1, ξ2, ξ3
). (2.23)
This immediately provides the relationship between the coordinate basis generated
by (ξ1, ξ2, ξ3) and X and that generated by (η1, η2, η3) and X:
Ei =∂
∂ξiX
(ξ1, ξ2, ξ3
)=
∂
∂ξiX
(η1, η2, η3
)
=∂ηj
∂ξi
∂X
∂ηj=
∂ηj
∂ξiEj. (2.24)
Inherent in this formula is the assumption that each coordinate ηi is treated as a
differentiable function of (ξ1, ξ2, ξ3). If, on the other hand, each ξi is a differentiable
function of (η1, η2, η3), then we have
Ei =∂
∂ηiX
(η1, η2, η3
)=
∂
∂ηiX
(ξ1, ξ2, ξ3
)
=∂ξj
∂ηi
∂X
∂ξj=
∂ξj
∂ηiEj. (2.25)
We define the dual basis vectors Ejso that E
j · Ei = δji . Hence, the matrix that
has Eias columns satisfies
(E
1E
2E
3)
=(E1 E2 E3
)−T
, (2.26)
where −T indicates the transpose of the inverse. Since
(E1 E2 E3
)= (E1 E2 E3)
∂ξ1
∂η1∂ξ1
∂η2∂ξ1
∂η3
∂ξ2
∂η1∂ξ2
∂η2∂ξ2
∂η3
∂ξ3
∂η1∂ξ3
∂η2∂ξ3
∂η3
, (2.27)
29
(E
1E
2E
3)
=(E1 E2 E3
)−T
=
(E1 E2 E3)
∂ξ1
∂η1∂ξ1
∂η2∂ξ1
∂η3
∂ξ2
∂η1∂ξ2
∂η2∂ξ2
∂η3
∂ξ3
∂η1∂ξ3
∂η2∂ξ3
∂η3
−T
= (E1 E2 E3)−T
∂ξ1
∂η1∂ξ1
∂η2∂ξ1
∂η3
∂ξ2
∂η1∂ξ2
∂η2∂ξ2
∂η3
∂ξ3
∂η1∂ξ3
∂η2∂ξ3
∂η3
−T
=(E1 E2 E3
)
∂η1
∂ξ1∂η2
∂ξ1∂η3
∂ξ1
∂η1
∂ξ2∂η2
∂ξ2∂η2
∂ξ2
∂η1
∂ξ3∂η3
∂ξ3∂η3
∂ξ3
, (2.28)
where in the second line we consider the ξi to be functions of ηj and in the last line
we consider the ηj to be functions of ξi. The conclusion is that
Ei=
∂ηi
∂ξjEj and Ei =
∂ξi
∂ηjE
j. (2.29)
Now consider a vector v expressed with respect to the various bases arising from
the various coordinate systems:
v = viEi = vjEj = vj ∂ηi
∂ξjEi, (2.30)
v = viEi= vjE
j = vj∂ξj
∂ηiE
i. (2.31)
We conclude that there are two distinct linear transformation rules for the coeffi-
cients:
vi = vj ∂ηi
∂ξjand vi = vj
∂ξj
∂ηi. (2.32)
The coefficients vi are called the contravariant components of the vector v, while
the coefficients vi are called the covariant components. A vector of the form viEi is
often called a contravariant vector, while viEi is called a covariant vector.
A multi-indexed object that is contravariant or covariant in each index is called
a tensor. This ensures that, regardless of coordinate system chosen, when a tensor
30
value is re-written in terms of the Cartesian tangent vectors i, j, and k, the result
is the same.
An object of this type with m covariant and n contravariant indices will be called
a tensor of type(
mn
).2 For example, the metric tensor of a coordinate system, which
we will explore in the next section, has the form
Eij Ei ⊗Ej, (2.33)
where Eij = Ei ·Ej. We will show that this is a tensor of type(02
)by considering
a change of variables. The coefficients transform as
Eij = Ei ·Ej =
(∂ηk
∂ξiEk
)·(
∂η`
∂ξjE`
)= Ek`
∂ηk
∂ξi
∂η`
∂ξj, (2.34)
where Ek` = Ek · E`. There is one summation for each index. The full object
transforms as
Eij Ei ⊗Ej =
(Ek`
∂ηk
∂ξi
∂η`
∂ξj
) (∂ξi
∂ηmE
m)⊗
(∂ξj
∂ηnE
n)
=
(∂ξi
∂ηm
∂ηk
∂ξi
∂η`
∂ξj
∂ξj
∂ηn
)Ek` E
m ⊗ En
= δkmδ`
n Ek` Em ⊗ E
n
= Emn Em ⊗ E
n. (2.35)
This ensures that the metric tensor has the same formula, in terms of basis vectors,
regardless of the coordinate system chosen.
To show how tensors with upper indices transform, consider the inverse of the
metric tensor:
Eij Ei ⊗Ej. (2.36)
This is a tensor of type(20
), and its coefficients transform as
Eij = Ei ·Ej =
(∂ξi
∂ηkE
k)·(
∂ξj
∂η`E
`)
= Ek` ∂ξi
∂ηk
∂ξj
∂η`.
2This convention is adopted from page 65 of Marsden and Hughes (1983), which is the opposite
of the convention defined on page 12 of Lee (1997).
31
2.4 Metric Tensor
Eij ei ⊗ Ej is a tensor of type(02
)and is called the metric tensor induced by the
position function X. We have seen an example in which the metric tensor for a
coordinate system provides a method to convert a triple of curvilinear velocities
into the square of the corresponding Euclidean speed. In fact, the metric tensor
provides a method for assigning a norm to any contravariant vector. If v = viEi is
such a vector, then
(viEi
) · (Ek` Ek ⊗E`) · (vjEj
)= Ek`v
ivj(Ei ·Ek
) (E` ·Ej
)
= Eklvivjδk
i δ`j
= Eijvivj. (2.37)
The metric tensor assigns size to vectors when it is used as a quadratic form. It
can also be used as a bilinear form, providing an inner product of pairs of vectors.
Let u = uiEi and v = viEi. Their inner product is
(uiEi
) · (Ek` Ek ⊗E`) · (vjEj
)= Ek`u
ivj(Ei ·Ek
) (E` ·Ej
)
= Ekluivjδk
i δ`j
= Eijuivj. (2.38)
We continue to treat the contravariant (viEi) and covariant (viEi) expressions
for a vector v as alternative descriptions of the same object. The metric tensor
provides a method for converting between such expressions. First note that
viEi = vjEj
(Ek`E
k ⊗E`) · (viEi
)=
(Ek`E
k ⊗E`) · (vjE
j)
Ek`viδ`
i Ek = Ek`vjE`jEk
Ek`v`Ek = Ek`E
`jvjEk. (2.39)
It is shown in Appendix A that the matrix [Eij] is the inverse of the matrix [Eij]
formed by the dot-products Eij. Hence the last line above can be re-written as
Ek`v`Ek = δj
kvjEk = vkE
k, (2.40)
32
from which we conclude
vk = Ek`v`. (2.41)
Similar manipulations show the inverse transformation,
vj = Ejivi. (2.42)
Even after we abandon the equality of viEi and viEi, there will remain a one-
to-one correspondence between contravariant and covariant vectors, and Eqs. (2.41)
and (2.42) will remain the formulas relating the coefficients.
2.5 Christoffel Symbols of the Second Kind
If v is a vector field on (X Ψ)(B), then its rate of change with position is expressed
in terms of the rates of change of its coefficients vi and also of the basis vectors Ei:
∂v
∂ξj=
∂vi
∂ξjEi + vi ∂Ei
∂ξj.
∂Ei/∂ξj is vector-valued, so it can be expressed as a linear combination of the basis
vectors. One way to define the Christoffel symbols of the second kind (for a given
coordinate system), denoted by Γkij, is by their role in the expression
∂Ei
∂ξj= Γk
jiEk. (2.43)
They can also be defined as follows. Since Ej ·Ei = δji ,
∂
∂ξj
(Ek ·Ei
)= 0
∂Ek
∂ξj·Ei + Ek · ∂Ei
∂ξj= 0
Ek · ∂Ei
∂ξj
︸ ︷︷ ︸Γk
ji
= −Ei · ∂Ek
∂ξj. (2.44)
33
The derivative of a vector field can now be written as
∂v
∂ξj=
∂vi
∂ξjEi + viΓk
jiEk
=∂vi
∂ξjEi + vkΓi
jkEi
=
(∂vi
∂ξj+ Γi
jkvk
)Ei. (2.45)
The coefficients in this expression are denoted by the shorthand ∇jvi or vi
||j:
∇jvi = vi
||j =∂vi
∂ξj+ Γi
jkvk. (2.46)
The tensor formed by ∇jvi is called the covariant derivative of the vector field v.
Note that ∇jvi has one upper and one lower index.
We can also derive the form of the covariant derivative of the corresponding
covariant expression:
∂
∂ξj
(viE
i)
=∂vi
∂ξjEi + vi
∂Ei
∂ξj
=∂vi
∂ξjEi + vk
∂Ek
∂ξj
=∂vi
∂ξjEi + vk
(Ei ⊗Ei
) · ∂Ek
∂ξj
=∂vi
∂ξjEi + vk
(Ei · ∂Ek
∂ξj
)Ei
=∂vi
∂ξjEi + vk
(−Γkji
)Ei
=
(∂vi
∂ξj− Γk
jivk
)Ei, (2.47)
where we have used the fact that Ei ⊗ Ei is the identity tensor among the con-
travariant vectors. The shorthand for the coefficients of this derivative is
∇jvi = vi||j =∂vi
∂ξj− Γk
jivk. (2.48)
34
The Christoffel symbols of the second kind are symmetric in their lower indices:
Γkji = Ek · ∂Ei
∂ξj= Ek · ∂2X
∂ξj∂ξi
= Ek · ∂2X
∂ξi∂ξj= Ek · ∂Ej
∂ξi
= Γkij. (2.49)
The Christoffel symbols of the first kind do not form a tensor, as their transfor-
mation under a change of variables, proved in Appendix A, lacks the proper form:
Γkji =
∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξiΓ`
mn +∂ξk
∂η`
∂2η`
∂ξj∂ξi. (2.50)
However, the difference Γkji − Γk
ij does transform properly:
Γkji − Γk
ij =∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξi
(Γ`
mn − Γ`nm
), (2.51)
so the difference Γkji − Γk
ij is a tensor of type(12
). It is called the torsion tensor (of
the connection provided by the Christoffel symbols Γkji; this will be discussed later).
Although this is the zero tensor here, it need not be so in more abstract geometries.
2.6 Christoffel Symbols of the First Kind
The Christoffel symbols of the first kind are produced by “lowering” the lone “upper”
index in Γkij:
Γji` = E`kΓkji = E`kE
k · ∂Ei
∂ξj(2.52)
These symbols are symmetric in their first two indices. The symbols alone do not
form a tensor, but the difference Γji` − Γij` is a tensor of type(03
):
Γjik − Γijk =∂η`
∂ξk
∂ηm
∂ξj
∂ηn
∂ξi
(Γmn` − Γnm`
). (2.53)
We will again exploit the equality of contravariant and covariant representations
of a vector:
Γji` = E`kΓkji = E`kE
k · ∂Ei
∂ξj= E` · ∂Ei
∂ξj. (2.54)
35
This equality and the symmetry in the first two indices allows us to express Γjik
completely in terms of the metric tensor and its derivatives:
Γjik =1
2
(Ek · ∂Ei
∂ξj+ Ek · ∂Ej
∂ξi
)
=1
2
(∂
∂ξj(Ek ·Ei)−Ei · ∂Ek
∂ξj+
∂
∂ξi(Ek ·Ej)−Ej · ∂Ek
∂ξi
)
=1
2
(∂Eki
∂ξj+
∂Ekj
∂ξi−Ei · ∂2X
∂ξj∂ξk−Ej · ∂2X
∂ξi∂ξk
)
=1
2
(∂Eki
∂ξj+
∂Ekj
∂ξi−Ei · ∂Ej
∂ξk−Ej · ∂Ei
∂ξk
)
=1
2
(∂Eki
∂ξj+
∂Ekj
∂ξi− ∂Eij
∂ξk
). (2.55)
Hence the Christoffel symbols of the second kind can be expressed completely in
terms of metric tensor coefficients, their derivatives, and inverse metric tensor coef-
ficients via
Γkji = Ek`Γji` =
1
2Ek`
(∂E`i
∂ξj+
∂E`j
∂ξi− ∂Eij
∂ξ`
). (2.56)
2.7 Ricci’s Theorem
Proposition 2.7.1 (Ricci’s Theorem) The covariant derivative of the metric tensor
is zero.
Proof
∂
∂ξk
(EijE
i ⊗Ej)
=∂Eij
∂ξkEi ⊗Ej + Eij
∂Ei
∂ξk⊗Ej + EijE
i ⊗ ∂Ej
∂ξk
=∂Eij
∂ξkEi ⊗Ej − EijΓ
ik`E
` ⊗Ej − EijΓjk`E
i ⊗E`
=
(∂Eij
∂ξk− EmjΓ
mki − EimΓm
kj
)Ei ⊗Ej
=
(∂Eij
∂ξk− Γkij − Γkji
)Ei ⊗Ej. (2.57)
However,
Γkij + Γkji =1
2
(∂Eji
∂ξk+
∂Ejk
∂ξi− ∂Eki
∂ξj
)+
1
2
(∂Eik
∂ξj+
∂Eij
∂ξk− ∂Ejk
∂ξi
)
=∂Eij
∂ξk. (2.58)
36
This can be expressed as
∇kEij = Eij||k =∂Eij
∂ξk− EmjΓ
mki − EimΓm
kj = 0, (2.59)
which illustrates the fact that the covariant derivative of a tensor with two lower
indices includes one sum with Christoffel symbols for each index, and that these
sums are subtracted.
Each coefficient in the covariant derivative of the metric tensor is zero, so
∂
∂ξk
(EijE
i ⊗Ej)
= O, (2.60)
the zero tensor.
Ricci’s Theorem ensures that the rate of change of the “size” of a vector field
is due entirely to the covariant derivatives of the components. For example, the
squared norm of a vector v is Eijvivj, and the derivative of this quantity is
∂
∂ξk
(Eijv
ivj)
=∂
∂ξk
((v`E`
) · (EijEi ⊗Ej
) · (vmEm))
=∂v`E`
∂ξk· (EijE
i ⊗Ej) · (vmEm)
+(v`E`
) ·(
∂
∂ξk
(EijE
i ⊗Ej)) · (vmEm)
+(v`E`
) · (EijEi ⊗Ej
) · ∂vmEm
∂ξk
=(v`||kE`
) · (EijEi ⊗Ej
) · (vmEm)
+(v`E`
) · (EijEi ⊗Ej
) · (vm||kEm
)
= Eij
(vi||kv
j + vivj||k
). (2.61)
2.8 Further Properties of the Covariant Derivative
2.8.1 Covariant Derivative with Respect to a Vector Field
We have already seen the definition and proper method for computing the total
derivative of a vector field with respect to coordinates ξi. This definition provides a
37
method for computing a derivative of a vector field with respect to another vector
field.
Suppose that γ : s 7→ γ(s) = (ξ1(s), ξ2(s), ξ3(s)) is a curve in Ψ(B). Then X γ
is its image in E3. Let v = viEi be a vector field on (XΨ)(B). Though v is a vector
field on the Euclidean body, it can still be expressed as a function of coordinates.
Its rate of change along X γ is
d
ds(v γ) (s) =
d
ds
(vi γ
)(s) (Ei γ) (s)
=(vi γ
)′(s) (Ei γ) (s) +
(vi γ
)(s) (Ei γ)′ (s), (2.62)
where
(vi γ
)′(s) = lim
ε→0
vi (γ(s + ε))− vi (γ(s))
ε
=∂vi
∂ξj
∣∣∣∣γ(s)
dγj
ds
∣∣∣∣γ(s)
, (2.63)
and
(Ei γ)′ (s) = limε→0
Ei (γ(s + ε))−Ei (γ(s))
ε
=∂Ei
∂ξj
∣∣∣∣γ(s)
dγj
ds
∣∣∣∣γ(s)
= ΓkjiEk
∣∣γ(s)
dγj
ds
∣∣∣∣γ(s)
. (2.64)
The full derivative is then
d
ds(v γ) (s) =
(∂vi
∂ξj+ Γi
jkvk
)dγj
dsEi = vi
||jdγj
dsEi. (2.65)
To compute the rate of change of v along this curve, the components vi||j of the
covariant derivative are weighted by components dγj/ds of the tangent vector to the
curve.
A continuously differentiable non-zero vector field u on (X Ψ)(B) and an initial
point in (XΨ)(B) can be used to define a curve γ that passes through that point; u
serves as the tangent vector to this curve ( dds
(X γ) = u). Based on the observation
38
above, we can define the covariant derivative of a vector field v with respect to the
vector field u:
∇uv =
(∂vi
∂ξj+ Γi
jkvk
)ujEi = vi
||jujEi. (2.66)
2.8.2 Covariant Derivative as a Tensor Field
If v = viEi is a vector field, then the coefficients vi||j form a tensor of type
(11
). The
full object is
vi||jEi ⊗Ej, (2.67)
which yields a contravariant vector only after contraction in the lower index:
(vi||jEi ⊗Ej
) ·Ek =(vi||jEi
)δjk = vi
||kEi. (2.68)
In other words, the covariant derivative of a contravariant vector field is a tensor
field of type(11
)that can be “fed” a contravariant vector to yield a contravariant
vector field. If the vector that it is fed is u = ukEk, then the result is
(vi||jEi ⊗Ej
) · (ukEk
)= vi
||jukδj
kEi = vi||ju
jEi = ∇uv, (2.69)
the covariant derivative of v with respect to u.
The proof that the coefficients vi||j transform properly can be found in Appendix
A.
2.9 Curvature
Consider the impact of repeated covariant differentiation on a vector field:
∇w∇uv = ∇w
((∂vi
∂ξj+ Γi
jkvk
)ujEi
)
=
(∂
∂ξ`
[(∂vi
∂ξj+ Γi
jkvk
)uj
]+ Γi
m`
(∂vm
∂ξj+ Γm
jkvk
)uj
)w`Ei
=
∂2vi
∂ξ`∂ξjuj +
∂vi
∂ξj
∂uj
∂ξ`+
∂Γijk
∂ξ`vkuj + Γi
jk
∂vk
∂ξ`uj + Γi
jkvk ∂uj
∂ξ`
+ Γim`
(∂vm
∂ξj+ Γm
jkvk
)uj
w`Ei. (2.70)
39
Computing repeated covariant differentiation in the opposite order and finding the
difference yields
∇w∇uv −∇u∇wv =
∂vi
∂ξj
(∂uj
∂ξ`w` − ∂wj
∂ξ`u`
)+ Γi
jkvk
(∂uj
∂ξ`w` − ∂wj
∂ξ`u`
)
+
(∂Γi
jk
∂ξ`− ∂Γi
`k
∂ξj+ Γi
m`Γmjk − Γi
mjΓm`k
)vkujw`
Ei. (2.71)
Next we define the Lie bracket of two vector fields:
[w,u] =[w`E`, u
jEj
]=
(wk ∂uj
∂ξk− uk ∂wj
∂ξk
)Ej, (2.72)
which is a contravariant vector field. The covariant derivative with respect to the
Lie bracket is then
∇[w,u]v = ∇(wk ∂uj
∂ξk−uk ∂wj
∂ξk
)Ej
v
= ∇wk ∂uj
∂ξk Ejv −∇
uk ∂wj
∂ξk Ejv
=
(∂vi
∂ξj+ Γi
j`v`
)wk ∂uj
∂ξkEi −
(∂vi
∂ξj+ Γi
j`v`
)uk ∂wj
∂ξkEi
=
∂vi
∂ξj
(∂uj
∂ξkwk − ∂wj
∂ξkuk
)+ Γi
j`v`
(∂uj
∂ξkwk − ∂wj
∂ξkuk
)Ei.(2.73)
The difference between Eq. (2.71) and Eq. (2.73) is
∇w∇uv −∇u∇wv −∇[w,u]v
=
(∂Γi
jk
∂ξ`− ∂Γi
`k
∂ξj+ Γi
m`Γmjk − Γi
mjΓm`k
)vkujw`Ei (2.74)
Note that the inputs are three contravariant vector fields (u, w, and v) and the
output is a contravariant vector field. There are three lower indices and one upper
index. We set
R(w,u)v = ∇w∇uv −∇u∇wv −∇[w,u]v, (2.75)
Rikj` =
∂Γijk
∂ξ`− ∂Γi
`k
∂ξj+ Γi
m`Γmjk − Γi
mjΓm`k. (2.76)
This defines a tensor field of type(13
)called the Riemann-Christoffel curvature ten-
sor. Before we explore the information carried in R, we consider an especially simple
case.
40
2.9.1 Example: Cartesian Coordinates
In Cartesian coordinates (X, Y, Z) with position function X i + Y j + Z k, each
Christoffel symbol is identically zero. Hence each coefficient of the Riemann-
Christoffel curvature tensor is also identically zero:
Rikj` ≡ 0, so R ≡ O. (2.77)
Since R is a tensor field, however, its value is invariant under a change of coordinates.
Hence R is identically zero in all coordinate systems on E3.
2.10 Abstract Manifolds
We have derived numerous objects directly from the position function X. But this is
just one example of a three-dimensional differentiable manifold. Many possibilities
arise when there is no position function.
2.10.1 Coordinates and Tangent Spaces
A three-dimensional differentiable manifold B is a set such that for each point p ∈ Bthere is an open neighborhood U(p) of p and a diffeomorphism Ψ between U(p)
and an open subset of R3. The image Ψ(U(p)) ⊂ R3 consists of coordinate points,
and the diffeomorphism Ψ is called a coordinate system for that neighborhood. If
p ∈ B is a point in a manifold and lies in the domain of two coordinate systems
Ψ = (ξ1, ξ2, ξ3) and Φ = (η1, η2, η3), then the change-of-coordinates maps Ψ Φ−1
and Φ Ψ−1 must be C∞.3
Each point in a differentiable manifold has a vector space of tangent vectors
attached to it. If p ∈ B, its tangent space is denoted TpB. For example, in the
presence of a position function, for each tangent vector at a point in E3 there is
a corresponding tangent vector in the space of coordinates, as well as an abstract
tangent vector at p. See Figure 2.6.
3In some applications, including solid mechanics, the requirement of infinite differentiability
relaxed is relaxed for these functions and for others, such as metric tensors. See the definitions
41
E3
ER
R3
¶
¶R
A BR
Π
2 Π
Q
v
Figure 2.6: When a position function exists, each tangent vector at (XΨ)(p) in theEuclidean body corresponds to a tangent vector at Ψ(p) and to an abstract tangentvector at p ∈ B.
When there is no position function X available, vectors may not be expressed
in terms of the partial derivatives ∂X/∂ξi. Instead, once a coordinate system is
assigned to a neighborhood, the vectors are expressed as linear combinations of the
partial derivative operators ∂/∂ξi. These vectors are contravariant, since under a
change of variables they transform as
∂
∂ηi=
∂ξj
∂ηi
∂
∂ξj. (2.78)
The linear functionals on the tangent space TpB also form a vector space, denoted
T ∗pB. If ∂/∂ξi, i = 1, 2, 3 form a basis for TpB, then a basis for T ∗
pB is provided by
the linear functionals dξj, j = 1, 2, 3, defined by
dξj
[∂
∂ξi
]= δj
i , (2.79)
where we use square brackets to indicate that the tangent vector ∂/∂ξi is the argu-
ment of dξj. The functionals dξj are called differential 1-forms, and T ∗pB is called a
cotangent space. The dξi are the covariant vectors on an abstract manifold, as
dηi =∂ηi
∂ξjdξj. (2.80)
In the case of cylindrical coordinates with position function X, the correspon-
dence between vectors is summarized in Tables 2.1 and 2.2.
in Ciarlet (2005).
42
Contravariant vector at X(R, Θ, Z) Tangent vector at (R, Θ, Z)
ER = ∂X∂R
∂∂R
EΘ = ∂X∂Θ
∂∂Θ
EZ = ∂X∂Z
∂∂Z
Table 2.1: Equivalent tangent vectors at X(R, Θ, Z) and at (R, Θ, Z)
Covariant vector at X(R, Θ, Z) 1-form at (R, Θ, Z)
ER dREΘ dΘEZ dZ
Table 2.2: Equivalent covariant vectors at X(R, Θ, Z) and 1-forms at (R, Θ, Z)
2.10.2 Metric Tensor
When a space of coordinates Ψ(B) has a position function X, there is a metric
tensor induced on Ψ(B). If M = Eij Ei⊗Ej is the metric tensor on the Euclidean
body, the corresponding metric tensor on Ψ(B) is
M = Eij dξi ⊗ dξj, (2.81)
where the Eij are now simply scalar functions of the coordinates (ξ1, ξ2, ξ3). Since
M is induced by X, the arclength of a differentiable curve γ in Ψ(B) computed
with M agrees with the Euclidean arclength of its image X γ. We will find that
deformations and growth change the metric tensor on Ψ(B) in such a way that the
arclengths computed with the new metric tensor do not necessarily agree with the
Euclidean arclengths of curves in (X Ψ)(B).
Without a position function X, a differentiable manifold does not automatically
come equipped with a metric tensor. A priori, each tangent space is a real vector
space only, not an inner-product space. Assigning a metric tensor field to a manifold
provides an inner product. A metric tensor has the form
g = gij dξi ⊗ dξj, gij = g
(∂
∂ξi,
∂
∂ξj
)=
⟨∂
∂ξi,
∂
∂ξj
⟩, (2.82)
where 〈 , 〉 indicates the inner product defined by g.
43
Definition 2.10.1 A(02
)tensor field is a Riemannian metric if
(i) it is C∞ (this degree of differentiability is relaxed in solid mechanics);
(ii) it is symmetric, i.e.
g(v,u) = g(u,v); (2.83)
(iii) and it is positive-definite, i.e.
g(v,v) ≥ 0 for all vectors v, and g(v,v) = 0 iff v = 0. (2.84)
2.10.3 Linear Connection
In the absence of a position function, there is no obvious coordinate-invariant
method for computing the derivative of a vector field. More concretely, there is
no method for computing the acceleration of a curve t 7→ γ(t) = (ξ1(t), ξ2(t), ξ3(t))
in the space of coordinates. When there is a position function X, one can compute
both the velocity and the acceleration of the curve’s Euclidean image X γ:
position: (X γ)(t) = X(ξ1(t), ξ2, ξ3(t)
),
velocity:d
dt(X γ)(t) =
∂X
∂ξi
∣∣∣∣∣γ(t)
(ξi
)′(t) = Ei|γ(t)
(ξi
)′(t),
acceleration:d2
dt2(X γ)(t) =
∂Ei
∂ξj
∣∣∣∣γ(t)
(ξj
)′(t)
(ξi
)′(t)
+ Ei|γ(t)
(ξi
)′′(t), (2.85)
where ′ indicates differentiation with respect to the parameter t. The tangent vec-
tors Ei are functions of coordinates (ξ1, ξ2, ξ3), so the derivatives ∂Ei/∂ξj can be
computed.
In the absence of a position function X, however, there is no Euclidean image
of the curve γ, and the tangent vector to the curve is
(ξi
)′(t)
∂
∂ξi
∣∣∣∣γ(t)
, (2.86)
44
whose derivative with respect to t is not defined without more information. We
must assign a method for computing derivatives of the vectors ∂/∂ξi.
A linear connection provides a definition for differentiation of a vector field. In
terms of differentiation of coordinate basis vectors with respect to other coordinate
basis vectors, a linear connection is a set of 33 = 27 smooth functions Γkji, known as
connection coefficients, with which we define the covariant derivative
∇ ∂
∂ξj
∂
∂ξi= Γk
ji
∂
∂ξk. (2.87)
Definition 2.10.2 A linear connection ∇ is a bilinear map (u,v) 7→ ∇uv on vector
fields that satisfies
(i) linearity over smooth real-valued functions in the first argument:
∇fu+hwv = f∇uv + h∇wv for f, h ∈ C∞(B); (2.88)
(ii) linearity over real numbers in the second argument:
∇u (av + bw) = a∇uv + b∇uw for a, b ∈ R; (2.89)
(iii) the following product rule in the second argument:
∇u (fv) = f∇uv +
(ui ∂f
∂ξi
)v for f ∈ C∞(B). (2.90)
The Christoffel symbols of the second kind, as derived in Eq. (2.56), are connec-
tion coefficients for a special case of a linear connection. A linear connection need
not be derived from the metric tensor. On an abstract manifold, the metric tensor
and connection may be assigned independently of each other. In fact, for a manifold
that can be completely covered by one coordinate chart, any collection of 27 smooth
functions Γkji can define a linear connection; see Lemma 4.4 in Lee (1997).
2.10.4 Riemannian Geometry
When connection coefficients are derived directly from a Riemannian metric via
Eq. (2.56), the resulting connection is called the Riemannian connection or the
45
Levi-Civita connection of the manifold. This connection is the unique connection
with two properties found in the connection derived from the position function X.
Theorem 2.10.3 “Fundamental Lemma of Riemannian Geometry” (Theorem 5.4
in Lee (1997)) The Riemannian connection on a manifold B with Riemannian metric
g is the sole linear connection that satisfies
(i) symmetry: the connection coefficients for a coordinate basis4 satisfy Γkij = Γk
ji,
or ∇uv −∇vu = [u,v] for all vector fields u, v; and
(ii) metricity: the covariant derivative (defined by the connection) of the metric
tensor is zero: ∇g = O.
See Lemma 5.2 and Theorem 5.4 in Lee (1997), or Theorem 4.24 in Marsden and
Hughes (1983).
While a three-dimensional Riemannian manifold may or may not “fit” into E3
(see below), it does have analogues of length, area, and volume, so lineal, areal, and
volumetric quantities can be defined on it.
2.11 Isometric Immersion and Embedding of a Riemannian Manifold
The position function X maps Ψ(B) into E3 in a special way. It is a diffeomorphism
from the manifold into E3, and for each curve γ in Ψ(B), the arclength of γ and the
arclength of its Euclidean image X γ agree. These properties earn X the status
of an isometric embedding of Ψ(B) into E3.
We have seen that starting with an isometric embedding X, computing from
it a Riemannian metric, and deriving the Riemannian connection from the metric,
produce a Riemann-Christoffel curvature tensor that is identically zero:
existence of isometric embedding into E3 =⇒ R ≡ O. (2.91)
4The connection coefficients of a symmetric connection are symmetric in the lower indices when
the coefficients are computed with respect to the coordinate basis ∂/∂ξi, i = 1, 2, 3. This need not
be true of coefficients for other bases. See the warning on the top of page 246 of Frankel (1997).
46
The three-dimensional version of a result dubbed the “Fundamental Theorem of
Riemannian Geometry” in Ciarlet (2005) gives a partial converse.
Definition 2.11.1 Let Ω be an open subset of R3. A mapping Y : Ω → E3 is
an immersion at (ξ1, ξ2, ξ3) ∈ Ω if it is differentiable at (ξ1, ξ2, ξ3) and the partial
derivatives ∂Y /∂ξi, i = 1, 2, 3, at (ξ1, ξ2, ξ3) are linearly independent.
Theorem 2.11.2 (Theorem 1.6-1 in Ciarlet (2005)) Let Ω be a simply connected set
in R3 and let C = [Eij] ∈ C2(Ω,S3>) be a symmetric positive-definite matrix-valued
function on Ω that satisfies
R ≡ O in Ω.
Then there exists an immersion Y ∈ C3(Ω,E3) such that
Eij =∂Y
∂ξi· ∂Y
∂ξjin Ω.
The isometric immersion promised by the theorem is a local position function.
It is a map from simply connected open subsets of the manifold into E3 that is
locally diffeomorphic and such that metric tensor-computed arclengths of curves in
coordinates and arclengths of their Euclidean images agree. An isometric immersion
may fail to be an embedding if it is not globally invertible.
2.11.1 Example: Immersion without Embedding
Consider the manifold B with coordinate system Ψ = (R, Θ, Z) satisfying
A ≤ R ≤ B, 0 ≤ Θ ≤ 3π
2, H1 ≤ Z ≤ H2, (2.92)
See Figure 2.7. Now consider the map
Y (R, Θ, Z) = R cos(2Θ) i + R sin(2Θ) j + Z k. (2.93)
47
The coefficients of the corresponding metric tensor are
ERR ERΘ ERZ
EΘR EΘΘ EΘZ
EZR EZΘ EZZ
=
1 0 0
0 4R2 0
0 0 1
. (2.94)
Computing the corresponding Christoffel symbols and Riemann-Christoffel curva-
ture tensor via Eq. (2.56) yields an identically zero curvature tensor.
While Y might seem to be a good candidate for a position function, Figure 2.7
shows that the Euclidean images of two disjoint simply connected open subsets of
Ψ(B) will have a nonempty intersection.
W1
R3
A B R
Π
2
Π
3 Π2
2 ΠQ
Y`HW1L
E3
-2 -1 1 2
-2
-1
1
2
W2
R3
A B R
Π
2
Π
3 Π2
2 ΠQ
Y`HW2L
E3
-2 -1 1 2
-2
-1
1
2
Figure 2.7: An isometric immersion that is not an embedding: the images Y (Ω1)
and Y (Ω2) overlap.
We have seen that Y is an isometric immersion of this manifold with this metric
into E3, but Y is not an embedding. Suppose we discard the would-be position
48
function Y and equip Ψ(B) with the metric tensor
dR⊗ dR + 4R2 dΘ⊗ dΘ + dZ ⊗ dZ, (2.95)
whose associated (through the Riemannian connection) Riemann-Christoffel curva-
ture tensor is identically zero. Is there an isometric embedding of the (open, simply
connected) interior of Ψ(B) with this metric into E3? A theorem on the “rigidity”
of isometric immersions shows that the answer is negative.
Theorem 2.11.3 Let Ω be an open connected subset of R3 and let Y ∈ C1(Ω,E3)
and Y ∈ C1(Ω,E3) be two isometric immersions such that their associated coordi-
nate bases satisfy
∂Y
∂ξi· ∂Y
∂ξj=
∂Y
∂ξi· ∂Y
∂ξjin Ω.
Then there are a vector c ∈ E3 and a proper rotation Q such that
Y (ξ1, ξ2, ξ3) = Q · Y (ξ1, ξ2, ξ3) + c
for each (ξ1, ξ2, ξ3) ∈ Ω.
See Theorem 1.7-1 in Ciarlet (2005).
A uniform rotation and translation of Y will not alleviate its global failure of
injectivity on the interior of Ψ(B). Since there is no isometric embedding of the
partial annulus, any complete annulus equipped with the same metric also lacks an
isometric embedding.
49
CHAPTER 3
KINEMATICS OF DEFORMATION
3.1 Deformation versus Change of Coordinates
A change of coordinates consists of two sets of coordinates and a position function
for each set. As shown in Figure 3.1, each occupied spatial point is mapped to a
coordinate point (η1, η2, η3) by the inverse position function X−1
; in Figure 3.1,
these are the Cartesian coordinates (X, Y, Z). Each coordinate point is mapped to
a coordinate point (ξ1, ξ2, ξ3) in a new coordinate system ((R, Θ, Z) in Figure 3.1)
in a continuously differentiable and invertible manner, at least locally. Each new
coordinate point is then mapped back into E3 by a new position function X. The
final results is that
X(η1, η2, η3
)= X = X
(ξ1
(η1, η2, η3
), ξ2
(η1, η2, η3
), ξ3
(η1, η2, η3
)). (3.1)
The same material points occupy the same spatial points. Only the coordinates
labeling the points have changed.
The transformation shown in Figure 3.2 is different. There is a diffeomorphism
f between the triples (R, Θ, Z) ∈ Ψ(B) and the triples (r, θ, z) ∈ (f Ψ)(B), but the
position function x sends (r, θ, z) to a point in E3 different from the point where X
sends the corresponding (R, Θ, Z). If this transformation satisfies certain conditions
to be discussed below, then it is called a deformation of the body.
Given coordinate system Ψ on B and position function X on Ψ(B), a deformation
of the body is a continuously differentiable and invertible function χ : (X Ψ)(B) →E3 : X 7→ x that has the form
x(ζ1, ζ1, ζ3) = (x f)(ξ1, ξ2, ξ3)
= (x f X−1
)(X)
= χ(X). (3.2)
50
FHBL
R3
0 1
0
12
1
X
Y
YHBL
R3
0 2
0
Π
4
Π
2
R
Q
X -1
X`
E3
HXëFLHBL = HX
`ëYLHBL
Figure 3.1: In a change from Cartesian to cylindrical coordinates, X−1
maps pointsin (X Φ)(B) ⊂ E3 to the coordinate points (X,Y, Z) ∈ Φ(B). Each Cartesiancoordinate point (X,Y, Z) is mapped to a cylindrical coordinate point (R, Θ, Z) ∈Ψ(B), which is then mapped, via X, to the point X(R, Θ, Z) = X(Z, Y, Z).
See Figure 3.2. We will refer to (X Ψ)(B) as the reference configuration and to
(χ X)(B) as the deformed configuration.
Note that computing a deformation requires three steps. First, the point
X ∈ (XΨ)(B) is mapped to the corresponding coordinate point (ξ1, ξ2, ξ3) ∈ Ψ(B)
by the inverse position function X−1
. Second, the coordinate point (ξ1, ξ2, ξ3) is
mapped to a coordinate point (ζ1, ζ2, ζ3) = f(ξ1, ξ2, ξ3). Finally, x maps the coor-
dinate point (ζ1, ζ2, ζ3) to a spatial point x(ζ1, ζ2, ζ3).
Just as X induces a metric tensor on Ψ(B), x induces a metric tensor on the
space of (ζ1, ζ2, ζ3)-coordinates. In the case of cylindrical coordinates (r, θ, z), we
51
R3
0
0
Π
2 Π
R
Q
f
R3
0
0
Π
2 Π
r
Θ
X` -1 x`
E3
E3
Figure 3.2: In a deformation, a point in the body may be assigned new coordinatesand be sent to a new point in E3. Further, the distances between material pointsmay change.
will use the position function
x(r, θ, z) = r cos(θ) i + r sin(θ) j + z k. (3.3)
The corresponding Euclidean metric tensor is
m = eij ei ⊗ ej
= er ⊗ er + r2 eθ ⊗ eθ + ez ⊗ ez, (3.4)
52
where eij = ei · ej and
er =∂x
∂r= cos(θ) i + sin(θ) j,
eθ =∂x
∂θ= r(− sin(θ) i + cos(θ) j),
ez =∂x
∂z= k. (3.5)
The metric tensor m corresponds to the metric tensor
m = dr ⊗ dr + r2 dθ ⊗ dθ + dz ⊗ dz (3.6)
on the space of (r, θ, z)-coordinates.
In a change of variables, the intrinsic arclengths of curves remain the same.
Deformation, on the other hand, changes the intrinsic arclengths. Local changes of
length and of direction are described via the deformation gradient.
3.2 Deformation Gradient
A local description of a deformation is provided by considering its impact on in-
finitesimal material lines. Consider the points X(ξ1, ξ2, ξ3) and X(ξ1 + c1ε, ξ2 +
c2ε, ξ3 + c3ε) in E3 and consider the difference quotient of χ evaluated at these two
points:
limε→0(χ X)(ξ1 + εc1, ξ2 + εc2, ξ3 + εc3)− (χ X)(ξ1, ξ2, ξ3)
ε
=∂x
∂ζj
∣∣∣∣X(ξ1,ξ2,ξ3)
ci ∂ζj
∂ξi
∣∣∣∣(ξ1,ξ2,ξ3)
, (3.7)
where (ζ1, ζ2, ζ3) = f(ξ1, ξ2, ξ3). The following is an equivalent but more illustrative
expression of this directional derivative:
∂x
∂ζjci ∂ζj
∂ξi=
∂ζj
∂ξi
∂x
∂ζj
(ckδi
k
)
=∂ζj
∂ξi
∂x
∂ζj
(ckEi ·Ek
)
=
(∂ζj
∂ξi
∂x
∂ζj⊗Ei
)
︸ ︷︷ ︸deformation gradient
· (ckEk
). (3.8)
53
The tensor in the last line maps a contravariant vector at X(ξ1, ξ2, ξ3) to a con-
travariant vector at x(ζ1, ζ2, ζ3). Because it maps in this way, this tensor, called
the deformation gradient, is called a two-point tensor.
The deformation gradient is usually expressed as follows:
F =∂ζj
∂ξiej ⊗Ei, (3.9)
where ej = ∂x/∂ζj is a tangent vector at the point x(ζ1, ζ2, ζ3). F is a tensor
field whose domain is the reference configuration (X Ψ)(B). Ei is evaluated at
(ξ1, ξ2, ξ3) and anchored at X(ξ1, ξ2, ξ3), while ej is evaluated at (η1, η2, η3) and
anchored at x(ζ1, ζ2, ζ3).
3.2.1 Example: Cartesian Coordinates
We consider a reference configuration with Cartesian coordinates (X, Y, Z) and po-
sition function
X(X,Y, Z) = X i + Y j + Z k. (3.10)
Note that
EX =∂X
∂X= i, EY =
∂X
∂Y= j, EZ =
∂X
∂Z= k. (3.11)
The deformed configuration is assigned Cartesian coordinates (x, y, z) and posi-
tion function
x(x, y, z) = x i + y j + z k, (3.12)
so that
ex =∂x
∂x= i, ey =
∂x
∂y= j, ez =
∂x
∂z= k. (3.13)
In the deformation, we consider x, y, and z to be functions of X, Y , and Z. The
54
deformation gradient is
F =∂x
∂Xex ⊗EX +
∂x
∂Yex ⊗EY +
∂x
∂Zex ⊗EZ
+∂y
∂Xey ⊗EX +
∂y
∂Yey ⊗EY +
∂y
∂Zey ⊗EZ
+∂z
∂Xez ⊗EX +
∂z
∂Yez ⊗EY +
∂z
∂Zez ⊗EZ
=∂x
∂Xi⊗ i +
∂x
∂Yi⊗ j +
∂x
∂Zi⊗ k
+∂y
∂Xj ⊗ i +
∂y
∂Yj ⊗ j +
∂y
∂Zj ⊗ k
+∂z
∂Xk ⊗ i +
∂z
∂Yk ⊗ j +
∂z
∂Zk ⊗ k. (3.14)
The vectors on the left-hand sides of the tensor products are anchored at the de-
formed point x(x, y, z), and the vectors on the right-hand sides are anchored at the
original point X(X,Y, Z) in the reference configuration.
Note that if we use the notation ∇x for the gradient of the function x, we can
write F as
F = i⊗ (∇x) + j ⊗ (∇y) + k ⊗ (∇z) . (3.15)
This gives an idea of the origin of the notation F = ∇⊗χ, which is used in Ogden
(1984). A clearer but still vague alternative notation would be F = (∇⊗ χ)T .
3.2.2 Example: Cylindrical Deformation
A cylindrical deformation of a body is expressed, in cylindrical coordinates (R, Θ, Z)
for the reference configuration and (r, θ, z) for the deformed configuration, by
(χ X)(R, Θ, Z) = x(r(R), θ(Θ), z(Z))
= r(R) (cos(θ(Θ)) i + sin(θ(Θ)) j) + z(Z) k, (3.16)
with θ(Θ) = Θ and z(Z) = Z. In this particular case, the deformation gradient has
only three terms:
F = r′er ⊗ER + eθ ⊗EΘ + ez ⊗EZ , (3.17)
where each vector on the right of a tensor product is anchored to the pre-deformation
point X(R, Θ, Z), and each vector on the left of a tensor product is anchored at
55
the post-deformation point x(r, θ, z) = (χ X)(R, Θ, Z).
Note the action of the deformation gradient on tangent vectors in the pre-
deformation body:
F ·ER = r′ er, F ·EΘ = eθ, F ·EZ = ez. (3.18)
3.3 Local Deformation of Euclidean Lines
The formula in Eq. (3.9) is equivalent to
F · ∂X
∂ξi=
∂ζj
∂ξi
∂x
∂ζj. (3.19)
If we allow ourselves the use of infinitesimals dξi and dζj, which we are not treating
as 1-forms, this can be written as
F · ∂X
∂ξidξi =
∂x
∂ζjdζj. (3.20)
Eq. (3.20) presents the deformation gradient as a linear map of “infinitesimal
material lines” in (X Ψ)(B) to their images in (χ X Ψ)(B). Infinitesimal
material lines are useful devices for describing the local strain of a deformation.
Consider an infinitesimal material line of the form
∂X
∂ξidξi = Ei dξi (3.21)
whose length is √(Ei dξi) · (Ej dξj) =
√Eijdξidξj. (3.22)
The image of this material line under the deformation is
F · (Ei dξi)
= F ·Ei dξi, (3.23)
and the length of this image is
√(F ·Ei dξi) · (F ·Ei dξi) =
√(F ·Ei)
T (F ·Ej) dξidξj
=√
Ei ·(F T · F ) ·Ejdξidξj. (3.24)
56
Note that the expression in Eq. (3.24) describes the length of the deformed infinites-
imal line in terms of the infinitesimals dξi on (X Ψ)(B).
While F is a linear map from the tangent space at X = X(ξ1, ξ2, ξ3) to the
tangent space at x = x(ζ1, ζ2, ζ3), F T ·F , known as the right Cauchy-Green tensor,
is a linear map from the tangent space at a point X = X(ξ1, ξ2, ξ3) in the reference
configuration to that same tangent space:
F : tangent space at X → tangent space at x, (3.25)
F T : tangent space at x → tangent space at X, (3.26)
F T · F : tangent space at X → tangent space at X. (3.27)
Such a tensor is called a Lagrangian tensor.
3.3.1 Example: Cylindrical Deformation
For the cylindrical deformation example, the length of an un-deformed infinitesimal
material line is √Eijdξidξj =
√(dR)2 + R2 (dΘ)2 + (dZ)2, (3.28)
while the length of its deformed image is
√Ei ·
(F T · F ) ·Ejdξidξj =
√(r′)2 (dR)2 + r2 (dΘ)2 + (dZ)2. (3.29)
The right Cauchy-Green tensor is
F T · F =(r′ ER ⊗ er + EΘ ⊗ eθ + EZ ⊗ ez
) · (r′ er ⊗ER + eθ ⊗EΘ + ez ⊗EZ)
= (r′)2err ER ⊗ER + eθθ EΘ ⊗EΘ + ezz EZ ⊗EZ
= (r′)2ER ⊗ER + r2 EΘ ⊗EΘ + EZ ⊗EZ . (3.30)
3.4 Local Deformation of Volumes
It is assumed that each subset of (X Ψ)(B) with positive volume is mapped to a
subset of (χ X Ψ)(B) with positive volume. This precludes deformations that
57
feature infinite mass density or inversion of the handedness of a neighborhood of
matter.
Consider an infinitesimal parallelepiped in the reference configuration, bounded
by infinitesimal material lines Eidξi, i = 1, 2, 3. The (infinitesimal) volume is
det(E1dξ1 E2dξ2 E3dξ3
)= det (E1E2E3) dξ1dξ2dξ3. (3.31)
The image of this parallelepiped will be bounded by infinitesimal material lines
F ·Eidξi, so the volume of the image will be
det(F ·E1dξ1 F ·E2dξ2 F ·E3dξ3
)
= det (F · (E1E2E3)) dξ1dξ2dξ3
= det (F ) det (E1E2E3) dξ1dξ2dξ3. (3.32)
Hence the deformation gradient must have positive determinant: det F > 0. This
will preclude not just the collapse of volumes to areas, but also reversal of orientation.
3.5 Local Deformation of Areas
An infinitesimal area is described by the cross-product of two noncollinear infinites-
imal lines. Consider an infinitesimal directed area of the form Ejdξj ×Ekdξk, and
let c be an arbitrary tangent vector in the same tangent space from which Ej and
Ek are drawn (so that it is acted upon by the same F ). Then the triple product
c · (Ej ×Ek) = det (c EjEk) , (3.33)
is transformed to
(F · c) · ((F ·Ej)× (F ·Ek)) = det ((F · c) (F ·Ej) (F ·Ek))
= det (F ) det (c EjEk)
= det (F ) c · (Ej ×Ek) . (3.34)
By the definition of tensor transpose F T ,
(F · c) · ((F ·Ej)× (F ·Ek)) = c · F T · ((F ·Ej)× (F ·Ek)) , (3.35)
58
so that
c · F T · ((F ·Ej)× (F ·Ek)) = det (F ) c · (Ej ×Ek) . (3.36)
Since c is arbitrary,
F T · ((F ·Ej)× (F ·Ek)) = det (F ) (Ej ×Ek) , (3.37)
so
(F ·Ej)× (F ·Ek) = det (F ) F−T · (Ej ×Ek) , (3.38)
where F−T is the inverse of the transpose F T . Hence the deformed area, expressed
in the coordinates ξi, is
det (F ) F−T · (Ej ×Ek) dξjdξk. (3.39)
The expression in Eq. (3.39) holds for each infinitesimal directed area, as Ej ×Ekdξjdξk, i, j = 1, 2, 3, i 6= j, form a basis for such areas. This transformation of
area, called Nanson’s formula, is sometimes summarized as
n ds = (det F ) F−T ·N dS, (3.40)
where N and n are unit vectors normal to the surface in the reference and deformed
configurations, respectively; and dS and ds are the infinitesimal scalar areas (or
area measures) in the reference and deformed configurations, respectively.
Eq. (3.40) is not technically correct as stated, however. The object on the left-
hand side of Eq. (3.40) is a vector-valued area measure on surfaces in the deformed
configuration, while the object on the right-hand side is an area measure on surfaces
in the reference configuration. The area measure on the right-hand side allows one to
compute surface integrals in the deformed configuration while still using coordinates
on the reference configuration:∮
∂(χX)(Ω)
n ds =
∮
∂X(Ω)
(det F ) F−T ·N dS, (3.41)
for every measurable subset Ω of Ψ (B) with well-behaved boundary ∂Ω.
59
Since Eq. (3.41) demonstrates equivalence of measures, it can be extended to
equality of integrals. For continuously differentiable scalar-valued functions ϕ,
∮
∂(χX)(Ω)
ϕ(χ−1(x)
)n(x) ds =
∮
∂X(Ω)
ϕ(X) (det F (X)) F (X)−T ·N(X) dS,
(3.42)
where x = χ(X). Let A be a continuously differentiable tensor field function whose
value at x ∈ (χX Ψ)(B) maps from the tangent space at x to that same tangent
space. Such a tensor is called an Eulerian tensor, and
∮
∂(χX)(Ω)
A (x) · n(x) ds =
∮
∂X(Ω)
(det F (X)) A(χ(X)) · F (X)−T ·N(X) dS.
(3.43)
3.6 Differential and Pullback
Before we explore the difference between growth and deformation, we delve into the
machinery derived from coordinate-to-coordinate maps between manifolds equipped
with positions functions.
Just as the metric tensor M on Ψ(B) is the analogue of M on (X Ψ)(B),
there are also analogues of F and F T · F . The primary object of study will be the
invertible map f that performs the change-of-coordinate step in a deformation, as
pictured in Figure 3.2.
Consider our original body manifold B with coordinate system Ψ = (ξ1, ξ2, ξ3)
and position function X on Ψ(B), and another three-dimensional manifold M with
coordinate system ψ = (ζ1, ζ2, ζ3) and position function x on ψ(M).
Definition 3.6.1 For a diffeomorphism f : Ψ(B) → (f Ψ)(B) ⊂ ψ(M) defined by
f(ξ1, ξ2, ξ3) =(ζ1(ξ1, ξ2, ξ3), ζ2(ξ1, ξ2, ξ3), ζ3(ξ1, ξ2, ξ3)
), (3.44)
there is a corresponding linear map f∗ : Tp(B) → Tf(p)M between tangent spaces
that is expressed in coordinates by
f∗ =∂ζ i
∂ξj
∂
∂ζ i⊗ dξi. (3.45)
60
f∗ is called the differential of f .1
The map f∗ sends tangent vectors ∂/∂ξj at (ξ1, ξ2, ξ3) ∈ Ψ(B) to tangent vectors
∂/∂ζ i at (ζ1, ζ2, ζ3) = f(ξ1, ξ2, ξ3) ∈ (f Ψ)(B) = ψ(M):
f∗
[vk ∂
∂ξk
]=
(∂ζ i
∂ξj
∂
∂ζ i⊗ dξj
)[vk ∂
∂ξk
]
= dξj
[vk ∂
∂ξk
]∂ζ i
∂ξj
∂
∂ζ i
= vk δjk
∂ζ i
∂ξj
∂
∂ζ i
= vj ∂ζ i
∂ξj
∂
∂ζ i, (3.46)
where vj and ∂ζ i/∂ξj are evaluated at (ξ1, ξ2, ξ3) = f−1(ζ1, ζ2, ζ3).
Just as the differential linearly maps tangent vectors at (ξ1, ξ2, ξ3) to tangent
vectors at f(ξ1, ξ2, ξ3), there is a linear map of cotangent vectors at (ζ1, ζ2, ζ3) to
cotangent vectors at f−1(ζ1, ζ2, ζ3).
Definition 3.6.2 The pull-back f ∗ of the diffeomorphism f : Ψ(B) → (f Ψ)(B) ⊂ψ(M) is the linear map satisfying, for each 1-form β ∈ T ∗
f(p)M and each vector
v ∈ TpB,
(f ∗β) [v] = β [f∗v] , (3.47)
where f∗v ∈ Tf(p)M.
1On page 38 of Marsden and Hughes (1983), the equivalent object is called a tangent map and
is written Tf .
61
In coordinates, Eq. (3.47) is written as
(f ∗βidζ i
) [vj ∂
∂ξj
]= βidζ i
[f∗vj ∂
∂ξj
]
= βidζ i
[vj ∂ζk
∂ξj
∂
∂ζk
]
= βi vj ∂ηk
∂ξjdζ i
[∂
∂ζk
]
= βi vj ∂ηk
∂ξjδik
= βk vj ∂ηk
∂ξj. (3.48)
See Section 2.3 of Frankel (1997).
The pull-back f ∗ can be applied to covariant tensors, as well. In particular, the
metric tensor m on ψ(M) can be pulled back, via f ∗, to a metric tensor on Ψ(B).
If
m = eij dζ i ⊗ dζj, (3.49)
then the pull-back of m is found by applying f ∗ to each covariant vector in m:
f ∗m = eij
(f ∗dζ i
)⊗ (f ∗dζj
)
= eij
(∂ζ i
∂ξkdξk
)⊗
(∂ζj
∂ξ`dξ`
)
= eij∂ζ i
∂ξk
∂ζj
∂ξ`dξk ⊗ dξ`, (3.50)
where the eij are evaluated at (ζ1, ζ2, ζ3) and the partial derivatives are evaluated
at (ξ1, ξ2, ξ3) = f−1(ζ1, ζ2, ζ3).
f ∗m is a metric tensor on Ψ(B). It is used to compute the intrinsic distances
between material points in B following a deformation. If the position function x
on ψ(M) maps (f Ψ)(B) invertibly into E3, then (x f Ψ)(B) is the deformed
Euclidean body. In this case, if γ is a curve in Ψ(B), then its arclength computed
with f ∗m matches the Euclidean arclength of its Euclidean image, (x f γ).
62
3.6.1 Example in Cylindrical Coordinates
We consider two manifolds with cylindrical coordinates, standard Euclidean metric
tensor for cylindrical coordinates, and standard position function for cylindrical
coordinates. See Tables 3.1 and 3.2 for details.
Manifold BCoordinates Ψ = (R, Θ, Z)Metric tensor M = dR⊗ dR + R2 dΘ⊗ dΘ + dZ ⊗ dZ
Position function X(R, Θ, Z) = R cos(Θ) i + R sin(Θ) j + Z k
Table 3.1: Body manifold with cylindrical coordinates
Manifold MCoordinates ψ = (r, θ, z)Metric tensor m = dr ⊗ dr + r2 dθ ⊗ dθ + dz ⊗ dzPosition function x(r, θ, z) = r cos(θ) i + r sin(θ) j + z k
Table 3.2: Secondary manifold with cylindrical coordinates
If f has the form
f(R, Θ, Z) = (r(R), θ(R, Θ), Z), (3.51)
then f∗ has the form
f∗ = r′∂
∂r⊗ dR +
∂θ
∂R
∂
∂θ⊗ dR +
∂θ
∂Θ
∂
∂θ⊗ dΘ +
∂
∂z⊗ dZ, (3.52)
where each ∂/∂ζ i is anchored at f(R, Θ, Z) ∈ ψ(M) and each dξj is anchored at
(R, Θ, Z) ∈ Ψ(B). The Euclidean analogue, under the position functions X and x,
is
F = r′ er ⊗ER +∂θ
∂Reθ ⊗EΘ +
∂θ
∂Θeθ ⊗EΘ + ez ⊗EZ . (3.53)
The pull-back of the metric tensor m is
f ∗m =
((r′)2
+ r2
(∂θ
∂R
)2)
dR⊗ dR + r2 ∂θ
∂R
∂θ
∂Θ(dR⊗ dΘ + dΘ⊗ dR)
+ r2
(∂θ
∂Θ
)2
dΘ⊗ dΘ + dZ ⊗ dZ. (3.54)
63
The Riemann-Christoffel curvature tensor associated with this metric is identically
zero. See the Mathematica code included in Appendix D to perform this computa-
tion.
The Euclidean analogue of f ∗m, using only the position function X, is
F T · F =
((r′)2
+ r2
(∂θ
∂R
)2)
ER ⊗ER + r2 ∂θ
∂R
∂θ
∂Θ
(ER ⊗EΘ + EΘ ⊗ eR
)
+ r2
(∂θ
∂Θ
)2
EΘ ⊗EΘ + EZ ⊗EZ . (3.55)
Just as the right Cauchy-Green tensor F T ·F gives a measure of the differential
arclength in the deformed Euclidean configuration, as a function of X ∈ (X Ψ)(B), the pull-back f ∗m gives a measure of the differential deformed arclength, as
a function of (R, Θ, Z) ∈ Ψ(B).
64
CHAPTER 4
INCOMPATIBLE GROWTH
While any two-point tensor with positive determinant at a fixed point X ∈ (XΨ)B)
can be the value of a deformation gradient, not every two-point tensor field with
positive determinant is equal globally to a deformation gradient. Further, some
two-point tensor fields are global deformation gradients on some bodies but not on
others.
For example, recall the (X Ψ)(B)-to-E3 map in Eq. (2.93):
(χ X)(R, Θ, Z) = x(r, θ, z)
= x(R, 2Θ, Z)
= R cos(2Θ) i + R sin(2Θ) j + Z k. (4.1)
Locally this performs an azimuthal expansion by a factor of 2, with no radial or
vertical deformation.
Figure 4.1: The map in Eq. (4.1) locally performs an azimuthal expansion. Thisneighborhood may also be translated and rotated by the deformation of the rest ofthe body.
If the body on the left-hand side of Figure 4.1 were the entire reference con-
figuration, then this would be a true deformation of the body. We have seen in
Figure 2.7, however, that this map is not globally injective on all Euclidean bodies.
In particular, an annulus whose pieces undergo such a transformation cannot remain
a solid annulus in E3.
65
If G is a continuously differentiable two-point tensor field on (X Ψ)(B) with
positive determinant but is not equal to the gradient of a deformation of (X Ψ)(B),
then G is called an incompatible growth tensor (field). In some treatments of
incompatible growth, the grown state is pictured as a disconnected collection of
stress-free neighborhoods (Rodriguez et al. (1994); Skalak et al. (1996); Taber and
Humphrey (2001); Klarbring et al. (2007)). When the body is considered this way,
incompatible growth is interpreted to describe how local neighborhoods of matter
would grow if they were not constrained by the presence of neighboring matter. See
Figure 4.2.
incompatible
growth
elasticresponse
HaL HbL HcL
Figure 4.2: The stress-free “virtual configuration” (b) is often drawn as a collectionof unconnected sets of matter.
However, we will show that incompatible growth amounts to a change of metric
on a Riemannian manifold, so we find it convenient to view such growth as chang-
ing the shape of the body manifold B in such a way that it does not “fit” into
three-dimensional Euclidean point space.1 The connectedness of the manifold is
unchanged by the growth. See Figure 4.3.
1We must specify that it is three-dimensional Euclidean point space that cannot accommo-
date the incompatibly grown body, as the first embedding theorem of Nash ensures that an n-
dimensional Riemannian manifold has a C1 isometric embedding into E2n+1. See Chapter 1 of
Han (2006) for a brief history of the subject.
66
HaL HbL HcL
Figure 4.3: We view incompatible growth as changing the metric of a Riemannianmanifold in such a way that the body cannot be isometrically embedded in E3. Allthe connectedness properties of the manifold remain unchanged by the growth.
4.1 Multiplicative Decomposition of Deformation Gradient
Regardless of the view taken of the intermediate, “grown” state, a solid body resides
in E3. It is assumed that there is an elastic response to the incompatible growth and
that this response acts on the disconnected neighborhoods or on the Riemannian
manifold in such a way that the body again occupies a closed subset of E3. The
elastic response is described locally by a two-point tensor A, which is analogous to
a deformation gradient. See Figures 4.2 and 4.3.
The result of the incompatible growth and the elastic response is a true defor-
mation of the body. The gradient of this deformation has the form
F = A ·G. (4.2)
If G represents incompatible growth, then G is not a deformation gradient. In
this case, A is also not a deformation gradient, and the elastic response is not a
deformation.
Multiplicative decompositions of the form in Eq. (4.2) arose after the introduc-
tion of Riemannian geometry into continuum mechanics in Eckart (1948). The form
F = F elastic · F plastic (4.3)
is sometimes labeled the Kroner-Lee decomposition, after Kroner (1959) and Lee
and Liu (1967), though the decomposition appears only in the latter. Kroner (1959)
67
is an exploration of linear connections other than the Levi-Civita connection, for
use in the theory of dislocations in crystalline solids. Lee and Liu introduced the
multiplicative decomposition after observing that
“[t]he combination of elastic and plastic strains, both finite, calls for a
more careful study of the kinematics than the usual assumption that the
total strain components are simply the sum of the elastic and plastic
components, as for infinitesimal strain theory.”
Rodriguez et al. (1994) is frequently cited as the introduction into bioelasticity
of the multiplicative decomposition in Eq. (4.2). It has since been used in an array
of models of soft tissue, but according to Chen and Hoger (2000),
“neither Rodriguez et al. (1994), nor Hoger (1999) provided a theoreti-
cally sound foundation for the two central ideas in their theory of growth:
the decomposition of the gradient of the total deformation; and the re-
lated assertion that the response of the material should depend only on
the elastic portion of the decomposed gradient.”
Chen and Hoger (2000) provided a foundation while considering a reference
configuration-free approach to elasticity with incompatible growth.
4.2 Change of Riemannian Metric
As in the introduction to deformation of a Euclidean body, the reference configu-
ration is described by a body manifold B with a coordinate system Ψ = (ξ1, ξ2, ξ3)
and an associated position function X, which induces a metric tensor M on Ψ(B)
as follows:
Ei =∂X
∂ξi, Eij = Ei ·Ej, M = Eij dξi ⊗ dξj. (4.4)
We will need a second three-dimensional manifold M with coordinate system
ψ = (ζ1, ζ2, ζ3) and associated position function x, which induces a metric tensor
68
m in an analogous fashion:
ei =∂x
∂ζ i, eij = ei · ej, m = eij dζ i ⊗ dζj. (4.5)
Each continuously differentiable2 two-point tensor field G with positive determi-
nant on the reference configuration (X Ψ)(B), can be used to induce a change in
the metric on Ψ(B), which amounts to a change in metric on B itself. If a two-point
tensor field of the form
G = Gij ei ⊗Ej, (4.6)
is completely specified, then the E3-point where ei is anchored, is specified. This
means that there is a local diffeomorphism f : Ψ(B) → ψ(M) such that the left-hand
side of each tensor product in Eq. (4.6) is the vector value
ei(ζ1, ζ2, ζ3) = ei(f(ξ1, ξ2, ξ3)) =
∂x
∂ζ i
∣∣∣∣(ζ1,ζ2,ζ3)=f(ξ1,ξ2,ξ3)
, (4.7)
and it is anchored at (x f)(ξ1, ξ2, ξ3).
This makes G a two-point tensor that maps vectors at X(ξ1, ξ2, ξ3) to vec-
tors at (x f)(ξ1, ξ2, ξ3). The corresponding object that maps tangent vectors at
(ξ1, ξ2, ξ3) ∈ Ψ(B) to tangent vectors at (ζ1, ζ2, ζ3) = f(ξ1, ξ2, ξ3) ∈ ψ(M) is
G = Gij
∂
∂ζ i⊗ dξj, (4.8)
We are assuming that the coefficients Gij form a matrix [Gi
j] with positive de-
terminant, so we can compute something akin to the pull-back of the metric tensor
m:
eij Gik Gj
` dξk ⊗ dξ`. (4.9)
Since [eij] is a symmetric, positive-definite matrix and [Gik] is a matrix with posi-
tive determinant, [eijGikG
j`] is another symmetric, positive-definite matrix, and the
object in Eq. (4.9) is Riemannian metric.
2We are relaxing the requirement that metric tensors be infinitely differentiable.
69
We ask, “Is there an embedding of Ψ(B) into E3 that has G as its gradient
and that is isometric with this new metric?” If the answer is affirmative, then that
embedding is a deformation of the body, and G is a deformation gradient. The map
from Ψ(B) has the form x f , where f(ξ1, ξ2, ξ3) = (ζ1, ζ2, ζ3) and x is the position
function from (ζ1, ζ2, ζ3)-space to E3.
There are several ways in which such a tensor field can fail to be a deformation
gradient. We consider simple examples in cylindrical coordinates. We will use the
notation introduced in Tables 3.1 and 3.2.
4.3 Example: Immersion without Embedding
We have already considered an example of two-point tensor field of the form
G = γ1 er ⊗ER +γ2
γ1
eθ ⊗EΘ + ez ⊗EZ , (4.10)
where γ1 and γ2 are constants. If G is a local deformation gradient, then it must
have the form
er ⊗GRAD r + eθ ⊗GRAD θ + ez ⊗GRAD z
= er ⊗(
∂r
∂RER +
∂r
∂ΘEΘ +
∂r
∂ZEZ
)
+ eθ ⊗(
∂θ
∂RER +
∂θ
∂ΘEΘ +
∂θ
∂ZEZ
)
+ ez
(∂z
∂RER +
∂z
∂ΘEΘ +
∂z
∂ZEZ
). (4.11)
If G is of this form, then
GRAD r = γ1 ER, GRAD θ = γ2 EΘ, GRAD z = EZ . (4.12)
To test whether these are actual gradients, we compute the curls of each. For
example,
CURL(γ1 ER
)= REΘ∂γ1
∂Z− EZ
R
∂γ1
∂Θ= 0. (4.13)
The curls of γ2 EΘ and EZ are also identically zero. This means that G is locally
a deformation gradient. We will see below that G induces an immersible metric
70
tensor on Ψ(B). See Appendix F in Marion and Thornton (1988) for gradient and
curl in cylindrical coordinates.
In the spaces of coordinates, G corresponds to
G = γ1∂
∂r⊗ dR +
γ2
γ1
∂
∂Θ⊗ dΘ +
∂
∂z⊗ dZ. (4.14)
This is the tangent map for
(r, θ, z) = f(R, Θ, Z) =
(γ1R,
γ2
γ1
Θ, Z
). (4.15)
The new metric tensor on Ψ(B) is then
eij Gik Gj
` dξk ⊗ dξ` = γ21 dR⊗ dR +
(γ2
γ1
)2
r2 dΘ⊗ dΘ + dZ ⊗ dZ
= γ21 dR⊗ dR + γ2
2 dΘ⊗ dΘ + dZ ⊗ dZ. (4.16)
The Riemann-Christoffel curvature tensor associated with this metric tensor is iden-
tically zero, which guarantees that Ψ(B) can be mapped locally into E3 in a way
that agrees with the new metric.
The immersion of Ψ(B) isometric with the new metric is
(x f)(R, Θ, Z) = γ1R cos
(γ2
γ1
Θ
)i + γ1R sin
(γ2
γ1
Θ
)i + Z k. (4.17)
Figure 4.4 shows the images of Ψ(B) under x f if γ1 6= γ2. While (x f) may
send Ψ(B) into E3 in a fashion isometric with the new metric, the result is not
an embedding. Theorem 2.11.3 and the failure of (x f) ensure that there is no
embedding of Ψ(B) into E3 that agrees with the new metric.
4.4 Example: Lack of Immersion
Consider a one-to-one map of the form
f(R, Θ, Z) = (r(R), θ(R, Θ), Z). (4.18)
The gradient of (x f X−1
) is
F = r′ er ⊗ER +∂θ
∂Reθ ⊗ER +
∂θ
∂Θeθ ⊗EΘ + ez ⊗EZ , (4.19)
71
may not be the deformation of a globally one-to-one (X Ψ)(B)-to-E3 map. This
gradient is a two-point tensor that corresponds to
f∗ = r′∂
∂r⊗ dR +
∂θ
∂R
∂
∂θ⊗ dR +
∂θ
∂Θ
∂
∂θ⊗ dΘ +
∂
∂z⊗ dZ. (4.20)
The pull-back of m is then
f ∗m =
((r′)2
+ r2
(∂θ
∂R
)2)
dR⊗ dR + r2 ∂θ
∂R
∂θ
∂Θ(dR⊗ dΘ + dΘ⊗ dR)
+ r2
(∂θ
∂Θ
)2
dΘ⊗ dΘ + dZ ⊗ dZ. (4.21)
Applying Eqs. (2.44) and (2.76) yields an identically zero Riemann-Christoffel curva-
ture tensor. See the Mathematica code in Appendix D to perform this computation.
By Theorem 2.11.2, Ψ(B) can be immersed in E3 isometrically with f ∗m.
We may create a metric tensor with non-zero Riemann-Christoffel curvature by
dropping some terms in f ∗m. For example, let M be the “diagonal” portion of f ∗m:
M =
((r′)2
+ r2
(∂θ
∂R
)2)
dR⊗ dR + r2
(∂θ
∂Θ
)2
dΘ⊗ dΘ + dZ ⊗ dZ (4.22)
Computing the Riemann-Christoffel curvature tensor associated with this metric
tensor has some rather messy nonzero terms (modify the Mathematica commands
in Appendix D), so we simplify yet again:
˜M = (r′)2
dR⊗ dR + r2
(∂θ
∂Θ
)2
dΘ⊗ dΘ + dZ ⊗ dZ. (4.23)
The Riemann-Christoffel curvature tensor associated with˜M (modify the Math-
ematica code in Appendix D yet again) has four terms that may be nonzero:
RΘRRΘ = −RΘRΘR = RRΘΘR = −RRΘRΘ
=rθΘ
r′
(2 (r′)2
θRΘ − rr′′θRΘ + rr′θRRΘ
). (4.24)
Since we assume r is a function of R alone, these terms are not identically zero
unlessr2θRΘ
r′= constant in R. (4.25)
72
4.5 Non-Metric Problems
Consider the identity deformation, in which each material point remains in the same
position. The deformation gradient is
F = er ⊗ER + eθ ⊗EΘ + ez ⊗EZ , (4.26)
where r = R, θ = Θ, and z = Z. The corresponding right Cauchy-Green tensor is
F T · F = err eR ⊗ER + eθθ EΘ ⊗EΘ + ezz EZ ⊗EZ
= eR ⊗ER + r2 EΘ ⊗EΘ + EZ ⊗EZ
= eR ⊗ER + R2 EΘ ⊗EΘ + EZ ⊗EZ (4.27)
because r = R.
Now consider
G = (cos φ er − sin φ ez)⊗ER + (− sin φ er + sin φ ez)⊗EZ
+ eθ ⊗EΘ, (4.28)
again with r = R, θ = Θ, and z = Z. φ is an R-dependent angle. This corresponds
to a rotation of the vectors er and ez in the θ = constant plane. The analogue of
the right Cauchy-Green tensor is
GT ·G =(err cos2 φ + ezz sin2 φ
)ER ⊗ER + eθθ EΘ ⊗EΘ
+(err sin2 φ + ezz cos2 φ
)EZ ⊗EZ
= ER ⊗ER + R2 EΘ ⊗EΘ + EZ ⊗EZ , (4.29)
which is the same as the Cauchy-Green tensor for the identity deformation. We
know that the associated Riemann-Christoffel curvature tensor will be identically
zero.
For a deformation with coordinates (R, Θ, Z) and (r, θ, z), the portion of the
deformation gradient corresponding to r is
∂r
∂Rer ⊗ER +
∂r
∂Θer ⊗EΘ +
∂r
∂Zer ⊗EZ
= er ⊗(
∂r
∂RER +
∂r
∂ΘEΘ +
∂r
∂ZEZ
)
︸ ︷︷ ︸gradient of r
. (4.30)
73
If G is the gradient of a deformation — even a local deformation — then the gradient
of r is
GRAD r = cos φ ER − sin φ EZ . (4.31)
We take the curl of this vector field. The components of the curl are
1
R
∂
∂Θ(− sin φ) = 0, (4.32)
∂
∂Zcos φ− ∂
∂R(− sin φ) = −φ′ cos φ, (4.33)
− 1
R
∂
∂Θcos φ = 0. (4.34)
At a spot where φ′ cos φ 6= 0, the expression in Eq. (4.31) is not the gradient of a
function, and G is not even a local deformation, even though the Cauchy-Green
tensor corresponds to a metric tensor with identically zero Riemann-Christoffel cur-
vature tensor. G fails to be even a local deformation gradient because it describes
a re-orientation of material fibers without any bending, stretching, or compression
of the body. If the material fibers were re-oriented in this fashion in a deformation,
the result of this microscopic re-orientation would be a macroscopic change in the
shape of the body.
The problem with G cannot be detected with the Riemann-Christoffel curvature
tensor because the problem with G is not metric in nature. The only difference
between G and F in Eq. (4.26) is a rotation: er and ez in F are rotated about eθ
to create G. GT · G, the associated metric tensor, and the associated Riemann-
Christoffel curvature tensor cannot detect the rotation. Note that if F = R ·U is
the polar decomposition of a deformation gradient, then
F T · F = (R ·U )T · (R ·U ) = UT ·U . (4.35)
This shows that all information about orientation is absent from the right Cauchy-
Green tensor. It contains information about local arclengths only. Hence F T ·F , GT · G, and the associated metric tensors on Ψ(B) determine only that local
arclengths in the body manifold are compatible, at least locally, with a Euclidean
image of the manifold. These metric tensors cannot be used to detect changes in
orientation.
74
Deeper study of a growth tensor such as G in Eq. (4.28) requires consideration
of linear connections other than the Levi-Civita connection and will not be pursued
here. We will consider the types of incompatible growths seen in the previous
sections: (1) those that are local deformations but not global deformations, and (2)
those that are not isometrically immersible and have non-zero Riemann-Christoffel
curvature tensor.
75
R3
0 A B
0
Π
2 Π
R
Q
fR
3
0 Γ1A Γ1B
0
Π
Γ2
Γ12 Π
r
Θ
X` -1 x`
E3
E3
R3
0 A B
0
Π
2 Π
R
Q
f
R3
0 Γ1A Γ1B
0
Π
2 ΠΓ2
Γ12 Π
r
Θ
X` -1 x`
E3
E3
Figure 4.4: Top: If γ1 < γ2, then the image of the annulus under the isometricimmersion double-covers part of E3. Bottom: If γ1 > γ2, the image of the annulusdoes not close.
76
CHAPTER 5
ELEMENTARY MECHANICS OF SOLIDS
5.1 Material vs. Spatial Formulation; Conservation of Mass
To discuss continuum mechanics we assume the existence of a one-parameter family
of deformations χ(t, ·) : (X Ψ)(B) → E3, where χ(0, ·) is the identity map on (X Ψ)(B). Hence, a deformed configuration is often called not the final configuration,
but the current configuration (at time t).1
All physical quantities can be expressed as functions of three types of arguments:
the curvilinear coordinates (ξ1, ξ2, ξ3) of material points, the Euclidean locations
X(ξ1, ξ2, ξ3) of the material points in the reference configuration, or the Euclidean
locations x(ζ1, ζ2, ζ3) of the current configuration. The notations used will be dif-
ferent.
For example, if there are no sources or fluxes of mass in or on the body, then the
mass can be expressed in three ways as the integral of a mass density. First, one can
integrate a mass density over B, the set of curvilinear coordinates corresponding to
material points:
m(B) =
∫
Ψ(B)
ρ(ξ1, ξ2, ξ3
)det (E1E2E3) dξ1dξ2dξ3. (5.1)
The mass is also equal to the integral of a density over (X Ψ)(B) ⊂ E3, the portion
of Euclidean space occupied by the body before any deformation:
m(B) =
∫
(XΨ)(B)
ρ0(X) dV, (5.2)
1Later on we will consider statics, and we will consider deformations that consist of a growth
step followed by an elastic step. The configuration formed after both steps are complete will be
called the final state.
77
where dV is volume measure on (X Ψ)(B). Finally, the mass can be considered as
an integral of a density over the current configuration:
m(B) =
∫
χ(t,(XΨ)(B))ρ(t, x) dv
=
∫
(XΨ)(B)
ρ(t, χ(t, X)) det F (t, X) dV, (5.3)
where dv is volume measure on χ(t, (X Ψ)(B)), F (t, ·) is the gradient of χ(t, ·),and dV is volume measure on (X Ψ)(B). Since the position x = χ(t, X) ∈ E3 is
time-dependent, the density ρ(t, ·) : E3 → [0,∞) must also change with time.
Since these expression are also equivalent for all measurable subsets Ω of Ψ(B),
we can write
∫
X(Ω)
ρ0(X) dV = m(Ω) =
∫
χ(t,X(Ω))ρ(t,x) dv
=
∫
X(Ω)
ρ(t,χ(t,X)) det F (t, X) dV, (5.4)
from which we conclude the pointwise equality
ρ0(X) = ρ(t,χ(t, X)) det F (t, X) (5.5)
for (almost) every X ∈ (X Ψ)(B). This is an expression of the conservation of
mass.
Since we will assume that there are no sources or fluxes of mass in or on the
body, we will express all quantities as functions whose domain is the reference con-
figuration, (X Ψ)(B). That is, the original spatial position (X Ψ)(p) of the
material point p ∈ B is the independent non-temporal variable. In some sense, the
coordinate point Ψ(p) = (ξ1, ξ2, ξ3) may be considered the independent variable.
This approach is known as the material or Lagrangian formulation of continuum
mechanics, as opposed to the spatial or Eulerian formulation, in which a stationary
spatial point x ∈ E3 is the independent spatial variable.
78
5.2 Other Integral Balance Laws
The balance laws for continuous bodies with no exchange of mass or momentum are
generalizations of those for collections of particles. It should be noted that these
are axioms of solid mechanics, as they cannot be derived from analogous results
for Newtonian mechanics of collections of point particles. See page 402 of Antman
(1995).
First, the rate of change of linear momentum is equal to the total force on the
body:
d
dt
∫
(XΨ)(B)
∂χ
∂t(t, X)ρ0(X) dV
=
∫
(XΨ)(B)
f(t, X) dV +
∮
∂(XΨ)(B)
t(t, X, ∂(X Ψ)(B)) dS, (5.6)
or, since ρ0 and the spatial variable X ∈ (X Ψ)(B) are time-independent,
∫
(XΨ)(B)
∂2χ
∂t2(t, X)ρ0(X) dV
=
∫
(XΨ)(B)
f(t, X) dV +
∮
∂(XΨ)(B)
t(t, X, ∂(X Ψ)(B)) dS. (5.7)
Here f is the body force per unit reference volume, which includes outside forces
such as gravity; and t is the traction applied to the body’s surface. The notation
indicates that t is a function of position and of the boundary ∂(X Ψ)(B).
The rate of change of angular momentum is equal to the applied torque:
d
dt
∫
(XΨ)(B)
χ(t, X)× ∂χ
∂t(t, X)ρ0(X) dV
=
∫
(XΨ)(B)
χ(t, X)× f(t, X) dV +
∮
∂(XΨ)(B)
χ(t, X)× t(t, X) dS. (5.8)
Note that
d
dt
(χ× ∂χ
∂t
)=
∂χ
∂t× ∂χ
∂t+ χ× ∂2χ
∂t2
= χ× ∂2χ
∂t2. (5.9)
79
Balance of angular momentum can now be written as∮
∂(XΨ)(B)
χ× t dS +
∫
(XΨ)(B)
(χ× f − ρ0χ× χtt) dV = 0. (5.10)
5.3 Stress Tensors
Local versions of the integral balance laws above can be found if we adopt Cauchy’s
postulate that the traction t depends on the surface ∂(X Ψ)(B) only through the
unit normal vector N at X ∈ ∂(X Ψ)(B); we drop the previous notation and write
t (t, X,N) instead. In such a case, Cauchy’s Stress Theorem ensures a relatively
simple expression for the traction.
Theorem 5.3.1 (Cauchy’s Stress Theorem) If t(t, ·, N(·)) and ρ0(·)χtt(t, ·) −f(t, ·) are continuous on (X Ψ)(B), then there exists a tensor field P (t, ·) on
(X Ψ)(B) such that
t (t, X,N ) = P (t, X) ·N .
See Section XII.7.14 in Antman (1995).
P is called the first Piola-Kirchhoff stress tensor. Note that it maps a vector in
the reference configuration (N) to a vector in the deformed configuration (t); see
Figure 5.1. Hence, P is a two-point tensor. The traction on the surface of the body
can now be expressed as ∮
∂(XΨ)(B)
P (t, X) ·N (X)dS. (5.11)
Recall Eq. (3.43), which described how to compute, in reference configuration coor-
dinates, a tensorial flux in the deformed configuration: for each measurable subset
Ω of Ψ(B) with well-behaved boundary ∂Ω,∮
∂(χX)(Ω)
A (x) · n(x) ds =
∮
∂X(Ω)
(det F (X)) A(χ(X)) · F−T (X)︸ ︷︷ ︸P (X)
·N(X) dS,
(5.12)
where A is an Eulerian tensor. If we let P be the tensor integrand on the right-hand
side, then we define another tensor by
T = (det F )−1 P · F T . (5.13)
80
T is called the Cauchy stress tensor. It maps a unit normal vector n in the de-
formed configuration to the stress vector on the surface that has n as its normal;
see Figure 5.1. Since T maps from the tangent space at x into the same tangent
space, it is an Eulerian tensor.
N
X`
HWL
P×N = t t
HΧëX`
LHWL
T×n = t n
HΧëX`
LHWL
Figure 5.1: Left: The first Piola-Kirchhoff stress tensor maps unit normal vectors inthe reference configuration to traction vectors in the deformed configuration. Right:The Cauchy stress tensor maps unit normal vectors in the deformed configurationto traction vectors in the deformed configuration.
5.4 Differential Balance Laws
5.4.1 Linear Momentum
The entire body (X Ψ)(B) is not special with respect to the balance laws, so
Eqs. (5.7) and (5.10) should hold for each measurable subset Ω of Ψ(B) with well-
behaved boundary ∂Ω. The balance of linear momentum, combined with Cauchy’s
stress theorem, yields
∫
X(Ω)
(χttρ0 − f) dV =
∮
∂X(Ω)
P ·N dS. (5.14)
The first Piola-Kirchhoff stress tensor can be written as
P = P iJ ei ⊗EJ = P iJ ei ⊗EJ , (5.15)
where we use an upper-case index for reference configuration and a lower-case index
for deformed configuration. We have also taken advantage of the fact that in E3
covariant and contravariant vectors are both traditional vectors of the same type.
81
Note that for a given deformation χ, the deformed coordinate vector ei is, ultimately,
a function of X ∈ (X Ψ)(B). We cannot simply apply the Divergence Theorem
to P iJEJ and ignore the change in ei as X varies. There is an applicable extension
of the Divergence Theorem for this situation.
Theorem 5.4.1 Divergence Theorem for a Two-Point Tensor
∮
∂X(Ω)
P ·N dS =
∮
∂X(Ω)
(P iJ ei ⊗EJ
) ·N dS
=
∫
X(Ω)
P iJ||J ei dV, (5.16)
where the covariant derivative for a two-point tensor is
P iJ||J =
∂P iJ
∂ξJ+ ΓJ
LJP iL + PmJF `Jγi
m`, (5.17)
where γim` are Christoffel symbols of the second kind for the coordinate system on
the deformed configuration, anchored at x = χ(X).
See Proposition 4.29 and Theorem 2.6 in Marsden and Hughes (1983).
The balance of linear momentum can thus be written in differential form as
ρ0χtt = f + DIVP , (5.18)
where
DIVP = P iJ||J ei. (5.19)
Note that all quantities in Eq. (5.18) have spatial argument X ∈ (X Ψ)(B) but
take vector values anchored in the current configuration.
The expression equivalent to Eq. (5.18) in the spatial formulation is
ρ(t, x)d
dtv(t, x) = f(t,x) + div T (t, x), (5.20)
where we now consider f to have spatial argument x instead of X, div is divergence
with respect to coordinates in the current configuration, and
v(t, x) = χt(t, χ−1(t, x)). (5.21)
82
5.4.2 Angular Momentum
The analogous local expression of balance of angular momentum is
P · F T = F · P T . (5.22)
A derivation is presented in Appendix B.
Eq. (5.22) implies that the Cauchy stress tensor T is symmetric:
T T =((det F )−1 P · F T
)T
= (det F )−1 F · P T
= (det F )−1 P · F T = T . (5.23)
This is a local expression of balance of angular momentum in the spatial description.
83
CHAPTER 6
HYPERELASTICITY
6.1 Constitutive Relations
Consider for a moment Eq. (5.18) expressed with Cartesian coordinates (X, Y, Z)
for the reference configuration and Cartesian coordinates (x, y, z) for the current
configuration. Then the first Piola-Kirchhoff stress tensor is expressed as
P = P `M i` ⊗ iM , where i1 = i, i2 = j, i3 = k, (6.1)
and its divergence is
DIV P =∂P `M
∂XMi` (6.2)
The remaining quantities in Eq. (5.18) are
χ = χ` i`, f = f ` i`. (6.3)
Eq. (5.18) amounts to three scalar equations:
ρ0χ`tt = f ` +
∂P `M
∂XM, ` = 1, 2, 3. (6.4)
In the absence of body forces, which we shall assume, f ` ≡ 0. Eq. (5.22) represents
three linear constraints on P `M and the spatial partial derivatives of the χ`, so
we have six equations for our three unknowns χ` and nine unknowns P `M . The
remaining six equations come in the form of a constitutive relation that describes the
six independent entries in the stress tensor P as functions of the strains ∂χ`/∂XM ,
the position X, and perhaps other quantities, such as temperature.
It is through the constitutive relation that the specific physics of a material
is encoded into the differential equations of solid mechanics. For example, while
a block of rubber and a block of cement can be described as solid continua that
satisfy conservation of mass, of linear momentum, and of angular momentum, their
responses to applied tractions are expected to be quite different.
84
6.2 Simplifications
Numerous simplifications are applied in the search for constitutive relations. A
simple material is one in which the stress tensor at a point X ∈ (X Ψ)(B) in
the reference configuration depends on time t, position X, and the history of the
deformation gradient at that point (as opposed to the history of the deformation
gradient in a neighborhood of X):
P (t, X) = P (t, X,F t(·,X)), (6.5)
where the superscript indicates that this is the history of F up to the time t. P is
called the response function of the material.
A special case of a simple material is an elastic material, in which the only value
of F that plays a role in determining the stress is the instantaneous value at time
t, and t is not explicitly an argument of the response function:
P (t,X) = P (X,F (t, X)). (6.6)
In a Cauchy elastic material, the point X is also not an explicit argument:
P (t, X) = P (F (t, X)). (6.7)
6.3 Stress Power and Hyperelasticity
The power spent in the deformation of a solid is found through the stress and the
rate of strain. In the material formulation, the rate of strain is measured by the time
derivative of the deformation gradient. Recall the form of the deformation gradient:
F =∂ζ i
∂ξjej ⊗Ei. (6.8)
Since we are considering a fixed point in the reference configuration, the time-
dependence is in the deformed coordinates ζ i, and the time derivative of F has the
form
F t =∂ζ i,t∂ξj
ei ⊗Ej. (6.9)
85
The first Piola-Kirchhoff stress tensor is almost the appropriate stress to combine
with this strain rate. But both F t and P map from tangent spaces in the reference
configuration to tangent spaces in the current configuration. We need to use the
transpose of the first Piola-Kirchhoff stress tensor, denoted by S and often called
the nominal stress tensor:
S =(P k` ek ⊗E`
)T= P k` E` ⊗ ek. (6.10)
S · F t = ekiPk` ∂ζ i,t
∂ξjE` ⊗Ej. (6.11)
The trace of S ·F t is equal to the total inner product S : F t and is the stress power
of the deformation:
S : F t = tr (S · F t) . (6.12)
In a Cauchy elastic material, S will be determined by the local, instantaneous
value of F :
S = S(F ) =(P (F )
)T
, (6.13)
and the stress power density can be expressed as
tr(S(F ) · F t
). (6.14)
A Green elastic or hyperelastic material is a Cauchy elastic material for which
there exists a strain-energy density function W whose value at reference point X at
time t is determined by F (t, X). In such a case, the stress power density satisfies
d
dtW (F (t, X)) = tr
(S(F (t, X)) · F t(t,X)
). (6.15)
For a scalar function of a time-dependent tensor argument,
d
dtW (F (t, X)) =
∂W
∂F
∣∣∣∣F (t,X)
: F t(t, X); (6.16)
see Section XI.2 of Antman (1995). We conclude that for a hyperelastic material
the nominal stress tensor has the form
S(t, X) =∂W
∂F
∣∣∣∣F (t,X)
. (6.17)
86
6.4 Objectivity
One of the properties that all constitutive relations must have is objectivity or frame-
indifference. If the current configuration is viewed through a time-dependent set of
coordinates, the motion is expressed in the form
χ(t− a, X) = Q(t) · χ(t, X) + c(t), (6.18)
where Q(t) is a time-dependent rotation, c(t) is a time-dependent spatial transla-
tion, and a is a fixed translation in time. The first Piola-Kirchhoff stress tensor from
the second point of view must have form
P (t− a, X) = Q(t) · P (t, X). (6.19)
In Cauchy elastic materials, this requirement proscribes the types of dependence P
can have on F .
Theorem 6.4.1 A Cauchy elastic constitutive relation is invariant under rigid mo-
tions and time shifts if and only if it has the form
P (F ) = R · P (U ), (6.20)
where F = R ·U is the polar decomposition of F .
See Theorem XII.11.13 in Antman (1995) or Sections 4.2.1 and 4.2.2 in Ogden
(1984). See Appendix D for a discussion of the standard proof of the existence of
the polar decomposition and its relation to Theorem 6.4.1.
6.5 Isotropic Hyperelasticity
6.5.1 Isotropic Functions
We simplify yet again and assume that the value of the strain-energy density W
is unchanged if the direction of a stretch or compression is changed. This isotropy
87
indicates that there is no preferred direction in the material, i.e. that the mechanical
properties of the material are the same in each direction. In this case,
W (Q · F ) = W (F ) (6.21)
for each (Eulerian) rotation Q. In particular, if F = R·U is the polar decomposition
of the deformation gradient, then
W (F ) = W (R ·U ) = W (RT ·R ·U ) = W (U ). (6.22)
The strain-energy density is thus equal to some function of the stretch tensor U . Due
to the complications mentioned in Appendix D, however, we prefer to use a strain-
energy density that is a function the right Cauchy-Green tensor F T · F = UT ·U :
W (F ) = W (F T · F ). (6.23)
Real-valued isotropic functions of symmetric tensor arguments have a special
form.
Theorem 6.5.1 ϕ : Sym → R is isotropic if and only if there exists some function
ϕ : (0,∞)3 → R such that for each symmetric C,
ϕ(C) = ϕ(I1, I2, I3),
where I1, I2, I3 are the principal invariants of the symmetric tensor C:
I1 = tr (C) , I2 = det C tr(C−1
), I3 = det C.
See Section 37 in Gurtin (1981) for a proof.
6.5.2 Form of the Nominal Stress
For a isotropic hyperelastic material, the nominal stress has the form
S(F ) =∂
∂FW (F T · F ) = 2
∂W
∂C
∣∣∣∣∣F T ·F
· F T , (6.24)
88
since∂
∂F
(F T · F )
= 2F T . (6.25)
We use C as the symbol of the Lagrangian symmetric tensor argument of W . Note
the direction of the tangent space-to-tangent space mapping:
F T : tangent space at x −→ tangent space at X, (6.26)
and since the argument of W is Lagrangian,
∂W
∂C
∣∣∣∣∣F T ·F
: tangent space at X −→ tangent space at X. (6.27)
It should also be noted that, by Theorem 6.5.1, there is a function˜W such that
W (F T · F ) =˜W (I1, I2, I3), (6.28)
where I1, I2, I3 are the principal invariants of F T · F . The derivative of the strain-
energy with respect to its symmetric tensor argument then has the form
∂W
∂C
∣∣∣∣∣F T ·F
=∂˜W
∂I1
∂I1
∂C
∣∣∣∣∣∣F T ·F
+∂˜W
∂I1
∂I2
∂C
∣∣∣∣∣∣F T ·F
+∂˜W
∂I1
∂I3
∂C
∣∣∣∣∣∣F T ·F
. (6.29)
6.6 Hyperelasticity with Growth
6.6.1 Units without Growth
In hyperelasticity without growth, the strain-energy density has units
[W (F )] =elastic energy
reference volume, (6.30)
so the nominal stress tensor has units
[S] =[W (F )]
[F ]=
elastic energy / reference volume
deformed length / reference length
=elastic energy / deformed length
reference area. (6.31)
If we recognize elastic energy per final length as elastic force in the deformed con-
figuration, then the units of S reflect the role of P , its transpose, as a linear map
89
converting area in the reference configuration into stress in the deformed configura-
tion.
The Cauchy stress tensor T is related to S by
T = (det F )−1 F · S, (6.32)
so the units of T are
[T ] = [det F ]−1 [F ] [S]
=
(deformed volume
reference volume
)−1deformed length
reference length· elastic energy / deformed length
reference area
=elastic energy / deformed length
deformed area. (6.33)
These units reflect T ’s role as a linear map converting area in the deformed config-
uration to force in the deformed configuration.
6.6.2 Units with Growth
In a body with growth included via the multiplicative decomposition in Eq. (4.2),
we distinguish three states of the body. The pre-growth state is the traditional
reference configuration; the post-growth, pre-elastic response state (described by a
non-embeddable Riemannian manifold) is called intermediate; and the(Euclidean)
configuration found after the elastic response to the growth is called the final con-
figuration. The lengths, areas, and volumes attached to these three states will be
considered distinct.
In this model, the elastic strain-energy density’s argument is A, the sole (first-
order) descriptor of the elastic response. Just as F maps from a tangent space
in the reference configuration to a tangent space in the deformed configuration in
traditional hyperelasticity, A maps from a tangent space in the intermediate state
to a tangent space in the final configuration. Considering the units of W (F ) in
Eq. (6.30), we conclude that the units of W (A) are
[W (A)] =elastic energy
intermediate volume. (6.34)
90
The analogue of the nominal stress has units
[∂
∂AW (A)
]=
elastic energy / intermediate volume
final length / intermediate length
=elastic energy / final length
intermediate area, (6.35)
while the analogue of the Cauchy stress has units
[(det A)−1 A · ∂
∂AW (A)
]
=
(final volume
intermediate volume
)−1
· final length
intermediate length· elastic energy / final length
intermediate area
=elastic energy / final length
final area. (6.36)
A value of this analogue of the Cauchy stress is a linear map that coverts area in
the final configuration to stress in the final configuration. Since the elastic strain-
energy depends only on the elastic response A, this is the true Cauchy stress for a
body in which the deformation gradient is decomposed as F = A ·G.
The formula above provides the form of the (Eulerian) Cauchy stress tensor for
a hyperelastic body with growth, but we will need the first Piola-Kirchhoff stress
tensor P to express balance laws. In hyperelasticity with growth, P remains a
linear map from tangent spaces in the reference configuration to tangent spaces in
the final (post-growth, post-elastic response) configuration, and it is related to the
Cauchy stress tensor in the traditional manner:
P = (det F ) T · F−T
=
(det F
det A
)A ·
(∂
∂AW (A)
)· F−T
= (det G) A ·(
∂
∂AW (A)
)· F−T . (6.37)
While pursuing a formulation of constitutive relations for bodies undergoing
incompatible growth, Chen and Hoger (2000) made the argument above rigorous
and applicable beyond hyperlasticity. They showed that the Cauchy stress in a
body that has undergone growth as described by a growth tensor G, should be
91
given by
T (t, x) = T (F (t, χ−1(t, x)) ·G−1(χ−1(t, x)))
= T (A(t, χ−1(t, x))), (6.38)
where T is a traditional Cauchy response function.
The relationship between Cauchy stress and first Piola-Kirchhoff stress is un-
changed by growth, so in this case the first Piola-Kirchhoff stress tensor has the
form
P = (det F ) T · F−T = (det F ) T (A) · F−T .
In the hyperelastic case, we find a traditional Cauchy response function by first
finding the nominal stress and then its transpose, the first Piola-Kirchhoff stress:
S =∂W
∂F, P =
(∂W
∂F
)T
. (6.39)
Then we employ Eq. (5.13):
T = (det F )−1 P · F T = (det F )−1
(∂W
∂F
)T
· F T . (6.40)
The Cauchy response function is then
T (A) = (det A)−1
(∂W
∂A
)T
·AT , (6.41)
where the partial derivative ∂W/∂A is evaluated at A.
The first Piola-Kirchhoff stress is then
P = (det F ) (det A)−1
(∂W
∂A
)T
·AT · F−T = (det G)
(∂W
∂A
)T
·G−T . (6.42)
This is expressed in Ben Amar and Goriely (2005) by
S = JG−1 ·WA, (J = det G) (6.43)
for the nominal stress, the transpose of the first Piola-Kirchhoff stress.
92
6.7 Example: Compressible Neo-Hookean
We consider here an isotropic strain-energy density of neo-Hookean type, expressed
as a function of eigenvalues:
W (λ1, λ2, λ3) = µ
(1
2
(λ2
1 + λ22 + λ2
3 − 3)− ln J
)+ λ (J − 1− ln J) , (6.44)
where λ21, λ2
2, λ23 are the eigenvalues of F T ·F (λ1, λ2, and λ3 are eigenvalues of F ),
and J = λ1λ2λ3. The eigenvalues and principal invariants of a symmetric matrix
are related by
I1 = λ21 + λ2
2 + λ23, (6.45)
I2 = λ21λ
22 + λ2
2λ23 + λ2
3λ21, (6.46)
I3 = λ21λ
22λ
23, (6.47)
where it must be remembered that λ21, λ2
1, and λ21 are the eigenvalues of F T ·F , not
the squares of the eigenvalues. See Chapter 1 of Gurtin (1981).
The given strain-energy density can be expressed as a function of the principal
invariants of F T · F :
˜W (I1, I2, I3) = µ
(1
2(I1 − 3)− ln
√I3
)+ λ
(√I3 − 1− ln
√I3
)
= µ
(1
2(I1 − 3)− 1
2ln I3
)+ λ
(√I3 − 1− 1
2ln I3
). (6.48)
The nonzero partial derivatives of˜W are
∂˜W
∂I1
=µ
2, (6.49)
∂˜W
∂I3
=1
2
(λI
−1/23 − (λ + µ) I−1
3
)(6.50)
The derivatives of the principal invariants are
∂I1
∂C= I, (6.51)
∂I3
∂C= (det C) C−1, (6.52)
93
where I is the Lagrangian identity tensor. See Section 3.5 of Marsden and Hughes
(1983). The full derivative of˜W is
∂˜W
∂C=
µ
2I +
1
2
(λI
−1/23 − (λ + µ) I−1
3
)(det C) C−1
=µ
2I +
1
2
(λI
1/23 − (λ + µ)
)C−1 (6.53)
because I3 = det C.
Evaluating this derivative at C = F T · F , we have
∂˜W
∂C
∣∣∣∣∣∣F T ·F
=µ
2I +
1
2(λJ − (λ + µ)) F−1 · F−T , (6.54)
where J = λ1λ2λ3 = I1/23 . The nominal stress tensor is
S = 2∂˜W
∂C
∣∣∣∣∣∣F T ·F
· F T = µF T + (λJ − (λ + µ)) F−1. (6.55)
The corresponding Cauchy stress tensor is
T (F ) = (det F )−1 F · S=
µ
JF · F T +
(λ− λ + µ
J
)I, (6.56)
where I is now the Eulerian identity tensor.
Now we consider the case with growth. The Cauchy stress determined by A is
T (A) =µ
JA
A ·AT +
(λ− λ + µ
JA
)I,
=1
JA
µ
(λ2
1 a1 ⊗ a1 + λ22 a2 ⊗ a2 + λ2
3 a3 ⊗ a3
)
+ (λJA − (λ + µ)) I , (6.57)
where JA = det A and we have expressed the symmetric Eulerian tensor A ·AT as
its spectral resolution:
A ·AT = λ21 a1 ⊗ a1 + λ2
2 a2 ⊗ a2 + λ23 a3 ⊗ a3, (6.58)
94
where the λ2i are the eigenvalues of A·AT and the ai = ai are the unit eigenvectors of
A ·AT , which can be chosen to form an orthonormal basis due to the real symmetry
of A ·AT .
The corresponding first Piola-Kirchhoff stress is
P = (det F ) T (A) · F−T
= JA (det G) T (A) ·A−T ·G−T (6.59)
= (det G)µA ·G−T + (λJA − (λ + µ)) F−T
. (6.60)
95
CHAPTER 7
CAVITATION IN THE HAUGHTON-OGDEN MEMBRANE
Elastic cavitation is the mechanism by which an empty cavity forms inside an elastic
solid under tension. The modern study of cavitation in elastic solids began with Gent
and Lindley’s experimental observation and theoretical consideration of rupture in
rubber cylinders Gent and Lindley (1959). The subject became a staple of nonlinear
elasticity after Ball (1982) put the subject on a solid mathematical foundation in
the context of finite hyperelasticity. In particular, Ball transformed the analysis
into a bifurcation problem and showed that a neo-Hookean sphere under hydrostatic
load supports both trivial (with no inner cavity) and cavitated solutions for a load
exceeding 5E/6, where E is Young’s modulus. Following Ball’s work, cavitation,
bifurcations, and solutions have been studied for a variety of specific strain energy
densities, geometries, loads, and constraints (Horgan and Polignone (1993), Horgan
and Polignone (1995), Sivaloganathan (1999), Horgan (2001), Sivaloganathan and
Spector (2002), Pericak-Spector et al. (2002)). Further experimental and numerical
works have validated the theoretical approaches of some of these theories (Tvergaard
(1996), Fond (2001), Dollhofer et al. (2004)). However, the role of elastic cavitation
as a mechanism for fracture initiation remains a controversial topic.
Elastic cavitation has also been studied for membranes. Interestingly, depend-
ing on the modeling assumptions on the membrane, cavitation may not be possible
Haughton (2001). More precisely, in a model of an elastic membrane with Varga
strain-energy density and with thickness negligible compared to the smallest curva-
ture of the “middle surface” of the membrane, no cavitated solution exists (Haughton
(1986)). However, if the membrane is considered to be a true two-dimensional object
with no thickness, cavitation is possible (Steigmann (1992)). It is therefore partic-
ularly interesting to study the case of a membrane subject to growth and see how
it allows for the existence of cavitated solutions, a drastic change of basic material
96
behavior.
7.0.1 Kinematics of an Axisymmetric Membrane with Growth
We consider an axisymmetric membrane, so we assign cylindrical coordinates to
both reference and deformed configurations. We assume r is a function of R alone,
θ = Θ, and z is a function of R and Z. The deformation gradient has the form
F =
(dr
dRer +
∂z
∂Rez
)⊗ER + eθ ⊗EΘ +
dz
dZez ⊗EZ
=
(dr
dRh1 +
∂z
∂Rk
)⊗ h1 +
r
Rh2 ⊗ h2 +
dz
dZk ⊗ k. (7.1)
We consider a growth tensor that is locally a deformation gradient but is not the
gradient of an embedding of the membrane:
G = γ1 h1 ⊗ h1 + γ2 h2 ⊗ h2 + k ⊗ k, (7.2)
where γ1 and γ2 are constants. The corresponding A is then
A = γ−11
(dr
dRh1 +
∂z
∂Rk
)⊗ h1 +
r
γ2Rh2 ⊗ h2 +
dz
dZk ⊗ k. (7.3)
The elastic strain quantity used in this model is the average of A over the
thickness of the membrane:
A =1
2H
∫ H
−H
A dZ
= γ−11
(dr
dRh1 +
1
2H
∫ H
−H
∂z
∂RdZ k
)⊗ h1 +
r
γ2Rh2 ⊗ h2
+1
2H
∫ H
−H
dz
dZdZ k ⊗ k (7.4)
A is simplified with the assumption of uniformity in the Z-direction of the membrane
and the assumption that the thickness is negligible compared to the smallest radius
of curvature found in the middle surface:
A = γ−11
dr
dRh1 ⊗ h1 +
r
γ2Rh2 ⊗ h2 +
h
Hk ⊗ k, (7.5)
97
where the R-dependent function h is the half-thickness of the deformed membrane.
See Haughton and Ogden (1978).
The eigenvalues of A are
λ1 = γ−11
dr
dR, λ2 =
r
γ2R, λ3 =
h
H. (7.6)
7.0.2 Thickness-Averaged Hyperelasticity
Haughton and Ogden (1978) showed that the mechanics of a hyperelastic membrane
can be approximated by treating the strain-energy density as a function of the
thickness-averaged strain tensor. We consider a traditional strain-energy density
but use A as its argument. Based on Eq. (6.42) for the first Piola-Kirchhoff stress
tensor, which is the transpose of the nominal stress tensor, the nominal stress arising
from this energy is
S = (det G) G−1 · ∂W
∂A
∣∣∣∣A
(7.7)
The corresponding Cauchy stress tensor is found via the thickness-averaged defor-
mation gradient:
T =(det F
)−1F · S
=(det A
)−1(det G)−1 (det G) A ·G ·G−1 · ∂W
∂A
∣∣∣∣A
=(det A
)−1A · ∂W
∂A
∣∣∣∣A
. (7.8)
If the strain-energy density is isotropic, then there is some real-valued W such
that W (A) =˜W (I1, I2, I3), where I1, I2, I3 are the principal invariants of of A
T ·A.
7.0.3 Equilibrium Equation
The assumptions of axial symmetry and of uniformity in the Z-direction reduce
the Eulerian equation of equilibrium (Eq. (5.20)) to a single ordinary differential
equation. The two-dimensional divergence of the radial force in the membrane is
the integral of T over the thickness of the membrane. In the deformed configuration,
98
this is 2hT , and the equation of equilibrium is
d
dr
(2ht11
)+
2h(t11 − t22
)
r= 0, (7.9)
to which a “plane stress” condition is added:
t33 = 0. (7.10)
Here, t11 is the radial Cauchy stress, t22 is the azimuthal Cauchy stress, and t33 is
the Cauchy stress normal to the middle surface of the sheet.
If we assume constant half-thickness H of the reference configuration, then since
h = Hλ3, Eq. (7.9) simplifies to
1
λ3
d
dr
(λ3t11
)+
t11 − t22
r= 0. (7.11)
Since this model assumes minimal curvature of the membrane’s middle surface,
r = r(R) is a monotonically increasing function. Thus Eq. (7.11) can be re-written
as
1
λ3
(dr
dR
)−1d
dR
(λ3t11
)+
t11 − t22
r= 0, (7.12)
or,
1
λ3λ1
d
dR
(λ3t11
)+
t11 − t22
r= 0, (7.13)
where all quantities are now considered to be functions of R.
7.0.4 Growth-Induced Cavitation
We consider the possibility of growth-induced cavitation. That is, we prescribe
growth parameters and zero applied loads and then seek an axisymmetric deformed
configuration such that r(0) > 0. Since the total radial traction at the cavity’s sur-
face is 2πrh t11
∣∣R=0
= 2πrλ3H t11
∣∣R=0
and since r(0) > 0, we impose the restriction
λ3 t11
∣∣R=0
= 0, (7.14)
with the same boundary condition at R = Rmax. In the following sections we
consider two types of elastic strain energy densities.
99
7.1 Absence of Cavitation in Neo-Hookean Material
The compressible neo-Hookean strain energy density has the form
W (A) = W(AT ·A)
=˜W (I1, I2, I3)
= µ
(1
2I1 − g(
√I3)
), (7.15)
where µ is constant, I1 = tr(AT ·A)
, and I3 = det(AT ·A)
= (det A)2.
The derivative of W is then
∂W
∂A= 2
∂W
∂C·AT
= 2
∂
˜W
∂I1
∂I1
∂C+
∂˜W
∂I3
∂I3
∂C
·AT
= 2µ
(1
2I − 1
2I−1/23 g′
(√I3
)(det C) C−1
)·AT
= 2µ
(1
2I − 1
2I−1/23 g′
(√I3
)(det A)2 A−1 ·A−T
)·AT
= µ(AT − g′
(√I3
)(det A) A−1
). (7.16)
As a function of A, T is
T =(det A
)−1A · ∂W
∂A
∣∣∣∣A
= µ(det A
)−1A ·
(A
T − g′(√
I3
) (det A
)A−1
)
= µ(det A
)−1(A ·AT − (
det A)g′
(det A
)I)
, (7.17)
where I is the Eulerian identity tensor.
The radial component of the Cauchy stress is
t11 = h1 · T · h1 = µJ−1(λ2
1 − Jg′(J))
= µ
(λ1
λ2λ3
− g′(J)
), (7.18)
where λ1, λ2, λ3 are the eigenvalues of A, and J = det A = λ1λ2λ3.
100
The azimuthal Cauchy stress is
t22 = h2 · T · h2 = µJ−1(λ2
2 − Jg′(J))
= µ
(λ2
λ1λ3
− g′(J)
). (7.19)
The third principal Cauchy stress is
t33 = k · T · k = µJ−1(λ2
3 − Jg′(J))
= µ
(λ3
λ1λ2
− g′(J)
). (7.20)
The plane stress condition (Eq. (7.10)) can now be expressed as
λ23 = Jg′(J). (7.21)
The equilibrium equation takes the form
1
λ3γ1λ1
d
dR
(λ3 t11
)+
µ
r
(λ1
λ2λ3
− λ2
λ1λ3
)= 0, (7.22)
so thatd
dR
(λ3 t11
)=
µγ1
r
(λ2 − λ2
1
λ2
). (7.23)
We consider two types of behavior as R → 0: λ2 À λ1 and otherwise. The latter
condition will be denoted λ1À∼ λ2, which indicates that, asymptotically, λ1 grows
at a rate greater than or equivalent to the rate of growth of λ2.
7.1.1 λ2 À λ1
We assume λ1 = ε λ2, where ε is continuous and ε(R) → 0 as R → 0. The
equilibrium equation, Eq. (7.23), can be re-written as
d
dR
(λ3 t11
)=
µγ1
rλ2
(1− ε2
)
=µγ1
r
r
γ2R
(1− ε2
)
=µγ1
γ2
(1− ε2
)R−1. (7.24)
Given δ ∈ (0, 1), for small-enough R > 0,
d
dR
(λ3 t11
)>
µγ1
γ2
(1− δ2
)R−1, (7.25)
101
so that for sufficiently small R2 > R1 > 0,
λ3 t11
∣∣R=R2
− λ3 t11
∣∣R=R1
>µγ1
γ2
(1− δ2
)ln
(R2
R1
). (7.26)
If we fix R2 and let R1 → 0, we see that the total force on the surface at r(R1)
diverges. In this case a cavity cannot be sustained, even with (finite) nonzero radial
traction at the inner radius.
7.1.2 λ1À∼ λ2
In this case we let λ2 = ελ1, where ε is continuous and
limR→0
ε(R) =
0 if λ1 À λ2,
c > 0 if λ1 and λ2 are on the same scale.
The definitions in Eq. (7.6) and the relation ε λ1 = λ2 imply
ε(R)
γ1
dr
dR=
r
γ2 R, (7.27)
so that, given any δ > 0, for sufficiently small R2 > R1 > 0,
∫ R2
R1
dr
r=
γ1
γ2
∫ R2
R1
dR
ε(R) R>
γ1
γ2
∫ R2
R1
dR
(c + δ)R. (7.28)
Here c = 0 if λ1 À λ2. Integration yields
ln
(r(R2)
r(R1)
)>
γ1
γ2(c + δ)ln
(R2
R1
), (7.29)
or
Rγ1/γ2(c+δ)1 r(R2) > R
γ1/γ2(c+δ)2 r(R1). (7.30)
If we fix R2 and let R1 → 0, we see that r(R1) → 0. Cavitation is precluded in this
scenario, as well.
7.2 Varga Material
Haughton (2001) proved that elastic materials with the Varga strain energy density
(without volumetric growth) do not have cavitated configurations without radial
102
traction at the cavity’s surface. We show that a particular class of Varga strain en-
ergy densities permits cavitation in certain cases of homogeneous anisotropic growth.
The compressible Varga strain energy density has the form
W (A) = W (λ1, λ2, λ3) = µ (λ1 + λ2 + λ3 − g(J)) , (7.31)
where µ is constant; λ1, λ2, λ3 are the eigenvalues of A; and J = det A = λ1λ2λ3.
This strain-energy density is an isotropic function of A, but it cannot be ex-
pressed simply in terms of the principal invariants of AT ·A. We appeal to more
advanced results to find the Cauchy stress.
If W is an isotropic function of A and A = U ·R is the polar decomposition of
A, then there is a function W such that W (A) = W (U ), and
∂W
∂A=
∂W
∂U·RT . (7.32)
See Section 4.2 of Ogden (1984).
The sum λ1 +λ2 +λ3 is the first principal invariant (I1) of U , and J is the third
principal invariant (I3) of U . Their derivatives with respect to U are
∂I1
∂U= I,
∂I3
∂U= I3 U−1, (7.33)
where I is the Lagrangian identity tensor. See Section 4.2 of Ogden (1984).
The derivative of the strain-energy density is
∂W
∂A=
∂W
∂U·RT
= µ
(∂I1
∂U− g′(I3)
∂I3
∂U
)·RT
= µ(I − g′(J)JU−1
) ·R−1
= µ(RT − Jg′(J)A−1
), (7.34)
where RT = R−1 because R is orthogonal.
103
The Cauchy stress T is
T =(det A
)−1A · ∂W
∂A
∣∣∣∣A
= JA · µ(R
T − Jg′(J)A−1
)
= µJ−1(A ·RT − Jg′(J) I
)
= µJ−1(R ·U ·RT − Jg′(J) I
), (7.35)
where A = R · U is a polar decomposition of A. The other polar decomposition
of A has the form A = V ·R, where V is a symmetric, positive-definite Eulerian
tensor in the same sense that U is a symmetric, positive-definite Lagrangian tensor;
see Appendix C.
The conclusion is that
T = µ(J−1V − g′(J)I
)
= µ(J−1 (λ1 h1 ⊗ h1 + λ2 h2 ⊗ h2 + λ3 k ⊗ k)− g′(J)I
). (7.36)
The principal Cauchy stresses are
t11 = h1 · T · h1 = µ
(1
λ2λ3
− g′(J)
),
t22 = h2 · T · h2 = µ
(1
λ1λ3
− g′(J)
),
t33 = k · T · k = µ
(1
λ1λ2
− g′(J)
), (7.37)
and the plane-stress condition implies
1
λ1λ2
= g′(J), or λ3 = Jg′(J). (7.38)
We impose three restrictions on the function g found in the strain energy density.
First, since there should be no elastic energy in the absence of elastic deformation,
we require W (1, 1, 1) = 0, which implies g(1) = 3. Second, there should be no
stress in the elastically unstrained state, so each partial derivative of W must be
zero-valued when λ1 = λ2 = λ3 = 1; this implies g′(1) = 1. Finally, the requirement
104
of a positive bulk modulus in the unstrained state implies (see Scott (2007))
λ
3
d
dλtii(λ, λ, λ)
∣∣∣∣λ=1
> 0. (7.39)
For the compressible Varga material, we have
λ
3µ
(− 2
λ3− 3λ3g′′(λ3)
)∣∣∣∣λ=1
= − µ
3(2g′′(1) + 3) > 0. (7.40)
In response, we require that g′′(1) < −2/3.
7.2.1 An Equivalent ODE System
The equation of equilibrium and the plane stress condition can be converted into a
pair of ordinary differential equations with independent variable R and dependent
variables r and J . The first step is to combine Eqs. (7.13) and (7.37):
µ
γ1λ1λ3
d
dR
(λ3
(1
λ2λ3
− g′(J)
))+
µ
r
(1
λ2λ3
− 1
λ1λ3
)= 0,
which simplifies to
1
γ1λ1λ3
d
dR
(1
λ2
− λ3g′(J)
)+
1
r
(1
λ2λ3
− 1
λ1λ3
)= 0. (7.41)
Note that
d
dRλ−1
2 =d
dR
(γ2R
r
)= γ2
(1
r− Rr′
r2
)=
γ2
r
(1− γ1λ1
γ2λ2
).
When this is plugged into Eq. (7.41) we find
1
γ1λ1λ3
(γ2
r
(1− γ1λ1
γ2λ2
)− d
dR(λ3g
′(J))
)+
1
r
(1
λ2λ3
− 1
λ1λ3
)= 0.
d
dR(λ3g
′(J)) =γ2
r
(1− γ1λ1
γ2λ2
)+
γ1λ1λ3
r
(1
λ2λ3
− 1
λ1λ3
)
=γ2
r− γ1λ1
λ2r+
γ1λ1
λ2r− γ1
r
=γ2 − γ1
r
105
The plane stress condition, Eq. (7.38), allows us to eliminate λ3:
d
dR
(J (g′(J))
2)
=γ2 − γ1
r. (7.42)
If J ′ is the derivative of J with respect to R, then we have
J ′ (g′(J))2+ 2J g′(J) g′′(J) J ′ =
γ2 − γ1
r,
or
J ′ =γ2 − γ1
r g′(J) (g′(J) + 2J g′′(J)),
which is our first ordinary differential equation.
To derive our second ordinary differential equation, we adapt the definition of
λ1 (see Eq. (7.6)) with the definition J = λ1λ2λ3.
dr
dR= γ1λ1 = γ1
J
λ2λ3
= γ1J
(r/γ2R)Jg′(J)=
γ1γ2R
rg′(J)(7.43)
In summary, we have a pair of ordinary differential equations:
J ′ =γ2 − γ1
r g′(J) (g′(J) + 2Jg′′(J)), (7.44)
r′ =γ1γ2 R
r g′(J). (7.45)
In light of these equations, we impose more restrictions on g. We want both J ′
and r′ to remain finite within the material, so we require g′(J) and g′(J) + 2Jg′′(J)
to remain nonzero. As such, both of these expressions have fixed sign within the
material. Recall that g′(1) = 1 and g′′(1) < −2/3; based on this, we assume that
g′(J) is positive for J > 0 and g′(J) + 2Jg′′(J) is negative for J > 0.
7.2.2 Boundary Conditions
We seek growth-induced cavitation, so we assume zero radial force at both the inner
and outer radii of the cavitated configuration. With the Varga strain energy density,
the left-hand side of Eq. (7.14) takes the form
λ3 t11 = λ3 µ
(1
λ2λ3
− g′(J)
)= µ
(1
λ2
− J (g′(J))2
), (7.46)
106
where the last equality holds because of the plane-stress condition, Eq. (7.38). In
the presence of a cavity, there is no reason to assume that the principal stretches
are well-behaved near the surface of the cavity, which is reached in the R → 0 limit.
We consider the possible behaviors of J in this limit.
limR→0 J(R) ∈ (0,∞)
In the case of a nonzero finite limit limR→0 J(R) = J0, we have
limR→0
λ3 t11 = limR→0
µ
(1
λ2
− J (g′(J))2
)= − µJ0 (g′(J0))
2, (7.47)
where λ−12 = γ2 R/r → 0 as R → 0. Since we assume that g′(J) is positive for J > 0,
this leaves us with a nonzero force at the inner radius of the deformed configuration,
and this case is ruled out.
limR→0 J(R) = 0
We divide this case into three distinct sub-cases, depending on the behavior of g′(J)
as J → 0.
Sub-Case I: g′(0) = 0 Suppose first that g′(0) = 0. Since we assume g to
be twice continuously differentiable for J > 0, we have
g′(J) =
∫ J
0
g′′(x) dx < 0 (7.48)
because g′′(J) < 0 for J > 0. But this contradicts our assumption that g′(J) > 0
for J > 0.
Sub-Case II: g′(0) ∈ (0,∞) Now suppose that g′(0) ∈ (0,∞). Then J → 0
and J (g′(J))2 → 0 as R → 0. Recall that
J ′ =γ2 − γ1
r g′(J) (g′(J) + 2Jg′′(J)), while
d
dRJ (g′(J))
2=
γ2 − γ1
r.
Since J(0) = 0 and since J(R) > 0 for R > 0, J ′ must be nonnegative at R = 0.
Since g′(J) > 0 and g′(J) + 2Jg′′(J) < 0 for J > 0 (and thus for R > 0), we have
sign (J ′) =sign (γ2 − γ1)
sign (r g′(J)) sign (g′(J) + 2Jg′′(J))= sign (γ1 − γ2) . (7.49)
107
Thus we require γ1 > γ2.
But J (g′(J))2 is also zero at R = 0 and must also be positive for R > 0. Now
note that
sign
(d
dRJ (g′(J))
2
)=
sign (γ2 − γ1)
sign(r)= sign (γ2 − γ1) . (7.50)
For positivity of J (g′(J))2, we require γ2 > γ1, which contradicts the last result.
Thus we eliminate the case in which g′(0) = 0.
Sub-Case III: limJ→0 g′(J) = ∞ Finally, we consider limJ→0 g′(J) = ∞.
Since
d
dRJ (g′(J))
2=
γ2 − γ1
r
and since the right-hand side of this equation converges to a finite value as R → 0,
J (g′(J))2 also converges as R → 0. As argued above, since J(0) = 0 and since
J(R) must be positive for R > 0, we require γ1 > γ2. As a result, the derivative of
J (g′(J))2 is negative at R = 0, so J (g′(J))2, must be positive at R = 0. Otherwise,
J (g′(J))2, would be negative for small positive values of R.
Let limR→0 J (g′(J))2 = β > 0. Then
J(R) (g′(J(R)))2 − β =
∫ R
0
γ2 − γ1
r(s)ds < 0. (7.51)
This implies
0 < β − J(R) (g′(J(R)))2
= (γ1 − γ2)
∫ R
0
ds
r(s)<
(γ1 − γ2)R
r(0), (7.52)
as r(0) < r(R) for R > 0. Hence J (g′(J))2 = β +O(R) as R → 0. The radial force
at r(0) is then proportional to
limR→0
λ3 t11 = limR→0
µ
(1
λ2
− J (g′(J))2
)= − µβ.
Because of this nonzero radial force at R = 0, we rule out this case.
108
limR→0 J(R) = ∞
The only possibility that remains is the divergence of J as R → 0. We consider two
distinct limiting behaviors of g′(J) as J →∞.
Sub-Case I: limJ→∞ g′(J) > 0 In this case, proceeding as above shows that
limR→0
λ3 t11 = µ
(limR→0
1
λ2
− limR→0
J (g′(J))2
)= −∞. (7.53)
This precludes the possibility that limJ→∞ g′(J) > 0.
Sub-Case II: limJ→∞ g′(J) = 0 In this category we have found an example
of growth-induced cavitation that is sustained with zero radial traction on the cav-
ity’s surface. This example is presented in the next section.
7.2.3 A Necessary Condition
Before we explore this very specific example, we note a physically meaningful con-
dition that is necessary for growth-induced cavitation. Since J →∞ as R → 0 and
since J must be finite within the body, we must have J ′ → −∞ as R → 0. However,
the form of J ′ (Eq. (7.44)) and the restrictions on g′(J) and g′(J) + 2Jg′′(J) ensure
that J ′ is negative throughout the body. And since
sign(J ′) = sign (γ1 − γ2) ,
cavitation with zero radial traction is possible only if γ2 > γ1.
7.2.4 Constructing an Appropriate g′(J)
The following class of functions satisfies all the requirements so far imposed on g(J):
g(J) = A
(− 1
αJ−α + 3 +
1
α
)+ (1− A)
(2J1/2 + 1
),
where α > 0 and A > (6α + 3)−1.
However, even this small class must be reduced before we have a physically
acceptable g. Since the thickness of the membrane is proportional to λ3, λ3 must
109
not diverge as R → 0. The plane-stress condition requires λ3 = Jg′(J) (Eq. (7.38)),
which in this case amounts to
λ3 = AJ−α + (1− A)J1/2.
Since we now assume J →∞ as R → 0, this would imply λ3 →∞ as R → 0. Hence
we set A = 1:
g(J) = − 1
αJ−α + 3 +
1
α, (7.54)
g′(J) = J−α−1, (7.55)
g′(J) + 2Jg′′(J) = −(2α + 1)J−α−1. (7.56)
Eqs. (7.44)-(7.45) now take the form
J ′ =(γ1 − γ2) J2α+2
(2α + 1) r, (7.57)
r′ =γ1γ2 R Jα+1
r. (7.58)
7.2.5 Converting to an Autonomous System
Thanks to the explicit presence of R in Eq. (7.58), Eqs. (7.57)-(7.58) do not define
an autonomous system. To produce an equivalent autonomous system, set τ =
− ln(R/B) for some B > 0, and set q = r/R.
dJ
dτ=
(γ2 − γ1
2α + 1
)J2α+2 q, (7.59)
dq
dτ= q − γ1γ2 Jα+1 q−1, (7.60)
and the behavior as R → 0 corresponds now to the behavior of the two dependent
variables as the new “time” τ →∞
7.2.6 Converting to a Lotka-Volterra System
Demonstrating the existence of a solution with the desired asymptotic behavior is
easier if the autonomous system above is converted yet again, to a Lotka-Volterra
110
system. Set
Z1 = J2α+1 q−1, (7.61)
Z2 = Jα+1 q−2, (7.62)
so that
dZ1
dτ= −Z1 + (γ2 − γ1) Z2
1 + γ1γ2 Z1Z2, (7.63)
dZ2
dτ= −2Z2 + 2γ1γ2 Z2
2 +(γ2 − γ1)(α + 1)
2α + 1Z1Z2 (7.64)
7.2.7 Asymptotics of the Lotka-Volterra System
We have established that in a configuration with a cavity, both J and q diverge as
R → 0. Hence as τ →∞,
Z2
Z1
=Jα+1q−2
J2α+1q−1= J−αq−1 → 0. (7.65)
Any orbit of the Lotka-Volterra system for which Z2/Z1 diverges or converges to a
positive constant, does not correspond to a cavitated configuration.
We introduce one more useful fact before addressing specific orbits. The defini-
tion of q and Eq. (7.60) reveal that
dr
dτ=
d
dτ
(q B e−τ
)= q′ B e−τ − q B e−τ = − γ1γ2 Jα+1q−1 B e−τ . (7.66)
Combining this with the definition q = r/R and Eq. (7.62) shows that
d
dτln r = −γ1γ2 Z2. (7.67)
Now we may consider the growth or decay of Z2 to determine whether ln r converges
as R → 0 (as τ →∞).
Unbounded Orbits
We consider only those unbounded orbits on which Z1 À Z2 asymptotically. It is
evident from Eq. (7.64) that if the orbit does not converge to a fixed point and
Z1 À Z2, then Z2 does not decay rapidly to zero as τ →∞. Hence,
limτ→∞
ln r = −∞, (7.68)
111
Coordinates (z1, z2) Linearization Df(z1, z2) Eigenvalues
(0, 0)
( −1 00 −2
)−1, −2
(0, 1
γ1γ2
) (0 0
(γ2−γ1)(α+1)γ1γ2(2α+1)
2
)0, 2
(1
γ2−γ1, 0
) (1 γ1γ2
γ2−γ1
0 − (3α+12α+1
))
1, − (3α+12α+1
)
Table 7.1: Fixed points in the in (z1, z2)-plane
and limR→0 r = 0 on this orbit. No unbounded orbit corresponds to a cavitated
configuration.
The system defined by Eqs. (7.63)-(7.64) has three fixed points. The coordinates,
linearizations of the equations, and eigenvalues of these fixed points are listed in the
table below.
Fixed Point (0, 0)
The eigenvalues of Df(0, 0) indicate that (0, 0) is a sink. We consider orbits attracted
to (0, 0). Note that by Eqs. (7.61)-(7.62),
J3α+1 =Z2
1
Z2
(7.69)
so that by Eqs. (7.63)-(7.64),
d
dτJ3α+1 =
(γ2 − γ1)(3α + 1)
2α + 1Z1 J3α+1, (7.70)
ord
dτln
(J3α+1
)=
(γ2 − γ1)(3α + 1)
2α + 1Z1, (7.71)
112
or, even more directly,d
dτln J =
(γ2 − γ1
2α + 1
)Z1. (7.72)
By Eq. (7.63), given any δ ∈ (0, 1), for (Z1, Z2) in the first quadrant and close
enough to (0, 0),dZ1
dτ< (−1 + δ) Z1. (7.73)
For large-enough τ0 < τ , then,
Z1(τ) < Z1(τ0) e(−1+δ)(τ−τ0). (7.74)
Integrating Eq. (7.72) yields
ln
(J(τ)
J(τ0)
)=
(γ2 − γ1
2α + 1
) ∫ τ
τ0
Z1(s)ds
<
(γ2 − γ1
2α + 1
)Z1(τ0)
∫ τ
τ0
e(−1+δ)(s−τ0)ds
=
(γ2 − γ1
2α + 1
)Z1(τ0)
1− δ
(1− e(−1+δ)(τ−τ0)
)(7.75)
Taking the τ →∞ limit of Eq. (7.75) shows
limτ→∞
ln
(J(τ)
J(τ0)
)≤
(γ2 − γ1
2α + 1
)Z1(τ0)
1− δ< ∞. (7.76)
Eq. (7.76) reveals that on an orbit attracted to the fixed point (0, 0), J does not
diverge as τ → ∞. Such orbits correspond to configurations with nonzero radial
traction at the surface of the cavity.
Fixed Point(0, γ−1
1 γ−12
)
The presence of one positive eigenvalue and one zero eigenvalue of Df(γ−11 γ−1
2 , 0)
ensures the existence of a one-dimensional center manifold for this fixed point; see,
e.g., Perko (1996). At the fixed point, the center manifold is tangent to the 0-
eigenspace of Df(γ−11 γ−1
2 , 0). If any orbit approaches this fixed point , it lies on the
center manifold.
Suppose that there is an orbit that approaches this fixed point as τ →∞. Since
Z2 converges to a positive constant, Jα+1 and q2 must have the same asymptotic
growth rates. Since Jα+1 diverges, q2 grows at the same rate. But then
113
Z1 =J2α+1
q=
(Jα+1)(2α+1)/(α+1)
(q2)1/2>
Jα+1
q2→ (γ1γ2)
−1 .
However, Z1 → 0 as the orbit approaches this fixed point, contradicting the above
inequality. Hence the orbit approaching this fixed point (along the center manifold)
as τ →∞ does not correspond to a cavitated configuration.
Fixed Point((γ2 − γ1)
−1 , 0)
The linearization at this fixed point has one positive and one negative eigenvalue,
so this fixed point has a one-dimensional stable manifold; see Perko (1996). We
consider the orbit that approaches the fixed point along the stable manifold from
the first quadrant. Z1 converges to (γ2 − γ2)−1, while Z2 converges to 0. We will
now show that the stable manifold corresponds to a function r(R) that converges
to a positive constant as R → 0 (i.e. as τ →∞).
Recall from Eq. (7.67) that
d
dτln r = −γ1γ2 Z2. (7.77)
Z2 → 0 on the stable manifold, and we will show that Z2 → 0 fast enough that
limτ→∞ ln r > −∞.
Since Z2 → 0 and Z1 → (γ2 − γ1)−1 on the stable manifold, for small δ > 0 and
(Z1, Z2) close enough to((γ2 − γ1)
−1 , 0),
dZ2
dτ<
(−2 + δ +
(γ2 − γ1)(α + 1)
2α + 1
(1
γ2 − γ1
+ δ
))Z2
=
(−
(3α + 1
2α + 1
)+
(1 +
(γ2 − γ1)(α + 1)
2α + 1
)δ
)Z2,
so that for τ > τ0,
Z2(τ)
Z2(τ0)< exp
(−
(3α + 1
2α + 1
)+
(1 +
(γ2 − γ1)(α + 1)
2α + 1
)δ
(τ − τ0)
). (7.78)
If δ is chosen small enough and τ0 is chosen large enough, then from Eqs. (7.77) and
(7.78) for τ > τ0 we have
− γ1γ2 Z2(τ0) e−η(τ−τ0) <d
dτln r < 0, (7.79)
114
where
η =
(3α + 1
2α + 1
)−
(1 +
(γ2 − γ1)(α + 1)
2α + 1
)δ > 0.
Integration yields
γ1γ2 Z2(τ0)
η
(e−ητ − e−ητ0
)< ln
(r(τ)
r(τ0)
)< 0, (7.80)
so that
limτ→∞
ln r(τ) ≥ ln r(τ0)− γ1γ2 Z2(τ0)
ηe−ητ0 > −∞. (7.81)
Hence r(R) converges to a positive constant as R → 0.
It can be shown that each orbit attracted to the fixed point (0, 0) also has a
radius converging to a positive number as τ → ∞ (R → 0). However, such orbits
have been discounted because J does not diverge as τ →∞. We now establish that
J does diverge on the stable manifold of((γ2 − γ1)
−1 , 0).
It can be shown from Eqs. (7.61)-(7.62) that
J3α+1 =Z2
1
Z2
. (7.82)
On the stable manifold of((γ2 − γ1)
−1 , 0), Z1 → (γ2 − γ1)
−1 > 0 and Z2 → 0 as
τ → ∞. Hence J diverges as τ → ∞, and the radial traction at the surface of the
cavity is zero.
We have established the existence of a solution R 7→ r(R) that converges to a
positive constant as R → 0 and has zero radial stress in the same limit. Now we
must demonstrate that this solution has zero radial stress at some positive value
of R. The corresponding value of r is the outer radius of a cavitated configuration
with zero radial stress at the inner and outer radii.
7.2.8 Boundary Condition at Outer Radius
From Eqs. (7.46) and (7.55) we see that
λ3 t11 = µ
(γ2
q− J−(2α+1))
). (7.83)
115
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Z1
Z1 = g
2−1
Figure 7.1: Phase portrait for γ1 = 1.0, γ2 = 1.2, α = 1: Each fixed point is markedby a black square. The verical line is Z1 = γ−1
2 . The solid lines are two orbits of thesystem. Between the slanted dashed lines, dZ1/dτ > 0 and dZ2/dτ < 0. The arrowemanating from the fixed point
((γ2 − γ1)
−1 , 0)
= (5, 0) points along the stableeigenspace at this point. The stable manifold at this fixed point is asymptoticallytangent to this line.
For R > 0 both J and q are positive and finite, so the zero-radial force condition
implies q/γ2 = J2α+1, or
Z1 =J2α+1
q=
1
γ2
. (7.84)
If the stable manifold of the fixed point ((γ2− γ1)−1, 0) intersects the line Z1 = γ−1
2
in the (Z1, Z2)-plane, then the portion of the orbit from the fixed point to this line
corresponds to a cavitated configuration that has zero radial traction applied at its
inner and outer radii.
dZ1
dτ= 0 for Z2 = −
(γ2 − γ1
γ2γ1
)Z1 +
1
γ1γ2
, (7.85)
dZ1
dτ= 0 for Z2 = −
(α + 1
4α + 2
)(γ2 − γ1
γ2γ1
)Z1 +
1
γ1γ2
. (7.86)
116
The slope of line in Eq. (7.85) is more severe than that of the line in Eq. (7.86), and
these lines intersect at the fixed point (0, γ−11 γ−1
2 ). It can be seen from Eqs. (7.63)-
(7.64) that dZ1/dτ > 0 and dZ2/dτ < 0 between these lines in the first quadrant
of the plane. Hence any orbit that is in this region for some τ0 is trapped in this
region for τ ≤ τ0 and approaches the fixed point (0, γ−11 , γ−1
2 ) as τ → −∞.
The stable manifold of the fixed point((γ2 − γ1)
−1 , 0)
is asymptotically tangent
to the “stable eigenspace” at this fixed point. The linearization of Eqs. (7.63)-(7.64)
at this fixed point reveals that
−(2α + 1)γ2γ2
(5α + 1)(γ2 − γ1)
(7.87)
is a representative vector from this eigenspace. When attached to the fixed point((γ2 − γ1)
−1 , 0), this vector points into the region between the lines described in
Eqs. (7.85)-(7.86); see Fig. 7.1. The stable manifold lies between the lines for all τ
and thus intersects the line Z1 = γ−12 . The orbit of interest to us approaches the
fixed point((γ2 − γ1)
−1 , 0)
as τ →∞ and the fixed point (0, γ−11 γ−1
2 ) as τ → −∞.
7.2.9 Numerical Construction of Solutions
To compute the radius of a cavity numerically, a point on the stable eigenspace
attached to the fixed point((γ2 − γ1)
−1 , 0)
was chosen very close to the fixed point.
With this point as initial condition, the system Eqs. (7.63)-(7.64) was integrated
numerically backward in τ until the numerical orbit reached the line Z1 = γ−12 . This
enforced a condition of zero radial traction at the maximum value of R, i.e. at the
outer surface of the membrane. The elasped τ was then used to compute
r = qR =
(Zα+1
1
Z2α+12
)1/(3α+1)
Be−τ . (7.88)
The coordinates of the initial condition provided the values of Z1 and Z2, and B was
set to 1. The results showing limR→0 r(R) as a function of γ2 are shown for various
values of γ1 in Fig. 7.2.
117
The same numerical integration was used to compute the outer radius of the
membrane after deformation. In Eq. (7.88), τ is set to 0, B is set to 1, Z1 is set to
γ−12 , and the value of Z2 is that found by numerical integration to the point where
the orbit crosses the line Z1 = γ−12 .
3.0
1.5
0.25
1.5
1.75
4.0
0.75
0.5 2.01.0
1.0
3.5
0.0
2.5
1.25
0.5
2.50.5 4.0
3.0
1.5 3.0
2.0
2.0
3.5
1.0
0.5
1.5
3.5
1.0
2.5
b=r(B)a=r(A)
γ2
γ1=0.5
γ 1=0.5 γ
1=1.0
γ 1=1.0
γ1=1.5
γ 1=1.5
γ 1=2.0
γ1=2.0
γ1=2.5
γ 1=2.5
γ1=3
γ1=3 γ
2
Figure 7.2: Cavity radius a = r(0) and outer radius b = r(B) as functions of γ2 forvarious values of γ1
On each curve the radius of the cavity increases with γ2 − γ1. But the curves
describing the cavity’s radius are not mere translates of one another; the growth
with γ2− γ1 is slower for larger values of γ1. On the other hand, the curves showing
the radius of the membrane do appear to be parallel.
7.3 Conclusion
We have added homogeneous anisoptropic growth to a model of a three-dimensional
elastic membrane. We have demonstrated that a membrane whose elastic behavior
is neo-Hookean cannot sustain a cavity without radial traction on the cavity. While
growthless elastically compressible Varga membranes do not permit cavity forma-
tion without radial traction on the cavity, we have found a class of growing Varga
membranes that do permit such solutions.
In this special class of growing Varga membranes, cavity formation with zero ra-
dial traction on the cavity occurs as long as the azimuthal growth factor γ2 exceeds
118
the radial growth factor γ1. Cavity formation is independent of the sizes of these
factors; in particular, cavitation is found regardless of whether the growth factors
indicate shrinkage or expansion in the growth stage. Numerical results show, how-
ever, that the cavity’s radius is a function of both γ1 and γ2 and not of γ2 − γ1
alone.
119
CHAPTER 8
GLOBAL CONSTRAINTS AND THE INDUCED THEORY OF SHELLS
The equations for the Haughton-Ogden membrane were derived via Taylor expansion
of various geometric and stress quantities and keeping finite numbers of terms. While
this approach is popular and has a long history, the assumptions at the foundation of
such a model are not necessarily consistent or physically realistic for large deflections.
Geometrically exact theories, in which explicit functional forms of deformations are
specified from the start, have none of the geometric inconsistencies found in older
theories. Further, geometrically exact theories allow the use of wide varieties of
constitutive relations for the same geometries.
We review briefly the assumptions of Kirchhoff’s classical theory of plates, con-
sider some objections to the popular Foppl-von Karman equations for deflections of
plates, and introduce the geometrically exact theory of plates induced by constrained
three-dimensional finite elasticity.
8.1 Kirchhoff’s Classical Theory of Plates
The mechanical theory of plates begins in 1766 with Euler’s model of elastic mem-
branes (Euler (1766)). Jakob Bernoulli (1759-1789), a student of Euler, extended
Euler’s theory to plates (Bernoulli (1789)). After the German physicist E. F. F.
Chladni’s presentation of vibration patterns on plates in Paris in 1809, the French
Academy of Sciences, per Napoleon’s suggestion, offered a prize for a mathematical
essay of plate vibrations that could predict Chladni’s experimental results. Soon
after, theoretical developments in the theory of plates were dominated by French
researchers, among them Sophie Germain, Navier, Poisson, de Saint Venant, and
Lagrange; see Szilard (2004). But it was Kirchhoff, a German, who first produced
a theory that included both bending and stretching (Kirchhoff (1876)). Kirchhoff
120
also introduced the use of virtual displacements into the theory.
Kirchhoff’s theory began with assumptions on geometry and mechanical behav-
ior, enumerated as follows in Szilard (2004):
1. The material is homogeneous, isotropic, and linearly elastic. That is, it obeys
Hooke’s law.
2. The plate is initially flat.
3. The middle surface of the plate remains unstrained during bending.
4. The constant thickness of the plate is small compared to its other dimensions.
That is, the smallest lateral dimension of the plate is at least ten times the
thickness.
5. The transverse deflections are small compared to the plate thickness. A max-
imum deflection of one tenth of the thickness is considered the limit of the
small-deflection theory.
6. Slopes of the deflected middle surface are small compared to unity.
7. Sections taken normal to the middle surface before deformation remain plane
and normal to the deflected middle surface. Consequently, shear deformations
are neglected.
8. The normal stress in the direction transverse to the plate surface can be ne-
glected.
The modern, geometrically exact theory that includes Kirchhoff plates preserves
some but not all of these assumptions. In particular, a class of allowed deformations
is specified by prescribing explicit functional forms. All assumptions and restrictions
are included in a consistent manner, which cannot be said for the classical theory
of plates. See, for example, Sections X.1 and XIV.15 of Antman (1995).
121
8.2 Foppl-von Karman Equations
A widely-used model for both static and dynamic large deflections of thin plates
began with the German August Foppl (1907) and was extended by the Hungarian
Theodore von Karman (1910). Their equilibrium partial differential equations for
transverse deflection w(x, y) and stress Φ(x, y) are
D
h∆2w = w2
xxΦ2yy + w2
yyΦ2xx − 2wxyΦxy +
pz
h, (8.1)
1
E∆2Φ = −1
2
(w2
xxw2yy + w2
yyw2xx − 2wxywxy
), (8.2)
where h is thickness, E is Young’s modulus, pz(x, y) is lateral load, and
D =Eh3
12(1− ν2)(8.3)
is the flexural rigidity of the plate, where ν is Poisson’s ratio. See Chapter 11 of
Szilard (2004) and Section XIV.14 of Antman (1995).
While the Foppl-von Karman equations (8.1)-(8.2) have found favor among both
applied scientists and mathematicians, numerous objections have arisen from the
school of rational mechanics. In the epilogue of Truesdell (1978), Truesdell com-
plained about an apparent lack of rigor in the derivation of the Foppl-von Karman
equations. He opined, however,
“[t]hat is not to say that theory is bad. Rather, whether it is good or bad
is assessable only after its predictions shall have been compared with a
rational theory.”
Truesdell subsequently asked Stuart Antman, an originator of many of the mod-
ern foundations and implementations of rational mechanics, to specify his objections.
According to Antman, the derivation of thr Foppl-von Karman equations relies upon
1. “ ‘approximate geometry’, the validity of which is assessable only
in terms of some other theory”.
2. “assumptions about the way the stress varies over a cross-section,
assumptions that could be justified only in terms of some other
theory”.
122
3. “commitment to some specific linear constitutive relation — linear,
that is, in some special measure of strain, while such approximate
linearity should be the outcome, not the basis, of a theory”.
4. “neglect of some components of strain — again, something that
should be proved mathematically from an overriding, self-consistent
theory”.
5. “an apparent confusion of the referential and spatial descriptions —
a confusion that is easily justified for classical linearized elasticity
but here is carried over unquestioned, in contrast with all recent
studies of the elasticity of finite deformations”.
Since Truesdell (1978), much rigorous analysis has been devoted to the Foppl-
von Karman equations, in particular by Ciarlet (2005). The equations still arise
only from asymptotics in the small-thickness limit, and the formal validity of the
equations depends on careful scalings of the loads with respect to the thickness.
Different types of scalings have been found to yield a hierarchy of plate models that
includes Foppl-von Karman (Friesecke et al. (2006)).
The end of Section XIV.14 of Antman (1995) points out that the Foppl-von
Karman equations have the advantage of semilinearity, so their analysis is simpler
than some geometrically exact approaches. However, for Cosserat shells, examples
of which we will study, the linearized equations are also semilinear, so the bifurcation
analysis is of the same complexity. For this and other reasons, we avoid the Foppl-
von Karman theory that has had such success in Efrati et al. (2007a), Efrati et al.
(2007c), Efrati and Kupferman (2009), and Efrati et al. (2009).
8.3 Plate Theory Induced from Three-Dimensional Elasticity
In intrinsic theories of shells and plates, the body is treated as intrinsically two-
dimensional, and balance laws are formulated for stress and torque in terms of
integrals along curves in the body. Such a theory requires the introduction of an ob-
jective constitutive relation for the two-dimensional body. We consider here induced
123
theories, which can be viewed as either approximations of or constrained versions of
full three-dimensional theory. In particular, we consider the special Cosserat theory
of shells, in which a shell consists of a middle surface from which material fibers
sprout, to form a three-dimensional body. See Figure 8.1. A plate is a shell that
has no curvature in its middle surface in its reference configuration.
Figure 8.1: A slice of a Cosserat shell, with a finite number of material fibers shown.The material fibers form a continuum in the model.
8.3.1 Global Constraints
The body is assumed to occupy only “plate-like” configurations. This requires spec-
ifying a specific class of allowed configurations. In the case of Kirchhoff constraints
on a plate, the deformed configuration must have the form
(x f)(R, Θ, Z) = r(R, Θ) + Z d(R, Θ), (8.4)
where the image of r is the middle surface of the deformed configuration, and the
director d is a unit vector anchored at r(R, Θ) and normal to the deformed middle
surface:
d =∂r
∂R× ∂r
∂Θ
/wwww∂r
∂R× ∂r
∂Θ
wwww . (8.5)
Eqs. (8.4) and (8.5) allow us to define tangent vectors to the set of allowed
configurations. If we view (xf) as a function of r and its partial derivatives alone,
124
then the tangent vector has the form
Mx = lim
ε→0
1
ε
(x f)
[r + ε
Mr,
∂r
∂R+ ε
∂Mr
∂R,∂r
∂Θ+ ε
∂Mr
∂Θ
]− (x f)
[r,
∂r
∂R,∂r
∂Θ
]
= limε→0
1
ε
(r + ε
Mr + Z
(r,R +εMr,R )× (r,Θ +ε
Mr,Θ )
‖(r,R +εMr,R )× (r,Θ +ε
Mr,Θ )‖
)−
(r + Z
r,R×r,Θ‖r,R×r,Θ ‖
)
=Mr + Z
Mr,R×r,Θ +r,R×M
r,Θ‖r,R×r,Θ ‖ . (8.6)
8.3.2 Stress Decomposition
This approach to shell theory includes a constitutive assumption. The first Piola-
Kirchhoff stress tensor is decomposed as
P = P act + P lat, (8.7)
where P lat satisfies ∫
(XΨ)(B)
P lat(X) :∂
Mx
∂XdV = 0 (8.8)
This is analogous to decomposing the Cauchy stress as T = T (F ) − pI in the
case of incompressibility, where T (F ) is a constitutive function and pI contributes
nothing to the stress power. However, the Kirchhoff constraints cannot be expressed
as pointwise constraints, so the condition in Eq. (8.8) is not a pointwise condition.1
P lat is called the latent or reactive or even Lagrange stress.
8.3.3 Weak Formulation of Balance Laws
The balance of linear momentum can be expressed as the Impulse-Momentum Law:∫ τ
0
∫
∂X(Ω)\S3
P (t, X) · ν(X) dS dt +
∫ τ
0
∫
S3∩∂X(Ω)
τ (t, X) dS dt
+
∫ τ
0
∫
X(Ω)
f(t, X) dV =
∫
X(Ω)
ρ(X) (χt(τ, X)− χt(0, X)) dV, (8.9)
1In Section XII.12 of Antman (1995), Antman does not prove that the Kirchhoff constraints
cannot be expressed as pointwise constraints on the deformation gradient or the right Cauchy-
Green tensor. Rather, he shows that the Kirchhoff constraints do not satisfy the criteria of the
theorems by which constraints such as incompressibility and inextensibility are shown to be local.
125
for well-behaved subsets Ω of Ψ(B), where S3 is the subset of the reference config-
uration on which traction τ is prescribed. In the case of Kirchhoff plates, that will
be the surface formed by the boundary of the middle surface and the material fibers
attached to that boundary.
Antman and Osborn (1979) proved the equivalence of the Impulse-Momentum
Law and the Principle of Virtual Power:
∫ ∞
0
∫
(XΨ)(B)
(P :
∂Mx
∂X− f · M
x− ρχt ·Mxt
)dV dt
−∫ ∞
0
∮
∂(XΨ)(B)
τ (t, X) · Mx(t, X) dS dt
−∫
(XΨ)(B)
ρ(X)χt(0,X) · Mx(0, X) dV = 0 (8.10)
for all well-behavedMx with compact support in X(B)× [0,∞) that satisfy bound-
ary conditions specified by the constraints. For an introduction and examples, see
Sections XII.8 and XII.9 of Antman (1995).
For statics of a Kirchhoff plate free of body forces, the Principle of Virtual Power
gives ∫
(XΨ)(B)
(P act + P lat) :∂
Mx
∂XdV =
∮
∂(XΨ)(B)
τ · Mx dS. (8.11)
By Eq. (8.8), Eq. (8.10) can be re-written as
∫
(XΨ)(B)
P act :∂
Mx
∂XdV =
∮
∂(XΨ)(B)
τ · Mx dS. (8.12)
This will be the basis for our differential equations of equilibrium in the following
chapters. It will be sufficient for deriving equations representing balance of linear
momentum and of angular momentum.
126
CHAPTER 9
THE FLAT KIRCHHOFF PLATE
9.1 Kinematics
9.1.1 Deformation Gradient
In this chapter we will be considering statics of cylindrical deformations of axisym-
metric plates. The reference configuration has the form
X(R, Θ, Z) = R cos(Θ) i + R sin(Θ) j + Z k, (9.1)
while the final configuration has the form
x(r, θ, z) = r cos(θ) i + r sin(θ) j + z k, (9.2)
where r is a function of R alone, θ(Θ) = Θ, and z(Z) = Z.
The deformation gradient is
F = r′ er ⊗ER + eθ ⊗EΘ + ez ⊗EZ
= r′ h1 ⊗ h1 +r
Rh2 ⊗ h2 + k ⊗ k. (9.3)
where the right vector in each tensor product is to be understood as a tangent vector
anchored at a point in the reference configuration, while the left vector is a tangent
vector anchored at a point in the final configuration.
We will need the following expression later:
F−T =1
r′h1 ⊗ h1 +
R
rh2 ⊗ h2 + k ⊗ k. (9.4)
9.1.2 Incompatible Growth
Inspired by the growth tensors used in Goriely and Ben Amar (2005) and Ben Amar
and Goriely (2005) for spherical shells, we consider growth tensors whose expressions
127
in “physical components” take the form
G(R, Θ, Z) = γ1(R) h1(θ)⊗ h1(Θ) + γ2(R) h2(θ)⊗ h2(Θ) + k ⊗ k. (9.5)
The possible (R, Θ)-dependence of the coefficients indicates that such G may not
even be the gradient of an isometric immersion, so it is unclear where the vectors
on the left-hand side of each tensor product are anchored. We seek to remove, or at
least clarify, this ambiguity.
If we have a map (x f) from Ψ(B) into E3, where
f(R, Θ, Z) = (r(R), θ(R, Θ), Z), (9.6)
then the gradient of χ = (x f X−1
) has the form
∂χ
∂X= r′ er ⊗ER +
∂θ
∂Reθ ⊗ER +
∂θ
∂Θeθ ⊗EΘ + ez ⊗EZ
= r′ h1 ⊗ h1 +∂θ
∂Rr h2 ⊗ h1 +
∂θ
∂Θ
r
Rh2 ⊗ h2 + k ⊗ k, (9.7)
where the vectors on the left-hand sides of the tensor products are anchored at
(x f)(R, Θ, Z) = r(R) cos(θ(R, Θ)) i + r(R) sin(θ(R, Θ)) j + Z k
= r(R) h2(θ(R, Θ)) + Z k, (9.8)
where r and θ are assigned by the function f .
The tensor field G in Eq. (9.5) may be produced from a gradient of the kind found
in Eq. (9.7) by dropping the “off-diagonal” term, i.e. the one involving eθ ⊗ ER.
Comparing Eqs. (9.5) and (9.7), we set
r′ = γ1(R) so r(R) = r(0) +
∫ R
0
γ1(s)ds, (9.9)
and∂θ
∂Θ=
R
r(R)γ2(R), so θ(R, Θ) =
R
r(R)γ2(R)Θ + θ(R, 0), (9.10)
where the function θ(·, 0) remains unknown.
Since the given G has no term corresponding to ∂θ/∂R, we lack the information
needed to specify θ(·, 0). even if that off-diagonal term were included, we could
128
specify θ(·, 0) only up to an additive constant, just as for r. This differs from the
case of constant γ1 and γ2 only in that the θ-value for Θ = 0 may be R-dependent.
If we choose an additive constant r(0) and a well-behaved function R 7→ θ(R, 0),
we have a completely specified map f(R, Θ, Z) = (r, θ, z) and the corresponding
differential f∗ and pull-back f ∗. To produce something analogous to G in Eq. (9.5),
we drop the off-diagonal term:
G = γ1∂
∂r⊗ dR +
R
rγ2
∂
∂θ⊗ dΘ +
∂
∂z⊗ dZ. (9.11)
This G, which is analogous to a pull-back but will be equal to a pull-back only if γ1
and γ2 satisfy certain differential equations, can be used to produce a new metric on
Ψ(B) by computing the analogue of the pull-back of m, the standard metric tensor
on the space of (r, θ, z)-points:
m = dr ⊗ dr + r2 dθ ⊗ dθ + dz ⊗ dz. (9.12)
Since this is not a true pull-back of a metric tensor, there is no standard notation
for this operation. We appeal to intuition to describe the operation. First we let m
act on the “left sides” of the tensor products in G:
m [G] = dr
[γ1
∂
∂r
]dr ⊗ dR + r2 dθ
[R
rγ2
∂
∂θ
]dθ ⊗ dΘ + dz
[∂
∂z
]dz ⊗ dZ
= γ1 dr ⊗ dR + rRγ2dθ ⊗ dΘ + dz ⊗ dZ. (9.13)
Then we let the transpose of the two-point tensor m[G] act on the left sides in G:
(m [G])T [G] = dr
[γ1
∂
∂r
]γ1 dR⊗ dRdθ
[R
rγ2
∂
∂θ
]Rrγ2 dΘ⊗ dΘ
+ dz
[∂
∂z
]dZ ⊗ dZ
= γ21 dR⊗ dR + R2γ2
2 dΘ⊗ dΘ + dZ ⊗ dZ. (9.14)
The Mathematica code in Appendix C can be modified to demonstrate that the
Riemann-Christoffel curvature tensor associated with this metric has four terms that
may be nonzero:
RRΘRΘ = −RRΘΘR = −RΘRRΘ = RΘRΘR
=Rγ2
γ1
(γ2γ′1 + Rγ′1γ
′2 − γ1 (2γ′2 + Rγ′′2 )) . (9.15)
129
Since we need a Riemannian metric (so that the grown image of set with positive
volume, has positive volume), we require γ1 > 0 and γ2 > 0. Hence the terms in the
Riemann-Christoffel curvature tensor are identically zero if and only if
γ2γ′1 + Rγ′1γ
′2 − γ1 (2γ′2 + Rγ′′2 ) = 0, (9.16)
orγ′1γ1
=(γ2 + Rγ′2)
′
γ2 + Rγ′2, (9.17)
which impliesγ2 + Rγ′2
γ1
= constant. (9.18)
In all other cases, the Riemann-Christoffel curvature tensor is not identically zero,
and the growth described by G precludes even isometric immersion.
9.1.3 Elastic Response
The local linear description of the body’s elastic response to incompatible growth
has the form
A = F ·G−1
=r′
γ1
h1 ⊗ h1 +r
γ2Rh2 ⊗ h2 + k ⊗ k, (9.19)
where the vectors on the left-hand sides of the tensor products are anchored at the
point in the final configuration, and the vectors on the right-hand side are anchored
at the position in Eq. (9.8).
The eigenvalues of√
F T · F are called stretches, but since A is not a deformation
gradient, we cannot use that label for the eigenvalues of√
AT ·A. We choose the
label pseudostretches. In this case they are
λ1 =r′
γ1
, λ2 =r
γ2R, λ3 = 1. (9.20)
9.1.4 Virtual Displacements
The Kirchhoff plate consists of a two-dimensional middle surface from which material
fibers extend perpendicularly. The deformation in Eq. (9.2) leaves the middle surface
130
flat and the material fibers unstretched and perpendicular to the middle surface.
The only possible virtual displacements are axially symmetric perturbations of the
middle surface, so in this caseMx has the form
Mx(R, Θ) =
Mr(R) h1(Θ), (9.21)
whereMr is any as-regular-as-needed real-valued function. The Principle of Virtual
Power for statics, Eq. (8.12), takes the form
∫
plate
((h1 · P act · h1)
Mr′+ (h2 · P act · h2)
Mr
R
)dV =
∮
∂(plate)
τ ·(Mrh1
)dS (9.22)
because all other projections of the gradient ∂Mx/∂X are zero.
It should be noted that
(h1 · P act · h1)Mr′+ (h2 · P act · h2)
Mr
R
=(Mr′h1
)· (P act · h1) +
( Mr
Rh2
)· (P act · h2)
=
(∂
Mx
∂R
)· (P act · h1) +
(1
R
∂Mx
∂Θ
)· (P act · h2) +
(∂
Mx
∂Z
)· (P act · k)
=1
R
(∂
∂RR
Mx
)· (P act · h1)− M
x · P act · h1
+
(1
R
∂Mx
∂Θ
)· (P act · h2) +
(∂
Mx
∂Z
)· (P act · k)
= Div(Mx · P act
)− M
x ·(
1
R
∂
∂R(RP act · h1) +
1
R
∂
∂Θ(P act · h2)
), (9.23)
where Div indicates divergence with respect to cylindrical coordinates. We have
assumed that since there is no extension or shear through the thickness of the plate,
(P act · k) is vertical, so that ∂ (P act · k) /∂Z is also vertical and is orthogonal toMx.
By the Divergence Theorem applied to the vector fieldMx · P act,
∫
plate
Div(Mx · P act
)dV =
∮
∂(plate)
Mx · P act · ν dS
=
∮
∂(plate)
Mx · τ dS (9.24)
131
where ∂ (plate) is the boundary of the plate and ν is the outward-pointing unit
normal on this boundary. After cancellation of the surface traction terms, the weak
equation of equilibrium is∫
plate
Mr h1 ·
(1
R
∂
∂R(RP act · h1) +
1
R
∂
∂Θ(P act · h2)
)dV = 0, (9.25)
which must hold for all admissibleMr.
Due to the constraints imposed on the body, we have no equations to determine
the rate of change of stress in the Z-direction. However, in an induced theory
such as this, we consider not the pointwise values of stress in the three-dimensional
body; rather, we consider the stress integrated across the thickness of the plate. The
integrated stress is a function of the planar coordinates R and Θ only.
We take the weak equation of equilibrium and write the integral over the plate
as an iterated integral, first in Z and then in the planar coordinates:∫
slice
Mr h1 ·
[∫ H2
H1
(1
R
∂
∂R(RP act · h1) +
1
R
∂
∂Θ(P act · h2)
)dZ
]R dR dΘ = 0
(9.26)
We change the order of differentiation and Z-integration:∫
slice
Mr h1 ·
(1
R
∂
∂R
∫ H2
H1
(RP act · h1) dZ +1
R
∂
∂Θ
∫ H2
H1
(P act · h2) dZ
)R dR dΘ = 0
(9.27)
By our assumptions of symmetry, P act ·h2 must be proportional to h2, so P act ·h2 = (h2 · P act · h2) h2. And by symmetry, the coefficient (h2 · P act · h2) is Θ-
independent, so
∂
∂Θ
∫ H2
H1
(P act · h2) dZ =∂
∂Θ
∫ H2
H1
(h2 · P act · h2) h2 dZ
=∂h2
∂Θ
∫ H2
H1
(h2 · P act · h2) dZ
= −h1
∫ H2
H1
(h2 · P act · h2) dZ. (9.28)
Our weak equation of equilibrium is now∫
slice
Mr
(∂
∂RR
∫ H2
H1
(h1 · P act · h1) dZ −∫ H2
H1
(h2 · P act · h2) dZ
)dR dΘ = 0.
(9.29)
132
Our symmetry assumption imply that the remaining mechanical quantities are Θ-
independent, so we can integrate in Θ automatically and view the functions in the
result as functions of R alone:
2π
∫
slice
Mr
(d
dRR
∫ H2
H1
(h1 · P act · h1) dZ −∫ H2
H1
(h2 · P act · h2) dZ
)dR = 0. (9.30)
If this is to hold for all admissibleMr, then the other factor of the R-integrand must
be identically zero:
d
dR
(R
∫ H2
H1
(h1 · P act · h1) dZ
)−
∫ H2
H1
(h2 · P act · h2) dZ = 0. (9.31)
This is the equation of equilibrium for our constrained axisymmetric plate. Closure
of the system of differential equations requires a constitutive relation.
9.2 Compressible Neo-Hookean Constitutive Relation
We consider a compressible neo-Hookean constitutive relation with strain energy
density introduced in Chapter 6:
W (λ1, λ2, λ3) = µ
(1
2
(λ2
1 + λ22 + λ2
3 − 3)− ln J
)+ λ (J − 1− ln J) ; (9.32)
In growthless elasticity, λ21, λ2
2, and λ23 are the eigenvalues of the Cauchy-Green
tensors F · F T and F T · F , and J = λ1λ2λ3 = det F .
Following the procedure seen in Section 6.7, we find Eq. (6.57) for the active
Cauchy stress as a function of A = F ·G−1:
T act(A) =1
JA
µ
(λ2
1 a1 ⊗ a1 + λ22 a2 ⊗ a2 + λ2
3 a3 ⊗ a3
)
+ (λJA − (λ + µ)) I , (9.33)
where ai = ai are eigenvectors of A ·AT that form an orthonormal basis. In this
case, the appropriate eigenvalues and eigenvectors are
a1 = h1, λ1 = r′/γ1, (9.34)
a2 = h2, λ2 = r/γ2R, (9.35)
a3 = k, λ3 = 1. (9.36)
133
The active portion of the first Piola-Kirchhoff stress tensor is found via Eq. (6.60):
P act = (det F ) T act(A) · F−T
Only the projections h1 · P act · h1 and h2 · P act · h2 of P act are needed for our
calculations.
h1 · P act · h1 = (det F ) h1 · T act · F−T · h1
=(det A) (det G)
J
(µλ2
1 + λJ − (λ + µ))h1 ·
(1
r′h1
)
= (det G)
(µ
(r′
γ1
)2
+ λJ − (λ + µ)
)1
r′
= γ1γ2
(µ
r′
γ21
+ λr/R
γ1γ2
− λ + µ
r′
)
=µγ2
γ1
r′ + λr
R− (λ + µ) γ1γ2
r′. (9.37)
A similar calculation shows that
h2 · P act · h2 =µγ1
γ2
r
R+ λr′ − (λ + µ) γ1γ2
r/R. (9.38)
The flatness of the middle surface and the lack of stretching or shear through the
thickness have ensured a complete lack of Z-dependence, so the thickness-integrated
stresses are∫ H2
H1
(h1 · P act · h1) dZ = (H2 −H1)
(µγ2
γ1
r′ + λr
R− (λ + µ) γ1γ2
r′
)(9.39)
∫ H2
H1
(h2 · P act · h2) dZ = (H2 −H1)
(µγ1
γ2
r
R+ λr′ − (λ + µ) γ1γ2
r/R
).(9.40)
This leaves us the ordinary differential equation
d
dR
(R
(µγ2
γ1
r′ + λr
R− (λ + µ) γ1γ2
r′
))=
µγ1
γ2
r
R+ λr′ − (λ + µ) γ1γ2
r/R. (9.41)
After differentiation and cancellation of λr′ from each side, we have(
µγ2
γ1
)′Rr′ +
µγ2
γ1
r′ +µγ2
γ1
Rr′′ − (λ + µ) γ1γ2
r′
− (λ + µ) R (γ1γ2)′
r′+
(λ + µ) Rγ1γ2
(r′)2 r′′
=µγ1
γ2
r
R− (λ + µ) Rγ1γ2
r/R. (9.42)
134
9.2.1 A Restriction on γ1 and γ2 at R = 0
We assume that r is at least twice continuously differentiable and that r′ is bounded
away from zero. By the boundedness of derivatives near R = 0 and the fact that
r(R)/R → r′(0) as R → 0, the limiting form of the equation above is, after cancel-
lation,γ2(0)
γ1(0)r′(0) =
γ1(0)
γ2(0)r′(0). (9.43)
The conclusion is that we need either r′(0) = 0 or γ1(0) = γ2(0) for a configuration
without a cavity.
We consider the first possibility. Suppose that near R = 0, r(R) = A(R)Rα,
where A is a slowly-varying function bounded above zero near R = 0. Since we
want r to be twice continuously differentiable, we assume α = 1 or α ≥ 2. If α = 1,
then r′(0) 6= 0, and we assume γ1(0) = γ2(0). If α ≥ 2, then
r′ = αARα−1 + A′Rα, r′′ = (α− 1)αARα−2 + 2αA′Rα−1 + A′′Rα, (9.44)
and the most singular terms in Eq. (9.42) are
− (λ + µ) (γ1γ2)′
r′/R≈ −(λ + µ) (γ1γ2)
′
αA
∣∣∣∣R=0
R2−α, (9.45)
−(λ + µ)γ1γ2
r/R2≈ −(λ + µ)γ1γ2
A
∣∣∣∣R=0
R2−α. (9.46)
Requiring that the coefficients agree at R = 0 would put undue restrictions on
the types of growth functions γi we may consider. We opt for the less restrictive
alternative: the growth functions we will employ will satisfy γ1(0) = γ2(0).
9.3 Conversion to a Dynamical System
We perform a change of variables found in Antman and Negron-Marrero (1987) and
in Section X.3 of Antman (1995). Let n be the radial stress,
n =µγ2
γ1
r′ + λr
R− (λ + µ) γ1γ2
r′, (9.47)
and let
τ =r
R. (9.48)
135
Finding r′ as a function of n yields two solutions, only one of which is positive. The
flatness of the middle surface requires that r′ be positive, so we choose the positive
solution:
r′ =γ1
2µγ2
(n− λ
r
R+
√(n− λ
r
R
)2
+ 4µ(λ + µ)γ22
)
=γ1
2µγ2
(n− λτ +
√(n− λτ)2 + 4µ(λ + µ)γ2
2
). (9.49)
The differential equation can now be stated in terms of n and τ :
d
dR(Rn) =
µγ1
γ2
r
R+ λr′ − (λ + µ)γ1γ2
r/R
=µγ1
γ2
τ +λγ1
2µγ2
(n− λτ +
√(n− λτ)2 + 4µ(λ + µ)γ2
2
)
︸ ︷︷ ︸λr′
− (λ + µ)γ1γ2
τ. (9.50)
Note also that
d
dR(Rτ) = r′ =
γ1
2µγ2
(n− λτ +
√(n− λτ)2 + 4µ(λ + µ)γ2
2
). (9.51)
We define a new independent variable s by
R = Rmaxes−1, or s = ln
(R
Rmax
)+ 1. (9.52)
If we set n(s) = n(R) and τ(s) = τ(R), then the R- and s-derivatives are related
by
d
dR(Rn) =
d
dR(Rn) = n + R
ds
dR
dn
ds= n +
dn
ds, (9.53)
d
dR(Rτ) =
d
dR(Rτ) = τ + R
ds
dR
dτ
ds= τ +
dτ
ds, (9.54)
where the prime on n and τ indicates differentiation with respect to s. The differ-
ential equations in n and τ are
dn
ds=
µγ1
γ2
τ +λγ1
2µγ2
(n− λτ +
√(n− λτ)2 + 4µ(λ + µ)γ2
2
)
− (λ + µ)γ1γ2
τ− n, (9.55)
136
dτ
ds=
γ1
2µγ2
(n− λτ +
√(n− λτ)2 + 4µ(λ + µ)γ2
2
)− τ , (9.56)
If we set N = n/µ and κ = λ/µ, then we can simplify these equations some:
dN
ds=
γ1
γ2
τ +κγ1
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)
− (1 + κ)γ1γ2
τ−N, (9.57)
dτ
ds=
γ1
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)− τ , (9.58)
where γi(s) = γi(R).
Were it not for the s-dependence of the γi, these equations would be autonomous,
and they could be analyzed from the point-of-view of dynamical systems. If the γi
satisfied some autonomous differential equations, then adding those equations to the
pair above would form an autonomous system. Below we consider several examples
of γi that satisfy autonomous equations.
To the differential equations in Eqs. (9.57) and (9.58) we add two boundary
conditions:
N(1) = 0, (zero radial stress at R = Rmax) (9.59)
lims→−∞
Rmaxes−1τ(s) = 0. (r(0) = 0) (9.60)
Now we consider examples with specific growth functions γ1 and γ2.
9.4 γ1(R) ≡ 1, γ′2(R) > 0
Suppose that γ1(R) ≡ 1 and γ2(R) = a+ b (R/Rmax)2. Since we want γ1(0) = γ2(0),
we set a = 1. Then γ2 satisfies
d
dR(Rγ2(R)) =
d
dR
(R +
b
R2max
R3
)= 1 + 3b
(R
Rmax
)2
= 1 + 3 (γ2(R)− 1) ,
(9.61)
so γ2 satisfies
γ2
ds+ γ2 = 1 + 3 (γ2 − 1) , or
γ2
ds= 2 (γ2 − 1) . (9.62)
137
We have the system of autonomous equations
dτ
ds=
1
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)− τ , (9.63)
dN
ds=
τ
γ2
+κ
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)
− (1 + κ)γ2
τ−N, (9.64)
dγ2
ds= 2 (γ2 − 1) . (9.65)
We will often express this system of differential equations by
dy
ds= f (y) , (9.66)
where y = (τ , N, γ2)T . Points y ∈ R3 will be described via τ -, N -, and γ2-
coordinates.
We will need some notation and basic results from the theory of ordinary differ-
ential equations and dynamical systems.
Definition 9.4.1 The domain of f is
D (f) =(τ , N, γ2) ∈ R3 : f (τ , N, γ2) is defined
. (9.67)
In the case at hand, D(f) consists of all of R3 except for the planes τ = 0 and γ2 = 0.
Definition 9.4.2 Each point y0 ∈ D(f) can be considered as the initial condition
of some solution of Eqs. (9.63)-(9.65):
dy
ds= f(y), y(0) = y0. (9.68)
The maximal interval of existence I(y0) is the largest interval of s-values (both
positive and negative) for which the solution of Eq. (9.68) exists and is unique.
Theorem 9.4.3 If f and the τ -, N-, and γ2-derivatives of f are continuous on the
“box” y ∈ R3 : ‖y − y0‖ < b, then Eq. (9.68) has at most one solution.
138
For a proof, see e.g. Section 3.3 of Brauer and Nohel (1989). Note that for
Eqs. (9.63)-(9.65), f is continuously differentiable in any finite box bounded away
from τ = 0 and γ2 = 0.
Definition 9.4.4 If s 7→ y(s) is a solution of Eq. (9.68), then the curve
y(s) ∈ R3 : s ∈ I(y0)
(9.69)
is called an orbit or trajectory of Eq. (9.68). We will often say that the solution
s 7→ y(s) “traces out” its orbit.
One of the most important properties of orbits of autonomous systems of dif-
ferential equations is that in regions of D(f) where f is continuously differentiable,
distinct orbits do not intersect. Hence, if s 7→ y(s) and s 7→ z(s) are solutions of
Eq. (9.68) with different initial conditions and y(s1) = z(s2) for some s1 6= s1, then
z(s) = y(s + s1 − s2), i.e. s 7→ z(s) is essentially s 7→ y(s) with a constant shift or
argument.
Definition 9.4.5 Much of the analysis will focus on the coordinates (τ , N). If
s 7→ (τ(s), N(s), γ2(s)) is a solution of Eqs. (9.63)-(9.65), then we will refer to the
function s 7→ (τ(s), N(s)) as the solution’s projection into the (τ , N)-plane. We will
also refer to the set of points
(τ(s), N(s)) : s ∈ maximal interval of existence , (9.70)
as the projection of the orbit.
9.4.1 Fixed Points and Linearization
The fixed points of Eqs. (9.63)-(9.65) form curves whose coordinates satisfy
γ2 = 1, N = (1 + κ)(τ + τ−1
). (9.71)
The coordinates of the fixed points with τ > 0 satisfy
γ2 = 1, τ =N +
√N2 + 4 (1 + κ)2
2 (1 + κ). (9.72)
139
At each fixed point, the derivative of the right-hand sides of Eqs. (9.63)-(9.65)
has the form
Df(τ , (1 + κ)
(τ + τ−1
), 1
)(9.73)
=
− (1+κ)(1+τ2)1+κ+τ2
τ2
1+κ+τ2
τ(1+κ−τ2)1+κ+τ2
(1+κ)(1+τ2)(1+κ−(−1+κ)τ2)τ2(1+κ+τ2)
−1−κ+(−1+κ)τ2
1+κ+τ2 −1+κτ − τ − κτ + 2κ(1+κ)τ
1+κ+τ2
0 0 2
.
This matrix has eigenvalues −2, 0, and 2, regardless of the values of κ and τ . This
indicates the presence of a stable manifold, a center manifold, and an unstable
manifold attached to each fixed point; see Perko (1996).
Each of these manifolds is traced out by a solution of Eqs. (9.63)-(9.65). The
unstable manifold is “unstable” in the sense that the solution that traces it out
approaches the fixed point at an exponential rate as s → −∞, i.e. as the dy-
namical system is run backwards. The eigenvector corresponding to the eigenvalue
2 describes the direction from which this solution approaches the fixed point as
s → −∞.
For a solution corresponding to a configuration of the plate, γ2(s) is non-constant
and γ2(s) ↓ 1 as s → −∞. The only eigenvector of the derivative matrix that has
nonzero γ2-component is the one associated with the eigenvalue 2:
(−−τ − κτ + 2τ 3
4 (1 + κ + τ 2), − 1 + 2κ + κ2 + 3τ 2 + 2κτ 2 − κ2τ 2 + 2τ 4 + 2κτ 4
4τ (1 + κ + τ 2), 1
)T
.
(9.74)
This means that the only orbit attached to a fixed point that could possibly cor-
respond to a configuration of the body is an unstable manifold. We will show in
the next section that there is an unstable manifold that corresponds to a plate with
zero radial stress at its periphery.
9.4.2 Existence of Solution
We will demonstrate that there is one solution that traces out an unstable manifold
and whose N -coordinate satisfies N(1) = 0.
140
For each s > −∞, we define the horizontal isocline
Γh(s) =
(τ , N) : τ =
N +√
N2 + 4(1 + κ) (κ + γ2(s)) γ2(s)
2 (κ + γ2(s))
(9.75)
and the vertical isocline
Γv(s) = (τ , N) : τ > 0 and (9.76)
1
2γ2(s)
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2(s)2
)− τ = 0
.
Γh(s) consists of the points in the (τ , N)-plane with τ > 0 where the vector field has
zero τ -component at s. Γv(s) consists of the points in the same half-plane where
the vector field has zero N -component at s. If κ > 1, then Γv(s) has two branches,
one above Γh(s) and one below. See Figure 9.1.
GvHsL
GvHsL
GhHsL
GH-¥L
0 1 2 3 4
-5
0
5
10
15
Τ
Figure 9.1: Isoclines in the (τ , N)-plane at some s > −∞. The presence of the upperbranch of Γv(s) indicates that κ > 1 in this example. The line segments attachedto Γ(−∞) show the directions of the unstable eigenvectors.
141
Proposition 9.4.6 At s = −∞, Γh(s) and Γv(s) coincide with Γ(−∞).
Proof In the defining equation of the horizontal isocline Γh(s), set γ2 = 1. This
coincides with the equation for the curve of fixed points. Plug this formula for τ
into the defining equation of the vertical isocline (with γ2 set to 1) and the equation
becomes a tautology.
Proposition 9.4.7 At s = 1, the projection of each solution that traces out an
unstable manifold lies in the region between Γ(−∞) and the lower branch of Γv(1).
Proof First we note that the N -component of the “unstable eigenvector” in
Eq. (9.74),
− (1 + κ) (2τ 4 + (3− κ2) τ 2 + (1 + κ))
4τ (1 + κ + τ 2)(9.77)
is negative for all real τ . Hence each fixed point is asymptotically approached from
below (as s → −∞), which means each fixed point is approached asymptotically
from the region between Γ(−∞) and the lower branch of Γv(s).
Figure 9.2 shows the vector field for the dynamical system run in reverse. If
(τ(s), N(s)) is to the left of Γ(−∞) for some s > −∞, then τ(s) → 0 as s → −∞,
and the orbit does not approach a fixed point. If (τ(s), N(s)) is to the right of thew
lower branch of Γv(s) for some s > −∞, then τ(s) →∞ as s → −∞, and the orbit
does not approach a fixed point.
In particular, each solution whose orbit is an unstable manifold, lies in the region
between Γ(−∞) and the lower branch of Γv(1) at s = 1.
Definition 9.4.8 If we view each point y0 ∈ D(f) as the initial condition of the
problemdy
ds= f(y), y(0) = y0, (9.78)
then the flow of that dynamical system is a function defined as
Φ(y0, s) = y(s), s ∈ I(y0). (9.79)
142
GvHsL
GvHsL
GhHsL
GH-¥L1 2 3 4 Τ
-5
5
10
15
N = n Μ
Figure 9.2: The vector field of the dynamical system run in reverse
Let E be an open subset of D(f) such that f ∈ C1 (E). Define
Ω = (s,y0) ∈ R× E : s ∈ I(y0) . (9.80)
Then Φ ∈ C1(Ω), i.e., Φ is continuously differentiable in both its s and y0 arguments.
See Section 2.5 of Perko (1996).
Proposition 9.4.9 At s = 1, the positions of the solutions tracing out the unstable
manifolds form a continuous curve in the region between Γ(−∞) and the lower
branch of Γv(1).
Proof Fix τ0 > 0. Define E by
E = (τ , N) : (τ , N) is between Γ(−∞)
and the lower branch of Γv(1), and τ > τ0 > 0 . (9.81)
Then f ∈ C1(E), which implies that the flow Φ of the dynamical system is also
continuously differentiable. The Center Manifold Theorem (see Section 2.12 of Perko
(1996)) guarantees the unstable manifolds can be locally (near the fixed points)
143
continuously parametrized by s and an arclength coordinate for the curve of fixed
points. Hence for some s0 > −∞ with |s0| >> 1 there is a continuous curve formed
by the positions of the solutions that trace out unstable manifolds. See Figure 9.3.
Since f ∈ C1(E), the maximal interval of existence for each unstable manifold
includes the half-infinite interval (−∞, 1]. Hence the continuous curve formed by
the solutions at s = s0 is mapped continuously via Φ to its image at s = 1. By
continuity of the s = s0 curve and of Φ, the positions of the solutions at s = 1 also
form a continuous curve.
Note that E is bounded below by τ0 > 0. τ0 can be re-defined to allow E to
extend to any proximity to the τ = 0 line, and the argument still holds. On the
curve of fixed points, N → −∞ as τ → 0, so the continuous curve of positions at
s = 1 is infinite in extent. This argument leaves open the possibility that the map
from the fixed points to this curve may not be uniformly continuous.
GvH1L
GH-¥Lcurve formedby solutions
at s = s0 << 0
curve formedby solutions
at s = 1
0.5 1.0 1.5 2.0
-1
0
1
2
3
4
5
6
Τ
Figure 9.3: The continuity of the curve of fixed points is maintained in the curvesformed by the unstable manifold orbits for s > −∞.
Proposition 9.4.10 The continuous curve of positions of solutions at s = 1 inter-
sects the line N = 0.
Proof (Sketch) This can be proved by considering the one-point compactification
of the (τ , N)-plane, which is a 2-sphere, and examining the stereographic projections
onto the sphere of Γ(−∞), the lower branch of Γv(1), and the curve of solution
144
Figure 9.4: The stereographic projections of the lower branch of Γv(1) (outer curve)and Γ(−∞) (inner curve) intersect at the north pole of the sphere. The dashedcurve that they both intersect is the projection of the N = 0 line. The curve of lociat s = 1 is between these two curves and evolves into Γ(−∞) as s → −∞.
positions at s = 1. Each of these continuous images of R is projected to a closed
loop on the sphere; see Figure 9.4. Each “end” of the projections of Γ(−∞) and the
lower branch of Γv(1) lie at the north pole of the sphere. The closed loop of solution
positions at s = 1 lies between these projections. If the loop of positions is to evolve
into Γ(−∞) as s → −∞, the “ends” of this loop must also lie at the north pole.
Hence the loop of positions must intersect the N = 0 line on the sphere, shown in
Figure 9.4.
9.5 Numerical Procedure
We use the form of the fixed points of Eq. (9.63)-(9.65) to compute numerical solu-
tions of the equations. The method is as follows.
1. Choose τ0 > 0.
2. Set N0 = (1 + κ)(τ0 − τ−1
0
), the N -coordinate of the fixed point with τ -
coordinate equal to τ0.
3. Compute the unstable eigenvector at the fixed point (τ0, N0, 1).
145
4. For some small ε > 0, perturb (τ0, N0, 1) by ε times the unstable eigenvector:
(τ1
N1
(γ2)1
)=
(τ0
N0
1
)+ ε
( ((1 + κ)τ0 − 2τ3
0
)/4
(1 + κ + τ2
0
)
−(1 + κ)(2τ4
0 + (3− κ2)τ20 + (1 + κ)
)/4
(1 + κ + τ2
0
)
1
). (9.82)
5. Set s1 = 1+ 12ln (ε/b2). The formula in the step above gives an approximation
for the position, at s = s1, of the solution tracing out the unstable manifold
attached to the fixed point (τ0, N0, 1).
6. Integrate the equations numerically (for increasing s) until N = 0.
If N reaches zero at some s-value other than 1, adjust τ0 and repeat the above steps
until |N(1)| is near zero to within some tolerance.
Results for Rmax = 1 are pictured in Figures 9.5-9.10. In each case, the radial
stress is zero at the periphery and positive at the center, as expected from the form
of the vector field run in reverse; see Figure 9.2. The azimuthal stress, which agrees
with the radial stress at R = 0, is negative at the periphery, indicating that the
outer edge is under azimuthal compression.
Recall that for κ > 1, Γv(s) has two branches. The impact of the change in the
vector field can be seen by comparing Figures 9.6 and 9.8. As R → 0 (s → −∞), the
terminal τ -value is approached from below in Figure 9.6, indicating that the fixed
point is approached from the left. In Figure 9.8, on the other hand, the terminal
τ -value is approached from larger values of τ , indicating that the fixed point is
approached from the right.
9.6 γ1(R) ≡ 1, γ′2(R) < 0
Now we consider a growth pattern that allows us to use much of the previous section’s
analysis but produces quite different results. We again let the radial growth γ1
be constant, but we assign it a value γ0 greater than one. Further, we assume
γ2(R) ≤ γ0, with equality only at R = 0.
146
0.2 0.4 0.6 0.8 1.0 R
-1.5
-1.0
-0.5
0.5
stressΜ
Γ1HRL º 1, Γ2HRL = 1 + HRRmaxL2, Κ = 1
Figure 9.5: The radial stress (solid curve) drops to zero at the periphery, while theazimuthal stress (dashed curve) is negative at the periphery.
0.2 0.4 0.6 0.8 1.0 R
1.20
1.22
1.23
1.24
1.25
Τ = rRΓ1HRL º 1, Γ2HRL = 1 + HRRmaxL
2, Κ = 1
Figure 9.6: τ = r/R decreases monotonically in this configuration.
If γ1(R) ≡ γ0 > 1 and γ2(R) = γ0 − b (R/Rmax)2, the corresponding differential
147
0.2 0.4 0.6 0.8 1.0 R
-2.0
-1.5
-1.0
-0.5
0.5
1.0
stressΜ
Γ1HRL º 1, Γ2HRL = 1 + HRRmaxL2, Κ = 3
Figure 9.7: Changing κ from 1 to 3 produces little change in the radial (solid) andazimuthal (dashed) stress profiles.
0.2 0.4 0.6 0.8 1.0 R
1.17
1.18
1.19
1.20
Τ = rRΓ1HRL º 1, Γ2HRL = 1 + HRRmaxL
2, Κ = 3
Figure 9.8: Changing κ from 1 to 3 changes the τ = r/R profile dramatically.
equations are
dN
ds=
γ0
γ2
τ +κγ0
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)
− (1 + κ)γ0γ2
τ−N, (9.83)
dτ
ds=
γ0
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)− τ , (9.84)
dγ2
ds= 2 (γ2 − γ0) . (9.85)
148
0.2 0.4 0.6 0.8 1.0 R
-2
-1
1
stressΜ
Γ1HRL º 1, Γ2HRL = 1 + 2HRRmaxL2, Κ = 1
Figure 9.9: Changing b from 1 to 2 produces little change in the stress profiles.
0.2 0.4 0.6 0.8 1.0 R
1.40
1.45
1.50
Τ = rRΓ1HRL º 1, Γ2HRL = 1 + 2HRRmaxL
2, Κ = 1
Figure 9.10: Changing b from 1 to 2 produces little change in the τ = r/R profile.
The fixed points of this system form curves whose coordinates satisfy
γ2 = γ0, τ =N ±
√N2 + 4 (1 + κ)2 γ2
0
2 (1 + κ). (9.86)
τ > 0 on one curve; τ < 0 on the other. We focus on positive τ . At each such fixed
149
point, the linearization Df is
−(1+κ)
(τ2+γ2
0
)
τ2+(1+κ)γ20
τ2
τ2+(1+κ)γ20
τ(−τ2+(1+κ)γ2
0
)
γ0(
τ2+(1+κ)γ20
)
(1+κ)(
τ2+γ20
)(−(−1+κ)τ2+(1+κ)γ2
0
)
τ2(
τ2+(1+κ)γ20
) (−1+κ)τ2−(1+κ)γ20
τ2+(1+κ)γ20
−(1+κ)τ4+(−2−κ+κ2
)τ2γ2
0−(1+κ)2γ40
τγ0(
τ2+(1+κ)γ20
)
0 0 2
, (9.87)
which has eigenvalues 2, 0, and −2, regardless of κ, γ0, and τ . As in the previous
case, the only eigenvector with nonzero γ2-component is the unstable eigenvector,
which in this case is(
τ (−2τ 2 + (1 + κ)γ20)
4γ0 (τ 2 + (1 + κ)γ20)
,−2(1 + κ)τ 4 + (−3− 2κ + κ2) τ 2γ2
0 − (1 + κ)2γ40
4τ γ0 (τ 2 + (1 + κ)γ20)
, 1
)T
.
(9.88)
In this example, γ2(s) ↑ γ0 as s → −∞, which indicates that the unstable mani-
fold approaches the fixed point from the direction opposite to this eigenvector; see
Figure 9.11.
GvH1L
GhH1L
GH-¥L0.5 1.0 1.5 Τ
-1
1
2
3
4
N = nΜ
Figure 9.11: The vector field of the dynamical system run in reverse. In this case,the solutions tracing out the unstable manifolds approach the fixed points from theleft as s → −∞.
As in the previous section, we define horizontal and vertical isoclines Γh(s) and
Γv(s), respectively, and observe that they coincide with Γ(−∞), the curve of fixed
points, as s → −∞. Arguments similar to those in the last section show the existence
of an unstable manifold that intersects the line N = 0 at s = 1. In this case, however,
150
it is possible for a solution tracing out an unstable manifold to satisfy N(1) = 0
as well as lims→−∞ N(s) < 0, which corresponds to compressive radial stress at the
center of the disc.
Using a numerical approach nearly identical to that of the last section (with
negative ε in step 4), we found results with negative radial stress throughout the
body and azimuthal stress that changes sign within the body. In these examples,
γ0 ≥ γ2(R) ≥ 1, with γ2(0) = γ0 and γ2(Rmax) = 1.
0.2 0.4 0.6 0.8 1.0 R
-0.5
0.5
1.0
stressΜ
Γ1HRL º 1.5, Γ2HRL = 1.5 - 0.5HRRmaxL2, Κ = 1
Figure 9.12: The radial stress (solid line) is negative throughout the body, while theazimuthal stress (dashed line) changes sign, leaving it positive at the periphery.
This type of growth tensor induces negative radial stress throughout the body;
it is largest at R = 0 and decays monotonically until it is zero at the periphery. At
R = 0, the azimuthal stress agrees with the radial stress. It grows monotonically
in R and is positive at the periphery. The presence of negative stresses through so
much of the body suggests that these configurations may be prone to buckling.
151
0.2 0.4 0.6 0.8 1.0 R
1.362
1.364
1.366
1.368
1.370
1.372
Τ
Γ1HRL º 1.5, Γ2HRL = 1.5 - 0.5HRRmaxL2, Κ = 1
Figure 9.13: τ = r/R is monotone decreasing with R, but no change in concavity ofthe graph is apparent.
0.4 0.6 0.8 1.0 R
-0.2
-0.1
0.1
0.2
0.3
0.4
0.5
stressΜ
Γ1HRL º 1.5, Γ2HRL = 1.5 - 0.5HRRmaxL2, Κ = 3
Figure 9.14: Changing κ from 1 to 3 changes the magnitudes of the radial (solid)and azimuthal (dashed) stresses.
152
0.4 0.6 0.8 1.0 R
1.155
1.160
1.165
1.170
Τ
Γ1HRL º 1.5, Γ2HRL = 1.5 - 0.5HRRmaxL2, Κ = 3
Figure 9.15: Unlike what was found in the γ′2 > 0 case, changing κ from 1 to 3 haslittle impact on the form of τ = r/R.
0.2 0.4 0.6 0.8 1.0 R
-1.0
-0.5
0.5
1.0
1.5
2.0
2.5
stressΜ
Γ1HRL º 2, Γ2HRL = 2 - HRRmaxL2, Κ = 1
Figure 9.16: Keeping κ = 1 but doubling the growth increment approximatelydoubles the radial (solid) and azimuthal (dashed) stresses.
153
0.2 0.4 0.6 0.8 1.0 R
1.71
1.72
1.73
1.74
Τ
Γ1HRL º 2, Γ2HRL = 2 - HRRmaxL2, Κ = 1
Figure 9.17: Keeping κ = 1 but doubling the growth increment approximatelydoubles the expansion.
154
CHAPTER 10
AXISYMMETRIC BUCKLING OF THE KIRCHHOFF PLATE
10.1 Kirchhoff Constraints
In a Kirchhoff plate, the final configuration is assumed to have the form
(x f)(R, Θ, Z) = r(R, Θ) + Z d(R, Θ), (10.1)
where the function r describes a two-dimensional “middle surface” of the plate, and
the director field d is a unit vector-valued function that satisfies
d =∂r
∂R× ∂r
∂Θ
/wwww∂r
∂R× ∂r
∂Θ
wwww . (10.2)
Thus d can be viewed as a function of r and its partial derivatives. The three-
dimensional Kirchhoff plate consists of a two-dimensional surface from which one-
dimensional material fibers sprout in a perpendicular direction. See Figure 10.1.
The tangent vector to this class of deformations was found in Eq. (8.6):
Mx =
Mr + Z
Mr,R×r,Θ +r,R×M
r,Θ‖r,R×r,Θ ‖ , (10.3)
whereMr is an as-regular-as-needed function of R and Θ.
We could derive the vector equation of equilibrium by considering the weak form
of the Method of Virtual Work with this virtual displacement, but there is an easier
but less direct method. The director field must satisfy the (local) constraints
d · d = 1 and d · r,α = 0, α = R, Θ. (10.4)
Encoding this into the expression P act : ∂Mx/∂X is complicated, even in the axisym-
metric case. If we introduce Lagrange multipliers λ, βR, and βΘ, we can consider
155
instead the integral of the following:
τ k ·(
Mr + Z
Md
),k − λd ·
Md + βα
(Md · r,α + d · M
r,α
)
= τ k · Mr,k + βαd · r,α + τ k ·
(Z
Md
),k + (βαr,α − λd) ·
Md
= (τα + βαd) · Mr,α + τα · Z
Md,α +
(τZ + βαr,α − λd
) ·Md, (10.5)
where k = R, Θ, Z and α = R, Θ. Thanks to the Lagrange multipliers, we can
treatMr and
Md as though they were independent virtual displacements. We will be
considering a plate with no body forces and no applied tractions, so the weak form
of the equations of equilibrium will be
∫
(XΨ)(B)
(τα + βαd) · M
r,α + τα · ZMd,α +
(τZ + βαr,α − λd
) ·Md
dV = 0
(10.6)
for all admissible virtual displacementsMr.
Note that
τR ·(
Mr + Z
Md
),R =
(P act ·ER
) · ∂
∂R
(Mr + Z
Md
)
= (P act · h1) · ∂
∂R
(Mr + Z
Md
)(10.7)
τΘ ·(
Mr + Z
Md
),Θ =
(P act ·EΘ
) · ∂
∂Θ
(Mr + Z
Md
)
=
(P act · h2
R
)· ∂
∂Θ
(Mr + Z
Md
). (10.8)
10.2 Kinematics of an Axisymmetric Kirchhoff Plate
We consider axisymmetric deformations of an axisymmetric Kirchhoff plate. The
reference configuration has the form
X(R, Θ, Z) = R cos(Θ) i + R sin(Θ) j + Z k = R h1(Θ) + Z k, (10.9)
where
h1(Θ) = cos(Θ) i + sin(Θ) j. (10.10)
156
h1
d
k
a
r
Ζ
Φ
Φ
Figure 10.1: A radial slice of an axisymmetric Kirchhoff plate. ζ is the height of apoint of the middle surface, and a material point’s height above the middle surfaceis measured along the unit vector d, which is perpendicular to the middle surface.
As a function of (R, Θ, Z), the final configuration will have the form
(x f)(R, Θ, Z) = r(R) h1(Θ) + ζ(R) k︸ ︷︷ ︸r(R,Θ)
+Z d(R, Θ). (10.11)
Here r is the cylindrical radius of a point in the middle surface, and ζ is the height
of a point in the middle surface. See Figure 10.1.
An outward-pointing radial tangent vector to the middle surface is
d
dR(r(R) h1(Θ) + ζ(R) k) = r′(R) h1(Θ) + ζ ′(R) k. (10.12)
We define a to be the corresponding unit vector:
a =r′ h1 + ζ ′ k√(r′)2 + (ζ ′)2
= cos φ h1 + sin φ k, (10.13)
where φ is the angle formed by a and the radial unit vector h1:
φ = arctan
(ζ ′
r′
). (10.14)
157
The director d is the “upward”-pointing unit vector orthogonal to a:
d = − sin φ h1 + cos φ k. (10.15)
The azimuthal unit vector h2 is the third member of the local orthonormal basis.
To construct the deformation gradient, we appeal to the relation
∂
∂ξi(χ X)(ξ1, ξ2, ξ3) = F ·Ei, (10.16)
where F and the Ei are evaluated at X(ξ1, ξ2, ξ3).
∂
∂R(r(R) h1(Θ) + ζ(R) k + Z d(R, Θ)) = r′(R) h1(Θ) + ζ ′(R) k + Z
∂d
∂R(R, Θ)
= r′ h1 + ζ ′ k − Zφ′ (cos φ h1 + sin φ k)
= r′ h1 + ζ ′ k − Zφ′ a. (10.17)
∂
∂Θ(r(R) h1(Θ) + ζ(R) k + Z d(R, Θ)) = r(R) h2(Θ) + Z
∂d
∂Θ(R, Θ)
= r h2 − Z sin φ∂h1
∂Θ
= (r − Z sin φ) h2. (10.18)
∂
∂Z(r(R) h1(Θ) + ζ(R) k + Z d(R, Θ)) = d(R, Θ) (10.19)
These results show that
F = (r′ h1 + ζ ′ k − Zφ′ a)⊗ER + (r − Z sin φ) h2 ⊗EΘ + d⊗EZ
= (r′ h1 + ζ ′ k − Zφ′ a)⊗ h1 +
(r − Z sin φ
R
)h2 ⊗ h2 + d⊗ k, (10.20)
where it must be understood that the vectors on the right-hand sides of the tensor
products are anchored at the reference point, while those on the left-hand sides are
anchored at the deformed point.
We will find it convenient to use the orthonormal basis a, h2,d in the deformed
configuration, so we will re-write the left-hand side of the first tensor product in F .
158
Note that a
d
=
cos φ sin φ
− sin φ cos φ
h1
k
=⇒
h1
k
=
cos φ − sin φ
sin φ cos φ
a
d
.
(10.21)
As a result,
r′ h1 + ζ ′ k = r′ (cos φ a− sin φ d) + ζ ′ (sin φ a + cos φ d)
= (r′ cos φ + ζ ′ sin φ) a + (ζ ′ cos φ− r′ sin φ) d. (10.22)
By definition of φ, though, ζ ′ = r′ tan φ, so
r′ cos φ + ζ ′ sin φ = r′ (cos φ + tan φ sin φ)
= r′(
cos φ +sin2 φ
cos φ
)
= r′ sec φ, (10.23)
ζ ′ cos φ− r′ sin φ = r′ (tan φ cos φ− sin φ) = 0. (10.24)
The deformation gradient can thus be written as
F = (r′ sec φ− Zφ′) a⊗ h1 +
(r − Z sin φ
R
)h2 ⊗ h2 + d⊗ k. (10.25)
We will need the following expression for later computations:
F−T = (r′ sec φ− Zφ′)−1a⊗ h1 +
(r − Z sin φ
R
)−1
h2 ⊗ h2 + d⊗ k. (10.26)
We will use incompatible growth tensors of the form presented in Eq. (4.6):
G = γ1 Er ⊗ER +γ2
γ1
Eθ ⊗EΘ + Ez ⊗EZ
= γ1 h1 ⊗ h1 +γ2
γ1
rh2 ⊗ h2
R+ k ⊗ k
= γ1 h1 ⊗ h1 + γ2 h2 ⊗ h2 + k ⊗ k, (r = γ1R) (10.27)
with γ1 and γ2 constant. It should be noted that in Eq. (10.27) the vectors on
the right-hand side of each tensor product are anchored at a point in the reference
159
configuration and each vector on the left-hand side of a tensor product is anchored
at the image under the incompatible growth discussed above.
With F and G so defined, the tensor describing the elastic response to growth
is
A = F ·G−1
=
(r′ sec φ− Zφ′
γ1
)a⊗ h1 +
(r − Z sin φ
γ2R
)h2 ⊗ h2 + d⊗ k. (10.28)
The vector on the right-hand side of each tensor product is anchored at a point
in the incompatibly grown state, and each vector on the left-hand side of a tensor
product is anchored at a point in the final configuration.
The most important tensor for the constitutive relation will be the Eulerian
tensor
A ·AT =
(r′ sec φ− Zφ′
γ1
)2
a⊗ a +
(r − Z sin φ
γ2R
)2
h2 ⊗ h2 + d⊗ d. (10.29)
It might be clearer that G maps from the tangent space at a point in a body in
E3 to the tangent space at a point in a body that cannot “fit” into E3, if G were
written as
G = γ1∂
∂r⊗ER +
γ2
γ1
∂
∂θ⊗EΘ +
∂
∂z⊗EZ , (10.30)
where ∂/∂r, ∂/∂θ, and ∂/∂z are tangent vectors at (r, θ, z) = f(R, Θ, Z) in (f Ψ)(B) ⊂ R3. Computing A = F ·G−1 would be just as simple, but computing A·AT
with this notation would require a detour into the form of invariant expressions for
transposes of tensors. Eqs. (10.27), (10.28), and (10.29) are sufficient for our goals.
10.3 Radial Virtual Displacement
First we consider∫
plate
(τα + βαd) · Mr,α dV
=
∫
plate
(P act · h1 + βRd
) · ∂Mr
∂R+
(P act · h2
R+ βΘd
)· ∂
Mr
∂Θ
dV
=
∫
plate
∂
Mr
∂R· (P act · h1 + βRd
)+
∂Mr
∂Θ·(
P act · h2
R+ βΘd
)dV.
160
It will be helpful to consider βR and βΘ as components of a vector β:
β = βRER + βΘEΘ = βR h1 + βΘR h2. (10.31)
With β so defined, we can consider the two-point tensor field P act + d⊗ β.
Consider the planar portion ofMr · (P act + d⊗ β):
Mr · (P act + d⊗ β)planar =
(Mr · P act · h1 +
(Mr · d
)(β · h1)
)h1
+(Mr · P act · h2 +
(Mr · d
)(β · h2)
)h2. (10.32)
In polar coordinates, the divergence of a planar vector field is
Div (f h1 + g h2) =1
R
∂
∂R(Rf) +
1
R
∂g
∂Θ, (10.33)
so the divergence ofMr · (P act + d⊗ β)planar is
Div(Mr · (P act + d⊗ β)planar
)
=1
R
∂
∂R
(R
Mr · P act · h1 + R
(Mr · d
)(β · h1)
)
+1
R
∂
∂Θ
(Mr · P act · h2 +
(Mr · d
)(β · h2)
)
=∂
Mr
∂R· (P act · h1 + βRd
)+
Mr · 1
R
∂
∂R
(R P act · h1 + R βRd
)
+1
R
∂Mr
∂Θ· (P act · h2 + (β · h2) d) +
Mr · 1
R
∂
∂Θ(P act · h2 + (β · h2) d)
=∂
Mr
∂R· (P act · h1 + βRd
)+
Mr · 1
R
∂
∂R
(R P act · h1 + R βRd
)
+1
R
∂Mr
∂Θ· (P act · h2 + βΘR d
)+
Mr · 1
R
∂
∂Θ
(P act · h2 + βΘR d
)
=∂
Mr
∂R· (P act · h1 + βRd
)+
Mr · 1
R
∂
∂R
(R P act · h1 + R βRd
)
+∂
Mr
∂Θ·(
P act · h2
R+ βΘ d
)+
Mr · ∂
∂Θ
(P act · h2
R+ βΘ d
). (10.34)
161
TheMr-portion of Eq. (10.6) can be written as
∫
plate
∂
Mr
∂R· (P act · h1 + βRd
)+
∂Mr
∂Θ·(
P act · h2
R+ βΘd
)dV
=
∫
plate
Div(Mr · (P act + d⊗ β)planar
)dV
−∫
plate
Mr ·
1
R
∂
∂R
(R P act · h1 + R βRd
)
+∂
∂Θ
(P act · h2
R+ βΘd
)dV. (10.35)
By the Divergence Theorem in the plane, the integral of the first integrand can be
re-written
∫
plate
Div(Mr · (P act + d⊗ β)planar
)dv
=
∫ H2
H1
∫
slice
Div(Mr · (P act + d⊗ β)planar
)R dR dΘ
dZ
=
∫ H2
H1
∮
∂(slice)
(Mr · (P act + d⊗ β)planar
)· ν d`
dZ
=
∮
∂(slice)
Mr ·
∫ H2
H1
(P act + d⊗ β)planar · ν dZ
d`, (10.36)
where the “slice” mentioned is a planar slice of the reference configuration of the
plate, ∂(slice) is the curve that forms the planar boundary of the slice, ν is the
outward-pointing unit normal on ∂(slice), and d` is arclength measure on ∂(slice).
In order to balance the boundary term in the weak form of the Principle of
Virtual Work, we must specify the value of
∫ H2
H1
(P act + d⊗ β)planar · ν dZ =
∫ H2
H1
(P act + d⊗ β) · h1 dZ (10.37)
on the boundary of the middle surface.
Now that boundary terms are balanced, we have
∫
plate
Mr ·
1
R
∂
∂R
(R P act · h1 + R βRd
)+
∂
∂Θ
(P act · h2
R+ βΘd
)dV = 0
(10.38)
162
for all as-smooth-as-needed functionMr of R and Θ. Since
∫
plate
Mr ·
1
R
∂
∂R
(R P act · h1 + R βRd
)+
∂
∂Θ
(P act · h2
R+ βΘd
)dv
=
∫
slice
∫ H2
H1
1
R
∂
∂R
(R P act · h1 + R βRd
)
+∂
∂Θ
(P act · h2
R+ βΘd
)dZ · M
r R dR dΘ
= 0, (10.39)
for such a large class ofMr, we conclude that the integral in Z is identically zero:
∫ H2
H1
1
R
∂
∂R
(R P act · h1 + R βRd
)+
∂
∂Θ
(P act · h2
R+ βΘd
)dZ = 0.
(10.40)
10.4 Director Virtual Displacement
We apply the same procedure to the variation inMd. The result will be a weak
version of the balance of angular momentum in the constrained plate with zero
applied moments.
∫
plate
τα · Z
Md,α +
(τZ + βαr,α−λd
) ·Md
dV
=
∫
plate
τR · Z ∂
Md
∂R+ τΘ · Z ∂
Md
∂Θ+
(τZ + βαr,α−λd
) ·Md
dV
=
∫
plate
Z Div
(Md · P act
)
planar
dV
+
∫
plate
Md ·
(τZ + βαr,α−λd
)− Z1
R
∂
∂R(R P act · h1)− Z
∂
∂Θ
(P act · h2
R
)dV
=
∮
∂(slice)
Md ·
(∫ H2
H1
Z P act dZ
)· ν d`
+
∫
slice
Md ·
∫ H2
H1
τZ + βαr,α−λd− Z
1
R
∂
∂R(R P act · h1)
− Z∂
∂Θ
(P act · h2
R
)dZ R dR dθ (10.41)
163
As before, ν = h1 is the outward-pointing unit normal to the planar boundary
∂(slice) of the slice in the reference configuration, and d` is arclength measure on
∂(slice).
To balance the boundary terms, we must set values of the moment∫ H2
H1
Z (P act · h1) dZ (10.42)
on the planar boundary of the middle surface.
We have found that∫
slice
Md ·
∫ H2
H1
τZ + βαr,α−λd− Z
1
R
∂
∂R(R P act · h1)
− Z∂
∂Θ
(P act · h2
R
)dZ R dR dθ = 0 (10.43)
for all as-smooth-as-needed functionsMd of R and Θ. We conclude that
∫ H2
H1
τZ + βαr,α−λd− Z
1
R
∂
∂R(R P act · h1)− Z
∂
∂Θ
(P act · h2
R
)dZ = 0.
(10.44)
10.5 Addressing Lagrange Multipliers
We are not yet ready to introduce the constitutive relation to close the system of
equations. The quantities λ, βR, and βΘ are not constitutively defined. However,
there is enough information in the equations above to get around this complication.
First we claim that if we take the cross-product with d, the term λd will vanish.
By the form of G we are assuming and the form of F−T found above, we can be
sure that F−T ·k = d. If we use an isotropic hyperelastic constitutive relation, then
the form of A = F ·G−1 in such a relation will guarantee that T · d points in the
direction d. As a result,∫ H2
H1
τZdZ =
∫ H2
H1
(P act · k) dZ
=
∫ H2
H1
(det F )(T · F−T · k)
dZ
=
∫ H2
H1
(det F ) (T · d) dZ ∝ d, (10.45)
164
which implies
d×∫ H2
H1
τZdZ = 0, (10.46)
the elimination we sought.
When we take the cross-product of d with Eq. (10.44), we have
d× 1
R
∂
∂R
(R
∫ H2
H1
Z (P act · h1) dZ
)
+ d× ∂
∂Θ
∫ H2
H1
Z
(P act · h2
R
)dZ − d× r,α
∫ H2
H1
βαdZ = 0. (10.47)
We will consider all these terms in detail to find expressions for βR and βΘ in terms
of kinematic and constitutively-defined quantities.
By the symmetry of the deformation, the vector field P act ·h1 has zero projection
onto h2. It is determined solely by its projections onto h1 and k:
(P act · h1) = (h1 · P act · h1) h1 + (k · P act · h1) k (10.48)
Since h1 and k are R-independent,
1
R
∂
∂RR (P act · h1) = h1
1
R
∂
∂RR (h1 · P act · h1) + k
1
R
∂
∂RR (k · P act · h1) .
(10.49)
In particular,
h1 · 1
R
∂
∂RR (P act · h1) =
1
R
∂
∂RR (h1 · P act · h1) , (10.50)
k · 1
R
∂
∂RR (P act · h1) =
1
R
∂
∂RR (k · P act · h1) . (10.51)
165
We have
d× 1
R
∂
∂R
(R
∫ H2
H1
Z (P act · h1) dZ
)
= (cos φ k − sin φ h1)
×
h11
R
∂
∂R
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+ k1
R
∂
∂R
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
= h2
cos φ
R
∂
∂R
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+sin φ
R
∂
∂R
(R
∫ H2
H1
Z (k · P act · h1) dZ
). (10.52)
The symmetry of the deformation also requires that P act · h2 point along h2:
(P act · h2) = (h2 · P act · h2) h2. (10.53)
Further, the coefficient (h2 · P act · h2) is Θ-independent, so the Θ-derivative of
(P act · h2) is
∂
∂Θ(P act · h2) =
∂
∂Θ(h2 · P act · h2) h2
= (h2 · P act · h2)∂h2
∂Θ
= − (h2 · P act · h2) h1. (10.54)
The cross-product with d is
d× ∂
∂Θ
∫ H2
H1
Z
(P act · h2
R
)dZ
= (cos φ k − sin φ h1)×(−h1
∫ H2
H1
Z
(h2 · P act · h2
R
)dZ
)
= −h2cos φ
R
∫ H2
H1
Z (h2 · P act · h2) dZ. (10.55)
166
The remaining terms are
d× r,α
∫ H2
H1
βαdZ
= d×
∂r
∂R
∫ H2
H1
βRdZ +∂r
∂Θ
∫ H2
H1
βΘdZ
= (cos φ k − sin φ h1)×
(r′ h1 + ζ ′ k)
∫ H2
H1
βRdZ + r h2
∫ H2
H1
βΘdZ
=
(r′ cos φ
∫ H2
H1
βRdZ
)h2 +
(ζ ′ sin φ
∫ H2
H1
βRdZ
)h2
−(
r cos φ
∫ H2
H1
βΘ
)h1 −
(r sin φ
∫ H2
H1
βΘ
)k. (10.56)
What we have shown is
cos φ
R
∂
∂R
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+sin φ
R
∂
∂R
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
− cos φ
R
∫ H2
H1
Z (h2 · P act · h2) dZ
h2
= (r′ cos φ + ζ ′ sin φ)
(∫ H2
H1
βRdZ
)h2
− r (cos φ h1 − sin φ k)
∫ H2
H1
βΘdZ. (10.57)
Note that the left-hand side is a scalar multiple of h2, and that∫ H2
H1βΘdZ on the
right-hand side is the coefficient of a linear combination of h1 and k, both of which
are orthogonal to h2. Since this holds for each (R, Θ), we see that∫ H2
H1βΘdZ = 0.
Computing the h2-projection reveals
cos φ
R
∂
∂R
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+sin φ
R
∂
∂R
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
− cos φ
R
∫ H2
H1
Z (h2 · P act · h2) dZ
= (r′ cos φ + ζ ′ sin φ)
(∫ H2
H1
βRdZ
). (10.58)
167
If we multiply by R and divide by cos φ, we have
∂
∂R
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+ tan φ∂
∂R
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
−∫ H2
H1
Z (h2 · P act · h2) dZ
= R (r′ + ζ ′ tan φ)
(∫ H2
H1
βRdZ
)
= R(r′ + r′ tan2 φ
) (∫ H2
H1
βRdZ
)
= Rr′ sec2 φ
(∫ H2
H1
βRdZ
). (10.59)
In the next section we will find another expression for the integral of βR,
and this equation will become a differential equation in terms of kinematical and
constitutively-defined quantities.
10.6 Problem As a Pair of ODEs
Since we have eliminated all differentiation with respect to Θ in the differential
equations, we can view them as a system of ordinary differential equations with
independent variable R.
Thanks to the discoveries of the last section, Eq. (10.40) can now be written as
1
R
d
dR
h1 R
∫ H2
H1
(h1 · P act · h1) dZ
+ k R
∫ H2
H1
(k · P act · h1) dZ + R d
∫ H2
H1
βRdZ
= h1
∫ H2
H1
(h2 · P act · h2
R
)dZ. (10.60)
The projection of this equation onto k is
1
R
d
dR
R
∫ H2
H1
(k · P act · h1) dZ + R cos φ
∫ H2
H1
βRdZ
= 0, (10.61)
168
so that
R
∫ H2
H1
(k · P act · h1) dZ + R cos φ
∫ H2
H1
βRdZ = constant. (10.62)
Recall that we must assign the value of the stress
∫ H2
H1
(P act · h1 + βRd
)dZ (10.63)
on the boundary (R = Rmax) of the middle surface. If this stress is zero, then in
particular its k-projection is zero. This implies that the constant on the right-hand
side of Eq. (10.62) is zero, and
∫ H2
H1
βRdZ = − sec φ
∫ H2
H1
(k · P act · h1) dZ. (10.64)
We will see later that the isotropic hyperelastic constitutive relation we choose
ensures that P act · h1 points in the direction a = cos φ h1 + sin φ k. As a result of
the direction of P act · h1,
P act · h1 = (a · P act · h1) a, (10.65)
so (h1 · P act · h1) and (k · P act · h1) satisfy
k · P act · h1 = (a · P act · h1) k · a = sin φ (a · P act · h1) , (10.66)
h1 · P act · h1 = (a · P act · h1) h1 · a = cos φ (a · P act · h1) , (10.67)
and
(k · P act · h1) = tan φ (h1 · P act · h1) . (10.68)
As a result,
∫ H2
H1
βRdZ = − sec φ tan φ
∫ H2
H1
(h1 · P act · h1) dZ. (10.69)
The h1-projection of Eq. (10.60) becomes
1
R
d
dR
R
∫ H2
H1
(h1 · P act · h1) dZ −R sin φ
∫ H2
H1
βRdZ
=
∫ H2
H1
(h2 · P act · h2
R
)dZ, (10.70)
169
or
d
dR
R sec2 φ
∫ H2
H1
(h1 · P act · h1) dZ
=
∫ H2
H1
(h2 · P act · h2) dZ (10.71)
We have one more scalar ordinary differential equation from the previous section:
d
dR
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+ tan φd
dR
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
−∫ H2
H1
Z (h2 · P act · h2) dZ
= Rr′ sec2 φ
(∫ H2
H1
βRdZ
)
= −Rr′ tan φ sec3 φ
∫ H2
H1
(h1 · P act · h1) dZ, (10.72)
where we have used Eq. (10.69) to re-write the right-hand side. Note that
tan φd
dR
(R
∫ H2
H1
Z (k · P act · h1) dZ
)
= tan φd
dR
(R tan φ
∫ H2
H1
Z (h1 · P act · h1) dZ
)
= φ′ tan φ sec2 φR
∫ H2
H1
Z (h1 · P act · h1) dZ
+ tan2 φd
dR
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
). (10.73)
Eq. (10.72) can be re-written as
(1 + tan2 φ)d
dR
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
+ φ′ tan φ sec2 φR
∫ H2
H1
Z (h1 · P act · h1) dZ
=
∫ H2
H1
Z (h2 · P act · h2) dZ
− Rr′ tan φ sec3 φ
∫ H2
H1
(h1 · P act · h1) dZ. (10.74)
170
Since 1 + tan2 φ = sec2 φ = (cos φ)−2, we can re-write this as
d
dR
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
= cos2 φ
∫ H2
H1
Z (h2 · P act · h2) dZ
− (Rφ′ + Rr′ sec φ) tan φ
∫ H2
H1
Z (h1 · P act · h1) dZ. (10.75)
In summary, the balance laws for linear momentum and angular momentum for
the plate are
d
dR
(R sec2 φ
∫ H2
H1
(h1 · P act · h1) dZ
)=
∫ H2
H1
(h2 · P act · h2) dZ, (10.76)
and
d
dR
(R
∫ H2
H1
Z (h1 · P act · h1) dZ
)
= cos2 φ
∫ H2
H1
Z (h2 · P act · h2) dZ
− (Rφ′ + Rr′ sec φ) tan φ
∫ H2
H1
Z (h1 · P act · h1) dZ. (10.77)
We have already ensured that∫ H2
H1
(k · P act · h1) dZ + cos φ
∫ H2
H1
βRdZ
= k ·∫ H2
H1
(P act · h1) dZ + d
∫ H2
H1
βRdZ
(10.78)
is identically zero. One of the boundary conditions is that the analogous h1-
projection is zero at the boundary of the middle surface:
h1 ·∫ H2
H1
(P act · h1) dZ + d
∫ H2
H1
βRdZ
=
∫ H2
H1
(h1 · P act · h1) dZ − sin φ
∫ H2
H1
βRdZ
=
∫ H2
H1
(h1 · P act · h1) dZ + sin φ sec φ
∫ H2
H1
(k · P act · h1) dZ
=(1 + tan2 φ
) ∫ H2
H1
(h1 · P act · h1) dZ
= sec2 φ
∫ H2
H1
(h1 · P act · h1) dZ. (10.79)
171
Hence the boundary condition for stress is
sec2 φ
∫ H2
H1
(h1 · P act · h1) dZ
∣∣∣∣R∈Rmin,Rmax
= 0, (10.80)
where we consider Rmax in every case and Rmin only in the case of an annulus.
The boundary condition for moment is
∫ H2
H1
Z (P act · h1) dZ
∣∣∣∣R∈Rmin,Rmax
= 0. (10.81)
We will find a scalar equation that ensures that the h1- and k-projections are both
zero.
10.7 Applying the Constitutive Relation
We use the isotropic hyperelastic constitutive relation seen in Section 6.7 and Chap-
ter 9:
W (λ1, λ2, λ3) = µ
(1
2
(λ2
1 + λ22 + λ3
3 − 3)− ln J
)+ λ (J − 1− ln J) . (10.82)
Following the method outlined in Section 6.7, we find that the active portion of the
Cauchy stress tensor in the presence of growth is
T act =1
J
(µ
(λ2
1 a⊗ a + λ22 h2 ⊗ h2 + λ2
3 d⊗ d)
+ (λJ − (λ + µ)) I), (10.83)
where the λi are the “pseudo-stretches”
λ1 =r′ sec φ− Zφ′
γ1
, λ2 =r − Z sin φ
γ2R, λ3 = 1, (10.84)
and J = det A = λ1λ2λ3.
The differential equations Eqs. (10.76) and (10.77) are written in terms of P act =
(det F ) T act · F−T , but we need only certain projections of this tensor. Note that
F−T · h1 = (r′ sec φ− Zφ′)−1a, (10.85)
F−T · h2 =
(r − Z sin φ
R
)−1
h2. (10.86)
172
We have
P act · h1 = (det F ) T act · F−T · h1
= Jγ1γ2 T act · (r′ sec φ− Zφ′)−1a
= γ1γ2
(µ
γ21
(r′ sec φ− Zφ′) +λ
γ1γ2
(r′ sec φ− Zφ′)
− λ + µ
r′ sec φ− Zφ′
)a, (10.87)
where
det F = (det A) (det G) = (J) (γ1γ2) . (10.88)
A similar calculation shows that
P act · h2 (10.89)
= γ1γ2
(µ
γ22
(r − Z sin φ
R
)+
λ
γ1γ2γ3
(r − Z sin φ
R
)− λ + µ
(r − Z sin φ) /R
)h2.
The projections pertinent for the differential equations are
h1 · P act · h1 (10.90)
= cos φ
(µγ2
γ1
(r′ sec φ− φ′Z) + λ
(r − Z sin φ
R
)− (λ + µ) γ1γ2
r′ sec φ− Zφ′
),
h2 · P act · h2 (10.91)
=µγ1
γ2
(r − Z sin φ
R
)+ λ (r′ sec φ− Zφ′)− (λ + µ) γ1γ2
(r − Z sin φ) /R.
We also assume that the middle surface occupies the true middle of the plate, so
that H1 = −H and H2 = H for some H > 0. The Z-integrals then have the forms∫ H
−H
(h1 · P act · h1) dZ (10.92)
= cos φ
2H
(µγ2
γ1
r′ sec φ + λr
R
)− 2 (λ + µ) γ1γ2
φ′arctanh
(Hφ′
r′ sec φ
)
∫ H
−H
(h2 · P act · h2) dZ (10.93)
= 2H
(µγ1
γ2
r
R+ λr′ sec φ
)− 2 (λ + µ) γ1γ2R
sin φarctanh
(H sin φ
r
)
173
∫ H
−H
Z (h1 · P act · h1) dZ (10.94)
= − cos φ
2
3H3
(µγ2
γ1
φ′ + λsin φ
R
)
+2 (λ + µ) γ1γ2
(φ′)2
(r′ sec φ arctanh
(Hφ′
r′ sec φ
)−Hφ′
)
∫ H
−H
Z (h2 · P act · h2) dZ (10.95)
= −2
3H3
(µγ1
γ2
sin φ
R+ λφ′
)
− 2 (λ + µ) γ1γ2R
sin2 φ
(r arctanh
(H sin φ
r
)−H sin φ
)
We get a further simplification if we divide by µ:
1
µ
∫ H
−H
(h1 · P act · h1) dZ (10.96)
= cos φ
2H
(γ2
γ1
r′ sec φ + κr
R
)− 2 (1 + κ) γ1γ2
φ′arctanh
(Hφ′
r′ sec φ
)
1
µ
∫ H
−H
(h2 · P act · h2) dZ (10.97)
= 2H
(γ1
γ2
r
R+ κr′ sec φ
)− 2 (1 + κ) γ1γ2R
sin φarctanh
(H sin φ
r
)
1
µ
∫ H
−H
Z (h1 · P act · h1) dZ (10.98)
= − cos φ
2
3H3
(γ2
γ1
φ′ + κsin φ
R
)
+2 (1 + κ) γ1γ2
(φ′)2
(r′ sec φ arctanh
(Hφ′
r′ sec φ
)−Hφ′
)
174
1
µ
∫ H
−H
Z (h2 · P act · h2) dZ (10.99)
= −2
3H3
(γ1
γ2
sin φ
R+ κφ′
)
− 2 (1 + κ) γ1γ2R
sin2 φ
(r arctanh
(H sin φ
r
)−H sin φ
),
where κ = λ/µ. The differential equations have the same form if each stress and
moment is replaced by the same quantity divided by µ.
10.8 The Flat Plate
If the plate is un-buckled, then there is no moment, P act ·h1 has no Z-dependence,
and the radial Piola-Kirchhoff stress is radial only, i.e. it has no vertical component.
In the case of constant γ1 and γ2, the equations for the un-buckled plate can be
converted into a pair of autonomous ordinary differential equations, as detailed in
Antman and Negron-Marrero (1987). If we set
R = Rmaxes−1, (10.100)
τ =r
R, (10.101)
N =γ2
γ1
r′ + κr
R− (1 + κ)γ1γ2
r′. (10.102)
When τ and N are viewed as functions of the independent variable s, they satisfy
dτ
ds=
γ1
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)− τ, (10.103)
dN
ds=
γ1
γ2
τ +κγ1
2γ2
(N − κτ +
√(N − κτ)2 + 4(1 + κ)γ2
2
)
−(1 + κ)γ1γ2
τ−N. (10.104)
We consider an annulus that has zero radial stress at its inner and outer faces.
We can find numerically an initial value for τ such that the boundary conditions
n(s0) = 0 and n(s1) = 0. The functions n(s) and τ(s) correspond to functions N(R)
175
and T (R) via
N(R) = n
(1 + ln
(R
Rmax
)), T (R) = τ
(1 + ln
(R
Rmax
)). (10.105)
This correspondence also gives us a function r(R):
r(R) = RT (R) = Rτ
(1 + ln
(R
Rmax
)), (10.106)
which also provides a function r′(R).
10.9 Differential Equations for Bifurcation
Finding numerical solutions of the boundary-value problem for the buckled plate
has proved a daunting task. To demonstrate the existence of buckled solutions, we
consider perturbation about the flat configuration.
Eqs. (10.76) and (10.77) for the buckled plate, i.e. with stresses and moments
given by Eqs. (10.93)-(10.96), can be expressed in the form
1 0 0 0
0 1 0 0
0 0 a33 a34
0 0 a43 a44
︸ ︷︷ ︸A(R,r,φ,r′,φ′)
d
dR
r
φ
r′
φ′
=
r′
φ′
f3
f4
︸ ︷︷ ︸f(R,r,φ,r′,φ′)
, (10.107)
with
a33 = 2HR
(γ2 sec2 φ
γ1
+γ1γ2(1 + κ)
(r′)2 −H2 (φ′)2 cos2 φ
), (10.108)
a34 = 2R
γ1γ2(1 + κ) sec φ
(φ′)2 arctanh
(Hφ′ cos φ
r′
)
− Hγ1γ2(1 + κ)
r′φ′(1− H2(φ′)2 cos2 φ
(r′)2
) (10.109)
176
a43 = 2R
−γ1γ2(1 + κ)
(φ′)2 arctanh
(Hφ′ cos φ
r′
)
+Hγ1γ2(1 + κ) cos φ
r′φ′(1− H2(φ′)2 cos2 φ
(r′)2
) (10.110)
a44 = 2R
−H3γ2 cos φ
3γ1
+Hγ1γ2(1 + κ) cos φ
(φ′)2
+2γ1γ2(1 + κ) cos φ
(arctanh
(Hφ′ cos φ
r′
)r′ sec φ−Hφ′
)
(φ′)3
− Hγ1γ2(1 + κ) cos φ
(φ′)2(1− H2(φ′)2 cos2 φ
(r′)2
) , (10.111)
f3 = 2Rγ1γ2(1 + κ)arctanh
(H sin φ
r
)csc φ− 2Hκr sec φ
R
+ 2Hκr′ sec φ− 2H
(γ1r
Rγ2
+ κr′ sec φ
)
+ sec φ
2H
(κr
R+
γ2r′ sec φ
γ1
)−
2γ1γ2(1 + κ)arctanh(
Hφ′ cos φr′
)
φ
+2HRγ2r
′φ′ sec2 φ tan φ
γ1
+ R sec φ tan φ
2H
(κr
R+
γ2r′ sec φ
γ1
)−
2γ1γ2(1 + κ)arctanh(
Hφ′ cos φr′
)
φ′
φ′
+2HRγ1γ2(1 + κ)φ′ tan φ
r′(1− H2(φ′)2 cos2 φ
(r′)2
) (10.112)
177
f4 =2H3κ cos φ sin φ
3R
+ Rr′ tan φ
2H
(κr
R+
γ2r′ sec φ
γ1
)−
2γ1γ2(1 + κ)arctanh(
Hφ′ cos φr′
)
φ′
−2Rγ1γ2(1 + κ)arctanh
(Hφ′ cos φ
r′
)r′ tan φ
φ′
− 2
3H3κφ′ cos2 φ +
2HRγ1γ2(1 + κ) sin φ
1− H2(φ′)2 cos2 φ
(r′)2
− cos φ
2γ1γ2(1 + κ)(arctanh
(Hφ′ cos φ
r′
)r′ sec φ−Hφ′
)
(φ′)2
+2
3H3
(κ sin φ
R+
γ2φ′
γ1
)
− cos φ
−2Rγ1γ2(1 + κ) csc2 φ
(rarctanh
(H sin φ
r
)−H sin φ
)
− 2
3H3
(γ1 sin φ
Rγ2
+ κφ′)
(10.113)
The matrix A is invertible symbolically, so the equations can be expressed in
quasilinear form:
d
dR
r
φ
r′
φ′
= A−1f . (10.114)
We consider expressions r = r0 +εr1 and φ = εφ1, where r0 is the radius function
in Eq. (10.106), i.e. the solution of the boundary-value problem for the un-buckled
178
plate. Inserting these into Eq. (10.114), we have
d
dR
r0
0
r′0
0
+ εd
dR
r1
φ1
r′1
φ′1
= A−1f∣∣
r = r0
r′ = r′0φ = 0
φ′ = 0
+ ε∂A−1f
∂(r, φ, r′, φ)
∣∣∣∣ r = r0
r′ = r′0φ = 0
φ′ = 0
·
r1
φ1
r′1
φ′1
+O (ε2
). (10.115)
Equating terms with equal powers of ε, we are left with a system of linear equations
for (r1, φ1, r′1, φ
′1):
d
dR
r1
φ1
r′1
φ′1
=∂A−1f
∂(r, φ, r′, φ)
∣∣∣∣ r = r0
r′ = r′0φ = 0
φ′ = 0
·
r1
φ1
r′1
φ′1
. (10.116)
179
10.10 Linear Boundary Conditions for Bifurcation
We employ first-order expansions of the scalar coefficient of the radial stress in
Eq. (10.93):
1
µ
∫ H
−H
(a · P act · h1) dZ
=1
µ
∫ H
−H
(a · P act · h1) dZ
∣∣∣∣r = r0, r′ = r′0φ = 0, φ′ = 0
+∂
(1µ
∫ H
−H(a · P act · h1) dZ
)
∂(r, φ, r′, φ′)
∣∣∣∣∣∣ r = r0, r′ = r′0φ = 0, φ′ = 0
· ε
r1
φ1
r′1
φ′1
+O (ε2
)
= 2H
(γ2
γ1
r′0 + κr0
R− (1 + κ)γ1γ2
r′0
)
+ ε
(2Hκ
Rr1 +
(2Hγ2
γ1
+2H(1 + κ)γ1γ2
(r′0)2
)r′1
)+O (
ε2). (10.117)
At R = R0 = Rmin and R = R1 = Rmax, the radial stress of the flat annulus is zero,
so for i = 0, 1,
1
µ
∫ H
−H
(a · P act · h1) dZ
∣∣∣∣R=Ri
= ε
(2Hκ
Rr1 +
(2Hγ2
γ1
+2H(1 + κ)γ1γ2
(r′0)2
)r′1
)∣∣∣∣R=Ri
+O (ε2
). (10.118)
This provides one linear boundary condition for R = R0 and one for R = R1:
κ
Rr1 +
(γ2
γ1
+(1 + κ)γ1γ2
(r′0)2
)r′1
∣∣∣∣R=Ri
= 0. (10.119)
180
We perform the same expansion procedure for the moment in Eq. (10.95):
1
µ
∫ H
−H
(a · P act · h1) Z dZ
=1
µ
∫ H
−H
(a · P act · h1) Z dZ
∣∣∣∣ r = r0
r′ = r′0φ = 0
φ′ = 0
+∂
(1µ
∫ H
−H(a · P act · h1) Z dZ
)
∂(r, φ, r′, φ′)
∣∣∣∣∣∣ r = r0
r′ = r′0φ = 0
φ′ = 0
· ε
r1
φ1
r′1
φ′1
+O (ε2
)
= 0 +2H3ε
3
(κ
Rφ1 +
(γ2
γ1
+(1 + κ)γ1γ2
(r′0)2
)φ′1
)+O (
ε2). (10.120)
Recall that the un-buckled plate has identically zero moment. This provides linear
boundary conditions at R = R0 and R = R1:
κ
Rφ1 +
(γ2
γ1
+(1 + κ)γ1γ2
(r′0)2
)φ′1
∣∣∣∣R=Ri
= 0. (10.121)
10.11 Evidence of Bifurcation
We now have a system of four linear ordinary differential equations and four lin-
ear boundary conditions. The linear boundary conditions at R = R0 have a two-
dimensional vector space of solutions. Considering that the boundary conditions at
R = R0 can be expressed as
κR
γ2
γ1+ (1+κ)γ1γ2
(r′0)20 0
0 0 κR
γ2
γ1+ (1+κ)γ1γ2
(r′0)2
r1
r′1
φ1
φ′1
∣∣∣∣∣∣∣∣∣∣∣R=R0
=
0
0
, (10.122)
181
we see that the two-dimensional solution space at R = R0 is spanned by, for example,
r1
r′1
φ1
φ′1
∣∣∣∣∣∣∣∣∣∣∣R=R0
=
γ2
γ1+ (1+κ)γ1γ2
(r′0)2
− κR
0
0
∣∣∣∣∣∣∣∣∣∣∣R=R0
(10.123)
and
r1
r′1
φ1
φ′1
∣∣∣∣∣∣∣∣∣∣∣R=R0
=
0
0
γ2
γ1+ (1+κ)γ1γ2
(r′0)2
− κR
∣∣∣∣∣∣∣∣∣∣∣R=R0
(10.124)
Let u and v be solutions of Eq. (10.116) with the initial (R = R0) conditions
above. Then u and v have the forms
u1
u2
u3
u4
=
r1
r′1
0
0
and
v1
v2
v3
v4
=
0
0
φ1
φ′1
. (10.125)
Every linear combination αu + βv is a solution of the differential equations that
satisfies the linear boundary conditions Eq. (10.119) and (10.121) at R = R0.
We seek a linear combination of this form that also satisfies the linear boundary
conditions at R = R1. That is, we want
κR
γ2
γ1+ (1+κ)γ1γ2
(r′0)20 0
0 0 κR
γ2
γ1+ (1+κ)γ1γ2
(r′0)2
αu1
αu2
βv3
βv4
∣∣∣∣∣∣∣∣∣∣∣R=R1
=
0
0
. (10.126)
We can seek the existence of such a linear combination by re-writing this condition
as
κRu1 +
(γ2
γ1+ (1+κ)γ1γ2
(r′0)2
)u2 0
0 κRv1 +
(γ2
γ1+ (1+κ)γ1γ2
(r′0)2
)v2
∣∣∣∣∣∣R=R1
α
β
=
0
0
(10.127)
182
If the matrix has zero determinant, then there is a non-trivial solution (α, β)T of
this pair of linear equations at R = R1.
We know that if γ1 = γ2, then the growth tensor is a true deformation gradient,
and there is no residual stress in the un-buckled annulus. We fix a value for γ2 and
let γ1 vary until the determinant of the matrix in Eq. (10.127) is zero. In truth, we
seek the first γ1-value at which either of the non-zero entries in the matrix is zero.
10.12 Numerical Results
In all numerical results found so far, the first term to drop to zero is the (2, 2)-entry
of the matrix in Eq. (10.127). This means that we can choose a perturbation of
the moment alone. Further, the (2, 2)-entry reaches zero at some γ1 > γ2, which
indicates buckled solutions for the case in which incompatible growth corresponds
to azimuthal contraction.
In the following example, we set γ2 = 1.01, H = 0.01, R0 = 0.01, R1 = 1, and
κ = 1. As seen in Figures 10.2 and 10.3, as functions of γ1, the stress perturbation
(dependent on r1 alone) at R = R1 has no zeros, while the moment perturbation
(dependent on φ1 alone) at R = R1 has multiple zeros, all greater than γ2. The first
four zeros are found at γ1 = 1.01086, 1.01348, 1.01793, and 1.0242.
1.012 1.014 1.016 1.018 1.020 1.022 1.024Γ1
405
410
415
420
Κ r1HR1L
R1+Γ2
Γ1+HΚ + 1L Γ1 Γ2
r0HR1L2
r1¢HR1L
Figure 10.2: The stress perturbation (dependent on r1) at R = R1 has no zeros.
183
1.012 1.014 1.016 1.018 1.020 1.022 1.024Γ1
-50
50
100
150
Κ Φ1HR1L
R1+Γ2
Γ1+HΚ + 1L Γ1 Γ2
r0¢HR1L
2Φ1¢HR1L
Figure 10.3: The moment perturbation (dependent on φ1) at R = R1 has multiplezeros.
0.02 0.04 0.06 0.08 0.10H
1.02
1.04
1.06
1.08
1.10
HΓ1Lfirst critical value
Figure 10.4: The first bifurcation value of γ1, as a function of half-thickness H
10.13 Numerical Example: Buckling Near a Bifurcation Point
Iterating the numerical scheme described in Section 9.5 of Chapter 9, we compute
solutions of Eqs. (9.57)-(9.58) for the flat annulus until a flat solution satisfying the
boundary conditions is found. We call this solution r0. We use r0 in the matrix on
the right-hand side of Eq. (10.116). We also use r0(Rmin) and r′0(Rmin) to choose
values at Rmin for the perturbations r1 and r′1. Since we want r1 and r′1 to satisfy
184
0.02 0.04 0.06 0.08 0.10H
1.1
1.2
1.3
1.4
1.5
1.6
1.7HΓ1Lcritical value
Figure 10.5: The first three critical values of γ1, as functions of H
the boundary condition in Eq. (10.119), we set
r1(Rmin) =γ2
γ1
+(1 + κ)γ1γ2
(r′0(Rmin))2 , r′1(Rmin) = − κ
Rmin
. (10.128)
We also set
φ1(Rmin) = 0, φ′1(Rmin) = 0. (10.129)
With these as initial conditions, we integrate Eq. (10.116).
Fixing r0(Rmin) and r′0(Rmin), we seek numerically for values for φ and φ′ so that
the nonlinear boundary conditions
2H
(γ2
γ1
r′ sec φ + κr
Rmin
)− 2(1 + κ)γ1γ2
φ′arctanh
(Hφ′
r′ sec φ
)= 0 (10.130)
and
2H3
3
(γ2
γ1
φ′ + κsin φ
Rmin
)− 2(1 + κ)γ1γ2
(φ′)2
(r′ sec φarctanh
(Hφ′
r′ sec φ
)−Hφ′
)= 0
(10.131)
are satisfied with r = r0(Rmin), r′ = r′0(Rmin), φ = φ and φ′ = φ′. With these
initial conditions, we integrate Eqs. (10.76)-(10.77) numerically. We find very good
numerical satisfaction of the boundary condition at R = Rmax.
The numerical results for the two methods are compared. The numerical results
for the linear perturbation are scaled to agree as much as possible with the results of
185
integrating Eqs. (10.76)-(10.77). The numerical results of both methods are shown
together below.
In this example, we use parameters
γ2 = 1.01, γ2 =56026090
55424431≈ 1.01086, H = 0.01, Rmin = 0.01, Rmax = 1, κ = 1.
(10.132)
0.2 0.4 0.6 0.8 1.0 R
1.´10-6
1.5´10-6
2.´10-6
2.5´10-6
Φ
Figure 10.6: The buckling angle φ found from the two methods. Solid: nonlinear.Dashed: linear perturbation.
0.2 0.4 0.6 0.8 1.0 R
5.´10-7
1.´10-6
1.5´10-6
2.´10-6
Ζ
Figure 10.7: The middle surface height ζ found from the two methods. Solid:nonlinear. Dashed: linear perturbation.
186
0.2 0.4 0.6 0.8 1.0 R
-0.00006
-0.00005
-0.00004
-0.00003
-0.00002
-0.00001
1Μà-H
H Ha×Pact×h1L âZ
Figure 10.8: Thickness-integrated radial stress found from the two methods. Solid:nonlinear. Dashed: linear perturbation.
0.2 0.4 0.6 0.8 1.0 R
5.´10-12
1.´10-11
1.5´10-11
2.´10-11
1Μà-H
H Ha×Pact×h1LZ âZ
Figure 10.9: Thickness-integrated radial moment found from the two methods.Solid: nonlinear. Dashed: linear perturbation.
By the nature of the bifurcation analysis, the buckling found at each bifurcation
value is small. We present in the next section an example of much larger buckling.
10.14 Numerical Example: Buckling Due to Immersion-Preventing Growth
We employ the growth tensor
G = 1.9 h1 ⊗ h1 +(1.9− 0.9R2
)h2 ⊗ h2 + k ⊗ k. (10.133)
187
This leads to a new metric tensor on Ψ(B) whose associated Riemann-Christoffel
curvature tensor is nonzero, so this growth precludes isometric immersion of the
grown body.
We fix the parameter values
Rmin = 0.2, Rmax = 0.709, H = 0.01, κ = 1, (10.134)
and solve Eqs. (10.76)-(10.77) numerically. We search for initial conditions for which
the boundary conditions at R = Rmin and R = Rmax are satisfied simultaneously.
The results are shown in Figures 10.10-10.13.
0.3 0.4 0.5 0.6 0.7 R
-0.014
-0.012
-0.010
-0.008
-0.006
-0.004
-0.002
Hintegrated stressLΜ
Figure 10.10: Thickness-integrated radial (solid) and azimuthal (dashed) stresses
ζ(0.709) ≈ 0.064, which is 9% of the radius of the reference configuration of the
annulus. This buckling is visible in a plot of the middle surface. See Figure 10.13.
10.15 Conclusion
We have derived geometrically exact equations for an axisymmetric flat and buckled
Kirchhoff plates that has undergone axisymmetric incompatible growth. Through
perturbation about the final (grown and elastically distorted) flat configuration, we
have shown that the equations for the buckled configuration have solutions free of
applied tractions. Further, we have found in the model that an increase of the
plate’s thickness makes the plate more resistant to buckling. We have demonstrated
188
0.3 0.4 0.5 0.6 0.7 R
6.0
6.2
6.4
6.6
6.8
7.0
Φ
Figure 10.11: Buckling angle φ
0.3 0.4 0.5 0.6 0.7 R
0.01
0.02
0.03
0.04
0.05
0.06
Ζ
Figure 10.12: Middle surface height ζ
that incompatible growth can induce buckling with an amplitude several times the
thickness of the plate.
189
Figure 10.13: Top view of middle surface
190
CHAPTER 11
CONCLUSION
Our presentation of finite elasticity has a special emphasis on the functional form
of deformation in order to show that the standard kinematical conventions of fi-
nite deformation can be adapted to describe the geometric impact of incompatible
growth. We have demonstrated that importing growth into finite hyperelasticity via
the multiplicative decomposition F = A · G permits the use of the geometrically
exact theory of shells with just a change in the form of the first Piola-Kirchhoff
stress tensor. We have found examples in which the elastic response to incompat-
ible growth distorts a non-immersible grown body into a Euclidean body without
the assistance of applied tractions. In another example, the elastic response to
incompatible growth induced buckling of a plate free of applied tractions.
The next step in this enterprise should be numerical solution of Eqs. (10.76) and
(10.77) in the regions between critical values of γ1. Though we have demonstrated
the presence of buckled solutions, the size such buckling can reach is of interest.
To test the value of this approach against that of Efrati et al. (2007a), etc., this
theory should be applied Kirchhoff discs that are allowed to buckle azimuthally as
well as radially. The kinematics of such a project will be more complicated, but still
within the realm presented in chapter of Antman (1995). The greater difficulty will
be posed by the partial differential equations that will arise.
The examples presented here have not included a shell, which differs from a
plate only in that its middle surface is curved in the reference configuration. The
locomotive “gel caterpillar” mentioned in chapter 1 (Simonite (2009)), for example,
looks like a bent beam and would require a shell model. Except in cases of special
symmetry, models of shells will produce partial differential equations.
Even more advanced models for shell-like bodies have been suggested. For ex-
ample, DiCarlo et al. (2001) proposed a method for introducing thickness distension
191
into shells. Compared to the complexities coming from such extensions of the theory,
including incompatible growth in the fashion discussed here is hardly a hardship at
all.
Various theories for geometry-independent dynamics of growth have been pro-
posed (Epstein and Maugin (2000), DiCarlo and Quiligotti (2002), Lubarda and
Hoger (2002)). Thin-walled tubes dominate the explicit models of dynamical
growth (Lin and Taber (1995), Taber and Eggers (1996), Taber (1998a), Taber
and Humphrey (2001), Vandiver (2009)). One of the foundational difficulties of
the subject is the derivation of balance laws in residually stressed bodies, including
those that have already undergone incompatible growth and the subsequent elastic
response (Hoger (1997), Chen and Hoger (2000), Klarbring et al. (2007)).
We hope that our combination of incompatible growth and the geometrically
exact theory of shells provides a guide for the construction of models and for the
understanding of the interplay between three-dimensional Riemannian geometry and
elasticity.
192
APPENDIX A
PROOFS OF SOME RESULTS IN CHAPTER 2
Proposition A.0.1 The matrix [Eij] of dot-products Eij = Ei · Ej is the inverse
of the matrix [Eij] of dot-products Eij = Ei ·Ej.
Proof Recall that
det (abc) = a · (b× c) = c · (a× b) = b · (c× a) , (A.1)
which is non-zero if a, b, and c are linearly independent. In particular, the denom-
inator in the definition of Ei is independent of i. Let (ijk) and (`mn) be cyclic
permutations of (123). Then
Ei ·E` =Ej ×Ek
det (E1E2E3)· Em ×En
det (E1E2E3)
=(Ej ×Ek) · (Em ×En)
det [E1E2E3]2
=(Ej ·Em) (Ek ·En)− (Ej ·En) (Ek ·Em)
det (E1E2E3)2
=EjmEkn − EjnEkm
det (E1E2E3)2
=1
det (E1E2E3)2
∣∣∣∣∣∣Ejm Ejn
Ekm Ekn
∣∣∣∣∣∣. (A.2)
The 2×2 determinant in the last line is the i, j-entry of the adjugate matrix of [Eij].
The determinant det (E1E2E3) is related to the determinant of the matrix [Eij] as
193
follows.
det
E11 E12 E13
E21 E22 E23
E31 E32 E33
= det
ET1
ET2
ET3
(E1 E2 E3)
= det
ET1
ET2
ET3
det
(E1 E2 E3
)
= det(E1E2E3)2. (A.3)
We now have
Ei` =(adj[Ejm])i`
det[Ejm], (A.4)
where adj[Ejm] is the adjugate of the matrix [Ejm]. By Cramer’s Rule, this shows
that [Eij] is the inverse of the matrix [Eij] (Strang (1980), page 170).
Proposition A.0.2 Christoffel symbols of the second kind transform as
Γkji =
∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξiΓ`
mn +∂ξk
∂η`
∂2η`
∂ξj∂ξi. (A.5)
Proof
Γkji = Ek · ∂Ei
∂ξj
=
(∂ξk
∂η`E
`)· ∂ηm
∂ξj
∂
∂ηm
(∂ηn
∂ξiEn
)
=∂ξk
∂η`
∂ηm
∂ξj
(∂ηn
∂ξiE
` · ∂En
∂ηm+ E
` · En∂
∂ηm
(∂ηn
∂ξi
))
=∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξiΓ`
mn +∂ξk
∂η`
∂ηm
∂ξjδ`n
∂ξp
∂ηm
∂
∂ξp
(∂ηn
∂ξi
)
=∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξiΓ`
mn +∂ξk
∂η`δ`nδ
pj
∂2ηn
∂ξp∂ξi
=∂ξk
∂η`
∂ηm
∂ξj
∂ηn
∂ξiΓ`
mn +∂ξk
∂η`
∂2η`
∂ξj∂ξi(A.6)
194
Proposition A.0.3 The coefficients vi||j transform as the coefficients of a tensor
of type(11
).
Proof Recall that
vi||j =
∂vi
∂ξj+ Γi
jkvk. (A.7)
The first term transforms as
∂vi
∂ξj=
∂ηm
∂ξj
∂
∂ηm
(vn ∂ξi
∂ηn
)
=∂ηm
∂ξj
∂vn
∂ηm
∂ξi
∂ηn+
∂ηm
∂ξjvn ∂2ξi
∂ηm∂ηn, (A.8)
while the Christoffel symbol of the second kind transforms as
Γijk =
∂ξi
∂η`
∂ηm
∂ξj
∂ηn
∂ξkΓ`
mn +∂ξi
∂η`
∂2η`
∂ξj∂ξk. (A.9)
Γijkv
k =
(∂ξi
∂η`
∂ηm
∂ξj
∂ηn
∂ξkΓ`
mn +∂ξi
∂η`
∂2η`
∂ξj∂ξk
)(∂ξk
∂ηpvp
)
=∂ξi
∂η`
∂ηm
∂ξj
∂ηn
∂ξk
∂ξk
∂ηpΓ`
mn vp +∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηpvp
=∂ξi
∂η`
∂ηm
∂ξjδnp Γ`
mn vp +∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηpvp
=∂ξi
∂η`
∂ηm
∂ξjΓ`
mn vn +∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηpvp (A.10)
Hence, we have
∂vi
∂ξj+ Γi
jkvk =
∂ηm
∂ξj
∂vn
∂ηm
∂ξi
∂ηn+
∂ηm
∂ξjvn ∂2ξi
∂ηm∂ηn
+∂ξi
∂η`
∂ηm
∂ξjΓ`
mn vn +∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηpvp
=∂ηm
∂ξj
∂vn
∂ηm
∂ξi
∂ηn+
∂ξi
∂η`
∂ηm
∂ξjΓ`
mn vn
+∂ηm
∂ξjvn ∂2ξi
∂ηm∂ηn+
∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηpvp
=∂ξi
∂ηn
∂ηm
∂ξj
(∂vn
∂ηm+ Γn
mpvp
)
+
(∂ηm
∂ξj
∂2ξi
∂ηm∂ηn+
∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηn
)vn. (A.11)
195
We will show that the last line vanishes. Note that
∂ηm
∂ξj
∂2ξi
∂ηm∂ηn=
∂
∂ξj
(∂ξi
∂ηn
)(A.12)
∂ξi
∂η`
∂2η`
∂ξj∂ξk
∂ξk
∂ηn=
∂ξi
∂η`
∂ξk
∂ηn
∂
∂ξj
(∂η`
∂ξk
). (A.13)
We will view the partial derivatives as functions of (ξ1, ξ2, ξ3). Let
Aij =
∂ξj
∂ηj,
(A−1
)i
j=
∂ηi
∂ξj, (A.14)
as these collections of partial derivatives do form inverse matrices. Then the coeffi-
cient of vn is∂Ai
n
∂ξj+ Ai
`Akn
∂ (A−1)`k
∂ξj. (A.15)
Since (A−1)`p Ap
q = δ`q,
∂
∂ξj
(A−1
)`
pAp
q = 0
Apq
∂
∂ξj
(A−1
)`
p= − (
A−1)`
p
∂
∂ξjAp
q
(A−1
)q
kAp
q
∂
∂ξj
(A−1
)`
p= − (
A−1)q
k
(A−1
)`
p
∂
∂ξjAp
q
δpk
∂
∂ξj
(A−1
)`
p= − (
A−1)q
k
(A−1
)`
p
∂
∂ξjAp
q
∂
∂ξj
(A−1
)`
k= − (
A−1)q
k
(A−1
)`
p
∂
∂ξjAp
q (A.16)
The coefficient of vn can be written as
∂Ain
∂ξj− Ai
`Akn
(A−1
)q
k
(A−1
)`
p
∂
∂ξjAp
q
=∂Ai
n
∂ξj− δi
p δqn
∂
∂ξjAp
q
=∂Ai
n
∂ξj− ∂Ai
n
∂ξj
= 0. (A.17)
The result is that
∂vi
∂ξj+ Γi
jkvk =
∂ξi
∂ηn
∂ηm
∂ξj
(∂vn
∂ηm+ Γn
mpvp
). (A.18)
196
APPENDIX B
SYMMETRY OF P · F T
In many introductions, such as Gurtin (1981) and Ogden (1984), this is derived
in terms of the Cauchy stress. We follow the derivation in Section X11.7 of Antman
(1995).
With the first Piola-Kirchhoff stress known, Eq. 5.10 can be written as
∮
∂(XΨ)(B)
χ× (P · ν) dS +
∫
(XΨ)(B)
(χ× f − ρ0χ× χtt) dV = 0, (B.1)
where ν is the outward-pointing unit normal to the surface. This holds for all
well-behaved subsets Ω ⊂ Ψ(B):
∮
∂X(Ω)
χ× (P · ν) dS +
∫
X(Ω)
(χ× f − ρ0χ× χtt) dV = 0. (B.2)
To localize the balance of angular momentum, we must convert the surface in-
tegral to a volume integral. However, χ × (P · ν) is not the flux of a vector field
through the surface, so the standard Divergence Theorem for vector fields is not
applicable. For a fixed vector b, though, b · P is a vector field, and b · P · ν is the
flux of that vector field. We can apply the Divergence Theorem to such vector fields.
We take the dot-product of this equation with a×b, where a and b are constant
vectors.
(a× b) · (χ× (P · ν)) = (a · χ) (b · P · ν)− (a · P · ν) (b · χ) (B.3)
= b · ((P · ν)⊗ χ) · a− a · ((P · ν)⊗ χ) · b= ((P · ν)⊗ χ) : (a⊗ b− b⊗ a) (B.4)
(a× b) · (χ× f) = (f ⊗ χ) : (a⊗ b− b⊗ a) (B.5)
(a× b) · (χ× χtt) = (χtt ⊗ χ) : (a⊗ b− b⊗ a) (B.6)
197
DIV ((a · χ) (b · P )) = (b · P ) ·GRAD (a · χ) + (a · χ) DIV (b · P )
= (b · P ) · (a · F ) + (a · χ) (b ·DIVP )
= b · (P · F T + (DIVP )⊗ χ) · a
=(P · F T + (DIVP )⊗ χ
): a⊗ b (B.7)
DIV ((b · χ) (a · P )) =(P · F T + (DIVP )⊗ χ
): b⊗ a (B.8)
∮
∂X(Ω)
(a× b) · (χ× (P · ν)) dS
=
∮
∂X(Ω)
((a · χ) (b · P · ν)− (a · P · ν) (b · χ)) dS
=
∫
X(Ω)
(DIV ((a · χ) (b · P ))−DIV ((b · χ) (a · P ))) dV
=
∫
X(Ω)
(P · F T + (DIVP )⊗ χ
): (a⊗ b− b⊗ a) dV (B.9)
The cross-product of Eq. (B.1) with a× b can now be written as∫
X(Ω)
(P · F T + (DIVP + f − ρ0χtt)⊗ χ
)dV : (a⊗ b− b⊗ a) = 0 (B.10)
for all fixed vectors a and b. By the balance of linear momentum as given in
Eq. (5.18),
DIVP + f − ρ0χtt ≡ 0, (B.11)
so we have ∫
X(Ω)
P · F T dV : (a⊗ b− b⊗ a) = 0 (B.12)
Since a and b are arbitrary,∫
X(Ω)
P · F T dV : B = 0 (B.13)
for each skew-symmetric Lagrangian tensor B. By the near-arbitrariness of the set
Ω, we conclude that almost every pointwise value of P ·F T is symmetric: P ·F T =
F · P T .
198
APPENDIX C
POLAR DECOMPOSITION OF A TWO-POINT TENSOR
The existence of a polar decomposition of an invertible linear transformation is
frequently used in finite elasticity.
Theorem C.0.4 If F is an arbitrary linear transformation on a finite-dimensional
inner-product space, then there is a uniquely determined positive semi-definite linear
transformation U and an isometry R such that F = R ·U. If F is invertible, then
R is also unique.
This is quoted, with a slight change of notation, from Halmos (1958). Given this
result, elementary linear algebra can be used to show that if the linear transformation
F is invertible, then U is positive-definite. It can also be shown that if F is a linear
transformation on a real inner-product space with positive determinant, then R is
an orthogonal matrix, i.e. RT ·R = I, the identity.
The proof involves the properties of FT ·F. In the case of a real linear transfor-
mation F with positive determinant, then FT ·F is symmetric and positive-definite,
and U is the unique positive-definite symmetric square root of FT · F.
It should be noted that this theorem holds for a linear transformation from an
inner-produce space into the same inner-product space, yet it is the deformation
gradient, a two-point tensor, that is most frequently subjected to the polar decom-
position in finite elasticity.
Each deformation gradient is a map from the tangent space at X ∈ (X Ψ)(B)
to the tangent space at the deformed point x = χ(X), and it is most often expressed
initially with respect to the dual basis Ej at X and the coordinate basis ei at x:
F =∂ζ i
∂ξjei ⊗Ej, ei =
∂x
∂ζ i, Ej dual to Ej =
∂X
∂ξj.
199
If we follow the proof of the existence of the polar decomposition, we consider
F T · F :
F T · F =
(∂ζk
∂ξ`E` ⊗ ek
)·(
∂ζ i
∂ξjei ⊗Ej
)
= eki∂ζk
∂ξ`
∂ζ i
∂ξjE` ⊗Ej.
F T · F is symmetric and positive-definite and maps from the tangent space at
X into that same tangent space, i.e. it is Lagrangian. However, F T · F actually
maps tangent vectors Ej (contravariant) to cotangent vectors E` (covariant). At a
point in E3, cotangent vectors and tangent vectors are of the same kind, but they
have opposite transformation properties and correspond to very different objects in
Ψ(B), the space of coordinates. F T · F can be expressed as a map from tangent
vectors to tangent vectors as follows:
F T · F = eki∂ζk
∂ξ`
∂ζ i
∂ξjE` ⊗Ej = eki
∂ζk
∂ξ`
∂ζ i
∂ξjE`m Em ⊗Ej.
Since F T · F is a symmetric, positive-definite, Lagrangian tensor, it has a sym-
metric, positive-definite Lagrangian square root. It would seem natural to let U be
this Lagrangian square root. However, Theorem 6.4.1 states that a response func-
tion P (the function that assigns values of the first Piola-Kirchhoff stress tensor)
must satisfy
P (F ) = R · P (U ),
where F = R · U is the polar decomposition. If F is a two-point tensor and U is
Lagrangian, they cannot both be used as arguments of P . It would seem that U
must be a two-point square root. But it is not clear in what sense a two-point tensor
can be symmetric. In addition, the commonly-used notation U 2 is meaningless if
U is a two-point tensor:
U ·U =(U i
j ei ⊗EJ) · (Uk
` ek ⊗E`)
is undefined, as Ej and ek belong to different tangent spaces, so the dot-product
Ej · ek is undefined.
200
APPENDIX D
MATHEMATICA CODE FOR THE
RIEMANN-CHRISTOFFEL CURVATURE TENSOR
The following Mathematica commands compute the Riemann-Christoffel curvature
tensor for metric tensor presented in Eq. (4.21). In our experience, Mathematica
simplifies expressions more rapidly when computing the coefficients
Rqijk =∂Γikq
∂ξj− ∂Γijq
∂ξk+ Γp
ijΓkqp − ΓpikΓjqp,
where Γijq are the Christoffel symbols of the first kind:
Γijq =1
2
(∂Eiq
∂ξj+
∂Ejq
∂ξi− ∂Eij
∂ξq
).
and Γkij are the Christoffel symbols of the second kind:
Γkij = EkqΓijq.
We use the symbol g for the metric tensor, as E is protected for other uses in
Mathematica.
(* Define cylindrical coordinates. *)
coords = R,Θ,Z(* Define the metric tensor with respect to basis dual to coordinate basis. *)
g[R , Θ , Z ] :=
(r′[R])2 + r[R]2θ(1,0)[R, Θ]2, r[R]2θ(1,0)[R, Θ]θ(0,1)[R, Θ], 0
,r[R]2θ(1,0)[R, Θ]θ(0,1)[R, Θ], r[R]2θ(0,1)[R, Θ]2, 0
,
0, 0, 1
201
(* Define inverse of metric tensor. *)
ginv[R , Θ , Z ] := Simplify[Inverse[g[R, Θ, Z]]]
(* Define Christoffel symbols of the first kind. *)
Γfirst[i , j , q , R , Θ , Z ] =1
2(D[g[R, Θ, Z][[i]][[q]], coords[[j]]]
+ D[ g[R, Θ, Z][[j]][[q]], coords[[i]] ]
− D[ g[R, Θ, Z][[i]][[j]], coords[[q]] ])
(* Define Christoffel symbols of the second kind. The last integer index is the
raised index. *)
Γsecond[i , j , q , R , Θ , Z ]
:= Sum[ ginv[R, Θ, Z][[k]][[q]] ∗ Γfirst[i, j, q, R, Θ, Z], q, 1, 3 ]
(* Define the Riemann-Christoffel curvature tensor. *)
Riem[q , i , j , R , Θ , Z ]
= D[ Γfirst[i, k, q, R, Θ, Z], coords[[j]] ]
− D[ Γfirst[i, j, q, R, Θ, Z], coords[[k]] ]
+ Sum[ Γsecond[i, j, p, R, Θ, Z] ∗ Γfirst[k, q, p, R, Θ, Z], p, 1, 3 ]
− Sum[ Γsecond[i, k, p, R, Θ, Z] ∗ Γfirst[j, q, p, R, Θ, Z], p, 1, 3 ]
(* Print out the components of the Riemann-Christoffel curvature tensor. *)
For[q = 1, m < 4, m++,
For[i = 1, i < 4, i++,
For[j = 1, j < 4, j++,
For[k = 1, k < 4, k++,
Print[”Riem[”, q, ”,”, i, ”,”, j, ”,”, k, ”,R,Θ,Z] = ”,Riem[q,i,j,k,R,Θ,Z]]
]
]
]
]
202
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