Functional and Structured Tensor Analysis for Engineers A casual (intuition-based) introduction to vector and tensor analysis with reviews of popular notations used in contemporary materials modeling R. M. Brannon University of New Mexico, Albuquerque Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. Contact author at rmbrann@sandia.gov UNM BOOK DRAFT September 4, 2003 5:21 pm NOTE: When using Adobe’s “acrobat reader” to view this document, the page numbers in acrobat will not coincide with the page numbers shown at the bottom of each page of this document. Note to draft readers: The most useful textbooks are the ones with fantastic indexes. The book’s index is rather new and still under construction. It would really help if you all could send me a note whenever you discover that an important entry is miss- ing from this index. I’ll be sure to add it. This work is a community effort. Let’s try to make this document helpful to others. FUNCTIONAL AND STRUCTURED TENSOR ANALYSIS FOR ENGINEERS A casual (intuition-based) introduction to vector and tensor analysis with reviews of popular notations used in contemporary materials modeling Rebecca M. Brannon † † University of New Mexico Adjunct professor rmbrann@sandia.gov Abstract Elementary vector and tensor analysis concepts are reviewed in a manner that proves useful for higher-order tensor analysis of anisotropic media. In addition to reviewing basic matrix and vector analysis, the concept of a tensor is cov- ered by reviewing and contrasting numerous different definition one might see in the literature for the term “tensor.” Basic vector and tensor operations are provided, as well as some lesser-known operations that are useful in materials modeling. Considerable space is devoted to “philosophical” discussions about relative merits of the many (often conflicting) tensor notation systems in popu- lar use. ii iii Acknowledgments An indeterminately large (but, of course, countable) set of people who have offered advice, encouragement, and fantastic suggestions throughout the years that I’ve spent writing this document. I say years because the seeds for this document were sown back in 1986, when I was a co-op student at Los Alamos National Laboratories, and I made the mistake of asking my supervisor, Norm Johnson, “what’s a tensor?” His reply? “read the appendix of R.B. “Bob” Bird’s book, Dynamics of Polymeric Liquids. I did — and got hooked. Bird’s appendix (which has nothing to do with polymers) is an outstanding and succinct summary of vector and tensor analysis. Reading it motivated me, as an under- graduate, to take my first graduate level continuum mechanics class from Dr. H.L. “Buck” Schreyer at the University of New Mexico. Buck Schreyer used multiple underlines beneath symbols as a teaching aid to help his students keep track of the different kinds of strange new objects (tensors) appearing in his lectures, and I have adopted his notation in this document. Later taking Buck’s beginning and advanced finite element classes further improved my command of matrix analysis and partial differential equations. Buck’s teach- ing pace was fast, so we all struggled to keep up. Buck was careful to explain that he would often cover esoteric subjects principally to enable us to effectively read the litera- ture, though sometimes merely to give us a different perspective on what we had already learned. Buck armed us with a slew of neat tricks or fascinating insights that were rarely seen in any publications. I often found myself “secretly” using Buck’s tips in my own work, and then struggling to figure out how to explain how I was able to come up with these “miracle instant answers” — the effort to reproduce my results using conventional (better known) techniques helped me learn better how to communicate difficult concepts to a broader audience. While taking Buck’s continuum mechanics course, I simulta- neously learned variational mechanics from Fred Ju (also at UNM), which was fortunate timing because Dr. Ju’s refreshing and careful teaching style forced me to make enlighten- ing connections between his class and Schreyer’s class. Taking thermodynamics from A. Razanni (UNM) helped me improve my understanding of partial derivatives and their applications (furthermore, my interactions with Buck Schreyer helped me figure out how gas thermodynamics equations generalized to the solid mechanics arena). Following my undergraduate experiences at UNM, I was fortunate to learn advanced applications of con- tinuum mechanics from my Ph.D advisor, Prof. Walt Drugan (U. Wisconsin), who intro- duced me to even more (often completely new) viewpoints to add to my tensor analysis toolbelt. While at Wisconsin, I took an elasticity course from Prof. Chen, who was enam- oured of doing all proofs entirely in curvilinear notation, so I was forced to improve my abilities in this area (curvilinear analysis is not covered in this book, but it may be found in a separate publication, Ref. [6]. A slightly different spin on curvilinear analysis came when I took Arthur Lodge’s “Elastic Liquids” class. My third continuum mechanics course, this time taught by Millard Johnson (U. Wisc), introduced me to the usefulness of “Rossetta stone” type derivations of classic theorems, done using multiple notations to make them clear to every reader. It was here where I conceded that no single notation is superior, and I had better become darn good at them all. At Wisconsin, I took a class on Greens functions and boundary value problems from the noted mathematician R. Dickey, who really drove home the importance of projection operations in physical applications, and instilled in me the irresistible habit of examining operators for their properties and iv Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. classifying them as outlined in our class textbook [12]; it was Dickey who finally got me into the habit of looking for analogies between seemingly unrelated operators and sets so that my strong knowledge. Dickey himself got sideswiped by this habit when I solved one of his exam questions by doing it using a technique that I had learned in Buck Schreyer’s continuum mechanics class and which I realized would also work on the exam question by merely re-interpreting the vector dot product as the inner product that applies for continu- ous functions. As I walked into my Ph.D. defense, I warned Dickey (who was on my com- mittee) that my thesis was really just a giant application of the projection theorem, and he replied “most are, but you are distinguished by recognizing the fact!” Even though neither this book nor very many of my other publications (aside from Ref. [6], of course) employ curvilinear notation, my exposure to it has been invaluable to lend insight to the relation- ship between so-called “convected coordinates” and “unconvected reference spaces” often used in materials modeling. Having gotten my first exposure to tensor analysis from read- ing Bird’s polymer book, I naturally felt compelled to take his macromolecular fluid dynamics course at U. Wisc, which solidified several concepts further. Bird’s course was immediately followed by an applied analysis course, taught by ____, where more correct “mathematician’s” viewpoints on tensor analysis were drilled into me (the textbook for this course [17] is outstanding, and don’t be swayed by the fact that “chemical engineer- ing” is part of its title — the book applies to any field of physics). These and numerous other academic mentors I’ve had throughout my career have given me a wonderfully bal- anced set of analysis tools, and I wish I could thank them enough. For the longest time, this “Acknowledgement” section said only “Acknowledgements to be added. Stay tuned ” Assigning such low priority to the acknowledgements section was a gross tactical error on my part. When my colleagues offered assistance and sugges- tions in the earliest days of error-ridden rough drafts of this book, I thought to myself “I should thank them in my acknowledgements section.” A few years later, I sit here trying to recall the droves of early reviewers. I remember contributions from Glenn Randers-Pher- son because his advice for one of my other publications proved to be incredibly helpful, and he did the same for this more elementary document as well. A few folks (Mark Chris- ten, Allen Robinson, Stewart Silling, Paul Taylor, Tim Trucano) in my former department at Sandia National Labs also came forward with suggestions or helpful discussions that were incorporated into this book. While in my new department at Sandia National Labora- tories, I continued to gain new insight, especially from Dan Segalman and Bill Scherz- inger. Part of what has driven me to continue to improve this document has been the numer- ous encouraging remarks (approximately one per week) that I have received from researchers and students all over the world who have stumbled upon the pdf draft version of this document that I originally wrote as a student’s guide when I taught Continuum Mechanics at UNM. I don’t recall the names of people who sent me encouraging words in the early days, but some recent folks are Ricardo Colorado, Vince Owens, Dave Dooli- nand Mr. Jan Cox. Jan was especially inspiring because he was so enthusiastic about this work that he spent an entire afternoon disscussing it with me after a business trip I made to his home city, Oakland CA. Even some professors [such as Lynn Bennethum (U. Colo- rado), Ron Smelser (U. Idaho), Tom Scarpas (TU Delft), Sanjay Arwad (JHU), Kaspar William (U. Colorado), Walt Gerstle (U. New Mexico)] have told me that they have v directed their own students to the web version of this document as supplemental reading. In Sept. 2002, Bob Cain sent me an email asking about printing issues of the web draft; his email signature had the Einstein quote that you now see heading Chapter 1 of this document. After getting his permission to also use that quote in my own document, I was inspired to begin every chapter with an ice-breaker quote from my personal collec- tion. I still need to recognize the many folks who have sent helpful emails over the last year. Stay tuned. vi Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. Contents Acknowledgments iii Preface xv Introduction 1 STRUCTURES and SUPERSTRUCTURES 2 What is a scalar? What is a vector? 5 What is a tensor? 6 Examples of tensors in materials mechanics 9 The stress tensor 9 The deformation gradient tensor 11 Vector and Tensor notation — philosophy 12 Terminology from functional analysis 14 Matrix Analysis (and some matrix calculus) 21 Definition of a matrix 21 Component matrices associated with vectors and tensors (notation explanation) 22 The matrix product 22 SPECIAL CASE: a matrix times an array 22 SPECIAL CASE: inner product of two arrays 23 SPECIAL CASE: outer product of two arrays 23 EXAMPLE: 23 The Kronecker delta 25 The identity matrix 25 Derivatives of vector and matrix expressions 26 Derivative of an array with respect to itself 27 Derivative of a matrix with respect to itself 28 The transpose of a matrix 29 Derivative of the transpose: 29 The inner product of two column matrices 29 Derivatives of the inner product: 30 The outer product of two column matrices 31 The trace of a square matrix 31 Derivative of the trace 31 The matrix inner product 32 Derivative of the matrix inner product 32 Magnitudes and positivity property of the inner product 33 Derivative of the magnitude 34 Norms 34 Weighted or “energy” norms 35 Derivative of the energy norm 35 The 3D permutation symbol 36 The ε-δ (E-delta) identity 36 The ε-δ (E-delta) identity with multiple summed indices 38 Determinant of a square matrix 39 More about cofactors 42 Cofactor-inverse relationship 43 vii Derivative of the cofactor 44 Derivative of a determinant (IMPORTANT) 44 Rates of determinants 45 Derivatives of determinants with respect to vectors 46 Principal sub-matrices and principal minors 46 Matrix invariants 46 Alternative invariant sets 47 Positive definite 47 The cofactor-determinant connection 48 Inverse 49 Eigenvalues and eigenvectors 49 Similarity transformations 51 Finding eigenvectors by using the adjugate 52 Eigenprojectors 53 Finding eigenprojectors without finding eigenvectors. 54 Vector/tensor notation 55 “Ordinary” engineering vectors 55 Engineering “laboratory” base vectors 55 Other choices for the base vectors 55 Basis expansion of a vector 56 Summation convention — details 57 Don’t forget what repeated indices really mean 58 Further special-situation summation rules 59 Indicial notation in derivatives 60 BEWARE: avoid implicit sums as independent variables 60 Reading index STRUCTURE, not index SYMBOLS 61 Aesthetic (courteous) indexing 62 Suspending the summation convention 62 Combining indicial equations 63 Index-changing properties of the Kronecker delta 64 Summing the Kronecker delta itself 69 Our (unconventional) “under-tilde” notation 69 Tensor invariant operations 69 Simple vector operations and properties 71 Dot product between two vectors 71 Dot product between orthonormal base vectors 72 A “quotient” rule (deciding if a vector is zero) 72 Deciding if one vector equals another vector 73 Finding the i-th component of a vector 73 Even and odd vector functions 74 Homogeneous functions 74 Vector orientation and sense 75 Simple scalar components 75 Cross product 76 Cross product between orthonormal base vectors 76 Triple scalar product 78 viii Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. Triple scalar product between orthonormal RIGHT-HANDED base vectors 79 Projections 80 Orthogonal (perpendicular) linear projections 80 Rank-1 orthogonal projections 82 Rank-2 orthogonal projections 83 Basis interpretation of orthogonal projections 83 Rank-2 oblique linear projection 84 Rank-1 oblique linear projection 85 Degenerate (trivial) Rank-0 linear projection 85 Degenerate (trivial) Rank-3 projection in 3D space 86 Complementary projectors 86 Normalized versions of the projectors 86 Expressing a vector as a linear combination of three arbitrary (not necessarily orthonormal) vectors 88 Generalized projections 90 Linear projections 90 Nonlinear projections 90 The vector “signum” function 90 Gravitational (distorted light ray) projections 91 Self-adjoint projections 91 Gram-Schmidt orthogonalization 92 Special case: orthogonalization of two vectors 93 The projection theorem 93 Tensors 95 Analogy between tensors and other (more familiar) concepts 96 Linear operators (transformations) 99 Dyads and dyadic multiplication 103 Simpler “no-symbol” dyadic notation 104 The matrix associated with a dyad 104 The sum of dyads 105 A sum of two or three dyads is NOT (generally) reducible 106 Scalar multiplication of a dyad 106 The sum of four or more dyads is reducible! (not a superset) 107 The dyad definition of a second-order tensor 107 Expansion of a second-order tensor in terms of basis dyads 108 Triads and higher-order tensors 110 Our V m n tensor “class” notation 111 Comment 114 Tensor operations 115 Dotting a tensor from the right by a vector 115 The transpose of a tensor 115 Dotting a tensor from the left by a vector 116 Dotting a tensor by vectors from both sides 117 Extracting a particular tensor component 117 Dotting a tensor into a tensor (tensor composition) 117 Tensor analysis primitives 119 [...]... product Fourth-order tensor inner product Fourth-order Sherman-Morrison formula Higher-order tensor inner product Self-defining notation The magnitude of a tensor or a vector Useful inner product identities Distinction between an Nth-order tensor and an Nth-rank tensor Fourth-order oblique tensor projections Leafing and palming operations... Symmetric and skew-symmetric tensors 155 Positive definite tensors 156 Faster way to check for positive definiteness 156 Positive semi-definite 157 Negative definite and negative semi-definite tensors 157 Isotropic and deviatoric tensors 158 Tensor operations 159 Second-order tensor inner product 159 ix A NON-recommended scalar-valued... Coordinate/basis transformations Change of basis (and coordinate transformations) EXAMPLE Definition of a vector and a tensor Basis coupling tensor Tensor (and Tensor function) invariance What’s the difference between a matrix and a tensor? Example of a “scalar rule” that satisfies tensor invariance Example of a “scalar rule” that violates tensor invariance... may be reproduced for profit T AFT DR ann ca Br Rebec July 11, 2003 1:03 pm Preface on xvi Copyright is reserved Individual copies may be made for personal use No part of this document may be reproduced for profit DRAF September 4, 2003 5:24 pm Introduction Rebec FUNCTIONAL AND STRUCTURED TENSOR ANALYSIS FOR ENGINEERS: a casual (intuition-based) introduction to vector and tensor analysis with reviews... Vector, tensor, and matrix analysis are subsets of a more general area of study called functional analysis One purpose of this book is to specialize several overly-general results from functional analysis into forms that are the more convenient for “real world” engineering applications where generalized abstract formulas or notations are not only not necessary, but also damned distracting Functional analysis. .. complementary projection tensors 143 Self-adjoint (orthogonal) projectors 143 Non-self-adjoint (oblique) projectors 144 Generalized complementary projectors 145 More Tensor primitives 147 Tensor properties 147 Orthogonal (unitary) tensors 148 Tensor associated with the cross product 151 Cross-products in left-handed and general bases...Three kinds of vector and tensor notation 119 REPRESENTATION THEOREM for linear forms 122 Representation theorem for vector-to-scalar linear functions 123 Advanced Representation Theorem (to be read once you learn about higher-order tensors and the Vmn class notation) 124 Finding the tensor associated with a linear function 125 Method... Quantities such as A or T with two ˜under-tildes are second˜ ˜ order tensors In general, the number of under-tildes beneath a symbol indicates to you the order of that tensor (for this reason, scalars are sometimes called zeroth-order tensors and vectors are called first-order tensors) Occasionally, we will want to make statements that apply equally well to tensors of any order In that case, we might... of the deformation gradient tensor Of course, this is only a qualitative description of the deformation gradient tensor A more classical (and quantified) definition of the deformation gradient tensor starts with the assertion that each point x in the currently deformed ˜ body must have come from some unique initial location X in the initial undeformed refer˜ ence configuration, you can therefore claim... meaning of the tensor too (i.e., how it shows how squares deform to parallelepipeds) All that is needed to determine the components of this (or any) tensor is knowledge of how that transformation changes any three linearly independent vectors Vector and Tensor notation — philosophy This section may be skipped You may go directly to page 21 without loss Tensor notation unfortunately remains non-standardized, . community effort. Let’s try to make this document helpful to others. FUNCTIONAL AND STRUCTURED TENSOR ANALYSIS FOR ENGINEERS A casual (intuition-based) introduction to vector and tensor analysis. made for personal use. No part of this document may be reproduced for profit. FUNCTIONAL AND STRUCTURED TENSOR ANALYSIS FOR ENGINEERS: a casual (intuition-based) introduction to vector and tensor. Functional and Structured Tensor Analysis for Engineers A casual (intuition-based) introduction to vector and tensor analysis with reviews of popular notations