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UNM BOOK DRAFT September 4, 2003 5:21 pm 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 [email protected] 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 [email protected] 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 covered 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 popular use. ii 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 undergraduate, 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 teaching 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 literature, 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 continuum mechanics from my Ph.D advisor, Prof. Walt Drugan (U. Wisconsin), who introduced 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 iii 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 relationship between so-called “convected coordinates” and “unconvected reference spaces” often used in materials modeling. Having gotten my first exposure to tensor analysis from reading 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 engineering” 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 suggestions 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 numerous 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 iv Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. 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. v 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 vi Copyright is reserved. Individual copies may be made for personal use. No part of this document may be reproduced for profit. 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 vii 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 n tensor “class” notation........................................................................ 111 m 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 viii Copyright is reserved. 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