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 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 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 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 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 iv Copyright is reserved.