Finite-Deformation Modeling of Elastodynamics and Smart Materials with Nonlinear Electro-Magneto-Elastic Coupling Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Robert Lindsey Lowe, B.S., M.S. Graduate Program in Mechanical Engineering The Ohio State University 2015 Dissertation Committee: Prof. Sheng-Tao John Yu, Advisor Prof. Marcelo J. Dapino Prof. Daniel A. Mendelsohn Prof. Amos Gilat Dr. Kelly S. Carney c Copyright by Robert Lindsey Lowe 2015 Abstract Eulerian formulations of the equations of finite-deformation solid dynamics are ideal for numerical implementation in modern high-resolution shock-capturing schemes. These powerful numerical techniques – traditionally employed in unsteady compress- ible flow applications – are becoming increasingly popular in the computational solid mechanics community. Their primary appeal is an exceptional ability to capture the evolution and interaction of nonlinear traveling waves. Currently, however, Eulerian models for the nonlinear dynamics of rods, beams, plates, membranes, and other elastic structures are currently unavailable in the literature. The need for these reduced-order (1-D and 2-D) Eulerian structural models moti- vates the first part of this dissertation, where a comprehensive perturbation theory is used to develop a 1-D Eulerian model for nonlinear waves in elastic rods. The leading- order equations in the perturbation formalism are (i) verified using a control-volume analysis, (ii) linearized to recover a classical model for longitudinal waves in ultrasonic horns, and (iii) solved numerically using the novel space-time Conservation Element and Solution Element (CESE) method for first-order hyperbolic systems. Numeri- cal simulations of several benchmark problems demonstrate that the CESE method effectively captures shocks, rarefactions, and contact discontinuities. ii The second part of this dissertation focuses on another emerging area of finite- deformation mechanics: magnetoelectric polymer composites (MEPCs). A distin- guishing feature of MEPCs is the tantalizing ability to electrically control their mag- netization, or, conversely, magnetically control their polarization. Leveraging this magnetoelectric coupling could potentially impact numerous technologies, including information storage, spintronics, sensing, actuation, and energy harvesting. Most of the research on MEPCs to date, however, has focused on optimizing the magnitude of the magnetoelectric coupling through iterative design. Substantially less activity has occurred in the way of mathematical modeling and experimental characterization at finite strains, which are needed to advance fundamental understanding of MEPCs and encourage their technological implementation. The aforementioned need motivates the second part of this dissertation, where a finite-strain theoretical framework is developed for modeling soft magnetoelectric composites. Finite deformations, electro-magneto-elastic coupling, and material non- linearities are incorporated into the model. A particular emphasis is placed on the development of tractable constitutive equations to facilitate material characteriza- tion in the laboratory. Accordingly, a catalogue of free energies and constitutive equations is presented, each employing a different set of independent variables. The ramifications of invariance, angular momentum, incompressibility, and material sym- metry are explored, and a representative (neo-Hookean-type) free energy with full electro-magneto-elastic coupling is posed. iii Dedicatedinlovingmemoryofmyfather iv Acknowledgments This dissertation represents the culmination of a rich and rewarding journey. I have reached this point by standing on the shoulders of others. I will never be able to repay my debt of gratitude in kind, but I will do my best to pay it forward. I would like to begin by expressing my sincere appreciation to my advisor, Pro- fessor Sheng-Tao John Yu, for his guidance, advice, support, and friendship through the years. I thoroughly enjoyed our conversations – both technical and otherwise – and the wisdom and perspective I gained through our interactions. Prof. Yu’s words of encouragement always gave me hope and inspiration when times were challenging. I profited greatly from his expertise in numerical methods and programming. I would also like to thank my former advisor, Professor Stephen Bechtel, who was forced to retire far too young due to the devastating effects of Parkinson’s dis- ease. Prof. Bechtel taught me the importance of rigour, clarity, and perfectionism in scientific work as well as the art of technical writing. These lessons have had an indelible impression on me. I feel privileged to have been selected by him to convert his prized set of continuum mechanics course notes into a published textbook. Prof. Bechtel’s unique perspective on the topic and legacy as a continuum mechanician live on through our book. I would like to express my appreciation to Professor Marcelo Dapino, Professor Dan Mendelsohn, Professor Amos Gilat, and Dr. Kelly Carney for serving on my v doctoral committee. Each of them has willingly offered support, guidance, and ad- vice when I needed it. Prof. Dapino’s input on the smart materials portion of this dissertation brought a much-needed shift toward developing theories that would be useful in practical applications. I am thankful for the opportunity to have taken Prof. Mendelsohn’s course in elastic wave propagation and Prof. Gilat’s course in plasticity, both of which were foundational in my research. Their help and friendship over the years are also acknowledged. Dr. Carney’s comments on the computational aspects of my work are also much appreciated; I look forward to working together more in the coming years. I would like to acknowledge several faculty members at OSU who have provided mentoring, guidance, support, and friendship, including Professors Carlos Castro, Brian Harper, Gary Kinzel, Cheena Srinivasan, and Denny Guenther. A hearty thank you is also due to Ms. Janeen Sands and Mr. Nick Breckenridge in the Graduate Advising Office. Briefly stated, both are tremendous assets to our Department. I am particularly thankful for Janeen’s compassion, willingness to listen, and helpful advice, which helped me weed my way though several challenging situations. I would like to express my gratitude to several friends with whom I’ve had the pleasure of collaborating: First, I’d like to thank Dr. Po-Hsien Lin for performing the numerical computations in Chapter 2. I’ve enjoyed and profited greatly from our conversations about hyperbolic systems and numerical methods. Secondly, I’d like to thank Professors Monon Mahboob and Md. Zahabul Islam for a fruitful inter- national collaboration that resulted in three journal publications on the mechanics of nanoscale structures. Thirdly, I’d like to thank Professor Sushma Santapuri for vi numerous discussions on the nuances of continuum electrodynamics and an enjoy- able collaboration in this field. I would also like to thank Dr. Lixiang Yang for a productive collaboration during his time at OSU and for many helpful conversations. Next, I would like to thank Professor Prasad Mokashi for his close friendship and support throughout the years. I would also like to thank Dr. Hafez Tari, who has become a dear friend to my wife and I in but a few short years in Columbus. I extend many thanks to my amigos Professor Chris Cooley, Mr. Tom Walters, Dr. Jeremy Seidt, Dr. David Bilyeu, and Dr. Krista Kecskemety for sharing the joys (mostly) and frustrations (some) of graduate life over hot and cold beverages alike. A special thanks as well to my friends outside of graduate school – Shawn, Kyle, Nate, Brad, and many others – for road trips, rounds of golf, and chatting about sports and other topics over the occasional frosty beverage. I acknowledge the generous financial support I received throughout my doctoral studies, most notably a Presidential Fellowship from the OSU Graduate School, a Graduate Teaching Fellowship from ASME, and a research assistantship from NSF. I am grateful to Department Chairman Ahmet Selamet and Associate Chairman Dan Mendelsohn for their financial support during several times of need. I would also like to thank former Department Chairman Cheena Srinivasan and former Associate Chairman Gary Kinzel for the opportunity to serve as an independent course in- structor. I also gratefully acknowledge travel support from The Honor Society of Phi Kappa Phi and the OSU Council of Graduate Students. I would like to extend a special thanks to my family, particularly my wife Karen and my mother Kristine. Karen – words cannot convey how grateful I am for your enduring love, support, patience, understanding, and encouragement during this long vii journey. We did it! Mom – I will never be able to repay you for your unwavering love, devotion, and selflessness, but know that it is felt and cherished. I truly hope I can be half the parent you are. A loving thank you is also due to my Aunt Doey, whose weekly phone calls always provided me with a few good laughs, some lively chat about sports, and all the latest news in Bryan, Ohio. A heartfelt thank you to my Uncle John & Aunt Diane and cousins Kirk & Amy, who lovingly looked out for me during my years in Columbus. And last, but certainly not least, much love and many thanks to my in-laws Ed, Barbara, Thomas, and Kim for their love, support, and unwavering belief in me. I am saddened that my father Bob,