This book is the first organized collection of some results that have been obtained by the authors, their collaborators, and other researchers in the variational approach to structured deformations. It sets the basis and makes more accessible the theoretical apparatus for assigning an energy to a structured deformation, thereby providing motivation to researchers in applied mathematics, continuum mechanics, engineering, and materials science to study the deformation of a solid body without committing at the outset to a specific mechanical theory. Researchers will benefit from an approach in which elastic, plastic, and fracture phenomena can be treated in a unified way.
?The book is intended for an audience acquainted with measure theory, the theory of functions of bounded variation, and continuum mechanics. Any students in their last years of undergraduate studies, graduate students, and researchers with a background in applied mathematics, the calculus of variations, and continuum mechanics will have the prerequisite to read this book.
1. Introduction.- 2. Mathematical preliminaries.- 3. Energetic
relaxation to first-order structured deformations.- 4. Energetic relaxation
to second-order structured deformations.-
5. Outlook for future research.
José Matias is an associate professor at the Department of Mathematics of Instituto Superior Técnico in Lisbon. He was awarded his PhD from Carnegie Mellon University in 1993 and his research interests lie in the field of applied mathematics and calculus of variations. Among other topics, he has studied extensively the mathematical aspects of models for continuum mechanics and differential inclusions. Marco Morandotti is an associate professor at the Department of Mathematical Sciences of Politecnico di Torino. He was awarded his PhD in Applied Mathematics from SISSA, Trieste in 2011 and has been a research scholar at Carnegie Mellon University, Instituto Superior Técnico, and the Technical University of Munich. His research spans some areas of applied mathematics, including models for continuum mechanics, models for defects in solids, the game-theoretic study of multi-agent systems, and the motion of micro-swimmers in viscous fluids. David R. Owen is an emeritus professor at the Department of Mathematical Sciences of Carnegie Mellon University. He was awarded his PhD from Brown University in 1968 in applied mathematics and has been in the faculty at Carnegie Mellon since 1967, apart from visiting position through the years in France, Germany, and Italy. Together with the late Gianpietro Del Piero, he published the first paper on the theory of structured deformations, which he has developed during its evolution from the mechanical to the variational context. His research interests include the mathematical foundations of thermodynamics, mathematical models of the plastic behavior of solids, and multiscale descriptions of geometrical changes in continuous bodies.
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