Applications of Tensor Analysis in Continuum Mechanics

Tensors, Tensor Analysis, Vectors, Vector Analysis, Differential Geometry, Engineering Mechanics, Elasticity, Theory of Shells, Generalized Media, Mathematical Physics


Authors : Victor A Eremeyev (Gdańsk University of Technology, Poland), Michael J Cloud (Lawrence Technological University, USA) and Leonid P Lebedev (National University of Colombia, Colombia)

Publisher : World Scientific

ISBN : 978-981-3238-96-1

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A tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. The tensorial nature of a quantity permits us to formulate transformation rules for its components under a change of basis. These rules are relatively simple and easily grasped by any engineering student familiar with matrix operators in linear algebra. More complex problems arise when one considers the tensor fields that describe continuum bodies. In this case general curvilinear coordinates become necessary. The principal basis of a curvilinear system is constructed as a set of vectors tangent to the coordinate lines. Another basis, called the dual basis, is also constructed in a special manner. The existence of these two bases is responsible for the mysterious covariant and contravariant terminology encountered in tensor discussions.

This book provides a clear, concise, and self-contained treatment of tensors and tensor fields. It covers the foundations of linear elasticity, shell theory, and generalized continuum media, offers hints, answers, and full solutions for many of the problems and exercises, and Includes a handbook-style summary of important tensor formulas.

The book can be useful for beginners who are interested in the basics of tensor calculus. It also can be used by experienced readers who seek a comprehensive review on applications of the tensor calculus in mechanics.
Victor A Eremeyev received his DSc degree from the Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences (RASci), and his PhD from the Department of Mechanics and Mathematics at Rostov State University, Russia. He currently holds a faculty position at Gdańsk University of Technology, Poland. In the past he held research and teaching positions at Rzeszów University of Technology, Poland, Otto von Guericke University Magdeburg, Germany, Martin Luther University Halle-Wittenberg, Germany, the Southern Scientific Center of RASci, and Southern Federal University in Rostov-on-Don, Russia. Dr. Eremeyev has coauthored or edited nine previous books and published more than 50 book chapters and 100 journal papers in the areas of shell theory, continuum mechanics, nonlinear elasticity, and nanomechanics. Dr Eremeyev is a member of the American Mathematical Society, and the International Research Center on Mathematics and Mechanics of Complex Systems (M&MoCS) in Italy. He serves on the editorial boards of ZAMM, Technische Mechanik (Technical/Applied Mechanics), Nanoscience and Technology: An International Journal, World Journal of Mechanics, Notices of South Sci Center of RASci, and Bulletin of PNRPU. He is coeditor of the book The Complete Works of Gabrio Piola, and has served as guest coeditor for the journals Continuum Mechanics and Thermodynamics, Mathematics and Mechanics of Solids, and International Journal of Engineering Science.

Michael J Cloud received his BS, MS, and PhD degrees from Michigan State University, USA, all in electrical engineering. He has been a faculty member in the Department of Electrical and Computer Engineering at Lawrence Technological University, USA since 1987, and currently holds the rank of Associate Professor. Dr Cloud has coauthored fourteen other books, and is a senior member of the Institute of Electrical and Electronics Engineers.

Leonid P Lebedev received his DSc and PhD degrees in physics and mathematics from the Division of Elasticity at Rostov State University (now Southern Federal University), Russia. He is currently professor of mathematics at the National University of Colombia at Bogota, Columbia and simultaneously holds a faculty appointment at Southern Federal University, Russia. Dr Lebedev has coauthored ten previous books, and published more than 40 journal papers pertaining to nonlinear solid mechanics, the thermodynamics of solid mechanics, and the theory of the finite element method in nonlinear mechanics. Dr Lebedev is a member of the Society for the Interaction of Mechanics and Mathematics.
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