Tensor Analysis and Nonlinear Tensor Functions Tensor Analysis and Nonlinear Tensor Functions by Yu. I. Dimitrienko Bauman Moscow State Technical University, Moscow. Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6169-0 ISBN 978-94-017-3221-5 (eBook) DOI 10.1007/978-94-017-3221-5 Printed on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 Softcover reprint of the hardcover 1st edition 2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. TABLE OF CONTENTS Preface vii Sources of Tensor Calculus ix Introduction xv Chapter 1. TENSOR ALGEBRA 1 1.1. Local Basis Vectors. Jacobian and Metric Matrices 1 1.2. Vector Product 12 1.3. Geometric Definition of a Tensor and Algebraic Opera- tions on Tensors 17 1.4. Algebra of Tensor Fields 34 1.5. Eigenvalues of a Tensor 40 1.6. Symmetric, Skew-Symmetric and Orthogonal Tensors 43 1. 7. Physical Components of Tensors 50 1.8. Tensors of Higher Orders 53 1.9. Pseudotensors 59 Chapter 2. TENSORS IN LINEAR SPACES 65 2.1. Linear n-Dimensional Space 65 2.2. Matrices of the nth Order 74 2.3. Linear Transformations of n-Dimensional Spaces 85 2.4. Dual Space 93 2.5. Algebra of Tensors inn-Dimensional Linear Spaces 98 2.6. Outer Forms · 116 Chapter 3. GROUPS OF TRANSFORMATIONS 129 3.1. Linear Transformations of Coordinates 129 3.2. Transformation Groups in Three-Dimensional Euclidean Space 142 3.3. Symmetry of Finite Bodies 146 3.4. Matrix Representation of Transformation Groups 164 Chapter 4. INDIFFERENT TENSORS AND INVARI- ANTS 169 4.1. Indifferent Tensors 169 4.2. A Number of Independent Components for Indifferent Tensors 180 4.3. Symmetric Indifferent Tensors 192 4.4. Scalar Invariants 202 4.5. Invariants of Symmetric Second-Order Tensors 212 Chapter 5. TENSOR FUNCTIONS 227 5.1. Linear Tensor Functions 227 5.2. Scalar Functions of a Tensor Argument 253 5.3. Potential Tensor Functions 265 5.4. Quasilinear Tensor Functions 274 5.5. Spectral Resolutions of Second-Order Tensors 283 5.6. Spectral Resolutions of Quasilinear Tensor Functions 298 5.7. Nonpotential Tensor Functions 310 5.8. Differentiation of a Tensor Function with respect to a Tensor Argument 325 v vi TABLE OF CONTENTS 5.9. Scalar Functions of Several Tensor Arguments 328 5.10. Tensor Functions of Several Tensor Arguments 342 Chapter 6. TENSOR ANALYSIS 347 6.1. Covariant Differentiation 34 7 6.2. Differentiation of Second-Order Tensors 357 6.3. Properties of Covariant Derivatives 361 6.4. Covariant Derivatives of the Second Order 367 6.5. Differentiation in Orthogonal Curvilinear Coordinates 372 Chapter 7. GEOMETRY OF CURVES AND SUR- FACES 385 7.1. Curves in Three-Dimensional Euclidean Space 385 7.2. Surfaces in Three-Dimensional Euclidean Space 394 7.3. Curves on a Surface 413 7.4. Geometry in a Vicinity of a Surface 426 7.5. Planar Surfaces in IR3 432 Chapter 8. TENSORS IN RIEMANNIAN SPACES AND AFFINELY CONNECTED SPACES 437 8.1. Riemannian Spaces 437 8.2. Affinely Connected Spaces 448 8.3. Riemannian Affinely Connected Spaces 456 8.4. The Riemann-Christoffel Tensor 462 Chapter 9. INTEGRATION OF TENSORS 475 9.1. Curvilinear Integrals of Tensors 475 9.2. Surface Integrals of Tensors 483 9.3. Volume Integrals of Tensors 488 Chapter 10. TENSORS IN CONTINUUM MECHAN- ICS 493 10.1. Deformation Theory 493 10.2. Velocity Characteristics of Continuum Motion 508 10.3. Co-rotational Derivatives 516 10.4. Mass, Momentum and Angular Momentum Balance Laws 524 10.5. Thermodynamic Laws 538 10.6. The Deformation Compatibility Equation 546 10.7. The Complete System of Continuum Mechanics Laws 553 Chapter 11. TENSOR FUNCTIONS IN CONTINUUM MECHANICS 555 11.1. Energetic and Quasienergetic Couples of Tensors 555 11.2. General Principles for Tensor Functions in Continuum Mechanics 572 11.3. The Material Indifference Principle 582 11.4. The Material Symmetry Principle 600 11.5. Tensor Functions for Nonlinear Elastic Continua 618 11.6. Tensor Functions for Nonlinear Hypoelastic Continua 646 References 653 Subject Index 655 PREFACE Tensor calculus appeared in its present-day form thanks to Ricci, who, first of all, suggested mathematical methods for operations on systems with indices at the close of the XIX century. Although these systems had been detected before, namely in investigations of non-Euclidean geometry by Gauss, Riemann, Christoffel and of elastic bodies by Cauchy, Euler, Lagrange, Poisson (see paragraph 'Sources of Tensor Calculus'), it was Ricci who developed the convenient compact system of symbols and concepts, which is widely used nowadays in different fields of mechan ics, physics, chemistry, crystallophysics and other sciences. At present tensor calculus goes on developing: advance directions appear and some concepts, introduced before, are re-interpreted. That is why, in spite of existing works on tensors (see References), there is an actual need of expounding these questions. To illustrate the above, we give one example. The following questions: 'May a second-order tensor be represented visually or graphically as well as a vector in three-dimensional space?' and ' What is a dyad?' - can cause difficulties even for readers experienced in studying of tensors. The present book is intended for a reader beginning to study methods of tensor calculus. That is why the introduction of the book gives the well-known concept of a vector as a geometric object in three-dimensional space. On the basis of the concept, the author suggests a geometric definition of a tensor. This definition allows us to see a tensor and main operations on tensors. And only after this acquaintance with tensors, there is a formal generalized definition of a tensor in an arbitrary linear n-dimensional space. According to the definition, a tensor is introduced as an element of a factor-space relative to the special equivalence. The book presents this approach in a mathematically rigorous form (the preceding works did not take into account the role of zero vectors in the equivalence relation). It should be noted that this approach introduces the notion of a tensor as an individual object, while other existing definitions introduce not a tensor itself but only concepts related to a tensor: tensor components, or linear transformations (for a second-order tensor), or bilinear functionals etc. The principal idea, that a tensor is an individual object, is the basis of the present book. I hope that the book is of interest also for investigators in continuum mechanics, solid physics, erystallophysics, quantum chemistry, because, besides chapters for beginners, the book expounds many problems of the tensor theory which were not resolved before. This concerns tensors specifying physical properties (they are called indifferent tensors in the book), tensor invariants relative to crystallographic groups, a theory of tensor functions and integration of tensors. The book pays great attention to the problems of construction of nonlinear tensor bases, besides the book is the first to present the construction methods for tensor bases in a systematized form. Then with their help, tensor anisotropic nonlinear functions for all crystallographic groups are constructed as well. The classification of tensor functions is given, and the representations are shown for Vll viii PREFACE most important classes of these functions. Theorems about a number of indepen dent components of tensors for all crystallographic groups and theorems about a number of functionally independent invariants of second-order tensors (including joint invariants) appear to be correctly formulated and proved in the present book for the first time. Several chapters are devoted to tensor analysis. Besides the traditional informa tion on covariant differentiation, there are results concerning nonlinear differential operators applied to nonlinear solid mechanics. New results, which are of interest for geometry and general relativity, are given in Chapter 8 devoted to tensors in Riemannian and affinely connected spaces. The last two chapters are devoted to application of tensors and tensor functions to continuum mechanics. The book is the first to give a systematized theory of co-rotational derivatives of tensors specified in moving continua and to present a systematized description of energetic and quasienergetic couples of stress and deformation tensors. These quasienergetic couples have been found by the author. With the help of these couples, four main types of continuum models are intro duced, which cover all known models of nonlinear elastic continua and contain new models of solids including hypoelastic continua and anisotropic continua with finite deformations. The book is constructed by the mathematical principle: there are definitions, theorems, proofs and exercises at the end of each paragraph. The beginning and the end of each proof are denoted by symbols Y and & , respectively. The indexless form of tensors is preferable in the book, that allows us to formu late different relationships in mechanics and physics compactly without overload ing a physical essence of phenomena. At the same time, there are corresponding component and matrix representations of tensor relationships, when they are ap propriate. I would like to thank Professor B.E.Pobedrya (Moscow Lomonosov State Uni versity), Professor A.G.Gorshkov and Professor D.V.Tarlakovskii (Moscow Avia tion Institute), Professor V.S.Zarubin (Moscow Bauman State Technical Univer sity) for fruitful discussions and valuable advice on different problems in the book. I am very grateful to Dr.lrina D .Dimitrienko (Department of Mechanics and Mathematics at Moscow Lomonosov State University), who translated the book into English and prepared the camera-ready typescript. I hope that the book proves to be useful for graduates and post-graduates of mathematical and natural-scientific departments of universities and for investigat ors and academic scientists working in mathematics and also in solid mechanics, physics, general relativity, crystallophysics and quantum chemistry of solids. Yuriy Dimitrienko SOURCES OF TENSOR CALCULUS • The predecessors of tensors were vectors, matrices and systems without indices. Archimedes (287-212 B.C.) added forces acting on a body by the parallelogram rule, i.e. he introduced intuitively special objects which were characterized not only by a value but also by a direction. This basic principle for the development of vector calculus remained the only one for a long time. The Holland mathematician and engineer S.Stevin (1548-1620), who is considered to be a creator of the concept of a vector value, actually re-discovered once again the law of addition of forces by the parallelogram rule. This law was also formulated by !.Newton (1642-1727) in 'Principia mathematica' side by side with the laws of a motion of bodies. The next important step in the development of vector calculus was made only in the XIX century by the Irish mathematician W.Hamilton (1805-1865), who extended a theory of quaternion-hypercomplex numbers, introduced in 1845 the term 'vector' (from Latin 'vector', i.e. carrying) and also the terms: 'scalar', 'scalar product', 'vector product', and gave a definition of these operations. At the same time, G.Grassmann (1809-1877) created a theory of outer products (this concept was introduced in 1844), which is known nowadays as Grassmann's algebra. The English scientist W.Clifford (1845-1879) merged Hamilton's and Grassmann's approaches, but a final connection of quaternions, Grassmann's al gebra and vector algebra was established only at the close of the XIX century by J.W.Gibbs (1839-1903). The geometric image of a vector as a straight-line segment with arrow appeared to be used for the first time thanks to Hamilton, and in 1853 the French mathe matician O.Cauchy (1789-1857) introduced the concept of a radius-vector and the corresponding notation i. In the XIX century, mathematicians actively began to use one more object, namely a matrix being the predecessor of a tensor. The first appearance of ma trices is connected with Old Chinese mathematicians, who in the II century B.C. applied matrices to writing systems of linear equations. The matrix expression of algebraic equations and the up-to-date matrix calculus were developed by the Eng lish mathematician A.Cayley (1821-1895), who introduced in 1841, in particular, *This brief historical sketch does not pretend to embrace the whole history of a development of tensor calculus and other sciences connected with tensor calculus; the purpose of the sketch is to acquaint a beginning reader with some stages of the development and with names of the scientists whose efforts promoted the creation of the up-to-date tensor calculus. IX X SOURCES OF TENSOR CALCULUS the notation for the determinant being used nowadays: Many basic results in the theory of linear algebraic equation systems were obtained by the German mathematician L.Kronecker (1823-1894). During the XIX century, systems with indices appeared in different fields of mathematics. For example, these were quadratic forms in algebra (this theory was developed by A.Cayley, S.Lie (1842-1899) and others), quadratic differential forms in geometry, which are known as the first and the second quadratic forms of a surface and the square of elementary segment length nowadays. The outstanding German scientist K.F.Gauss (1777-1855) is rightfully consid ered to be a founder of the surface theory. Many important results in this field were obtained by B.Riemann (1826-1866), who extended the surface theory for then-dimensional case, and also by E.Beltrami (1835-1900), F.Klein (1849-1925), G.Lame (1795-1870). In 1869 E.B.Christoffel (1829-1900) considered transforma = tions of the quadratic forms ds2 I:JL,v gJLvdx!Ldxv and established a tensor law of their transformation for the first time, and then introduced the concept of deriva tives of vector values, which were transformed by the tensor law (they are called covariant derivatives nowadays). In the XVIII century the efforts of outstanding mathematicians and mechani cians: L.Euler (1707-1783), J.Lagrange (1736-1813), P.Laplace (1749-1827), S.Poisson (1781-1840), O.Cauchy (1789-1857), M.V.Ostrogradskii (1801-1861) re sulted in the creation of a theory of a motion and equilibrium of elastic bodies (elasticity theory), which became one more source for the appearance of systems with indices (components of stresses and strains). Stress components were denoted by Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz, and they were considered as projections of forces, acting on the sides of an elementary cube, onto coordinate axes. The operations on such systems with indices were rather awkward, contained many repetitions up to the cyclic change of notation. However, only at the close of the XIX century scientists succeeded in understanding the internal unity of formu lae containing systems with indices and in finding a new mathematical apparatus which would made the operations on the systems compact and suitable. For the first time, for vector values this problem was solved by the American physicist and mathematician J.W.Gibbs, who created the vector algebra with the operations of addition, scalar and vector multiplication and showed its connec tion with the theory of quaternions and Grassmann's algebra. Moreover, Gibbs developed the up-to-date vector analysis (the theory of differential calculus of vector fields) and the language of vector calculus, where there were both com ponent and indexless forms of relationships. In particular, he gave appropriate representations for the operations of divergence and curl on vector fields. These distinguished results obtained by Gibbs can be compared with the introduction of algebraic symbolics by F.Vieta (1540-1603), which has been used during last