tensor algebra and tensor analysis for engineers with applications to continuum mechanics second edition pdf

www.TechnicalBooksPDF.com Tensor Algebra and Tensor Analysis for Engineers Second edition www.TechnicalBooksPDF.com Mikhail Itskov Tensor Algebra and Tensor Analysis for Engineers With Applications to Continuum Mechanics Second edition 123 www.TechnicalBooksPDF.com Prof Dr.-Ing Mikhail Itskov Department of Continuum Mechanics RWTH Aachen University Eilfschornsteinstr 18 D 52062 Aachen Germany Itskov@km.rwth-aachen.de ISBN 978-3-540-93906-1 e-ISBN 978-3-540-93907-8 DOI 10.1007/978-3-540-93907-8 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926098 c Springer-Verlag Berlin Heidelberg 2007, 2009 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: eStudio Calamar S.L Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) www.TechnicalBooksPDF.com Moim roditel m www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com Preface to the Second Edition This second edition is completed by a number of additional examples and exercises In response of comments and questions of students using this book, solutions of many exercises have been improved for a better understanding Some changes and enhancements are concerned with the treatment of skewsymmetric and rotation tensors in the first chapter Besides, the text and formulae have thoroughly been reexamined and improved where necessary Aachen, January 2009 Mikhail Itskov www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com Preface to the First Edition Like many other textbooks the present one is based on a lecture course given by the author for master students of the RWTH Aachen University In spite of a somewhat difficult matter those students were able to endure and, as far as I know, are still fine I wish the same for the reader of the book Although the present book can be referred to as a textbook one finds only little plain text inside I tried to explain the matter in a brief way, nevertheless going into detail where necessary I also avoided tedious introductions and lengthy remarks about the significance of one topic or another A reader interested in tensor algebra and tensor analysis but preferring, however, words instead of equations can close this book immediately after having read the preface The reader is assumed to be familiar with the basics of matrix algebra and continuum mechanics and is encouraged to solve at least some of numerous exercises accompanying every chapter Having read many other texts on mathematics and mechanics I was always upset vainly looking for solutions to the exercises which seemed to be most interesting for me For this reason, all the exercises here are supplied with solutions amounting a substantial part of the book Without doubt, this part facilitates a deeper understanding of the subject As a research work this book is open for discussion which will certainly contribute to improving the text for further editions In this sense, I am very grateful for comments, suggestions and constructive criticism from the reader I already expect such criticism for example with respect to the list of references which might be far from being complete Indeed, throughout the book I only quote the sources indispensable to follow the exposition and notation For this reason, I apologize to colleagues whose valuable contributions to the matter are not cited Finally, a word of acknowledgment is appropriate I would like to thank Uwe Navrath for having prepared most of the figures for the book Further, I am grateful to Alexander Ehret who taught me first steps as well as some “dirty” tricks in LATEX, which were absolutely necessary to bring the www.TechnicalBooksPDF.com X Preface to the First Edition manuscript to a printable form He and Tran Dinh Tuyen are also acknowledged for careful proof reading and critical comments to an earlier version of the book My special thanks go to the Springer-Verlag and in particular to Eva Hestermann-Beyerle and Monika Lempe for their friendly support in getting this book published Aachen, November 2006 Mikhail Itskov www.TechnicalBooksPDF.com Solutions 233 7.2 (7.5)1 , (S.14), (S.15): s s U= Λi ⊗ = e3 ⊗ e3 λi Pi = i=1 i=1 + Λ1 √ e1 + + Λ1 Λ1 e2 + Λ1 ⊗ √ e1 + + Λ1 Λ1 e2 + Λ1 + Λ2 √ e1 − + Λ2 Λ2 e2 + Λ2 ⊗ √ e1 − + Λ2 Λ2 e2 + Λ2 = γ2 + γ e1 ⊗ e1 + γ2 + (e1 ⊗ e2 + e2 ⊗ e1 ) + γ2 + γ2 + e2 ⊗ e2 + e3 ⊗ e3 7.3 The proof of the first relation (7.21) directly results from the definition of the analytic tensor function (7.15) and is obvious In order to prove (7.21)2 we first write 2πi f (A) = Γ f (ζ) (ζI − A) −1 dζ, h (A) = 2πi −1 Γ h (ζ ) (ζ I − A) dζ , where the closed curve Γ of the second integral lies outside Γ which, in turn, includes all eigenvalues of A Using the identity −1 (ζI − A) (ζ I − A) −1 = (ζ − ζ) −1 (ζI − A) −1 −1 − (ζ I − A) valid both on Γ and Γ we thus obtain −1 f (A) h (A) = f (ζ) h (ζ ) (ζI − A)−1 (ζ I − A) dζdζ (2πi) Γ Γ = + 2πi 2πi f (ζ) Γ 2πi h (ζ ) Γ Since the function f (ζ) (ζ − ζ ) (see, e.g [5]) implies that 2πi Γ Γ 2πi −1 h (ζ ) −1 dζ (ζI − A) dζ ζ −ζ Γ f (ζ) −1 dζ (ζ I − A) dζ ζ −ζ is analytic in ζ inside Γ the Cauchy theorem f (ζ) dζ = ζ−ζ Noticing further that 2πi Γ h (ζ ) dζ = h (ζ) ζ −ζ we obtain www.TechnicalBooksPDF.com 234 Solutions f (A) h (A) = = 2πi Γ 2πi Γ f (ζ) 2πi Γ h (ζ ) −1 dζ (ζI − A) dζ ζ −ζ −1 f (ζ) h (ζ) (ζI − A) dζ −1 g (ζ) (ζI − A) dζ = g (A) 2πi Γ Finally, we focus on the third relation (7.21) It implies that the functions h and f are analytic on domains containing all the eigenvalues λi of A and h (λi ) (i = 1, 2, , n) of B = h (A), respectively Hence (cf [25]), = f (h (A)) = f (B) = 2πi −1 Γ f (ζ) (ζI − B) dζ, (S.31) where Γ encloses all the eigenvalues of B Further, we write (ζI − B)−1 = (ζI − h (A))−1 = 2πi −1 Γ (ζ − h (ζ )) −1 (ζ I − A) dζ , (S.32) where Γ includes all the eigenvalues λi of A so that the image of Γ under h lies within Γ Thus, inserting (S.32) into (S.31) delivers −1 −1 f (h (A)) = f (ζ) (ζ − h (ζ )) (ζ I − A) dζ dζ (2πi) Γ Γ = = f (ζ) (ζ − h (ζ )) (2πi)2 Γ 2πi f (h (ζ )) (ζ I − A) Γ Γ −1 −1 −1 dζ (ζ I − A) dζ dζ −1 g (ζ ) (ζ I − A) dζ = g (A) 2πi Γ 7.4 Inserting into the right hand side of (7.54) the spectral decomposition in terms of eigenprojections (7.1) and taking (4.46) into account we can write similarly to (7.17) ⎛ ⎞−1 s 1 −1 ⎝ζI − (ζI − A) dζ = λj Pj ⎠ dζ 2πi Γi 2πi Γi j=1 = ⎡ = 2πi s = j=1 ⎣ Γi 2πi s ⎤−1 (ζ − λj ) Pj ⎦ dζ = j=1 Γi 2πi s Γi j=1 −1 (ζ − λj ) (ζ − λj )−1 dζ Pj www.TechnicalBooksPDF.com Pj dζ Solutions 235 In the case i = j the closed curve Γi does not include any pole so that 2πi −1 Γi (ζ − λj ) dζ = δij , i, j = 1, 2, s This immediately leads to (7.54) 7.5 By means of (7.43) and (7.83) and using the result for the eigenvalues of A by (S.17), λi = 6, λ = −3 we write ρ1p Ap = − P1 = p=0 λ 1 I+ A = I + A, (λi − λ) (λi − λ) I − A Taking symmetry of A into account we further obtain by virtue of (7.56) and (7.84) P2 = I − P1 = υ1pq (Ap ⊗ Aq ) s P1 ,A = p,q=0 =− 2λλi (λi − λ) 3I s + λi + λ (λi − λ) s (I ⊗ A + A ⊗ I) − (λi − λ) s (A ⊗ A) s s s I + (I ⊗ A + A ⊗ I) − (A ⊗ A) 81 243 729 The eigenprojection P2 corresponds to the double eigenvalue λ = −3 and for this reason is not differentiable = 7.6 Since A is a symmetric tensor and it is diagonalizable Thus, taking double coalescence of eigenvalues (S.17) into account we can apply the representations (7.77) and (7.78) Setting there λa = 6, λ = −3 delivers exp (A) = e6 + 2e−3 e6 − e−3 I+ A, exp (A) ,A = + Inserting ⎡ −2 A=⎣ 21 24 13e6 + 32e−3 s 10e6 − 19e−3 I + (A ⊗ I + I ⊗ A)s 81 243 7e6 + 11e−3 (A ⊗ A)s 729 ⎤ ⎦ ei ⊗ ej www.TechnicalBooksPDF.com 236 Solutions into the expression for exp (A) we obtain ⎡ ⎤ e + 8e−3 2e6 − 2e−3 2e6 − 2e−3 exp (A) = ⎣ 2e6 − 2e−3 4e6 + 5e−3 4e6 − 4e−3 ⎦ ei ⊗ ej , 2e6 − 2e−3 4e6 − 4e−3 4e6 + 5e−3 which coincides with the result obtained in Exercise 4.12 7.7 The computation of the coefficients series (7.89), (7.91) and (7.96), (7.97) with the precision parameter ε = · 10−6 has required 23 iteration steps and has been carried out by using MAPLE-program The results of the computation are summarized in Tables S.1 and S.2 On use of (7.90) and (7.92) we thus obtain exp (A) = 44.96925I + 29.89652A + 4.974456A2, exp (A) ,A = 16.20582Is + 6.829754 (I ⊗ A + A ⊗ I)s + 1.967368 (A ⊗ A)s s s +1.039719 I ⊗ A2 + A2 ⊗ I + 0.266328 A ⊗ A2 + A2 ⊗ A +0.034357 A2 ⊗ A2 s Taking into account double coalescence of eigenvalues of A we can further write A2 = (λa + λ) A − λa λI = 3A + 18I Inserting this relation into the above representations for exp (A) and exp (A) ,A finally yields exp (A) = 134.50946I + 44.81989A, s s exp (A) ,A = 64.76737Is + 16.59809 (I ⊗ A + A ⊗ I) + 3.87638 (A ⊗ A) Note that the relative error of this result in comparison to the closed-form solution used in Exercise 7.6 lies within 0.044% Exercises of Chapter 8.1 By (8.2) we first calculate the right and left Cauchy-Green tensors as ⎡ ⎤ ⎡ ⎤ −2 520 C = FT F = ⎣ −2 ⎦ ei ⊗ ej , b = FFT = ⎣ ⎦ ei ⊗ ej , 01 001 √ with the Λ1 = 1, Λ2 = 4, Λ3 = Thus, λ1 = Λ1 = 1, √ following eigenvalues √ λ2 = Λ2 = 2, λ3 = Λ3 = By means of (8.11-8.12) we further obtain ϕ0 = 35 , ϕ1 = 12 , ϕ2 = − 60 and www.TechnicalBooksPDF.com Solutions (r) Table S.1 Recurrent calculation of the coefficients ωp (r) (r) (r) r ar ω0 a r ω1 a r ω2 23 ·10−6 ϕp 0 9.0 12.15 4.05 3.394287 44.96925 4.5 2.25 6.075 4.05 2.262832 29.89652 0 0.5 1.125 0.45 1.0125 0.377134 4.974456 (r) Table S.2 Recurrent calculation of the coefficients ξpq (r) (r) (r) (r) (r) (r) r ar ξ00 ar ξ01 ar ξ02 ar ξ11 ar ξ12 ar ξ22 23 ·10−6 ηpq 0 4.5 4.05 2.284387 16.20582 0.5 1.125 0.9 1.0125 1.229329 6.829754 0 0.166666 0.225 0.15 0.197840 1.039719 0 0.166666 0.45 0.15 0 0.041666 0.075 0 0 0.008333 0.623937 0.099319 0.015781 1.967368 0.266328 0.034357 ⎡ ⎤ 11 −2 1 U = I + C − C2 = ⎣ −2 14 ⎦ ei ⊗ ej , 12 60 0 ⎡ ⎤ 11 1 v = I + b − b2 = ⎣ 14 ⎦ ei ⊗ ej 12 60 0 = − 14 , ς2 = 60 and ⎡ ⎤ 340 1⎣ −4 ⎦ ei ⊗ ej = 005 Eqs (8.16-8.17) further yield ς0 = R=F 37 1 I − C + C2 30 60 37 30 , ς1 8.2 (4.44), (5.33), (5.47), (5.55), (5.85)1 : www.TechnicalBooksPDF.com 237 238 Solutions Pij : Pkl = (Pi ⊗ Pj + Pj ⊗ Pi )s : (Pk ⊗ Pl + Pl ⊗ Pk )s s s = [(Pi ⊗ Pj + Pj ⊗ Pi ) : (Pk ⊗ Pl + Pl ⊗ Pk )] = t Pi ⊗ Pj + Pj ⊗ Pi + (Pi ⊗ Pj ) + (Pj ⊗ Pi ) : (Pk ⊗ Pl + Pl ⊗ Pk )} t s s = (δik δjl + δil δjk ) (Pi ⊗ Pj + Pj ⊗ Pi ) , i = j, k = l In the case i = j or k = l the previous result should be divided by 2, whereas for i = j and k = l by 4, which immediately leads to (8.65) 8.3 Setting f (λ) = ln λ in (8.50) and (8.56)1 one obtains s ˙ (0) = (ln U)˙ = E i=1 s ln λi − ln λj ˙ i+ ˙ j Pi CP Pi CP 2 − λ2 2λi λ i j i,j=1 i=j www.TechnicalBooksPDF.com References Baásar Y, Kră atzig WB (1985) Mechanik der Flă achentragwerke Vieweg Verlag, Braunschweig Boehler JP (1977) Z Angew Math Mech 57: 323–327 de Boer R (1982) Vektor- und Tensorrechnung fă ur Ingenieure Springer, Berlin Heidelberg New York Boulanger Ph, Hayes M (1993) Bivectors and Waves in Mechanics and Optics Chapman & Hall, London Bronstein IN, Semendyayev KA, Musiol G, Muehlig H (2004) Handbook of Mathematics Springer, Berlin Heidelberg New York Brousse P (1988) Optimization in Mechanics: Problems and Methods Elsevier Science, Amsterdam Carlson DE, Hoger A (1986) J Elasticity 16:221–224 Chen Y, Wheeler L (1993) J Elasticity 32:175–182 Cheng H, Gupta KC (1989) J Appl Mech 56:139–145 10 Chrystal G (1980) Algebra An elementary text-book Part I Chelsea Publishing Company, New York 11 Dui G, Chen Y-C (2004) J Elasticity 76:107–112 12 Friedberg SH, Insel AJ, Spence LE (2003) Linear Algebra Pearson Education, Upper Saddle River, New Jersey 13 Gantmacher FR (1959) The theory of matrices Chelsea Publishing Company, New York 14 Guo ZH (1984) J Elasticity 14:263–267 15 Halmos PR (1958) Finite-Dimensional Vector Spaces Van Nostrand, New York 16 Hill R (1968) J Mech Phys Solids 16:229–242 17 Hill R (1978) Adv Appl Mech 18:1–75 18 Hoger A, Carlson DE (1984) J Elasticity 14:329–336 19 Itskov M (2002) Z Angew Math Mech 82:535–544 20 Itskov M (2003) Comput Meth Appl Mech Engrg 192:3985–3999 21 Itskov M (2003) Proc R Soc Lond A 459:1449–1457 22 Itskov M (2004) Mech Res Commun 31:507–517 23 Itskov M, Aksel N (2002) Int J Solids Struct 39:5963–5978 24 Kaplan W (2003) Advanced calculus Addison Wesley, Boston 25 Kato T (1966) Perturbation theory for linear operators Springer, New York 26 Kreyszig E (1991) Differential geometry Dover Publication, New York www.TechnicalBooksPDF.com 240 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 References Lax PD (1997) Linear algebra John Wiley & Sons, New York Lew JS (1966) Z Angew Math Phys 17:650–653 Lubliner J (1990) Plasticity theory Macmillan Publishing Company, New York Ogden RW (1984) Non-Linear Elastic Deformations Ellis Horwood, Chichester Ortiz M, Radovitzky RA, Repetto EA (2001) Int J Numer Meth Engrg 52:14311441 Papadopoulos P, Lu J (2001) Comput Methods Appl Mech Engrg 190:4889– 4910 Pennisi S, Trovato M (1987) Int J Engng Sci 25:1059–1065 Rinehart RF (1955) Am Math Mon 62:395–414 Rivlin RS (1955) J Rat Mech Anal 4:681–702 Rivlin RS, Ericksen JL (1955) J Rat Mech Anal 4:323–425 Rivlin RS, Smith GF (1975) Rendiconti di Matematica Serie VI 8:345–353 Rosati L (1999) J Elasticity 56:213–230 Sansour C, Kollmann FG (1998) Comput Mech 21:512–525 Seth BR (1964) Generalized strain measures with applications to physical problems In: Reiner M, Abir D (eds) Second-order effects in elasticity, plasticity and fluid dynamics Academic Press, Jerusalem Smith GF (1971) Int J Engng Sci 9:899–916 Sokolnikoff IS (1964) Tensor analysis Theory and applications to geometry and mechanics of continua John Wiley & Sons, New York Spencer AJM (1984) Constitutive theory for strongly anisotropic solids In: Spencer AJM(ed) Continuum theory of the mechanics of fibre-reinforced composites Springer, Wien, New York Steigmann DJ (2002) Math Mech Solids 7:393–404 Ting TCT (1985) J Elasticity 15:319–323 Truesdell C, Noll W (1965) The nonlinear field theories of mechanics In: Flă ugge S (ed) Handbuch der Physik, Vol III/3 Springer, Berlin Wheeler L (1990) J Elasticity 24:129–133 Xiao H (1995) Int J Solids Struct 32:3327–3340 Xiao H, Bruhns OT, Meyers ATM (1998) J Elasticity 52:1–41 Zhang JM, Rychlewski J (1990) Arch Mech 42:267–277 Further Reading 51 Abraham R, Marsden JE, Ratiu T (1988) Manifolds, Tensor Analysis and Applications Springer, Berlin Heidelberg New York 52 Akivis MA, Goldberg VV (2003) Tensor Calculus with Applications World Scientific Publishing, Singapore 53 Anton H, Rorres C (2000) Elementary linear algebra: application version John Wiley & Sons, New York 54 Ba¸sar Y, Weichert D (2000) Nonlinear Continuum Mechanics of Solids Fundamental Mathematical and Physical Concepts Springer, Berlin Heidelberg New York 55 Bertram A (2005) Elasticity and Plasticity of Large Deformations An Introduction Springer, Berlin Heidelberg New York 56 Betten J (1987) Tensorrechnung fă ur Ingenieure Teubner-Verlag, Stuttgart www.TechnicalBooksPDF.com References 241 57 Bishop RL, Goldberg SI (1968) Tensor Analysis on Manifolds The Macmillan Company, New York 58 Borisenko AI, Tarapov IE (1968) Vector and Tensor Analysis with Applications Prentice-Hall, Englewood Cliffs 59 Bowen RM, Wang C-C (1976) Introduction to vectors and tensors Plenum Press, New York 60 Brillouin L (1964) Tensors in Mechanics and Elasticity Academic Press, New York 61 Chadwick P (1976) Continuum Mechanics Concise Theory and Problems George Allen & Unwin, London 62 Dimitrienko Yu I (2002) Tensor Analysis and Nonlinear Tensor Functions Kluwer Academic Publishers, Dordrecht 63 Flă ugge W (1972) Tensor Analysis and Continuum Mechanics Springer, Berlin Heidelberg New York 64 Golub GH, van Loan CF (1996) Matrix computations The Johns Hopkins University Press, Baltimore 65 Gurtin ME (1981) An Introduction to Continuum Mechanics Academic Press, New York 66 Lebedev LP, Cloud MJ (2003) Tensor Analysis World Scientific Publishing, Singapore 67 Lă utkepohl H (1996) Handbook of matrices John Wiley & Sons, Chichester 68 Narasimhan MNL (1993) Principles of Continuum Mechanics John Wiley & Sons, New York 69 Noll W (1987) Finite-Dimensional Spaces Martinus Nijhoff Publishers, Dordrecht 70 Renton JD (2002) Applied Elasticity: Matrix and Tensor Analysis of Elastic Continua Horwood Publishing, Chichester 71 Ru´ız-Tolosa JR, Castillo (2005) From Vectors to Tensors Springer, Berlin Heidelberg New York 72 Schade H (1997) Tensoranalysis Walter der Gruyter, Berlin, New York 73 Schey HM (2005) Div, grad, curl and all that: an informal text on vector calculus W.W.Norton & Company, New York 74 Schouten JA (1990) Tensor analysis for physicists Dover Publications, New York ˇ 75 Silhav´ y M (1997) The Mechanics and Thermodynamics of Continuous Media Springer, Berlin Heidelberg New York 76 Simmonds JG (1997) A Brief on Tensor Analysis Springer, Berlin Heidelberg New York 77 Talpaert YR (2002) Tensor Analysis and Continuum Mechanics Kluwer Academic Publishers, Dordrecht www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com Index algebraic multiplicity of an eigenvalue, 85, 89, 93, 96 analytic tensor function, 149 anisotropic tensor function, 119 arc length, 60 asymptotic direction, 71 axial vector, 29, 55, 98 basis of a vector space, binomial theorem, 150 binormal vector, 62 Biot strain tensor, 175 Cardano formula, 87 Cartesian coordinates, 47, 49, 52, 53, 56 Cauchy integral, 150 integral formula, 148 strain tensor, 103 stress tensor, 15, 75, 103, 180 stress vector, 15, 55, 75 theorem, 16, 55 Cayley-Hamilton equation, 161, 167, 171 Cayley-Hamilton theorem, 99, 149 characteristic equation, 84 polynomial, 84–86, 93, 98 Christoffel symbols, 46–48, 52, 57, 68, 79 coaxial tensors, 130 commutative tensors, 20 complex conjugate vector, 82 number, 81 vector space, 81 complexification, 81 compliance tensor, 103 components contravariant, 41 covariant, 41 mixed variant, 41 of a vector, composition of tensors, 20 cone, 79 contravariant components, 41 derivative, 46 coordinate line, 39, 67 system, 37 transformation, 39 coordinates Cartesian, 47, 49, 52, 53, 56 cylindrical, 37, 40, 42, 48, 52, 57 linear, 38, 42, 44, 57 spherical, 57 covariant components, 41 derivative, 46 on a surface, 69 curl of a vector field, 53 curvature directions, 70 Gaussian, 71 mean, 71 normal, 69 www.TechnicalBooksPDF.com 244 Index of the curve, 61 radius of, 62 curve, 59 left-handed, 62 on a surface, 66 plane, 62 right-handed, 62 torsion of, 62 cylinder, 66 cylindrical coordinates, 37, 40, 42, 48, 52, 57 Darboux vector, 64 defective eigenvalue, 90 tensor, 90 deformation gradient, 138, 158, 165 derivative contravariant, 46 covariant, 46 directional, 122, 136 Gateaux, 122, 136 determinant of a tensor, 86 deviatoric projection tensor, 113 tensor, 29 diagonalizable tensor, 89, 148, 152 dimension of a vector space, 3, directional derivative, 122, 136 divergence, 49 dual basis, dummy index, Dunford-Taylor integral, 147, 152 eigenprojection, 90 eigentensor, 111 eigenvalue, 83 defective, 90 problem, 83, 111 left, 83 right, 83 eigenvector, 83 left, 83 right, 83 Einstein’s summation convention, elasticity tensor, 103 elliptic point, 71 Euclidean space, 6, 81, 82 Euler-Rodrigues formula, 15 Eulerian strains, 146 exponential tensor function, 21, 91, 129, 158, 163 fourth-order tensor, 103 deviatoric projection, 113 spherical projection, 113 super-symmetric, 109 trace projection, 113 transposition, 112 Frenet formulas, 63 functional basis, 115 fundamental form of the surface first, 67 second, 69 Gateaux derivative, 122, 136 Gauss coordinates, 66, 68 formulas, 69 Gaussian curvature, 71 generalized Hooke’s law, 113 Rivlin’s identity, 141 strain measures, 146 geometric multiplicity of an eigenvalue, 85, 89, 93, 96 gradient, 44 Gram-Schmidt procedure, 7, 93, 96, 100 Green-Lagrange strain tensor, 133, 139, 146 Hill’s strains, 146 Hooke’s law, 113 hydrostatic pressure, 56 hyperbolic paraboloidal surface, 79 point, 71 hyperelastic material, 117, 132, 139 identity tensor, 18 invariant isotropic, 115 principal, 85 inverse of the tensor, 23 inversion, 23 invertible tensor, 23, 91 irreducible functional basis, 116 isotropic www.TechnicalBooksPDF.com Index invariant, 115 material, 117, 132, 139 symmetry, 119 tensor function, 115 Jacobian determinant, 39 Kronecker delta, Lagrangian strains, 146 Lam´e constants, 113, 133 Laplace expansion rule, 99 Laplacian, 54 left Cauchy-Green tensor, 138, 165 eigenvalue problem, 83 eigenvector, 83 mapping, 16, 17, 20, 104–107 stretch tensor, 146, 165 left-handed curve, 62 length of a vector, Levi-Civita symbol, 10 linear combination, coordinates, 38, 42, 44, 57 mapping, 12, 28, 29, 103, 112, 113 linear-viscous fluid, 56 linearly elastic material, 103, 133 logarithmic tensor function, 147 major symmetry, 109 mapping left, 16, 17, 20, 104–107 right, 16, 104, 106 material hyperelastic, 117, 132, 139 isotropic, 117, 132, 139 linearly elastic, 103, 133 Mooney-Rivlin, 117 Ogden, 117 orthotropic, 142 St.Venant-Kirchhoff, 133 time derivative, 173, 175 transversely isotropic, 119, 134, 139 mean curvature, 71 mechanical energy, 55 membrane theory, 78 metric coefficients, 19, 68 middle surface of the shell, 73 minor symmetry, 110 mixed product of vectors, 10 mixed variant components, 41 moment tensor, 76 momentum balance, 52 Mooney-Rivlin material, 117 moving trihedron of the curve, 62 multiplicity of an eigenvalue algebraic, 85, 89, 93, 96 geometric, 85, 89, 93, 96 Navier-Stokes equation, 56 Newton’s formula, 86 normal curvature, 69 plane, 68 section of the surface, 68 yield stress, 179 Ogden material, 117 orthogonal spaces, 29 tensor, 25, 94, 97 vectors, orthonormal basis, orthotropic material, 142 parabolic point, 71 permutation symbol, 10 plane, 66 plane curve, 62 plate theory, 78 point elliptic, 71 hyperbolic, 71 parabolic, 71 saddle, 71 polar decomposition, 165 positive-definite tensor, 94, 100 principal curvature, 70 invariants, 85 material direction, 119, 142 normal vector, 62, 68 stretches, 146, 167, 168, 170 traces, 86 proper orthogonal tensor, 98 Pythagoras formula, radius of curvature, 62 www.TechnicalBooksPDF.com 245 246 Index rate of deformation tensor, 56 representation theorem, 131, 132 residue theorem, 150, 151 Ricci’s Theorem, 49 Riemannian metric, 68 right Cauchy-Green tensor, 117, 120, 132, 138, 165 eigenvalue problem, 83 eigenvector, 83 mapping, 16, 104, 106 stretch tensor, 146, 163, 165 right-handed curve, 62 Rivlin’s identities, 140 rotation, 13 tensor, 14, 165 Rychlewski’s theorem, 134 saddle point, 71 scalar field, 42 product, of tensors, 26 second Piola-Kirchhoff stress tensor, 132, 143, 175 viscosity coefficient, 56 second-order tensor, 12 Seth’s strains, 146 shear viscosity, 56 yield stress, 181 shell continuum, 73 shifter, 73 similar tensors, 224 simple shear, 85, 158, 163, 168 skew-symmetric generator, 134 tensor, 23, 96, 98 spectral decomposition, 89, 111 mapping theorem, 83 sphere, 66 spherical coordinates, 57 projection tensor, 113 tensor, 29 spin tensor, 55 St.Venant-Kirchhoff material, 133 straight line, 59 strain energy function, 117 strain tensor Biot, 175 Cauchy, 103 Green-Lagrange, 133, 139, 146 strains Eulerian, 146 Hill’s, 146 Lagrangian, 146 Seth’s, 146 stress resultant tensor, 76 stress tensor Cauchy, 15, 75, 103 second Piola-Kirchhoff, 132 stretch tensors, 146, 165 structural tensor, 119 summation convention, super-symmetric fourth-order tensor, 109 surface, 66 hyperbolic paraboloidal, 79 Sylvester formula, 91, 152 symmetric generator, 134 tensor, 23, 92, 94 symmetry major, 109 minor, 110 symmetry group, 119 anisotropic, 120 isotropic, 119 of fiber reinforced material, 143 orthotropic, 142 transversely isotropic, 119, 134 triclinic, 119 tangent moduli, 139 vectors, 39 tensor defective, 90 deviatoric, 29 diagonalizable, 89, 148, 152 field, 42 function, 35 analytic, 149 anisotropic, 119 www.TechnicalBooksPDF.com Index exponential, 21, 91, 129, 158, 163 isotropic, 115 logarithmic, 147 identity, 18 invertible, 23, 91 left Cauchy-Green, 138, 165 left stretch, 146, 165 monomial, 21, 148 of the fourth order, 103 of the second order, 12 of the third order, 29 orthogonal, 25, 94, 97 polynomial, 21, 90, 129 positive-definite, 94, 100 power series, 21, 146 product, 16 proper orthogonal, 98 right Cauchy-Green, 117, 120, 132, 138, 165 right stretch, 146, 165 rotation, 14, 165 skew-symmetric, 23, 96, 98 spherical, 29 structural, 119 symmetric, 23, 92, 94 tensors coaxial, 130 commutative, 20 composition of, 20 scalar product of, 26 third-order tensor, 29 torsion of the curve, 62 torus, 71 trace, 27 trace projection tensor, 113 transposition, 22 transposition tensor, 112 transverse shear stress vector, 76 transversely isotropic material, 119, 134, 139 triclinic symmetry, 119 unit vector, vector axial, 29, 98 binormal, 62 complex conjugate, 82 components, Darboux, 64 field, 42 function, 35 length, product of vectors, 10, 13 space, basis of, complex, 81 dimension of, 3, Euclidean, zero, vectors mixed product of, 10 orthogonal, tangent, 39 velocity gradient, 145, 158, 163 Vieta theorem, 70, 86, 87, 149 von Mises yield function, 178 Weingarten formulas, 69 yield stress normal, 179 shear, 181 zero tensor, 12 zero vector, www.TechnicalBooksPDF.com 247 .. .Tensor Algebra and Tensor Analysis for Engineers Second edition www.TechnicalBooksPDF.com Mikhail Itskov Tensor Algebra and Tensor Analysis for Engineers With Applications to Continuum Mechanics. .. symmetric and skew-symmetric tensors, spherical and deviatoric tensors form orthogonal subspaces of Linn 1.12 Tensors of Higher Orders Similarly to second- order tensors we can define tensors of... vectors in En into secondorder tensors in Linn The tensors of the third order can likewise be represented with respect to a basis in Linn e.g by www.TechnicalBooksPDF.com 30 Vectors and Tensors