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Solid Mechanics and Its Applications Ciprian D Coman Continuum Mechanics and Linear Elasticity An Applied Mathematics Introduction Solid Mechanics and Its Applications Volume 238 Founding Editor G M L Gladwell, University of Waterloo, Waterloo, ON, Canada Series Editors J R Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids The scope of the series covers the entire spectrum of solid mechanics Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design The median level of presentation is the ﬁrst year graduate student Some texts are monographs deﬁning the current state of the ﬁeld; others are accessible to ﬁnal year undergraduates; but essentially the emphasis is on readability and clarity Springer and Professors Barber and Klarbring welcome book ideas from authors Potential authors who wish to submit a book proposal should contact Dr Mayra Castro, Senior Editor, Springer Heidelberg, Germany, e-mail: mayra.castro@springer.com Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink More information about this series at http://www.springer.com/series/6557 Ciprian D Coman Continuum Mechanics and Linear Elasticity An Applied Mathematics Introduction 123 Ciprian D Coman School of Computing and Engineering University of Huddersﬁeld Huddersﬁeld, UK ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-94-024-1769-2 ISBN 978-94-024-1771-5 (eBook) https://doi.org/10.1007/978-94-024-1771-5 © Springer Nature B.V 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, 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 The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations This Springer imprint is published by the registered company Springer Nature B.V The registered company address is: Van Godewijckstraat 30, 3311 GX Dordrecht, The Netherlands To the loving memory of my father and to my mother Preface Continuum Mechanics represents a logical evolutionary step in the development of several engineering disciplines involved with the mechanics of deformable objects It provides a uniﬁed high-level description for the behaviour of liquids, solids and gases, by dealing only with the observed, macroscopic effects experienced by these substances in response to various external agents What distinguishes Continuum Mechanics from other physical theories like, for example, Quantum Mechanics or Statistical Physics, is that all the ﬁne microscopic details are disregarded; in particular, it is assumed that the highly discontinuous atomic structure of matter can be replaced by a smoothed hypothetical body, usually referred to as a ‘continuum’.1 The upshot of introducing this simplifying hypothesis is intimately connected with the powerful tools of Vector Calculus involved in formulating the classical mathematical models of deformable continua This book represents an outgrowth of more than 15 years of teaching various courses related to mathematical modelling at several universities in the UK My audiences have largely been ﬁnal-year undergraduates on Mathematics Honours programmes; on several occasions, I also delivered similar material to ﬁrst-year (applied maths) Ph.D students through a couple of regional Doctoral Training Centres (in Glasgow and Nottingham) An embryonic collection of supplementary classroom notes was ﬁrst written around 2002/04, and covered most of the material in the ﬁrst four chapters As time went by, the scope of the narrative evolved from a mere outline to a more detailed treatment of the subject, which included additional topics related to Linear Elasticity and several other adjacent areas The ﬁnal result, the printed material within these pages, represents a scaled-down attempt to produce the kind of introductory textbook I wish I had as an undergraduate student A question likely to arise in the prospective reader’s mind is, probably, why another book on these particular topics? It is also reasonable to ask in what way this text is different from others available The answers lie partly in the subtitle: one The word continuous comes from the Latin root (‘together with’) and tenere (‘hold’), meaning ‘uninterrupted’ or ‘holding together’ ‘Continuum Mechanics’ is the mechanics of continuous media vii viii Preface of the intended goals was to produce a reasonably self-contained text that promotes the elementary use of coordinate-free equations and capitalises on two staple courses in undergraduate Mathematics curricula: Linear Algebra and Vector Calculus In the UK, courses that cover these topics are taken by most students in the ﬁrst years of their studies In a sense, both Continuum Mechanics and Linear Elasticity can be regarded as natural extensions of the aforementioned core subjects, and allow a mix of calculational work (both routine and advanced) with more theoretical material Continuum Mechanics helps students develop physical intuition, while Linear Elasticity builds conﬁdence in the mathematical techniques studied in other courses Because the two main topics of the book are well-established academic disciplines and have so many separate branches, I chose to concentrate on those chapters that best reveal the spirit of Rational Mechanics and are still within the grasp of the intended undergraduate audience By restricting the selection of topics discussed, an attempt was made to develop the ideas clearly, logically and without gaps Many of the derivations included in the next chapters give deliberately more details than strictly necessary or than most other similar books at this level give In particular, ample amount of support is provided for many of the formal vector and tensor manipulations, so that beginning students can develop a conﬁdent understanding and an appreciation for a range of calculations in different formats There is also less emphasis on stereotyped problems, which I believe not facilitate much personal discovery during the learning process To this end, a generous supply of exercises of varying degrees of difﬁculty is included at the end of each chapter (53 pages in total); these are regarded an integral part of the text and form one of its unique features The student who wants to achieve a thorough grasp of the material discussed in this book must invest some non-trivial amount of time in solving at least some of them before moving on to the next chapter The exercises included have been gathered over a period of years for various classes I have given; some are taken from others’ textbooks and papers, while some are modiﬁed or original It is no longer possible to properly acknowledge the original sources My debt to earlier writers is certainly substantial As already mentioned, the text assumes familiarity with elementary Linear Algebra and standard Vector Calculus; prior exposure to Vector Mechanics in three dimensions is desirable, but not absolutely essential Since it is my experience that almost no student retains much detailed information from one semester to the next, the ﬁrst chapter includes a convenient outline of matrix algebra, linear spaces, vectors and Euclidean point spaces Complex Analysis makes several ‘guest appearances’ in the second part of the book It is ﬁrst mentioned briefly in the chapter on torsion, but it plays only a marginal role and can be avoided (if one so desires) However, the last chapter deals with the application of Fourier integral transforms to a particular class of Linear Elasticity problems Although Complex Analysis does have a stronger presence in that chapter, most of our manipulations will be formal, and the inverse transforms will be carried out with the help of a small number of standard integrals One of the appendices at the end of the text summarises all the information that is needed for making that chapter reasonably Preface ix self-contained There is also another appendix in which the solution of the biharmonic equation is discussed from a number of different perspectives, one of which hinges on the use of more advanced Complex Analysis; again, that part can be skipped without any interference with the rest of the text I should perhaps emphasise that this is not a text on ‘Applied Analysis’ or ‘PDEs’, and I permit a certain degree of carelessness in restrictions and conditions Although some results are stated as ‘theorems’ and ‘propositions’, I omit rigorous proofs and offer instead informal arguments and examples that either shed light or amplify the formal statements That is partly because I intended this book to be no more than an introduction, one that could act as a bridge for the gap between the simple-minded accounts and the really advanced treatments of Continuum Mechanics and Linear Elasticity available in the literature I hope it will be read in conjunction with the classics in these areas, and that it will encourage further exploration The only consolation I can offer to readers who feel frustrated about my lack of rigorousness, or the number of times I seem to be cutting corners, is that the book is not addressed to them The manuscript was developed on and off over a period of years, mostly by compiling a series of handouts and hand-written notes used in my past classes A ﬁrst draft of the book was produced around 2010/12 Due to objective circumstances, I was unable to continue with this project for many years afterwards; it was eventually ﬁnalised during the academic year 2018/19 (in short bursts of activity, whenever my day job permitted) Given the prolonged gestation period of the manuscript and the long hiatus, it would be rather difﬁcult to accurately track down all sources that I consulted at various times For this reason, I have refrained from adding many bibliographical references in the main text, which would only have served as a distraction in a book of this type; a list of general references that have helped me clarify various points regarding the material presented is mentioned at the end of each chapter Of course, I make no claim for originality in the basic theory, other than in the organisation and details of presentation I would like to express my sincere thanks to the publisher, Springer Media, for not giving up on this project Their patience and understanding in waiting for the delivery of a long-overdue manuscript is deeply appreciated As I have learned through my own experience, the best lectures are a two-way conversation between the lecturer and his class By its very nature, a book limits the dialogue and some of that elusive quality is lost Nevertheless, I would be very pleased to receive questions, comments and criticism at cdc3p@yahoo.com An errata list will be available on my personal website, www.cipriancoman.net, and will be updated regularly Clifton Grove, Nottingham, UK June 2019 Ciprian D Coman Contents Part I Elements of Continuum Mechanics Vector, Tensors, and Related Matters 1.1 Matrices 1.2 Vector Spaces 1.3 Euclidean Vector Spaces 1.4 Euclidean Point Spaces 1.5 Second-Order Tensors: Fundamentals 1.6 Examples of Tensors: Elementary Projections 1.7 Basic Properties of Tensors 1.8 Linear Mappings as Geometric Transformations 1.9 Transformation Rules for a Change of Basis 1.10 Higher Order Tensors 1.11 A Special Property: Isotropy 1.12 Pseudo-scalars/vectors/tensors 1.13 Other Uses of Vector and Tensor Products 1.14 Fourth-Order Tensors 1.15 Finite Rotations 1.16 On the Relationship Between Skw and V 1.17 More General Bases 1.17.1 Cylindrical Polar Coordinates 1.17.2 Spherical Polar Coordinates 1.17.3 Physical Components for Vectors and Tensors 1.18 Invariants 1.19 Eigenvalues 1.20 Some Important Theorems 1.21 Projections in Lin 3 11 18 20 25 26 32 35 38 40 41 43 50 56 59 61 63 65 65 67 71 76 81 xi 504 Appendix E: The Bi-harmonic Equation Same annular domain as above, the edge |z| = b is held fixed, while the edge |z| = a undergoes a rigid-body infinitesimal displacement d > in the positive x-direction: ϕ(z) = A log z + Bz , χ (z) = C z log z + Dz + F/z (A, B, C, D, F ∈ R) Infinite plate with a traction-free hole |z| = a and a uniform pressure P applied at infinity: ϕ(z) = Az, χ (z) = B log z (A, B ∈ R), A := − P , B := a P Infinite plate with a traction-free hole |z| = a subjected to a uniaxial tension T11 = T0 , T22 = T12 = applied at infinity: ϕ(z) = Az + B/z, χ (z) = C z + D log z + F/z (A, B, C, D, F ∈ R), A = −C := T0 , a T0 , B = −D := F := − a T0 Same infinite plate with a circular hole as above, but subjected to a pure shear T11 = T22 = 0, T12 = S0 > at infinity: ϕ(z) = i A/z, χ (z) = i Bz + i C/z (A, B, C ∈ R), A := a S0 , B := S0 , C := − a S0 Elastic half-plane y ≥ 0, subjected to a concentrated force of magnitude Q at the origin and directed along the positive direction of the y-axis, while the boundary y = is traction-free: ϕ(z) = A log z, χ (z) = Bz log z, A = B := − iQ 2π Infinite elastic plate subjected to an in-plane concentrated force P + i Q at the origin: ϕ(z) = A log z, χ (z) = Bz log z, A := − E.3 P +iQ , 2π(1 + κ) B := κ(P − i Q) 2π(1 + κ) Polar Coordinates We explore various particular solutions of the bi-harmonic equation (E.1) when the Airy stress function is expressed in polar coordinates, i.e Φ = Φ(r, θ ) We recall that, in this case, ∇ ≡ er ∂ ∂ + eθ , ∂r r ∂θ ∇2 ≡ ∇ · ∇ = ∂2 ∂2 ∂ + + ∂r r ∂r r ∂θ (E.40) Appendix E: The Bi-harmonic Equation 505 As a first observation, let us note that the functions r cos θ F(r, θ ) , r F(r, θ ) , r sin θ F(r, θ ) , (E.41) are bi-harmonic in Σ ⊂ C provided that ∇ F(r, θ ) = for (r, θ ) ∈ Σ; this follows directly from (E.3) The linear function θ is trivially harmonic, so by taking F → θ in (E.41) we find a new set of bi-harmonic functions r θ cos θ , r θ sin θ , r 2θ (E.42) Similarly, with F → log r in (E.41), it transpires that r log r cos θ , r log r sin θ , r log r (E.43) are also particular solutions of the bi-harmonic equation (E.1) The Laplacian of the product of two real-valued functions F j ( j = 1, 2) can be readily calculated with the help of the formula ∇ (F1 F2 ) = F1 ∇ F2 + F2 ∇ F1 + ∇ F1 · ∇ F2 Letting F1 → θ , F2 → log r in this result confirms that θ log r is a harmonic function, and then F → θ log r in (E.41) generates three further bi-harmonic functions, θr log r cos θ , θr log r sin θ , θr log r (E.44) The Almansi representation (E.2) can also be adapted to the polar-coordinate setting In its modified form it asserts that any bi-harmonic function Φ can be expressed in terms of two arbitrary harmonic functions Φ j = Φ j (r, θ ) ( j = 1, 2), according to Φ(r, θ ) = Φ (r, θ ) + r Φ (r, θ ) (E.45) Straightforward direct calculations indicate that ∂ ∂ ∇ ( ) = ∇ ( ) , ∂θ ∂θ ∇2 r ∂ ∂ ( ) = + ∂r ∂r ∇ ( ) , where the ‘dots’ stand for any expression depending on the polar coordinates (r, θ ) An immediate consequence is that, if F = F(r, θ ) is a harmonic function, then ∂F ∂θ and r ∂F ∂r represent harmonic functions as well Furthermore, by using mathematical induction one can check that the same is true for r n (∂ n F/∂r n ) for n ≥ A more systematic way of obtaining harmonic functions is by looking for solutions with separable variables for the Laplace equation ∇ F = By writing 506 Appendix E: The Bi-harmonic Equation F(r, θ ) = R(r )Θ(θ ) , we find d2 R d 2Θ dR r2 =− + R(r ) dr r dr Θ(θ ) dθ (E.46) = ±k , (E.47) for some arbitrary k ∈ R There are three cases to be discussed If the common value of the first two expressions is taken to be +k (k = 0), then d 2Θ + k2Θ = , dθ dR d2 R k2 + R = 0, − dr r dr r2 whence R(r ) = C1 r k + C2 r −k , Θ(θ ) = C3 cos(kθ ) + C4 sin(kθ ) , (E.48) with C j ∈ R ( j = 1, 2, 3, 4) arbitrary constants If k = 0, R(r ) = C1 + C2 log r , Θ(θ ) = C2 + C4 θ (E.49) Considering (−k ) for the last term in (E.47), the equations become k2 dR d2 R + R = 0, + dr r dr r d 2Θ − k2Θ = , dθ with solutions R(r ) = C1 cos(k log r ) + C2 sin(k log r ) , Θ(θ ) = C3 ekθ + C4 e−kθ (E.50) Not all separable solutions found above have practical value in constructing Airy stress functions For example, the solutions (E.48) with k ≥ a positive integer, and (E.49) are the ones that play an important role in the examples discussed in Chap It should also be clear that in this case one can formally consider the superposition of the k-dependent solutions to generate yet another solution Up until now, we have constructed solutions of the bi-harmonic equation by taking advantage of the Almansi representation and a number of simple observations However, this is just one of several approaches available Since all the solutions found so far are in fact separable, it makes sense to look for such a solution directly in the bi-harmonic equation ∇ Φ = Let us note that by expanding the product of the Laplacian operators, the equation to be solved becomes ∂ 4Φ ∂ 3Φ ∂ 2Φ ∂Φ + − 2 + ∂r r ∂r r ∂r r ∂r ∂ 4Φ ∂ 3Φ + + − r ∂r ∂θ r ∂r ∂θ r ∂ 4Φ ∂ 2Φ + ∂θ ∂θ = (E.51) Appendix E: The Bi-harmonic Equation 507 Radially symmetric solutions correspond to Φ being independent of the azimuthal coordinate θ , so (E.51) reduces to d 4Φ d 3Φ d 2Φ dΦ + − + = 0, dr r dr r dr r dr (E.52) which is easily seen to be an Euler differential equation in which the unknown can be taken to be (dΦ/dr ) Such equations are routinely solved by making the substitution r = e ζ , which transforms (E.52) into a constant-coefficient equation for a new function of ζ (e.g., see [45]) Without going into details, after returning to the original independent variable ζ , the final result turns out to be Φ(r ) = C1 log r + C2 r + C3r log r + C4 , (Ci ∈ R, i = 1, 2, 3, 4) (E.53) If the situation of interest involves a configuration in which the θ -dependence is periodic, i.e Φ(r, θ + 2π ) = Φ(r, θ ), it makes sense to look for solutions of the form Φ(r, θ ) = RC (r ) cos(nθ ) or Φ(r, θ ) = R S (r ) sin(nθ ), with RC , R S functions to be determined and n a positive integer (n ≥ 1) Substituting this assumed form of solution into (E.51) yields the same equations for both RC and R S (identified as R below), d4 R d3 R + − dr r dr 2n + r2 d2 R + dr 2n + r3 n (n − 4) dR + R = dr r4 This is, again, an Euler differential equation, and we write r = e ζ , so that it becomes dR d3 R d2 R d4 R + n (n − 4) R = − − 2(n − 2) + 4n dζ dζ dζ dζ Next, we look for an exponential solution R(ζ ) = eλζ Routine calculations show that the characteristic equation is (λ2 − n )(λ2 − 4λ + − n ) = , whence the characteristic roots are λ1,2 = ±n and λ3,4 = ± n For n ≥ these roots lead to the four distinct solutions r n cos(nθ ) , r −n cos(nθ ) , r n+2 cos(nθ ) , r −n+2 cos(nθ ) , (E.54) and another similar set corresponding to sin(nθ ) For n = the situation is complicated by the fact that two of the characteristic roots coalesce; this leads to the appearance of a logarithmic term (e.g., see [46]), so in this case (E.54) are replaced by r cos(θ ) , r log r cos(θ ) (E.55) r cos(θ ) , r −1 cos(θ ) , 508 Appendix E: The Bi-harmonic Equation The formal superposition of a linear combinations of the functions (E.54) for n = 1, 2, leads to a (particular) solution of the Airy stress function Of course, the resulting solution can be combined with the radially symmetric expression (E.53), as well as a linear combination of (E.55), to obtain an even more general solution of the bi-harmonic equation We state below such a representation result, which includes a few additional (multi-valued) terms, Φ(r, θ ) = A + Br cos θ + Cr sin θ + Dr θ + A0 + B0 r + C0 log r + D0 r log r + A1c r + B1c r + C1c r −1 + D1c r log r cos θ + A1s r + B1s r + C1s r −1 + D1s r log r sin θ ∞ + n=2 ∞ + Anc r n + Bnc r n+2 + Cnc r −n + Dnc r −n+2 cos (nθ ) Ans r n + Bns r n+2 + Cns r −n + Dns r −n+2 sin (nθ ) , (E.56) n=2 where A, B, C, D and their indexed counterparts are all arbitrary constants The above result was originally derived by J H Michell in 1899 and has sufficient flexibility to accommodate the solution of a fairly large number of particular problems As seen in Chap 8, many interesting problems involving configurations whose boundaries are very simple in cylindrical polar coordinates can be solved by taking only a few of the terms of expression (E.56) The arbitrary constants that appear therein are determined by expanding the given boundary conditions in suitable Fourier θ -series, and then using the definitions of the stresses or displacements based on the Airy stress function given above; recall that the former can be calculated from Trr = ∂ 2Φ ∂Φ + , r ∂r r ∂θ Tθθ = ∂ 2Φ , ∂r Tr θ = − ∂ ∂r ∂Φ r ∂θ (E.57) Choosing the right form of Φ for a specific situation is usually a trial-and-error process This step is facilitated by knowledge of the type of stresses generated by each term in (E.56); for convenience, these expressions are summarised in Table E.1 Also, we include the expressions of the corresponding displacements in Tables E.2 and E.3—this latter piece of information is particularly useful if the boundary conditions or any symmetry considerations are directly linked to kinematics (the information in these tables is adapted from Sobrero’s book [47]) A less obvious feature of the polar-coordinate formulation of Plane Elasticity stems from the fact that such an approach is also relevant for various unbounded + ≡ {(x, y) ∈ E2 | y ≥ 0} can be domains For example, the upper half-plane Ω∞ + regarded as the “limit” of the semi-disk ΩΔ ≡ {(x, y) ∈ E2 | x + y < Δ2 , y ≥ 0} Appendix E: The Bi-harmonic Equation 509 Table E.1 Stress components associated with the various individual terms in the expression of the Airy stress function (E.56) Φ(r, θ) Trr Tθθ log r r −2 −r −2 r log r log r + log r + r log r cos θ r −1 cos θ r −1 cos θ r −1 sin θ r log r sin θ r −1 sin θ r −1 sin θ −r −1 cos θ θ 0 r −2 θr 2θ 2θ −1 θr cos θ −2r −1 sin θ 0 θr sin θ 2r −1 cos θ 0 r −1 cos θ −2r −3 cos θ 2r −3 cos θ −2r −3 sin θ r −1 sin θ −2r −3 sin θ 2r −3 sin θ 2r −3 cos θ r cos θ 2r cos θ 6r cos θ 2r sin θ r sin θ 2r sin θ 6r sin θ −2r cos θ r n cos(nθ) −n(n − 1)r n−2 cos(nθ) n(n − 1)r n−2 cos(nθ) n(n − 1)r n−2 sin(nθ) r n sin(nθ) −n(n − 1)r n−2 sin(nθ) n(n − 1)r n−2 sin(nθ) −n(n − 1)r n−2 cos(nθ) r n+2 cos(nθ) −(n + 1)(n − 2)r n cos(nθ) (n + 2)(n + 1)r n cos(nθ) n(n + 1)r n sin(nθ) r n+2 sin(nθ) −(n + 1)(n − 2)r n sin(nθ) (n + 2)(n + 1)r n sin(nθ) −n(n + 1)r n cos(nθ) r −n cos(nθ) −n(n + 1)r −n−2 cos(nθ) n(n + 1)r −n−2 cos(nθ) −n(n + 1)r −n−2 sin(nθ) Tr θ r −n sin(nθ) −n(n + 1)r −n−2 sin(nθ) n(n + 1)r −n−2 sin(nθ) n(n + 1)r −n−2 cos(nθ) r −n+2 cos(nθ) −(n + 2)(n − 1)r −n cos(nθ) (n − 2)(n − 1)r −n cos(nθ) −n(n − 1)r −n sin(nθ) r −n+2 sin(nθ) −(n + 2)(n − 1)r −n sin(nθ) (n − 2)(n − 1)r −n sin(nθ) n(n − 1)r −n cos(nθ) as Δ → ∞; similar statements can be made about an elastic plane or a quarter-plane (with the obvious modifications) An important caveat regarding the applicability of formula (E.56) concerns the class of infinite wedge-shaped two-dimensional domains For such configurations it is only in a few isolated cases that we can still resort to the Michell solution in its original form (as stated above) Strictly speaking, the separation of variables that led to (E.3) was based on taking Φ(r, θ ) = R(r ) exp(inθ ) with a positive n ∈ Z; by changing n → s ∈ R the corresponding solutions can be cast as, Φ(r, θ ) = r s A sin(sθ ) + B cos(sθ ) + C sin((s − 2)θ ) + D cos((s − 2)θ ) , where A, B, C, D ∈ R Several simple examples on how this solution is applied in concrete examples can be found in [22, 38, 48]; a more sophisticated approach based on a special type of (Mellin) integral transforms (e.g., see [25, 29, 49]) is included in [4, 24] 510 Appendix E: The Bi-harmonic Equation Table E.2 The components of plane stress displacement fields associated with the various individual terms in the expression of the Airy stress function (E.56) The corresponding components for plane strain are obtained by making the substitutions ν → ν/(1 − ν) and E → E/(1 − ν ) in the formulae recorded below (see Chap 8) Φ(r, θ) ur uθ log r + ν −1 − r E 2(1 − ν) 1+ν r log r − r E E 1−ν log r cos θ + θ sin θ E E 1+ν cos θ − 2E 1−ν log r sin θ − θ cos θ E E 1+ν sin θ − 2E θr E 1−ν − log r sin θ+ θ cos θ E E 1+ν sin θ − 2E 1−ν log r cos θ + θ sin θ E E 1+ν cos θ + 2E − 2(1 − ν) θr E 1−ν − log r sin θ + θ cos θ E E 1+ν sin θ + 2E 1−ν log r cos θ + θ sin θ E E 1+ν − cos θ 2E r log r E 1−ν − log r cos θ − θ sin θ E E 1+ν cos θ − 2E 1−ν − log r sin θ + θ cos θ E E 1+ν sin θ − 2E r −1 cos θ + ν −2 r cos θ E + ν −2 r sin θ E r −1 sin θ + ν −2 r sin θ E − r cos θ − 3ν r cos θ E 5+ν r sin θ E r sin θ − 3ν r sin θ E − r log r r log r cos θ r log r sin θ θ θr θr cos θ θr sin θ + ν −1 r E − + ν −2 r cos θ E 5+ν r cos θ E Appendix E: The Bi-harmonic Equation 511 Table E.3 Same as Table E.2 Φ(r, θ) ur r n cos(nθ) − r n sin(nθ) r n+2 cos(nθ) r n+2 sin(nθ) r −n cos(nθ) r −n sin(nθ) r −n+2 cos(nθ) r −n+2 sin(nθ) n(1 + ν) n−1 r cos(nθ) E n(1 + ν) n−1 − sin(nθ) r E − n − (2 + n)ν n+1 cos(nθ) r E − n − (2 + n)ν n+1 sin(nθ) r E n(1 + ν) −n−1 cos(nθ) r E n(1 + ν) −n−1 sin(nθ) r E + n + (n − 2)ν −n+1 cos(nθ) r E + n + (n − 2)ν −n+1 sin(nθ) r E uθ n(1 + ν) n−1 r sin(nθ) E n(1 + ν) n−1 − cos(nθ) r E + n + nν n+1 sin(nθ) r E + n + nν n+1 − cos(nθ) r E n(1 + ν) −n−1 sin(nθ) r E n(1 + ν) −n−1 − cos(nθ) r E − n − νn −n+1 − sin(nθ) r E − n − νn −n+1 cos(nθ) r E References Chen HC (1985) Theory of electromagnetic waves: a coordinate-free approach McGraw-Hill Book Company, New York Gibbs JW, Wilson EB (1901) Vector analysis Yale University Press, New Haven Gurtin ME (1972) Theory linear theory of elasticity In: Truesdell C (ed) Handbuch der Physik Springer, Berlin, pp 1–273 Lurie AI (2005) Theory of elasticity Springer, Berlin Pach K, Frey T (1964) Vector and tensor analysis Terra, Budapest Schumann W, Dubas M (1979) Holographic interferometry Springer, Berlin Schumann W, Zürcher J-P, Cuche D (1985) Holography and deformation analysis Springer, Berlin Weatherburn CE (1960) Advanced vector analysis G Bell and Sons Ltd, London 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Problems of mathematical physics PrenticeHall Inc, Englewood Cliffs 21 Tolstov GP (1976) Fourier series Dover Publications, Mineola 22 Little WR (1973) Elasticity, Prentice-Hall Inc, Englewood Cliffs, New Jersey 23 Teodorescu PP (2013) Treatise on classical elasticity Springer, Dordrecht 24 Uflyand YaS (1968) Survey of articles on the applications of integral transforms in the theory of elasticity Nauka, Leningrad, URSS (in Russian) 25 Debnath L, Bhatta D (2014) Integral transforms and their applications, 2nd edn CRC Press, Boca Raton 26 Kanwal RP (2004) Generalized functions: theory and applications, 3rd edn Birkhäuser, Boston 27 Gelfand IM, Shilov GE (1964) Generalized functions: properties and operations, vol Academic Press, New York © Springer Nature B.V 2020 C D Coman, Continuum Mechanics and Linear Elasticity, Solid Mechanics and Its Applications 238, https://doi.org/10.1007/978-94-024-1771-5 513 514 References 28 Vladimirov VS (1979) Generalized functions in mathematical physics Mir Publishers, Moscow 29 Zemanian AH (2003) Distribution theory and transform analysis Dover Publications, New York 30 Brown JW, Churchill RV (2008) Complex variables and applications McGraw-Hill, New York 31 Ablowitz MJ, Fokas AS (1997) Complex variables Cambridge University Press, Cambridge 32 Lavrentiev M, Shabat B (1977) Méthodes de la Théorie des Fonctions d’une Variable Complexe Mir Publishers, Moscow 33 Sidorov YuV, Fedoryuk MV, Shabunin MI (1985) Lectures on the theory of functions of a complex variable Mir Publishers, Moscow 34 Saff EB, Snider AD (1976) Fundamentals of complex analysis for mathematics, science and engineering, Prentice Hall Inc, Englewood Cliffs 35 Biezeno CB, Grammel R (1955) Engineering dynamics, vol (Theory of Elasticity) Blackie & Son Ltd., London 36 Girkmann K (1959) Flächentragwertke, 2nd edn Springer, Wien (in German) 37 Filonenko-Borodich M (1963) Theory of elasticity Mir Publishers, Moscow 38 Timoshenko SP, Goodier JN (1970) Theory of elasticity, International edn McGraw-Hill Book Company, Auckland 39 Muskhelishvili NI (1963) Some basic problems of the mathematical theory of elasticity P.Noordhoof Ltd., Groningen 40 Amenzade YuA (1979) Theory of elasticity Mir Publishers, Moscow 41 Leipholz H (1974) Theory of elasticity Noordhoff International Publishing, Leyden 42 Saada AS (1993) Elasticity: theory and applications Krieger Publishing Company, Malabar 43 Sokolnikoff IV (1956) Mathematical theory of elasticity McGraw-Hill Book Company Inc, New York 44 England AE (1971) Complex variable methods in elasticity Wiley-Interscience, London 45 Boyce WE, DiPrima RC (1996) Elementary differential equations and boundary value problems Wiley, New York 46 Coddington EA (1989) An introduction to ordinary differential equations Dover Publications, New York 47 Sobrero L (1942) Elasticidade Livraria Boffoni, Rio de Janeiro (in Portuguese) 48 Barber JR (2002) Elasticity Kluwer Academic Publishers, Dordrecht 49 Churchill RV (1972) Operational methods, 3rd edn McGraw-Hill Book Company, New York Index A Acceleration field, 128 Airy stress function, 379 Almansi Representation Theorem, 491 Alternator, see Ricci’s permutation symbol Analytic function (in C), 483 Angular momentum (of Pt ), 170 Anisotropic (Cauchy-elastic material), 214 Anticlastic curvature, 279 Antiplane stress/strain, 274 Axial vector, 31 B Bar bending plane, 268 flexural rigidity, 269 neutral axis, 268 neutral plane, 268 Beltrami–Michell equation, 299 Bi-harmonic complex potentials, 497 equation, 380 inhomogeneous, 409 Michell solution, 508 Goursat representation, 496 Blatz-Ko elastic material, 221 Boundary conditions linear elasticity, 253 plane elasticity, 373 Boundary-value problem linear elasticity, 253, 254 plane elasticity, 373 Bulk modulus, 251 C Cauchy-elastic materials, 209 Cauchy–Green left deformation tensor, 141 right deformation tensor, 139 Cauchy–Riemann equations, 483 Cauchy’s Stress Theorem, 172 Cauchy traction vector, 168 Cayley–Hamilton equation, 74 Centroid, 468 axial symmetry, 471 polar symmetry, 472 Cesàro–Volterra formula, 293 Change of observer, 198 Co-axial tensors, 78 Compatible (infinitesimal) strain field, 289 Complex torsion potential, 350 Complex variable methods plane elasticity, 495–504 torsion, 350–354 Compliance tensor, 250 Concentrated force problems, 399, 425, 427, 432, 434–437, 441–442, 453, 457 Configuration current, 122 reference, 121 Conservative fields, 288 Constitutive Principle for Elastic Materials with Constraints, 228 Constitutive relation, 197 Contracted product definition, 43 double-dot, 44 triple-dot, 45 Convected rates of change, 237 © Springer Nature B.V 2020 C D Coman, Continuum Mechanics and Linear Elasticity, Solid Mechanics and Its Applications 238, https://doi.org/10.1007/978-94-024-1771-5 515 516 Conversion between plane stress and plane strain, 375–376, 410 Convolution, see Fourier transforms Coordinate transformation, 35–38, 411 Curl, 93, 95 D Deformation, 122 gradient, 133 rate, 150 Del operator, 91 Dilatation, 126 Dirac delta function, 479 integral representation, 482 sifting property, 481 Displacement field, 136 Divergence operator, 92, 94 Divergence Theorem basic form, 102 generalised, 104 Double-dot (contracted) products, 44 Double-vector product, 49 Dyad, dyadic, 21 E Edge dislocation, 311 Eigenvalues and eigenvectors definition of, 71 for symmetric tensors, 73 Elastic stored energy, 219 Elastostatics, linear, 254 Energy-balance equation, 219 Equation of continuity, 165 Equipollent, see forces, statically equivalent Equivalent motions, 199 Euclidean point space, 18 Euler–Bernoulli bending law, 269 Euler differential equations, 391, 409, 507 Eulerian description, 123 principal axes, 143 Exponential of a second-order tensor, 239 External power, 218 Extra stress, 228 Extremal tangential stresses, 183 F Flamant solution (half-plane), 399–403, 427 Forces body & surface, 166 statically equivalent, 262 Index Fourier transforms, 475 evaluation using integrals in C, 485 examples, 487 for multivariate functions, 477 properties convolution, 477 differentiation, 477 shifting, 477 useful integrals, 488 G Galilean transformation, 199 Generalised plane stress, 376 Gradient, 89 Green’s identities, 369 Green’s Theorem, 369 H Hadamard–Green elastic material, 220 Half-plane, elastic, 421, 423, 425 Heisenberg delta function, 426 Hessian, 90 Hooke’s Law anisotropic solid, 247 inverted form, 250 isotropic solid, 248 plane strain, 373–374 plane stress, 375–376 Huygens–Steiner Theorem, see inertia tensor Hydrostatic stress, 184 Hyperelastic (or Green-elastic) material, 220 I Ideal fluid, 233 Incompatibility operator definition, 285 properties, 287 Incompressibility, 229 Inertia tensor, 467 centroidal, 468 Huygens–Steiner Theorem, 468 moment of inertia Euler, 467 geometrical vs mechanical, 473 Parallel-Axis Theorem, 469 polar, 467 products of inertia, 469 relative to an axis, 468 principal axes, 470 second moment of area, 473 Index 517 Inextensibility, 230 Infinitesimal rotation tensor, 245 Infinitesimal strain tensor, 244 Internal constraint, 227 Invariants, principal, 68 Isochoric, see motion Isotropic Cauchy-elastic material, 214 fields, 216 functional linear, 216 representation theorems, 216 Isotropic tensor, see tensor (second-order) J Jacobian matrix, 87 of the motion, 134 derivative of, 149 Jaumann stress rate, 238 K Kelvin problem (2D), 434–437 Kinematics, 121 Kinetic energy, 218 Kirsch problem, 389, 393 Kolosov complex potentials, see harmonic, complex potentials Kolosov’s formulae for displacements, 499 for stresses, 498 Kronecker delta, L Lagrangian description, 123 principal axes, 142 strain tensor, 141 Lamé constants, 248 parameters, 63 Laplacian, 94, 95 Linear momentum (of Pt ), 170 Localization Theorem, 165 M Mass density, 164 Material symmetry, 211 Material time derivative, 128 Matrix adjugate, cofactor of an element, determinant, identity, minor determinant, (multiplicative) inverse, singular, Michell solution, see bi-harmonic equation Modified stress function (for the torsion problem), 342 Moduli, elastic, 247 Modulus of compression, see bulk modulus Moment of inertia, see inertia tensor Mooney–Rivlin material, 231 Motion, 122 homogeneous, 143 isochoric, 138 N Nanson’s formula, 138 Navier–Lamé equation, 259 Navier–Stokes equation, 236 Neo-Hookean material, 231 Newtonian viscous fluid, 235 Normal stress, 179 bi- O Objective fields, 200 Objectively equivalent motions, see equivalent motions Observer invariance, see Principle of Material Frame-Indifference Octahedral planes/stresses, 193 Ogden material, 231 Orthogonal tensor, 27 P Papkovitch–Neuber representation, 444, 445 plane elasticity, 447 stress tensor, 447 Parallel-Axis Theorem, see inertia tensor Piola–Kirchhoff stress tensor, 186 Planar reversal, 58 Plane strain, 373 Plane stress, 375 Poisson’s ratio, 248 Polar Decomposition Theorem, 79 Pole (of order m ≥ 1), 484 Prandtl stress function, 339 Principal stress, 178 Principal stretches, 142 518 Principle of Action and Reaction, 171 of Conservation of Angular Momentum, 170 of Conservation of Linear Momentum, 170 of Determinism for the Stress, 206 of Local Action, 207 of Mass Conservation, 162 of Material Frame-Indifference, 202 of Superposition, 261 Projection of a vector along a given direction, 25 of a vector onto a given plane, 25 Pseudo-scalars, vectors, tensors, 41 Pure bending (cuboid), 127 Q Quadratic form, 257 Quarter-plane, elastic, 437 R Rate-of-strain tensor, see stretching tensor Reference triad, 20 Reiner–Rivlin fluid, 235 Representation theorems, see isotropic Residue, 484 Resultant force (on Pt ), 169 Resultant moment (on Pt ), 170 Reynolds’ Transport Theorem, 153 Ricci’s permutation symbol, 15 Rigid-body motion, 123 Rivlin–Ericksen tensors, 238 Rotation finite, 56 representation (involving two reference triads), 33 Rodrigues’ formula, 58 Rotation tensor, 142 S Saint-Venant’s compatibility equations, 290 Principle, 262 warping function, 328 Scalar triple product, 12, 13, 17 Screw dislocation, 311 Second moment of area, see inertia tensor Semi-inverse method of solution, 263 Shear modulus, 249 Shear (or tangential) stress, 179 Index Simple fluid, 232 materials, 207 shear, 125 torsion, 126 Simple extension, 125 Simply and multiply connected domains, 283 Solid (Cauchy-elastic) material, 214 Spectral Representation Theorem, 76 Spin tensor, 149 State of stress, 169 Stiffness tensor, 246 Stokes’ Theorem, 104 complex version, 369 Strain infinitesimal distortional, 245 volumetric/spherical, 245 Stress concentration, 392, 394 infinitesimal distortional, 250 spherical, 250 power, 218 Stretch in a given direction, 139 tensors (left & right), 142 Stretching tensor, 149 Summation convention, 14 Superposition Principle, see Principle of Superposition Sylvester’s criterion, 257 Symmetry group, 212 System of coordinates Cartesian, 20 curvilinear, 61 cylindrical, 63 spherical, 65 T Taylor’s expansion, 90 Tensor cofactor, 27 fourth-order, 50 conjugation (or square tensor) product, 54 generalised tensor product, 53 identity in Lin , 51 identity in SSym , 52 invertible, 52 positive-definite, 276 Index SSym , 52 strongly elliptic, 276 Sym , 51 transpose, 51 identities, see vector & tensor identities second-order, 20 invertible, 27 isotropic, 40 orthogonal, 27 positive (semi-) definite, 27 skew-symmetric, 27 skew-symmetric part, 30 Square-Root Theorem, 78 symmetric, 27 symmetric part, 30 transpose, 27 third-order, 38 Tensor of inertia, see inertia tensor Torsion constant, 333 Torsion of cylinders annular cross section circular, 362–364 elliptical, 364–365 elliptical cross section, 354–356 multiply connected cross section, 345– 347 rectangular cross section, 358–362 triangular cross section, 356–358 Torsional rigidity, 333 Trace, 44, 68 Trajectory of shear stress, 347 Transport formulae, 151 Triclinic material, 213 Twist per unit length, 325 519 U Uniaxial tension test, 249 Uniqueness, 255 V Vector & tensor identities, 96 Vector of a tensor, 32 Vector product between a vector and a tensor, 46 between two tensors, 48 between two vectors, 12, 17 Vector space definition, Euclidean, 11 finite-dimensional, linear transformation, standard basis, subspace, 11 Vector triple product, 17 Velocity field, 128 Velocity gradient, 148 Volumetric dilatation, 245 W Weingarten-Volterra dislocation, 308 Y Young’s modulus, 248 ... 2020 C D Coman, Continuum Mechanics and Linear Elasticity, Solid Mechanics and Its Applications 23 8, https://doi.org/10.1007/978-94-024-1771-5_1 Vector, Tensors, and Related Matters is called... to be defined in a natural way By definition, the difference of A, B ∈ Mm×n (R ), denoted by A − B, is obtained by adding to A the additive inverse of B, or A − B := A + (−B) The addition and subtraction... More generally, three vectors a, b, c ∈ V form a basis if [a, b, c] = The basis is right-handed if [a, b, c] > and left-handed if [a, b, c] < If one or all vectors of such a right-handed system have
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