This page intentionally left blank P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 AN INTRODUCTION TO CONTINUUM MECHANICS This textbook on continuum mechanics reflects the modern view that scientists and engineers should be trained to think and work in multidisciplinary environments A course on continuum mechanics introduces the basic principles of mechanics and prepares students for advanced courses in traditional and emerging fields such as biomechanics and nanomechanics This text introduces the main concepts of continuum mechanics simply with rich supporting examples but does not compromise mathematically in providing the invariant form as well as component form of the basic equations and their applications to problems in elasticity, fluid mechanics, and heat transfer The book is ideal for advanced undergraduate and beginning graduate students The book features: derivations of the basic equations of mechanics in invariant (vector and tensor) form and specializations of the governing equations to various coordinate systems; numerous illustrative examples; chapter-end summaries; and exercise problems to test and extend the understanding of concepts presented J N Reddy is a University Distinguished Professor and the holder of the Oscar S Wyatt Endowed Chair in the Department of Mechanical Engineering at Texas A&M University, College Station, Texas Dr Reddy is internationally known for his contributions to theoretical and applied mechanics and computational mechanics He is the author of over 350 journal papers and 15 books, including Introduction to the Finite Element Method, Third Edition; Energy Principles and Variational Methods in Applied Mechanics, Second Edition; Theory and Analysis of Elastic Plates and Shells, Second Edition; Mechanics of Laminated Plates and Shells: Theory and Analysis, Second Edition; and An Introduction to Nonlinear Finite Element Analysis Professor Reddy is the recipient of numerous awards, including the Walter L Huber Civil Engineering Research Prize of the American Society of Civil Engineers (ASCE), the Worcester Reed Warner Medal and the Charles Russ Richards Memorial Award of the American Society of Mechanical Engineers (ASME), the 1997 Archie Higdon Distinguished Educator Award from the American Society of Engineering Education (ASEE), the 1998 Nathan M Newmark Medal from the ASCE, the 2000 Excellence in the Field of Composites from the American Society of Composites (ASC), the 2003 Bush Excellence Award for Faculty in International Research from Texas A&M University, i 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 and the 2003 Computational Solid Mechanics Award from the U.S Association of Computational Mechanics (USACM) Professor Reddy is a Fellow of the American Institute of Aeronautics and Astronautics (AIAA), the ASME, the ASCE, the American Academy of Mechanics (AAM), the ASC, the USACM, the International Association of Computational Mechanics (IACM), and the Aeronautical Society of India (ASI) Professor Reddy is the Editorin-Chief of Mechanics of Advanced Materials and Structures, International Journal of Computational Methods in Engineering Science and Mechanics, and International Journal of Structural Stability and Dynamics; he also serves on the editorial boards of over two dozen other journals, including the International Journal for Numerical Methods in Engineering, Computer Methods in Applied Mechanics and Engineering, and International Journal of Non-Linear Mechanics ii 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 An Introduction to Continuum Mechanics WITH APPLICATIONS J N Reddy Texas A&M University iii 14:8 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521870443 © Cambridge University Press 2008 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2008 ISBN-13 978-0-511-48036-2 eBook (NetLibrary) ISBN-13 978-0-521-87044-3 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 ‘Tis the good reader that makes the good book; in every book he finds passages which seem confidences or asides hidden from all else and unmistakenly meant for his ear; the profit of books is according to the sensibility of the reader; the profoundest thought or passion sleeps as in a mine, until it is discovered by an equal mind and heart Ralph Waldo Emerson You cannot teach a man anything, you can only help him find it within himself Galileo Galilei v 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 vi 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 Contents Preface page xiii Introduction 1.1 Continuum Mechanics 1.2 A Look Forward 1.3 Summary problems Vectors and Tensors 2.1 Background and Overview 2.2 Vector Algebra 2.2.1 Definition of a Vector 2.2.2 Scalar and Vector Products 2.2.3 Plane Area as a Vector 2.2.4 Components of a Vector 2.2.5 Summation Convention 2.2.6 Transformation Law for Different Bases 2.3 Theory of Matrices 2.3.1 Definition 2.3.2 Matrix Addition and Multiplication of a Matrix by a Scalar 2.3.3 Matrix Transpose and Symmetric Matrix 2.3.4 Matrix Multiplication 2.3.5 Inverse and Determinant of a Matrix 2.4 Vector Calculus 2.4.1 Derivative of a Scalar Function of a Vector 2.4.2 The del Operator 2.4.3 Divergence and Curl of a Vector 2.4.4 Cylindrical and Spherical Coordinate Systems 2.4.5 Gradient, Divergence, and Curl Theorems 9 11 16 17 18 22 24 24 25 26 27 29 32 32 36 36 39 40 vii 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 viii October 3, 2007 Contents 2.5 Tensors 2.5.1 Dyads and Polyads 2.5.2 Nonion Form of a Dyadic 2.5.3 Transformation of Components of a Dyadic 2.5.4 Tensor Calculus 2.5.5 Eigenvalues and Eigenvectors of Tensors 2.6 Summary problems 42 42 43 45 45 48 55 55 Kinematics of Continua 61 3.1 Introduction 3.2 Descriptions of Motion 3.2.1 Configurations of a Continuous Medium 3.2.2 Material Description 3.2.3 Spatial Description 3.2.4 Displacement Field 3.3 Analysis of Deformation 3.3.1 Deformation Gradient Tensor 3.3.2 Isochoric, Homogeneous, and Inhomogeneous Deformations 3.3.3 Change of Volume and Surface 3.4 Strain Measures 3.4.1 Cauchy–Green Deformation Tensors 3.4.2 Green Strain Tensor 3.4.3 Physical Interpretation of the Strain Components 3.4.4 Cauchy and Euler Strain Tensors 3.4.5 Principal Strains 3.5 Infinitesimal Strain Tensor and Rotation Tensor 3.5.1 Infinitesimal Strain Tensor 3.5.2 Physical Interpretation of Infinitesimal Strain Tensor Components 3.5.3 Infinitesimal Rotation Tensor 3.5.4 Infinitesimal Strains in Cylindrical and Spherical Coordinate Systems 3.6 Rate of Deformation and Vorticity Tensors 3.6.1 Definitions 3.6.2 Relationship between D and E˙ 3.7 Polar Decomposition Theorem 3.8 Compatibility Equations 3.9 Change of Observer: Material Frame Indifference 3.10 Summary problems 61 62 62 63 64 67 68 68 71 73 77 77 78 80 81 84 89 89 89 91 93 96 96 96 97 100 105 107 108 Stress Measures 115 4.1 Introduction 4.2 Cauchy Stress Tensor and Cauchy’s Formula 115 115 14:8 P1: JzG References CUFX197-Reddy 978 521 87044 340 October 3, 2007 References [21] L E Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice Hall, Englewood Cliffs, NJ (1997) [22] G T Mase and G E Mase, Continuum Mechanics for Engineers, 2nd ed., CRC Press, Boca Raton, FL (1999) [23] J C Maxwell, “On the Calculation of the Equilibrium and the Stiffness of Frames,” Philosophical Magazine Serial 4, 27, 294 (1864) [24] N I Mushkelishvili, Some Basic Problems of the Mathematical Theory of Elas- ă ticity, Noordhoff, Groningen, the Netherlands (1963) [25] A W Naylor and G R Sell, Linear Operator Theory in Engineering and Sci- ence, Holt, Reinhart and Winston, New York (1971) [26] W Noll, The Non-Linear Field Theories of Mechanics, 3rd ed., Springer- Verlag, New York (2004) [27] R W Ogden, Non-Linear Elastic Deformations, Halsted (John Wiley & Sons), New York (1984) [28] W Prager, Introduction to Mechanics of Continua, Dover, Mineola, NY (2004) [29] J N Reddy, Applied Functional Analysis and Variational Methods in Engi- [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] neering, McGraw-Hill, New York (1986); reprinted by Krieger, Malabar, FL (1991) J N Reddy, Theory and Analysis of Elastic Plates and Shells, 2nd ed., Taylor & Francis, Philadelphia (2007) J N Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd ed., John Wiley & Sons, New York (2002) J N Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed., CRC Press, Boca Raton, FL (2004) J N Reddy, An Introduction to the Finite Element Method, 3rd ed., McGrawHill, New York (2006) J N Reddy and D K Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, 2nd ed., CRC Press, Boca Raton, FL (2001) J N Reddy and M L Rasmussen, Advanced Engineering Analysis, John Wiley, New York (1982); reprinted by Krieger, Malabar, FL (1991) H Schlichting, Boundary Layer Theory (translated from German by J Kestin), 7th ed., McGraw-Hill, New York (1979) L A Segel, Mathematics Applied to Continuum Mechanics, Dover, Mineola, New York (1987) ă W S Slaughter, The Linearized Theory of Elasticity, Birkhaser, Boston (2002) D R Smith and C Truesdell, Introduction to Continuum Mechanics, Kluwer, the Netherlands (1993) I S Sokolnikoff, Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York; reprinted by Krieger, Melbourne, FL (1956) A J M Spencer, Continuum Mechanics, Dover, Mineola, NY (2004) V L Streeter, E B Wylie, and K W Bedford Fluid Mechanics, 9th ed., McGraw-Hill, New York (1998) R M Temam and A M Miranville, Mathematical Modelling in Continuum Mechanics, 2nd ed., Cambridge University Press, New York (2005) S P Timoshenko and J N Goodier, Theory of Elasticity, 3rd ed., McGrawHill, New York (1970) C A Truesdell, The Elements of Continuum Mechanics, Springer-Verlag, New York (1984) C Truesdell and R A Toupin, “The Classical Field Theories,” in Encyclopedia ¨ of Physics, III/1, S Flugge (ed.), Springer-Verlag, Berlin (1965) C Truesdell and W Noll, “The Non-Linear Field Theories of Mechanics, in ă Encyclopedia of Physics, III/3, S Flugge (ed.), Springer-Verlag, Berlin (1965) 10:51 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 Answers to Selected Problems Chapter 1.1 The equation of motion is dv + αv = g, dt 1.2 c m The energy balance gives − 1.4 α= d ( Aq) + β P(T∞ − T) + Ag = dx The conservation of mass gives d( Ah) = qi − q0 , dt where A is the area of cross section of the tank (A = π D2 /4) Chapter 2.1 The equation of (or any multiple of it) the required line is C · [A − (A · eˆ B) eˆ B] = 2.2 The equation for the required plane is (A − B) × (B − C) · (A − C) = 2.6 (a) Sii = 12 (b) Si j S ji = 240 (e) Si j A j = {18 15 34}T 2.8 The vectors are linearly dependent 2.10 (a) The transformation is defined by    √1    eˆ   √2 eˆ =  14     −4 eˆ √ 42 −1 √ √3 14 √1 42 √1 √1 14 √5 42      eˆ    eˆ     eˆ 341 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 342 Answers to Selected Problems 2.12 Follows from the definition  [L] = 2.17 Note that ∂r ∂ xi √1   − 12 √1 √1 √1 − 12    = xi /r grad(r ) = 2r eˆ i ∂r = 2eˆ i xi = 2r ∂ xi Use of the divergence theorem gives the required result 2.18 Use the divergence theorem to obtain the required result 2.19 The integral relations are obvious (a) The identity is obtained by substituting A = φ ∇ψ for A into Eq (2.4.34) 2.20 See Problem 2.10(a) for the basis vectors of the barred coordinate system in terms of the unbarred system; the matrix of direction cosines [L] is given there Then the components of the dyad in the barred coordinate system are   − √1442   15 37 ¯ =  − √14 √  [ S] − 14  14  42 13 37 − 14√3 14 2.24 Begin with [(S · A) × (S · B)] · (S · C) = ei jk Si p S jq Skr A p Bq Cr obtain |S|e pqr − ei jk Si p S jq Skr = 2.25 Use the del operator from Table 2.4.2 to compute the divergence of the tensor S √ √ The 2.30 (a) λ1 = 3.0, λ2 = 2(1 + 5) = 6.472, λ3 = 2(1 − 5) = −2.472 (3) ˆ = ±(0.5257, eigenvector components Ai associated with λ3 are A 0.8507, 0) (c) The eigenvalues are λ1 = 4, λ2 = 2, (d) The eigenvalues are λ1 = 3, λ2 = 2, ˆ (1) = ± √1 (1, 0, 1) ated with λ1 is A λ3 = λ3 = −1 The eigenvector associ- (f) The eigenvalues are λ1 = 3.24698, λ2 = 1.55496, λ3 = 0.19806 The ˆ (2) = ±(0.591, −0.328, ˆ (1) = ±(0.328, −0.737, 0.591); A eigenvectors are A (3) ˆ = ±(0.737, 0.591, 0.328) −0.737); A 2.31 The inverse is [ A]−1   −2 1  = −2 −2  12 −2 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 Answers to Selected Problems 343 Chapter x t v = 1+t , a = (1+t) v   1 3.3 (c)    k1 3.4 (b) [C] =  k22 0  k1 e0 k2  3.5 (a) [F] = k2 0  cos At 3.6 (c) [F] =  − sin At 3.1 3.7 3.9  0  k32  0  k3 sin At cos At  0  + Bt (a) u1 (X) = AX2 , u2 (X) = BX1 , u3 (X) =   B2 A+ B (c) 2[E] =  A + B 0 A2 0   cosh t sinh t  (c) [F] = sinh t cosh t  0 3.11 (b) The angle ABC after deformation is 90 − β, where cos β = √ µ   3.12 (a) [E] = ([C] − [I]) = 3.13 u1 = e0 b 3.14 u1 = 3.15 E11 = 1+µ2  0 X2 , u2 = e0 b2 X22 , u2 = e1 X2 a b e12 +e22 X1 X2 a b2 + 12 X22 e12 +e22 a b2 , E22 = e2 X1 b a + 12 X12 e12 +e22 a b2 , 2E12 = e1 X1 b a + e2 X2 a b + 3.16 u1 = −0.2X1 + 0.5X2 , u2 = 0.2X1 − 0.1X2 + 0.1X1 X2 3.17 εrr = A, εrθ = 0, εr z = 0, εθθ = A, εzθ = Br + C r cos θ , εzz = 3.19 The linear components are given by ε11 = 3x12 x2 + c1 2c23 + 3c22 x2 − x23 , ε22 = − 2c23 + 3c22 x2 − x23 + 3c1 x12 x2 , 2ε12 = x1 x12 + c1 3c22 − 3x22 3.20 (b) The strain field is not compatible 3.21 (b) E11 (= Enn ) ≈ ae0 , a +b2 E12 (= Ens ) ≈ e0 2b a −b2 a +b2 − 3c1 x1 x22 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 344 October 3, 2007 Answers to Selected Problems 3.22 The principal strains are ε1 = and ε = 10−3 in./in The principal direction associated with ε1 = is A1 = eˆ − 2eˆ and that associated with ε = 10−3 is A2 = 2eˆ + eˆ 3.23 (c) u1 = cX1 X22 , u2 = cX12 X2 3.26 Use the definition (3.6.3) and Eqs (3.6.14) and (3.7.1) as well as the symmetry of U to establish the result 3.29 The function f (X2 , X3 ) is of the form f (X2 , X3 ) = A + BX2 + C X3 , where A, B, and C are arbitrary constants 5.0 0.40 2.2313 0.1455 , [U] = 0.4 1.16 0.1455 1.0671     2.121 0.707 0.707 0.707 3.36 [U] =  0.707 2.121  , [V] =  0.707 0.707  0 1.0 0 1.0 3.35 [C] = Chapter 4.3 (i)(a) tnˆ = 2(eˆ + 7eˆ + eˆ ) (c) σn = −7.33 MPa, σs = 12.26 MPa 4.4 (a) tnˆ = √13 (5eˆ + 5eˆ + 9eˆ )103 psi (b) σn = 6, 333.33 psi, σs = 1, 885.62 psi (c) σ p1 = 6, 656.85 psi, σp2 = 1, 000 psi, σ p3 = −4, 656.85 psi 4.5 σn = −2.833 MPa, σs = 8.67 MPa 4.6 σn = 0.3478 MPa, σs = 4.2955 MPa 4.9 σn = 3.84 MPa, σs = −17.99 MPa 4.11 σn = −76.60 MPa, σs = 32.68 MPa 4.12 σs = 90 MPa 4.13 σ p1 = 972.015 kPa, σ p2 = −72.015 kPa 4.14 σ p1 = 121.98 MPa, σ p2 = −81.98 MPa 4.15 σ1 = 11.824 × 106 psi, n(1) = ±(1, 0.462, 0.814) 4.17 λ1 = 23 , λ2 = 53 , λ3 = − 73 ; nˆ (1) = −0.577eˆ + 0.577eˆ + 0.577eˆ ˆ (1) = ±(0.42, 0.0498, −0.906) 4.18 λ1 = 6.856, A 4.19 (b) tn = −16.67 MPa, ts = 52.7 MPa 4.20 σ1 = 25 MPa, σ2 = 50 MPa, σ3 = 75 MPa; nˆ (1) = ± eˆ − 45 eˆ , nˆ (2) = ±eˆ , nˆ (3) = ± Chapter 5.6 (a) Satisfies (b) Satisfies 5.7 Q= b (3v0 − c) m3 /(s.m) eˆ + 35 eˆ 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 Answers to Selected Problems 345 5.8 (a) F = 24.12N (b) F = 12.06 N (c) Fx = 45 N 5.9 v(t) = v(t0 ) = g (x L − x02 ), a(t) = g (L2 L − x02 ) ≈ √ g x(t); L gL when L >> x0 5.14 The proof of this identity requires the following identities (here A is a vector and φ is a scalar function): ∇ · (∇ × A) = (1) ∇ × (∇φ) = (2) v · grad v = ∇ v2 − v × ∇ × v ∇ × (A × B) = B · ∇A − A · ∇B + A divB − B divA (3) (4) 5.17 v2 = 9.9 m/s, Q = 19.45 Liters/s 5.18 ρ f1 = 0, ρ f2 = a b2 + 2x1 x2 − x22 , ρ f3 = −4abx3 5.20 σ12 = − 2IP3 h2 − x22 , σ22 = I3 = 2bh3 5.21 (a) T = 0.15 N-m (b) When T = 0, ω0 = 477.5 rpm 5.22 ω = 16.21 rad/s = 154.8 rpm 5.24 v1 = 0.69 m/s, v2 = 2.76 m/s, loss = 5.3665 N · m/kg Chapter 6.2   37.8       σ   11       43.2    σ       27.0   22   σ23 = 10 Pa    21.6          σ      13   0.0      σ12   5.4 6.3 I1 = 108 MPa, I2 = 2, 507.76 MPa2 , I3 = 25, 666.67 MPa3 ; J1 = 500 × 10−6 , J2 = 235 × 10−9 , J3 = −32 × 10−12 6.4 I1 = 78.8 MPa, I2 = 1, 062.89 MPa2 , I3 = 17, 368.75 MPa3 6.5 J1 = 66.65 × 10−6 , J2 = 63, 883.2 × 10−12 , J3 = 244, 236 × 10−18 6.6 τ11 = 0, τ22 = 6.8 (1) Physical admissibility, (2) determinism, (3) equipresence, (4) local action, (5) material frame indifference, (6) material symmetry, (7) dimensionality, (8) memory, and (9) causality 2µk , 1+kt τ12 = µ 4tk x (1+kt)2 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 346 Answers to Selected Problems Chapter 7.1 σ11 = 96.88 MPa, σ22 = 64.597 MPa, σ13 = MPa, σ23 = 7.4 w0 7.6 w A = 0.656 in 7.7 wc = q0 a 64D wc = q0 a (1+ν)D 7.8 7.9 L wc = 7.10 wcb = 5F0 L3 48EI =− 43 q0 a 4800 D b Q0 16π D 5+ν 64 − 6+ν 150 log ab + 1− 4µ 3K 7.13 σrr = − + τ0 b2 2µa 7.11 uz(r ) = − ρga 4µ 7.15 uθ (r ) = σ12 = 4.02 MPa, 5+ν 1+ν 17q0 L4 384EI + σ33 = 48.443 MPa, r a − a2 b2 r2 a2 −1 , σθ z = 0, σzr = ρg r p, σθθ = σφφ = − − a r , b2 τ0 r2 σr θ = 2µ 3K p 7.18 σxx = 2D 3x y − 2y , σ yy = 2Dy3 , σxy = −6Dxy2 7.21 y 5a x y y3 + − b 2b2 a b b3 σxx = 3q0 10 σ yy = q0 σxy = 3q0 a x y2 1− 4b a b −2 − y y3 + b b , , 7.22 σxx = ∂2 τ0 = ∂y σxy = − 7.24 σrr = − 2πrf0 sin θ, − ∂2 2x 6xy 2a 6ay = 0, − + + , σ yy = b b b b ∂ x2 ∂2 τ0 =− ∂ x∂ y σθθ = 0, 1− 2y 3y2 − b b σr θ = 7.25 σ31 = ∂ µθ = x2 (x1 − a) ∂ x2 a σ32 = − The angle of twist is θ = ∂ µθ = x + 2ax1 − x22 ∂ x1 2a √ 3T 27µa 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 Answers to Selected Problems 347 7.27 The Euler equations are δw : − dw d GA φ + dx dx δφ : − d dx EI dφ dx −q =0 dw dx + GA φ + = q0 L 7.29 w(0) = u1 = − 24EI+kL 7.32 1 L = m1 l θ˙ + x˙ − 2l x˙ θ˙ sin θ + m2 x˙ 2 + m1 g(x − l cos θ ) + m2 gx + k(x + h)2 where h is the elongation in the spring due to the masses h = kg (m1 + m2 ) 7.33 ρl xă x or xă gl x = 7.34 ∂ ∂t ρA ∂w ∂t ∂2 ∂ x2 + EI ∂ 2w ∂ x2 − ∂2 ∂ x∂t ρI 2w xt = q 7.35 m(xă + ă cos θ − θ˙ sin θ ) + kx = F, m xă cos + ( + )ă + mg sin = 2a F cos θ 7.36 δu0 : − ∂ Nxx ∂ − f+ ∂x ∂t δw : − ∂ Qx ∂ −q+ ∂x ∂t δφ : − ∂ ∂ Mxx + Qx + ∂x ∂t ∂u ∂t = 0, ∂w ∂t = 0, m0 m0 m2 ∂φ ∂t = Chapter 8.2 (a) The pressure at the top of the sea lab is P = 1.2 MN/m2 8.3 ρ = 1.02 kg/m3 8.4 P = P0 + 8.5 P(y) = P0 + ρgh cos α − 8.7 The shear stress is given by mx3 θ0 τr z = − −g/mR , ρ = ρ0 + ¯ r dP + c1 dz r ¯ where d P/dz = ¯ dP dz + ρg y h mx3 θ0 , U(y) = =− −g/mR ρgh2 sin α 2µ hy − y2 h2 ¯ R r dP + (1 − α ) dz R log α R r , 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 348 Answers to Selected Problems 8.8 The velocity field is vθ (r ) = r12 r12 r2 r− 2 r − r2 If r1 = R and r2 = α R with < α < 1, we have vθ (r ) = R − α2 R r − α2 R r The shear stress distribution is given by τr θ = −2µ ρ −η 8.10 P = −ρgz + 8.12 vx (y, t) = U0 e R r r + c, where c = P0 + ρgz0 2 cos(nt − η) R r 3µV∞ 2R α2 1−α 2 R ∞ sin φ, P = P0 − ρgz − 3µV cos φ, where P0 is the pres8.15 τrφ = 2R r sure in the plane z = far away from the sphere and −ρgz is the contribution of the fluid weight (hydrostatic effect) 8.16 − drd (rqr ) + rρ Qe = 8.17 T(r ) = T0 + ∞ n=1 8.18 θ(x, t) = 8.20 T(r ) = T0 − 8.21 v y (x) = ρ Qe R2 4k 1− r R Bn sin λn x e−αλn t , Bn = µα 9k R03 1− ρr βr ga (T2 −T1 ) 12µ r R0 x a − L L f (x) sin λn x dx x a Chapter 9.1 −2H(t) + 2.5e−t + 0.5e−3t 9.2 J (t) = 9.3 J (t) = t η1 +η2 α2 = + 9.4 k1 G2 η1 Y(t) = − k2 e−t/τ , k1 (k1 +k2 ) + G2 η2 η1 +η2 Y(t) = k1 + k2 e−t/τ (1 − e−α2 t ) , Y(t) = η1 δ(t) + G2 e−t/τ2 , τ2 = G2 η2 k1 k2 k1 +k2 − e−λt + k1 et , = k1 +k2 ă where p0 = 9.5 q1 + q2 ă = p0 σ + p1 σ˙ + p2 σ, k1 p2 = k2 q1 = µ1 q2 = 9.6 Y(t) = k1 k2 µ2 λ1 −λ2 J (t) = k1 µ1 µ2 , p1 = + µ1 + , q0 = k1 η2 , k1 k2 µ1 µ2 , (λ1 − α)e−λ1 t − (λ2 − α)e−λ2 t ˙ where p0 = 9.7 q0 ε + q1 + q2 ă = p0 + p1 σ, q1 = + kk12 + ηη12 , q2 = kη12 9.9 q2 + p1 p0 η2 , p1 = k2 e−αt e−βt − + αβ α(β − α) β(β − α) e−βt βe−βt e−αt αe−αt − + p2 − + (β − α) (β − α) (β − α) (β − α) η2 G2 , 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 Answers to Selected Problems 349 The relaxation response is Y(t) = k1 + k2 e−αt + η1 δ(t) 9.10 σ(t) = [k1 + k2 e−αt + η1 δ(t)] ε0 + tk1 + 9.11 Y(t) = k1 + k2 e−t/τ , ¯ 9.12 E(s) = ¯ G(s) ¯ K(s) ¯ ¯ K(s)+ G(s) 9.13 ε(t) = σ1 t k1 + τ= , s ν(s) ¯ = −t/τ e k2 η k2 k2 α (1 − e−αt ) + η1 H(t) ε0 ¯ ¯ K(s)−2 G(s) ¯ ¯ 2[3 K(s)+ G(s)] , for t > t0 9.14 (a) 2G(t) = 2G0 [H(t) + τ δ(t)] dεi j (t ) t dt (c) σi j (t) = 2G(t)εi j (0) + G(t − t ) dt 9.15 σ(t) = ln(1 + t/C) 9.16 (a) wv (L, t) = 9.17 wv (L, t) = 9.18 w v (x, t) = 2τ E0 B e −A p0 q0 H(t) + P0 L3 3I q0 L4 360I 1− − e−t/τ + 9.19 P(t) = 2L − Aα E t P0 L3 3E0 I τ E0 x L t τ + E0 A q0 p1 −q0 p1 q1 q0 − 10 − t τ H(t) (b) w v (L, t) = P0 L3 3E0 I e − Aα E t e−(q1 / p1 )t x L +3 1− x L h(t), v − , σ(x, t) = −Ez ∂∂ xw2 = h(t) = where q0 L z 60I 1− x L x L h(t) δ0 E(t) + (δ1 − δ0 )E(t − t0 ) 9.20 The Laplace transformed viscoelastic solutions for the displacements and stresses are obtained from B¯ i (s) , u¯ r (r, s) = A¯ i (s)r + r2 4µ σrr (r, s) = (2µ + 3λ) A¯ i (s) − B¯ i (s), r ¯ σθθ (r, s) = σφφ (r, s) = [2s µ(s) ¯ + 3s λ(s)] A¯ i (s) + 4s µ(s) ¯ B¯ i (s), r3 where A¯ i (s) and B¯ i (s) are the same as Ai and Bi with ν and E replaced by ¯ s ν(s) ¯ and s E(s), respectively 10:53 P1: JZP Answers CUFX197-Reddy 978 521 87044 October 3, 2007 350 10:53 P1: draft index CUFX197-Reddy 978 521 87044 October 3, 2007 Index Absolute Temperature, 195 Airy Stress Function, 230, 231, 236, 269 Algebraic Multiplicity, 52 Almansi-Hamel Strain Tensor, 82 Ampere Law, 205 Analytical Solution, 224, 292, 297 Angular Displacement, 67 Angular Momentum, 5, 162 Angular Velocity, 13, 96 Anisotropic, 178 Approximate Solution, 224, 242 Axial Vector, 96 Axisymmetric Boundary Condition, 223 Axisymmetric Flow, 286 Axisymmetric Body, 114 Axisymmetric Heat Conduction, 295 Balance Equations, 132 Barotropic, 195 Beam Theory, 2, 103, 218, 238 Beltrami-Michell Equations, 212 Beltrami Equations, 214 Bernoulli Equations, 175 Betti’s Reciprocity Relations, 219, 222 Biaxial State of Strain, 113 Biharmonic Equation, 231, 235 Biharmonic Operator, 231 Bingham Model, 199 Body Couple, 162 Caloric Equation of State, 276 Canonical Relations, 166 Cantilever Beam, 103 Carreau Model, 198 Cartesian, 22 Cartesian Basis, 22, 33 Cartesian Coordinate System, 22, 27, 56, 62, 277, 280 Castigliano’s Theorem, 251, 254 Cauchy Elastic, 179 Cauchy Strain Tensor, 82, 115, 117 Cauchy–Green Deformation, 77 Cauchy–Green Deformation Tensor, 202 Cauchy-Green Strain Tensor, 99, 194 Cauchy’s Formula, 115, 118, 216, 247 Cayley–Hamilton Theorem, 55, 60 Chain, 157 Characteristic Equation, 48, 55, 124 Characteristic Value, 48 Characteristic Vectors, 48 Clapeyron’s Theorem, 216 Classical Beam Theory, 236 Clausius–Duhem, Clausius–Duhem Inequality, 170 Cofactor, 31 Collinear, 11 Compatibility Conditions, 101, 211 Compatibility Equations, 100, 103 Compliance, 182 Composite, Conduction, 203 Configuration, 62 Conservation of Angular Momentum, 161 Conservation of Energy Conservation of Linear Momentum Conservation of Mass, 143 Consistency, 198, Constant Strain Triangle, 253 Constitutive Equations, 4, 178, 211 Continuity Equation, 146, 147, 152 Continuum, Continuum Mechanics, 1, 61 Contravariant Components, 33 Control Volume, 147 Convection, 203 Convection Heat Transfer Coefficient, 203 Cooling Fin, 293, 294 Coordinate Transformations, 8, 115 Coplanar, 11 Corotational Derivative, 200 Correspondence Principle, 327, 328 Couette Flow, 285 Couple Partial Differential Equations, 227 Covariant Components, 34 Creep Compliance, 311 Creep Response, 313 Creep Test, 305 Creeping Flows, 289 Cross-Linked Polymer, 318 Curl, 36, 40 Cylindrical Coordinates, 39, 94, 149, 159, 278 Damping Coefficient, 261 Deformation, 62 Deformation Gradient Tensor, 68, 83, 91, 97, 152, 180 193 Deformation Mapping, 63, 73 351 10:55 P1: draft index CUFX197-Reddy 978 521 87044 October 3, 2007 352 Index Deformed Configuration, 63, 85 Del Operator, 36, 46 Density, 147 Deviatoric Components, 326 Deviatoric Components of Stress, 196 Deviatoric Stress, 124, 140 Deviatoric Tensor, 50 Diagonal Matrix, 25 Dielectric Materials, 206 Differential Models, 192 Differentials, 34 Dilatant, 198 Dilatation, 84 Dirac Delta Function, 307 Direction Cosines, 183 Directional Derivative, 35 Dirichlet Boundary Condition, 247 Displacement Field, 67 Dissipation Function, 281 Divergence, 36, 40 Dot Product, 11 Dual Basis, 34 Dummy Index, 18 Dyadics, 42 Eigenvalues, 48 Eigenvectors, 48, 99 Elastic, 179, 331 Elastic Stiffness Coefficients, 181 Elasticity Tensor, 190 Electric Field Intensity, 205 Electric Flux, 205 Electromagnetic, 205, 207 Elemental Surface, 158 Energetically Conjugate, 177 Energy Equation, 164 Engineering Constants, 185 Engineering Notation, 182 Engineering Shear Strains, 89 Entropy Equation of State, 170 Entropy Flux, 170 Entropy Supply Density, 170 Equation of State, 275 Equilibrium Equations, 211 Error Function, 302 Essential Boundary Condition, 247 Euclidean Space, 62 Euler–Bernoulli Beam Theory, 218, 235, 238, 249, 263 Euler Equations, 246 Euler Strain Tensor, 81, 82 Euler–Bernoulli Beam, 238 Euler–Bernoulli Hypothesis, Eulerian Description, 63, 66 Euler–Lagrange Equations, 259, 263, 273 Exact solution, 218, 242, 254, 286 Faraday’s Law, 205 Fiber-Reinforced Composite, 121 Film Conductance, 203, 204 Finger Tensor, 82, 202 First Law of Thermodynamics, 164 First Piola-Kirchhoff Stress Tensor, 128 First-Order Tensor, 8, 44 Fixed Region, 147 Fluid, 2, 275, Fourier’s Law, 2, 203, 203 Frame Indifference, 106, 195 Free Energy Function, 193 Free Index, 19 Generalized Displacements, 254 Generalized Forces, 254 Generalized Hooke’s Law, 180 Generalized Kelvin Voigt Model, 316, 324 Generalized Maxwell Model, 315 Geometric Multiplicity, 52 Gibb’s Energy, 166 Gradient, 40 Gradient Vector, 35 Gravitational Acceleration, 168 Green Elastic Material, 180 Green Strain Tensor, 71, 194 Green–Lagrange Strain, 79, 96 Green’s First Theorem, 58 Green’s Second Theorem, 58 Green–St Venant Strain Tensor, 79 Hamilton’s Principle, 257, 261, 263 Heat Conduction, 293 Heat Transfer, 3, 203, 276 Heat Transfer Coefficient, 204 Helmhotz Free Energy, 171 Hereditary Integrals, 323 Hermite Cubic Polynomials, 255 Herschel–Buckley Fluid, 199 Heterogeneous, 178 Homogeneous, 178 Homogenous Deformation, 71 Homogenous Stretch, 108 Hookean Solids, 179, 180 Hooke’s Law, Hydrostatic Pressure, 180, 194, 197 Hydrostatic Stress, 159, 124, 275 Hyperelastic, 180, 193 Ideal Elastic Body, 262 Ideal Fluid, 195 Ideal Gas, 167 Incompressible Fluid, 167, 286, 287 Incompressible Material, 166, 193, 281, Inelastic Fluids, 197 Infinitesimal Rotation Tensor, 91 Infinitesimal Strain Tensor, 89 Infinitesimal Strain Tensor Components, 89 Inhomogeneity, 183 Inner Product, 11 Integral Constitutive Equations, 323 Integral Models, 202 Internal Dissipation, 171 Internal Energy, 166 Interpolation Functions Invariant, 8, 44, 194 Invariant Form of Continuity Equation, 150 Invariants of Stress Tensor, 120 Inverse Methods, 224 Inviscid, 167, 195 Inviscid Fluids, 197, 282 Irreversible Process, 163, 170 Isochoric Deformation, 71 Isothermal, 193 Isotropic Body, 265 Isotropic Material, 178, 187 Isotropic Tensor, 45 Jacobian of a Matrix, 69 Jaumann Derivative, 200 Johnson–Segalman Model, 200 Kaye–Bkz Fluid, 202 Kelvin Voigt Model, 306, 315 10:55 P1: draft index CUFX197-Reddy 978 521 87044 October 3, 2007 Index Kinematic, 178 Kinematically Infinitesimal, 136 Kinematics, 4, 61 Kinetic Energy, 257 Kinetics, Kronecker Delta, 19, 44 Lagrange Equations of Motion, 259 Lagrangian Description, 63 Lagrangian Function, 259 Lagrangian Stress Tensor, 129 Lame´ Constants, 196 ´ Lame–Navier Equations, 212 Laplace Transform, 307 Laplace Operator, 41 Laplacian Operator, 36 Left Cauchy Stretch Tensor, 98 Left Cauchy–Green Deformation Tensor, 78 Leibnitz Rule, 146 Linear Displacement, 67 Linear Momentum, 5, 154 Linearly Independent, 11, 127 Lorentz Body Force, 207 Lower-Convected Derivative, 200 Magnetic Field Intensity, 205 Magnetic Flux Density, 205 Mapping, 66 Material Coordinate, 63 Material Coordinate System, 182 Material Derivative, 65 Material Description, 63 Material Frame Indifference, 105 Material Homogeneity, 111 Material Objectivity, 106 Material Plane of Symmetry, 183 Material Symmetry, 182 Material Time Derivative, 144, 148 Matrices, 24 Matrix Addition, 25 Matrix Determinant, 29 Matrix Inverse, 29 Matrix Multiplication, 25 Maxwell Element, 312 Maxwell Fluid, 200 Maxwell Model, 306 Maxwell’s Equation, 205 Maxwell’s Reciprocity Theorem, 222 Mechanics, 11 Method of Partial Fractions, 334 Method of Potentials, 225 Michell’s Equations, 213 Minimum Total Potential Energy, 249 Minor, 31 Moment, 12 Monoclinic, 183 Mooney–Rivlin Material, 193, 194 Multiplicative of Vector, 10 Nanson’s Formula, 109, 129 Natural Boundary Conditions, 246 Navier–Stokes Equations, 277, 289 Neo-Hookean Material, 193, 194 Neumann Boundary Condition, 247 Newtonian Constitutive Equations, 196 Newtonian Fluids, 5, 179, 195, 196, 197 Newtonian Viscosity, 305 Newton’s Law of Cooling, 203 Newton’s Laws, Nominal Stress Tensor, 129 353 Noncircular Cylinders, 240 Nonhomogeneous Deformation, 72 Nonion Form, 43 Nonisothermal, 198 Nonlinear Elastic, 193 Non-Newtonian, 195, 197, 198 Non-Viscous, 167 Normal Components, 120 Normal Derivative, 37 Normal Stress, 116 Null Vector, Numerical Solutions, 224 Observer Transformation, 107 Oldroyd A Fluid, 201 Oldroyd B Fluid, 201 Oldroyd Model, 201 Orthogonal, 11 Orthogonal Matrix, 56 Orthogonal Rotation Tensor, 98 Orthogonal Tensor, 45, 121 Orthogonality Property, 34 Orthotropic Material, 184, 186 Outer Product, 13 Outflow, 146 Parallel Flow, 284 Pendulum, 273 Perfect Gas, 195 Permanent Deformation, 193 Permeability, 207 Permittivity, 206 Permutation Symbol, 20 Phan Thien–Tanner Model, 201 Plane Strain, 227 Plane Stress, 227, 229 Plunger, 151 Poiseuille Flow, 285 Poisson’s Ratio, 186, 187 Polar Decomposition Theorem, 97 Polyadics, 42 Postfactor, 43 Potential Energy, 244, 257 Power-Law Index, 198 Power-Law Model, 198 Prandtl Stress Function, 240, 242 Prefactor, 43 Pressure Vessel, 123 Primary Field Variables, 178 Primary Variable, 247 Principal Directions of Strain, 84 Principal Invariants, 198 Principal Planes, 124 Principal Strains, 84 Principal Stresses, 124 Principal Stretch, 71, 93 Principle of Minimum Total Potential Energy, 245 Principle of Superposition, 185 Problem Coordinates, 182, 189 Pseudo Stress Tensor, 129, 133 Pseudoplastic, 198 Pure Dilation, 71 Radiation, 203 Rate of Deformation, 96, 196 Rate of Deformation Tensor, 96, 196 Reciprocal Basis, 34 Relaxation Modulus, 311 Relaxation Response, 306, 311, 321, 334 Relaxation Test, 305 10:55 P1: draft index CUFX197-Reddy 978 521 87044 October 3, 2007 354 Index Residual Stress, 181 Retardation Time, 201 Reynolds’s Transport Theorem, 153, 168 Right Cauchy Stretch Tensor, 98 Right Cauchy–Green Deformation Tensor, 77 Rigid-Body Motion, 67, 105 Rotation Tensor, 89, 233 Saint-Venant’s Principle, 233 Scalar Components, 18 Scalar Product, 11 Scalar Triple Product, 14 Second Law of Thermodynamics, 170 Second Piola–Kirchhoff Stress Tensor, 130 Secondary Field Variables, 178 Secondary Variable, 247 Second-Order Tensor, 44 Semi-Inverse Method, 225 Shear Components, 120 Shear Deformation, 75 Shear Extensional Coupling, 184 Shear Stress, 116, 126, 275 Shear Thickening, 198 Shear Thinning, 197, 198 Simple Fluids, 199 Simple Shear, 70, 86 Singular, 31 Skew Product, 13 Skew-Symmetric, 26, 91 Slider Bearing, 291 Small Deformation, 134 Solid, 275 Spatial Coordinates, 64 Spatial Description, 64 Specific Enthalpy, 166 Specific Entropy, 170 Specific Internal Energy, 164 Specific Volume, 166 Spherical Coordinate, 39, 94, 49, 160, 279 Spherical Stress Tensor, 140 Spin Tensor, 96 Spring-and-Dashpot Model, 306 St Venant’s Compatibility, 100 Stefan–Boltzmann Law, 203, 204 Stieljes Integral, 324 Stiffness Matrix, 253 Stokes Condition, 196 Stokes Equations, 289 Stoke’s Law, 261 Strain Energy, 244 Strain Energy Density, 180, 263 Strain Measures, 77 Strain–Displacement Relations, 211 Stream Function, 278 Stress Dyadic, 117 Stress Measures, 115 Stress–Strain, 211 Stretch, 80 Summation Convention, 18 Superposition Principle, 215 Surface Change, 74 Surface Forces, 158 Surface Stress Tensor, 158 Symmetric, 26, 130 Symmetry Transformations, 183 Taylor’s Series, 180 Tensor, Tetrahedral Element, 117 Thermal Conductivity, 203 Thermal Expansion, 167 Thermodynamic Form, 165 Thermodynamic Pressure, 194, 196 Thermodynamic Principles, 163 Thermodynamic State, 198 Thermodynamics, Thermomechanics, 4, 5, 195 Thick-Walled Cylinder, 225 Third-Order Tensor, 44 Timoshenko Beam, 238, 272, 274 Torsion, 271 Trace of a Dyadic, 44 Transformation of Strain Components, 188, 190 Transformation of Stress Components, 120, 188 Transformation Rule, 23 Transformations, 183 Triclinic Materials, 183 Triple Product of Vectors, 14 Two-Dimensional Heat Transfer, 297 Uniform Deformation, 70 Unit Impulse, 307 Unit Vector, Unitary System, 32 Upper-Convected Derivative, 200 Variational Methods, 225 Variational Operator, 245 Vectors, 8, Vector Addition, 10 Vector Calculus, 32 Vector Components, 18, 24 Vector Product, 12 Vector Triple Product, 15 Velocity Gradient Tensor, 97 Virtual Displacement, 262 Virtual Kinetic Energy, 258 Virtual Work, 258, 262 Viscoelastic, 305, 331 Viscoelastic Constitutive Models, 199 Viscoelastic Fluids, 197 Viscosity, 3, 195, 197 Viscous, 179 Viscous Dissipation, 166 Viscous Incompressible Fluids, 196, 277 Viscous Stress Tensor, 166, 276 Voigt–Kelvin Notation, 182 Volume Change, 73 Von Mises Yield Criterion, 199 Vorticity Tensor, 96 Vorticity Vector, 172 Warping Function, 242 Weissenberg Effect, 197 White-Metzner Model, 201 Young’s Modulus, Zero Vector, Zeroth-Order Tensor, 44 10:55 ... Applied Mechanics and Engineering, and International Journal of Non-Linear Mechanics ii 14:8 P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 An Introduction to Continuum Mechanics. .. left blank P1: JZP CUFX197-FM CUFX197-Reddy 978 521 87044 October 3, 2007 AN INTRODUCTION TO CONTINUUM MECHANICS This textbook on continuum mechanics reflects the modern view that scientists and... 521 87044 October 3, 2007 Preface formulations and applications to specific problems from heat transfer, fluid mechanics, and solid mechanics The motivation and encouragement that led to the writing