Elasticity_ Barber

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J.R. BarberElasticity3rd Revised Edition
J.R. BarberDepartment of Mechanical Engineeringand Applied MechanicsUniversity of MichiganAnn Arbor, MI 48109-2125USAjbarber@umich.eduISSN 0925-0042ISBN 978-90-481-3808-1 e-ISBN 978-90-481-3809-8DOI 10.1007/978-90-481-3809-8Springer Dordrecht Heidelberg London New YorkLibrary of Congress Control Number: 2009940788© Springer Science + Business Media B.V. 2010No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specifically for the purposeofbeing entered and executed on a computer system, for exclusive use by the purchaser of the work.Printed on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)
Page 1 and 2: J.R. BarberSolid Mechanicsand its A
Page 3: SOLID MECHANICS AND ITS APPLICATION
Page 7 and 8: viContentsPart II TWO-DIMENSIONAL P
Page 9 and 10: viiiContents10.3.1 Beam with sinuso
Page 11 and 12: xContents17 Shear of a prismatic ba
Page 13 and 14: xiiContents23 Singular solutions .
Page 15 and 16: xivContents30.2.3 Basic forms and s
Page 17 and 18: PrefaceThe subject of Elasticity ca
Page 19 and 20: Prefacexixproblems. This necessaril
Page 21 and 22: 1INTRODUCTIONThe subject of Elastic
Page 23 and 24: 1.1 Notation for stress and displac
Page 25 and 26: σ ii ≡1.1 Notation for stress an
Page 33 and 34: 1.2 Strains and their relation to d
Page 35 and 36: (anticlockwise positive).Proceeding
Page 37 and 38: 1.2 Strains and their relation to d
Page 39 and 40: 1.3 Stress-strain relations 21strai
Page 41 and 42: Problems 23have other more convenie
Page 43 and 44: 2EQUILIBRIUM AND COMPATIBILITYWe ca
Page 45 and 46: 2.2 Compatibility equations 27For t
Page 47 and 48: 2.2 Compatibility equations 29Figur
Page 49 and 50: 2.3 Equilibrium equations in terms
Page 51 and 52: Problems 33Show that this is possib
Page 53 and 54: 3PLANE STRAIN AND PLANE STRESSA pro
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3.1 Plane strain 39an unwanted norm
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for all x, y, z.It then follows tha
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( ) ( )( κ + 1 3 − κκ + 1e xx
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4STRESS FUNCTION FORMULATION4.1 The
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4.3 The Airy stress function 47so a
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4.4 The governing equation 49∂ 2
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Problems 51PROBLEMS1. Newton’s la
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Problems 53u = ∇φto develop a po
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56 5 Problems in rectangular coörd
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78 6 End effectsthe corrective solu
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80 6 End effectsin which case, (6.8
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82 6 End effectsapproximate solutio
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84 6 End effectsproblems therefore
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86 6 End effectswhich defines a lin
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88 6 End effectsAlternative series
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7BODY FORCESA body force is defined
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7.2 Particular cases 937.1.2 The co
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7.2 Particular cases 95to an elasti
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7.3 Solution for the stress functio
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7.3 Solution for the stress functio
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7.4 Rotational acceleration 101body
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7.4 Rotational acceleration 103We f
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Problems 1057.4.3 Weak boundary con
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Problems 1075. The beam −b < y <
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8PROBLEMS IN POLAR COÖRDINATESPola
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8.3 Fourier series expansion 111(
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8.3 Fourier series expansion 113σ
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8.3 Fourier series expansion 115(σ
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8.3 Fourier series expansion 117The
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8.4 The Michell solution 119Table 8
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Problems 121PROBLEMS1. A large plat
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9CALCULATION OF DISPLACEMENTSSo far
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9.1 The cantilever with an end load
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9.2 The circular hole 127from equat
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9.3 Displacements for the Michell s
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9.3 Displacements for the Michell s
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Problems 1332. The rectangular plat
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10CURVED BEAM PROBLEMSIf we cut the
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10.1 Loading at the ends 137Da 2 =
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10.2 Eigenvalues and eigenfunctions
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10.3 The inhomogeneous problem 1412
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10.3 The inhomogeneous problem 143A
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10.3 The inhomogeneous problem 145o
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Problems 147equation system resulti
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11WEDGE PROBLEMSIn this chapter, we
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11.1 Power law tractions 151Figure
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11.1 Power law tractions 153It is i
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11.2 Williams’ asymptotic method
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11.3 General loading of the faces 1
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Problems 167PROBLEMS1. Figure 11.9
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Problems 1696. Figure 11.12 shows a
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12PLANE CONTACT PROBLEMS 1In the pr
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12.3 The half-plane 173cancels in e
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12.4 Distributed normal tractions 1
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12.5 Frictionless contact problems1
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12.5 Frictionless contact problems
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12.5.3 The cylindrical punch (Hertz
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12.6 Problems with two deformable b
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12.6 Problems with two deformable b
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12.8 Combined normal and tangential
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Problems 1952. The disk 0 ≤ r < a
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Problems 1977. Express the stress f
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13FORCES, DISLOCATIONS AND CRACKSIn
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F +∫ 2π013.1 The Kelvin solution
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2µ µ(1 + ν)= = E (κ + 1) 2 4µ=
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13.2 Dislocations 205increases. Thi
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13.3 Crack problems 207Applied Mech
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13.3 Crack problems 20913.3.2 Plane
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13.3 Crack problems 211(κ + 1)SξB
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W = 1 2and hence∫ ∆S0σ yy (s)
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13.4 Method of images 215ExampleAs
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Problems 217We now introduce a crac
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14THERMOELASTICITYMost materials te
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14.2 Heat conduction 22114.2 Heat c
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14.3 Steady-state problems 22314.3.
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Problems 225unit volume, where r is
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228 15 Antiplane shearIn the absenc
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230 15 Antiplane shearwith solution
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232 15 Antiplane shearDescribing th
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234 15 Antiplane shearsurface being
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Part IIIEND LOADING OF THE PRISMATI
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240z-axis will generate the uniform
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242 16 Torsion of a prismatic barco
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244 16 Torsion of a prismatic barEx
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246 16 Torsion of a prismatic baran
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248 16 Torsion of a prismatic bar
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250 16 Torsion of a prismatic barfo
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252 16 Torsion of a prismatic barMe
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254 16 Torsion of a prismatic bar7.
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256 16 Torsion of a prismatic bar12
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17SHEAR OF A PRISMATIC BAR 1In this
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which imply that17.3 The boundary c
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17.4 Methods of solution 26317.4.1
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17.4 Methods of solution 265Substit
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Problems 267where ω z is the rotat
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Part IVCOMPLEX VARIABLE FORMULATION
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272the several classical works on t
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274 18 Preliminary mathematical res
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P 1●280 18 Preliminary mathematic
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294 19 Application to elasticity pr
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Part VTHREE DIMENSIONAL PROBLEMS
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322 20 Displacement function soluti
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334 21 The Boussinesq potentials21.
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336 21 The Boussinesq potentialsand
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338 21 The Boussinesq potentialsTab
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340 21 The Boussinesq potentialsTab
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342 21 The Boussinesq potentialsfro
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344 21 The Boussinesq potentialsσ
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22THERMOELASTIC DISPLACEMENTPOTENTI
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22.1 Plane problems 34922.1 Plane p
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22.1 Plane problems 351∂u∂ ¯ζ
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22.1 Plane problems 353The homogene
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22.3 Steady-state temperature : Sol
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22.3 Steady-state temperature : Sol
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d 2dR 2 + 2 dR dR ≡ 1 (dR 2 dRR 2
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Problems 361If the half-space remai
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364 23 Singular solutionswith a poi
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366 23 Singular solutionsFigure 23.
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368 23 Singular solutionsFigure 23.
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370 23 Singular solutionsF (1 − 2
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372 23 Singular solutions23.4 Image
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374 23 Singular solutionsand hence,
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376 23 Singular solutions6. Use Ade
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378 24 Spherical harmonics24.1 Four
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380 24 Spherical harmonicsIn view o
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382 24 Spherical harmonicstractions
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384 24 Spherical harmonicsThe atten
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386 24 Spherical harmonicsA conveni
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388 24 Spherical harmonicsψ 2k (ζ
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390 24 Spherical harmonicsχ 0 0 =
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392 25 Cylinders and circular plate
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406 26 Problems in spherical coörd
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420 27 Axisymmetric torsionConsider
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422 27 Axisymmetric torsionφ = r 2
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424 27 Axisymmetric torsion27.5 Cyl
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426 27 Axisymmetric torsionσ θR =
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428 27 Axisymmetric torsion(i) the
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430 28 The prismatic barwhere f, g
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432 28 The prismatic bar28.1.3 Prop
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434 28 The prismatic barwhere φ,
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436 28 The prismatic bar28.5 Proble
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438 28 The prismatic bar28.5.2 The
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440 28 The prismatic bar28.7.1 Airy
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442 28 The prismatic barThis comple
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444 28 The prismatic barIn-plane co
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446 28 The prismatic barσ zz (0,
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448 28 The prismatic baron the surf
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450 29 Frictionless contactwhere we
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452 29 Frictionless contacthowever
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454 29 Frictionless contact— i.e.
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456 29 Frictionless contactPROBLEMS
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30THE BOUNDARY-VALUE PROBLEMThe sim
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30.2 Collins’ Method 461For examp
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∫∂ϕ a∂z = R0g(t)dt√r2 + (z
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30.2 Collins’ Method 465Table 30.
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30.2 Collins’ Method 467In genera
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and hence∫ artg 1 (t)dt√t2 −
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30.3 Non-axisymmetric problems 4713
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30.3 Non-axisymmetric problems 473T
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Problems 475PROBLEMS1. A harmonic p
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Problems 477Figure 30.5: Flat punch
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31THE PENNY-SHAPED CRACKAs in the t
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31.1 The penny-shaped crack in tens
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Condition (31.27) then defines the
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Problems 485from Table 30.1 and equ
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32THE INTERFACE CRACKThe problem of
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32.2 The corrective solution 48932.
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32.2 The corrective solution 491Rea
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— i.e.32.3 The penny-shaped crack
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g 1 (x) +β 2 ∫ aπ 2 (1 − β 2
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32.5 Implications for Fracture Mech
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33VARIATIONAL METHODSEnergy or vari
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33.3 Potential energy of the extern
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33.5 Approximate solutions — the
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33.5 Approximate solutions — the
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33.7 Approximations using Castiglia
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33.7 Approximations using Castiglia
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A = − S4a 2 ;33.8 Uniqueness and
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33.8 Uniqueness and existence of so
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Problems 515Using this displacement
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34THE RECIPROCAL THEOREMIn the prev
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34.3 Use of the theorem 519equal to
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34.3 Use of the theorem 521Figure 3
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34.3 Use of the theorem 523However,
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34.4 Thermoelastic problems 525forc
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δ = νρgLhE,Problems 527where ρ,
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AUSING MAPLE AND MATHEMATICAThe alg
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INDEXAbel integral equations, 463-4
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Index533hydrostatic stress, 10, 518
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