Elasticity_ Barber

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ContentsxiiiPROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41427 Axisymmetric torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41927.1 The transmitted torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42027.2 The governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42027.3 Solution of the governing equation . . . . . . . . . . . . . . . . . . . . . . . . . 42127.4 The displacement field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42327.5 Cylindrical and conical bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42427.5.1 The centre of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42527.6 The Saint Venant problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42728 The prismatic bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42928.1 Power series solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42928.1.1 Superposition by differentiation . . . . . . . . . . . . . . . . . . . . . 43028.1.2 The problems P 0 and P 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43128.1.3 Properties of the solution to P m . . . . . . . . . . . . . . . . . . . . . 43228.2 Solution of P m by integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43228.3 The integration process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43328.4 The two-dimensional problem P 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 43528.5 Problem P 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43628.5.1 The corrective antiplane solution . . . . . . . . . . . . . . . . . . . . 43728.5.2 The circular bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43828.6 The corrective in-plane solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 43928.7 Corrective solutions using real stress functions . . . . . . . . . . . . . . 43928.7.1 Airy function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44028.7.2 Prandtl function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44028.8 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44128.9 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44228.9.1 Problem P 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44228.9.2 Problem P 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44328.9.3 End conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44729 Frictionless contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44929.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44929.1.1 Mixed boundary-value problems . . . . . . . . . . . . . . . . . . . . . 45029.2 Determining the contact area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45129.3 Contact problems involving adhesive forces . . . . . . . . . . . . . . . . . 45430 The boundary-value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45930.1 Hankel transform methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45930.2 Collins’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46030.2.1 Indentation by a flat punch . . . . . . . . . . . . . . . . . . . . . . . . . 46030.2.2 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
xivContents30.2.3 Basic forms and surface values . . . . . . . . . . . . . . . . . . . . . . 46330.2.4 Reduction to an Abel equation . . . . . . . . . . . . . . . . . . . . . . 46530.2.5 Smooth contact problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 46830.2.6 Choice of form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47130.3 Non-axisymmetric problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47130.3.1 The full stress field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47531 The penny-shaped crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47931.1 The penny-shaped crack in tension . . . . . . . . . . . . . . . . . . . . . . . . 47931.2 Thermoelastic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48532 The interface crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48732.1 The uncracked interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48732.2 The corrective solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48932.2.1 Global conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48932.2.2 Mixed conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49032.3 The penny-shaped crack in tension . . . . . . . . . . . . . . . . . . . . . . . . 49232.3.1 Reduction to a single equation . . . . . . . . . . . . . . . . . . . . . . 49432.3.2 Oscillatory singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49532.4 The contact solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49632.5 Implications for Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . 49733 Variational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49933.1 Strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50033.1.1 Strain energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50033.2 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50133.3 Potential energy of the external forces . . . . . . . . . . . . . . . . . . . . . . 50133.4 Theorem of minimum total potential energy . . . . . . . . . . . . . . . . 50233.5 Approximate solutions — the Rayleigh-Ritz method . . . . . . . . . 50333.6 Castigliano’s second theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50633.7 Approximations using Castigliano’s second theorem . . . . . . . . . . 50733.7.1 The torsion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50733.7.2 The in-plane problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50933.8 Uniqueness and existence of solution . . . . . . . . . . . . . . . . . . . . . . . 51133.8.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51434 The reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51734.1 Maxwell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51734.2 Betti’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51834.3 Use of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51934.3.1 A tilted punch problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52034.3.2 Indentation of a half-space . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Page 1 and 2: J.R. BarberSolid Mechanicsand its A
Page 3 and 4: SOLID MECHANICS AND ITS APPLICATION
Page 5 and 6: J.R. BarberDepartment of Mechanical
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: xiiContents23 Singular solutions .
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
Page 55 and 56: 3.1 Plane strain 39an unwanted norm
Page 57 and 58: for all x, y, z.It then follows tha
Page 59 and 60: ( ) ( )( κ + 1 3 − κκ + 1e xx
Page 61 and 62: 4STRESS FUNCTION FORMULATION4.1 The
Page 63 and 64: 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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